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17 Data Visualization Techniques All Professionals Should Know

• 17 Sep 2019

There’s a growing demand for business analytics and data expertise in the workforce. But you don’t need to be a professional analyst to benefit from data-related skills.

Becoming skilled at common data visualization techniques can help you reap the rewards of data-driven decision-making , including increased confidence and potential cost savings. Learning how to effectively visualize data could be the first step toward using data analytics and data science to your advantage to add value to your organization.

Several data visualization techniques can help you become more effective in your role. Here are 17 essential data visualization techniques all professionals should know, as well as tips to help you effectively present your data.

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What Is Data Visualization?

Data visualization is the process of creating graphical representations of information. This process helps the presenter communicate data in a way that’s easy for the viewer to interpret and draw conclusions.

There are many different techniques and tools you can leverage to visualize data, so you want to know which ones to use and when. Here are some of the most important data visualization techniques all professionals should know.

Data Visualization Techniques

The type of data visualization technique you leverage will vary based on the type of data you’re working with, in addition to the story you’re telling with your data .

Here are some important data visualization techniques to know:

• Gantt Chart
• Box and Whisker Plot
• Waterfall Chart
• Scatter Plot
• Pictogram Chart
• Highlight Table
• Bullet Graph
• Choropleth Map
• Network Diagram
• Correlation Matrices

1. Pie Chart

Pie charts are one of the most common and basic data visualization techniques, used across a wide range of applications. Pie charts are ideal for illustrating proportions, or part-to-whole comparisons.

Because pie charts are relatively simple and easy to read, they’re best suited for audiences who might be unfamiliar with the information or are only interested in the key takeaways. For viewers who require a more thorough explanation of the data, pie charts fall short in their ability to display complex information.

2. Bar Chart

The classic bar chart , or bar graph, is another common and easy-to-use method of data visualization. In this type of visualization, one axis of the chart shows the categories being compared, and the other, a measured value. The length of the bar indicates how each group measures according to the value.

One drawback is that labeling and clarity can become problematic when there are too many categories included. Like pie charts, they can also be too simple for more complex data sets.

3. Histogram

Unlike bar charts, histograms illustrate the distribution of data over a continuous interval or defined period. These visualizations are helpful in identifying where values are concentrated, as well as where there are gaps or unusual values.

Histograms are especially useful for showing the frequency of a particular occurrence. For instance, if you’d like to show how many clicks your website received each day over the last week, you can use a histogram. From this visualization, you can quickly determine which days your website saw the greatest and fewest number of clicks.

4. Gantt Chart

Gantt charts are particularly common in project management, as they’re useful in illustrating a project timeline or progression of tasks. In this type of chart, tasks to be performed are listed on the vertical axis and time intervals on the horizontal axis. Horizontal bars in the body of the chart represent the duration of each activity.

Utilizing Gantt charts to display timelines can be incredibly helpful, and enable team members to keep track of every aspect of a project. Even if you’re not a project management professional, familiarizing yourself with Gantt charts can help you stay organized.

5. Heat Map

A heat map is a type of visualization used to show differences in data through variations in color. These charts use color to communicate values in a way that makes it easy for the viewer to quickly identify trends. Having a clear legend is necessary in order for a user to successfully read and interpret a heatmap.

There are many possible applications of heat maps. For example, if you want to analyze which time of day a retail store makes the most sales, you can use a heat map that shows the day of the week on the vertical axis and time of day on the horizontal axis. Then, by shading in the matrix with colors that correspond to the number of sales at each time of day, you can identify trends in the data that allow you to determine the exact times your store experiences the most sales.

6. A Box and Whisker Plot

A box and whisker plot , or box plot, provides a visual summary of data through its quartiles. First, a box is drawn from the first quartile to the third of the data set. A line within the box represents the median. “Whiskers,” or lines, are then drawn extending from the box to the minimum (lower extreme) and maximum (upper extreme). Outliers are represented by individual points that are in-line with the whiskers.

This type of chart is helpful in quickly identifying whether or not the data is symmetrical or skewed, as well as providing a visual summary of the data set that can be easily interpreted.

7. Waterfall Chart

A waterfall chart is a visual representation that illustrates how a value changes as it’s influenced by different factors, such as time. The main goal of this chart is to show the viewer how a value has grown or declined over a defined period. For example, waterfall charts are popular for showing spending or earnings over time.

8. Area Chart

An area chart , or area graph, is a variation on a basic line graph in which the area underneath the line is shaded to represent the total value of each data point. When several data series must be compared on the same graph, stacked area charts are used.

This method of data visualization is useful for showing changes in one or more quantities over time, as well as showing how each quantity combines to make up the whole. Stacked area charts are effective in showing part-to-whole comparisons.

9. Scatter Plot

Another technique commonly used to display data is a scatter plot . A scatter plot displays data for two variables as represented by points plotted against the horizontal and vertical axis. This type of data visualization is useful in illustrating the relationships that exist between variables and can be used to identify trends or correlations in data.

Scatter plots are most effective for fairly large data sets, since it’s often easier to identify trends when there are more data points present. Additionally, the closer the data points are grouped together, the stronger the correlation or trend tends to be.

10. Pictogram Chart

Pictogram charts , or pictograph charts, are particularly useful for presenting simple data in a more visual and engaging way. These charts use icons to visualize data, with each icon representing a different value or category. For example, data about time might be represented by icons of clocks or watches. Each icon can correspond to either a single unit or a set number of units (for example, each icon represents 100 units).

In addition to making the data more engaging, pictogram charts are helpful in situations where language or cultural differences might be a barrier to the audience’s understanding of the data.

11. Timeline

Timelines are the most effective way to visualize a sequence of events in chronological order. They’re typically linear, with key events outlined along the axis. Timelines are used to communicate time-related information and display historical data.

Timelines allow you to highlight the most important events that occurred, or need to occur in the future, and make it easy for the viewer to identify any patterns appearing within the selected time period. While timelines are often relatively simple linear visualizations, they can be made more visually appealing by adding images, colors, fonts, and decorative shapes.

12. Highlight Table

A highlight table is a more engaging alternative to traditional tables. By highlighting cells in the table with color, you can make it easier for viewers to quickly spot trends and patterns in the data. These visualizations are useful for comparing categorical data.

Depending on the data visualization tool you’re using, you may be able to add conditional formatting rules to the table that automatically color cells that meet specified conditions. For instance, when using a highlight table to visualize a company’s sales data, you may color cells red if the sales data is below the goal, or green if sales were above the goal. Unlike a heat map, the colors in a highlight table are discrete and represent a single meaning or value.

13. Bullet Graph

A bullet graph is a variation of a bar graph that can act as an alternative to dashboard gauges to represent performance data. The main use for a bullet graph is to inform the viewer of how a business is performing in comparison to benchmarks that are in place for key business metrics.

In a bullet graph, the darker horizontal bar in the middle of the chart represents the actual value, while the vertical line represents a comparative value, or target. If the horizontal bar passes the vertical line, the target for that metric has been surpassed. Additionally, the segmented colored sections behind the horizontal bar represent range scores, such as “poor,” “fair,” or “good.”

14. Choropleth Maps

A choropleth map uses color, shading, and other patterns to visualize numerical values across geographic regions. These visualizations use a progression of color (or shading) on a spectrum to distinguish high values from low.

Choropleth maps allow viewers to see how a variable changes from one region to the next. A potential downside to this type of visualization is that the exact numerical values aren’t easily accessible because the colors represent a range of values. Some data visualization tools, however, allow you to add interactivity to your map so the exact values are accessible.

15. Word Cloud

A word cloud , or tag cloud, is a visual representation of text data in which the size of the word is proportional to its frequency. The more often a specific word appears in a dataset, the larger it appears in the visualization. In addition to size, words often appear bolder or follow a specific color scheme depending on their frequency.

Word clouds are often used on websites and blogs to identify significant keywords and compare differences in textual data between two sources. They are also useful when analyzing qualitative datasets, such as the specific words consumers used to describe a product.

16. Network Diagram

Network diagrams are a type of data visualization that represent relationships between qualitative data points. These visualizations are composed of nodes and links, also called edges. Nodes are singular data points that are connected to other nodes through edges, which show the relationship between multiple nodes.

There are many use cases for network diagrams, including depicting social networks, highlighting the relationships between employees at an organization, or visualizing product sales across geographic regions.

17. Correlation Matrix

A correlation matrix is a table that shows correlation coefficients between variables. Each cell represents the relationship between two variables, and a color scale is used to communicate whether the variables are correlated and to what extent.

Correlation matrices are useful to summarize and find patterns in large data sets. In business, a correlation matrix might be used to analyze how different data points about a specific product might be related, such as price, advertising spend, launch date, etc.

Other Data Visualization Options

While the examples listed above are some of the most commonly used techniques, there are many other ways you can visualize data to become a more effective communicator. Some other data visualization options include:

• Bubble clouds
• Circle views
• Dendrograms
• Dot distribution maps
• Open-high-low-close charts
• Polar areas
• Radial trees
• Ring Charts
• Sankey diagram
• Span charts
• Streamgraphs
• Wedge stack graphs
• Violin plots

Tips For Creating Effective Visualizations

Creating effective data visualizations requires more than just knowing how to choose the best technique for your needs. There are several considerations you should take into account to maximize your effectiveness when it comes to presenting data.

Related : What to Keep in Mind When Creating Data Visualizations in Excel

One of the most important steps is to evaluate your audience. For example, if you’re presenting financial data to a team that works in an unrelated department, you’ll want to choose a fairly simple illustration. On the other hand, if you’re presenting financial data to a team of finance experts, it’s likely you can safely include more complex information.

Another helpful tip is to avoid unnecessary distractions. Although visual elements like animation can be a great way to add interest, they can also distract from the key points the illustration is trying to convey and hinder the viewer’s ability to quickly understand the information.

Finally, be mindful of the colors you utilize, as well as your overall design. While it’s important that your graphs or charts are visually appealing, there are more practical reasons you might choose one color palette over another. For instance, using low contrast colors can make it difficult for your audience to discern differences between data points. Using colors that are too bold, however, can make the illustration overwhelming or distracting for the viewer.

Related : Bad Data Visualization: 5 Examples of Misleading Data

Visuals to Interpret and Share Information

No matter your role or title within an organization, data visualization is a skill that’s important for all professionals. Being able to effectively present complex data through easy-to-understand visual representations is invaluable when it comes to communicating information with members both inside and outside your business.

There’s no shortage in how data visualization can be applied in the real world. Data is playing an increasingly important role in the marketplace today, and data literacy is the first step in understanding how analytics can be used in business.

Are you interested in improving your analytical skills? Learn more about Business Analytics , our eight-week online course that can help you use data to generate insights and tackle business decisions.

This post was updated on January 20, 2022. It was originally published on September 17, 2019.

About the Author

What is visual representation?

In the vast landscape of communication, where words alone may fall short, visual representation emerges as a powerful ally. In a world inundated with information, the ability to convey complex ideas, emotions, and data through visual means is becoming increasingly crucial. But what exactly is visual representation, and why does it hold such sway in our understanding?

Defining Visual Representation:

Visual representation is the act of conveying information, ideas, or concepts through visual elements such as images, charts, graphs, maps, and other graphical forms. It’s a means of translating the abstract into the tangible, providing a visual language that transcends the limitations of words alone.

The Power of Images:

The adage “a picture is worth a thousand words” encapsulates the essence of visual representation. Images have an unparalleled ability to evoke emotions, tell stories, and communicate complex ideas in an instant. Whether it’s a photograph capturing a poignant moment or an infographic distilling intricate data, images possess a unique capacity to resonate with and engage the viewer on a visceral level.

Facilitating Understanding:

One of the primary functions of visual representation is to enhance understanding. Humans are inherently visual creatures, and we often process and retain visual information more effectively than text. Complex concepts that might be challenging to grasp through written explanations can be simplified and clarified through visual aids. This is particularly valuable in fields such as science, where intricate processes and structures can be elucidated through diagrams and illustrations.

Visual representation also plays a crucial role in education. In classrooms around the world, teachers leverage visual aids to facilitate learning, making lessons more engaging and accessible. From simple charts that break down historical timelines to interactive simulations that bring scientific principles to life, visual representation is a cornerstone of effective pedagogy.

Data Visualization:

In an era dominated by big data, the importance of data visualization cannot be overstated. Raw numbers and statistics can be overwhelming and abstract, but when presented visually, they transform into meaningful insights. Graphs, charts, and maps are powerful tools for conveying trends, patterns, and correlations, enabling decision-makers to glean actionable intelligence from vast datasets.

Consider the impact of a well-crafted infographic that distills complex research findings into a visually digestible format. Data visualization not only simplifies information but also allows for more informed decision-making in fields ranging from business and healthcare to social sciences and environmental studies.

Cultural and Artistic Expression:

Visual representation extends beyond the realm of information and education; it is also a potent form of cultural and artistic expression. Paintings, sculptures, photographs, and other visual arts serve as mediums through which individuals can convey their emotions, perspectives, and cultural narratives. Artistic visual representation has the power to transcend language barriers, fostering a shared human experience that resonates universally.

Conclusion:

In a world inundated with information, visual representation stands as a beacon of clarity and understanding. Whether it’s simplifying complex concepts, conveying data-driven insights, or expressing the depth of human emotion, visual elements enrich our communication in ways that words alone cannot. As we navigate an increasingly visual society, recognizing and harnessing the power of visual representation is not just a skill but a necessity for effective communication and comprehension. So, let us embrace the visual language that surrounds us, unlocking a deeper, more nuanced understanding of the world.

• X (Twitter)

Painting Pictures with Data: The Power of Visual Representations

Picture this. A chaotic world of abstract concepts and complex data, like a thousand-piece jigsaw puzzle. Each piece, a different variable, a unique detail.

Alone, they’re baffling, nearly indecipherable.

But together? They’re a masterpiece of visual information, a detailed illustration.

American data pioneer Edward Tufte , a notable figure in the graphics press, believed that the art of seeing is not limited to the physical objects around us. He stated, “The commonality between science and art is in trying to see profoundly – to develop strategies of seeing and showing.”

It’s in this context that we delve into the world of data visualization. This is a process where you create visual representations that foster understanding and enhance decision making.

It’s the transformation of data into visual formats. The information could be anything from theoretical frameworks and research findings to word problems. Or anything in-between. And it has the power to change the way you learn, work, and more.

And with the help of modern technology, you can take advantage of data visualization easier than ever today.

What are Visual Representations?

Think of visuals, a smorgasbord of graphical representation, images, pictures, and drawings. Now blend these with ideas, abstract concepts, and data.

You get visual representations . A powerful, potent blend of communication and learning.

As a more formal definition, visual representation is the use of images to represent different types of data and ideas.

They’re more than simply a picture. Visual representations organize information visually , creating a deeper understanding and fostering conceptual understanding. These can be concrete objects or abstract symbols or forms, each telling a unique story. And they can be used to improve understanding everywhere, from a job site to an online article. University professors can even use them to improve their teaching.

But this only scratches the surface of what can be created via visual representation.

Types of Visual Representation for Improving Conceptual Understanding

Graphs, spider diagrams, cluster diagrams – the list is endless!

Each type of visual representation has its specific uses. A mind map template can help you create a detailed illustration of your thought process. It illustrates your ideas or data in an engaging way and reveals how they connect.

Here are a handful of different types of data visualization tools that you can begin using right now.

1. Spider Diagrams

Spider diagrams , or mind maps, are the master web-weavers of visual representation.

They originate from a central concept and extend outwards like a spider’s web. Different ideas or concepts branch out from the center area, providing a holistic view of the topic.

This form of representation is brilliant for showcasing relationships between concepts, fostering a deeper understanding of the subject at hand.

2. Cluster Diagrams

As champions of grouping and classifying information, cluster diagrams are your go-to tools for usability testing or decision making. They help you group similar ideas together, making it easier to digest and understand information.

They’re great for exploring product features, brainstorming solutions, or sorting out ideas.

3. Pie Charts

Pie charts are the quintessential representatives of quantitative information.

They are a type of visual diagrams that transform complex data and word problems into simple symbols. Each slice of the pie is a story, a visual display of the part-to-whole relationship.

Whether you’re presenting survey results, market share data, or budget allocation, a pie chart offers a straightforward, easily digestible visual representation.

4. Bar Charts

If you’re dealing with comparative data or need a visual for data analysis, bar charts or graphs come to the rescue.

Bar graphs represent different variables or categories against a quantity, making them perfect for representing quantitative information. The vertical or horizontal bars bring the data to life, translating numbers into visual elements that provide context and insights at a glance.

Visual Representations Benefits

1. deeper understanding via visual perception.

Visual representations aren’t just a feast for the eyes; they’re food for thought. They offer a quick way to dig down into more detail when examining an issue.

They mold abstract concepts into concrete objects, breathing life into the raw, quantitative information. As you glimpse into the world of data through these visualization techniques , your perception deepens.

You no longer just see the data; you comprehend it, you understand its story. Complex data sheds its mystifying cloak, revealing itself in a visual format that your mind grasps instantly. It’s like going from a two dimensional to a three dimensional picture of the world.

2. Enhanced Decision Making

Navigating through different variables and relationships can feel like walking through a labyrinth. But visualize these with a spider diagram or cluster diagram, and the path becomes clear. Visual representation is one of the most efficient decision making techniques .

Visual representations illuminate the links and connections, presenting a fuller picture. It’s like having a compass in your decision-making journey, guiding you toward the correct answer.

3. Professional Development

Whether you’re presenting research findings, sharing theoretical frameworks, or revealing historical examples, visual representations are your ace. They equip you with a new language, empowering you to convey your message compellingly.

From the conference room to the university lecture hall, they enhance your communication and teaching skills, propelling your professional development. Try to create a research mind map and compare it to a plain text document full of research documentation and see the difference.

4. Bridging the Gap in Data Analysis

What is data visualization if not the mediator between data analysis and understanding? It’s more than an actual process; it’s a bridge.

It takes you from the shores of raw, complex data to the lands of comprehension and insights. With visualization techniques, such as the use of simple symbols or detailed illustrations, you can navigate through this bridge effortlessly.

5. Enriching Learning Environments

Imagine a teaching setting where concepts are not just told but shown. Where students don’t just listen to word problems but see them represented in charts and graphs. This is what visual representations bring to learning environments.

They transform traditional methods into interactive learning experiences, enabling students to grasp complex ideas and understand relationships more clearly. The result? An enriched learning experience that fosters conceptual understanding.

6. Making Abstract Concepts Understandable

In a world brimming with abstract concepts, visual representations are our saving grace. They serve as translators, decoding these concepts into a language we can understand.

Let’s say you’re trying to grasp a theoretical framework. Reading about it might leave you puzzled. But see it laid out in a spider diagram or a concept map, and the fog lifts. With its different variables clearly represented, the concept becomes tangible.

Visual representations simplify the complex, convert the abstract into concrete, making the inscrutable suddenly crystal clear. It’s the power of transforming word problems into visual displays, a method that doesn’t just provide the correct answer. It also offers a deeper understanding.

How to Make a Cluster Diagram?

Ready to get creative? Let’s make a cluster diagram.

First, choose your central idea or problem. This goes in the center area of your diagram. Next, think about related topics or subtopics. Draw lines from the central idea to these topics. Each line represents a relationship.

While you can create a picture like this by drawing, there’s a better way.

Mindomo is a mind mapping tool that will enable you to create visuals that represent data quickly and easily. It provides a wide range of templates to kick-start your diagramming process. And since it’s an online site, you can access it from anywhere.

With a mind map template, creating a cluster diagram becomes an effortless process. This is especially the case since you can edit its style, colors, and more to your heart’s content. And when you’re done, sharing is as simple as clicking a button.

A Few Final Words About Information Visualization

To wrap it up, visual representations are not just about presenting data or information. They are about creating a shared understanding, facilitating learning, and promoting effective communication. Whether it’s about defining a complex process or representing an abstract concept, visual representations have it all covered. And with tools like Mindomo , creating these visuals is as easy as pie.

In the end, visual representation isn’t just about viewing data, it’s about seeing, understanding, and interacting with it. It’s about immersing yourself in the world of abstract concepts, transforming them into tangible visual elements. It’s about seeing relationships between ideas in full color. It’s a whole new language that opens doors to a world of possibilities.

The correct answer to ‘what is data visualization?’ is simple. It’s the future of learning, teaching, and decision-making.

Keep it smart, simple, and creative! The Mindomo Team

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The Epistemology of Visual Thinking in Mathematics

Visual thinking is widespread in mathematical practice, and has diverse cognitive and epistemic purposes. This entry discusses potential roles of visual thinking in proving and in discovering, with some examples, and epistemic difficulties and limitations are considered. Also discussed is the bearing of epistemic uses of visual representations on the application of the a priori–a posteriori distinction to mathematical knowledge. A final section looks briefly at how visual means can aid comprehension and deepen understanding of proofs.

1. Introduction

2. historical background, 3.1 the reliability question, 3.2 visual means in non-formal proving, 3.3 a dispute: diagrams in proofs in analysis., 4.1 propositional discovery, 4.2 discovering a proof strategy, 4.3 discovering properties and kinds, 5. visual thinking and mental arithmetic, 6.1 evidential uses of visual experience, 6.2 an evidential use of visual experience in proving, 6.3 a non-evidential use of visual experience, 7. further uses of visual representations, 8. conclusion, other internet resources, related entries.

Visual thinking is a feature of mathematical practice across many subject areas and at many levels. It is so pervasive that the question naturally arises: does visual thinking in mathematics have any epistemically significant roles? A positive answer begets further questions. Can we rationally arrive at a belief with the generality and necessity characteristic of mathematical theorems by attending to specific diagrams or images? If visual thinking contributes to warrant for believing a mathematical conclusion, must the outcome be an empirical belief? How, if at all can visual thinking contribute to understanding abstract mathematical subject matter?

Visual thinking includes thinking with external visual representations (e.g., diagrams, symbol arrays, kinematic computer images) and thinking with internal visual imagery; often the two are used in combination, as when we are required to visually imagine a certain spatial transformation of an object represented by a diagram on paper or on screen. Almost always (and perhaps always) visual thinking in mathematics is used in conjunction with non-visual thinking. Possible epistemic roles include contributions to evidence, proof, discovery, understanding and grasp of concepts. The kinds and the uses of visual thinking in mathematics are numerous and diverse. This entry will deal with some of the topics in this area that have received attention and omit others. Among the omissions is the possible explanatory role of visual representations in mathematics. The topic of explanation within pure mathematics is tricky and best dealt with separately; for this an excellent starting place is the entry on explanation in mathematics (Mancosu 2011). Two other omissions are the development of logic diagrams (Euler, Venn, Pierce and Shin) and the nature and use of geometric diagrams in Euclid’s Elements , both of which are well treated in the entry diagrams (Shin et al. 2013). The focus here is on visual thinking generally, which includes thinking with symbol arrays as well as with diagrams; there will be no attempt here to formulate a criterion for distinguishing between symbolic and diagrammatic thinking. However, the use of visual thinking in proving and in various kinds of discovery will be covered in what follows. Discussions of some related questions and some studies of historical cases not considered here are to be found in the collection Diagrams in Mathematics: History and Philosophy (Mumma and Panza 2012).

“Mathematics can achieve nothing by concepts alone but hastens at once to intuition” wrote Kant (1781/9: A715/B743), before describing the geometrical construction in Euclid’s proof of the angle sum theorem (Euclid, Book 1, proposition 32). In a review of 1816 Gauss echoes Kant:

anybody who is acquainted with the essence of geometry knows that [the logical principles of identity and contradiction] are able to accomplish nothing by themselves, and that they put forth sterile blossoms unless the fertile living intuition of the object itself prevails everywhere. (Ewald 1996 [Vol. 1]: 300)

The word “intuition” here translates the German “ Anschauung ”, a word which applies to visual imagination and perception, though it also has more general uses.

By the late 19 th century a different view had emerged, at least in foundational areas. In a celebrated text giving the first rigorous axiomatization of projective geometry, Pasch wrote: “the theorem is only truly demonstrated if the proof is completely independent of the figure” (Pasch 1882), a view expressed also by Hilbert in writing on the foundations of geometry (Hilbert 1894). A negative attitude to visual thinking was not confined to geometry. Dedekind, for example, wrote of an overpowering feeling of dissatisfaction with appeal to geometric intuitions in basic infinitesimal analysis (Dedekind 1872, Introduction). The grounds were felt to be uncertain, the concepts employed vague and unclear. When such concepts were replaced by precisely defined alternatives without allusions to space, time or motion, our intuitive expectations turned out to be unreliable (Hahn 1933).

In some quarters this view turned into a general disdain for visual thinking in mathematics: “In the best books” Russell pronounced “there are no figures at all” (Russell 1901). Although this attitude was opposed by a few mathematicians, notably Klein (1893), others took it to heart. Landau, for example, wrote a calculus textbook without a single diagram (Landau 1934). But the predominant view was not so extreme: thinking in terms of figures was valued as a means of facilitating grasp of formulae and linguistic text, but only reasoning expressed by means of formulae and text could bear any epistemological weight.

By the late 20 th century the mood had swung back in favour of visualization: Mancosu (2005) provides an excellent survey. Some books advertise their defiance of anti-visual puritanism in their titles, for example Visual Geometry and Topology (Fomenko 1994) and Visual Complex Analysis (Needham 1997); mathematics educators turn their attention to pedagogical uses of visualization (Zimmerman and Cunningham 1991); the use of computer-generated imagery begins to bear fruit at research level (Hoffman 1987; Palais 1999), and diagrams find their way into research papers in abstract fields: see for example the papers on higher dimensional category theory by Joyal et al. (1996), Leinster (2004) and Lauda (2005, Other Internet Resources). But attitudes to the epistemology of visual thinking remain mixed. The discussion is mostly concerned with the role of diagrams in proofs.

3. Visual thinking and proof

In some cases, it is claimed, a picture alone is a proof (Brown 1999: ch. 3). But that view is rare. Even the editor of Proofs without Words: Exercises in Visual Thinking , writes “Of course, ‘proofs without words’ are not really proofs” (Nelsen 1993: vi). Expressions of the other extreme are rare but can be found:

[the diagram] has no proper place in the proof as such. For the proof is a syntactic object consisting only of sentences arranged in a finite and inspectable array. (Tennant 1986)

Between the extremes we find the view that, even if no picture alone is a proof, visual representations can have a non-superfluous role in reasoning that constitutes a proof. (This is not to deny that there may be another proof of the same conclusion which does not involve any visual representation.) Geometric diagrams, graphs and maps, all carry information. Taking valid deductive reasoning to be the reliable extraction of information from information already obtained, Barwise and Etchemendy (1996:4) pose the following question: Why cannot the representations composing a proof be visual as well as linguistic? The sole reason for denying this role to visual representations is the thought that, with the possible exception of very restricted cases, visual thinking is unreliable, hence cannot contribute to proof. Is that right?

Our concern here is thinking through the steps in a proof, either for the first time (a first successful attempt to construct a proof) or following a given proof. Clearly we want to distinguish between visual thinking which merely accompanies the process of thinking through the steps in a proof and visual thinking which is essential to the process. This is not always straightforward as a proof can be presented in different ways. How different can distinct presentations be and yet be presentations of the same proof? There is no context-invariant answer to this. Often mathematicians are happy to regard two presentations as presenting the same proof if the central idea is the same in both cases. But if one’s main concern is with what is involved in thinking through a proof, its central idea is not enough to individuate it: the overall structure, the sequence of steps and perhaps some other factors affecting the cognitive processes involved will be relevant.

Once individuation of proofs has been settled, we can distinguish between replaceable thinking and superfluous thinking, where these attributions are understood as relative to a given argument or proof. In the process of thinking through a proof, a given part of the thinking is replaceable if thinking of some other kind could stand in place of the given part in a process that would count as thinking through the same proof. A given part of the thinking is superfluous if its excision without replacement would be a process of thinking through the same proof. Superfluous thinking may be extremely valuable in facilitating grasp of the proof text and in enabling one to understand the idea underlying the proof steps; but it is not necessary for thinking through the proof.

It is uncontentious that the visual thinking involved in symbol manipulations, for example in following the “algebraic” steps of proofs of basic lemmas about groups, can be essential, that is neither superfluous nor replaceable. The worry is about thinking visually with diagrams, where “diagram” is used widely to include all non-symbolic visual representations. Let us agree that there can be superfluous diagrammatic thinking in thinking through a proof. This leaves several possibilities.

• (a) All diagrammatic thinking in a process of thinking through a proof is superfluous.
• (b) Not all diagrammatic thinking in a process of thinking through a proof is superfluous; but if not superfluous it will be replaceable by non-diagrammatic thinking.
• (c) Some diagrammatic thinking in a process of thinking through a proof is neither superfluous nor replaceable by non-diagrammatic thinking.

The negative view stated earlier that diagrams can have no role in proof entails claim (a). The idea behind (a) is that, because diagrammatic reasoning is unreliable, if a process of thinking through an argument contains some non-superfluous diagrammatic thinking, that process lacks the epistemic security to be a case of thinking through a proof.

This view, claim (a) in particular, is threatened by cases in which the reliability of the diagrammatic thinking is demonstrated non-visually. The clearest kind of example would be provided by a formal system which has diagrams in place of formulas among its syntactic objects, and types of inter-diagram transition for inference rules. Suppose you take in such a formal system and an interpretation of it, and then think through a proof of the system’s soundness with respect to that interpretation; suppose you then inspect a sequence of diagrams, checking along the way that it constitutes a derivation in the system; suppose finally that you recover the interpretation to reach a conclusion. (The order is unimportant: one can go through the derivation first and then follow the soundness proof.) That entire process would constitute thinking through a proof of the conclusion; and the diagrammatic thinking involved would not be superfluous.

Shin et al. (2013) report that formal diagrammatic systems of logic and geometry have been proven to be sound. People have indeed followed proofs in these systems. That is enough to refute claim (a), the claim that all diagrammatic thinking in thinking through a proof is superfluous. For a concrete example, Figure 1 presents a derivation of Euclid’s first theorem, that on any straight line segment an equilateral triangle is constructible, in a formal diagrammatic system of a part of Euclidean geometry (Miller 2001).

What about Tennant’s claim that a proof is “a syntactic object consisting only of sentences” as opposed to diagrams? A proof is never a syntactic object. A formal derivation on its own is a syntactic object but not a proof. Without an interpretation of the language of the formal system the end-formula of the derivation says nothing; and so nothing is proved. Without a demonstration of the system’s soundness with respect to the interpretation, one may lack sufficient reason to believe that all derivable conclusions are true. A formal derivation plus an interpretation and soundness proof can be a proof of the derived conclusion, but that whole package is not a syntactic object. Moreover, the part of the proof which really is a syntactic object, the formal derivation, need not consist solely of sentences; it can consist of diagrams.

With claim (a) disposed of, consider again claim (b) that, while not all diagrammatic thinking in a process of thinking through a proof is superfluous, all non-superfluous diagrammatic thinking will be replaceable by non-diagrammatic thinking in a process of thinking through that same proof. The visual thinking in following the proof of Euclid’s first theorem using Miller’s formal system consists in going through a sequence of diagrams and at each step seeing that the next diagram results from a permitted alteration of the previous diagram. It is clear that in a process that counts as thinking through this proof, the diagrammatic thinking is neither superfluous nor replaceable by non-diagrammatic thinking. That knocks out (b), leaving only (c): some thinking that involves a diagram in thinking through a proof is neither superfluous nor replaceable by non-diagrammatic thinking (without changing the proof).

Mathematical practice almost never proceeds by way of formal systems. Outside the context of formal diagrammatic systems, the use of diagrams is widely felt to be unreliable. A diagram can be unfaithful to the described construction: it may represent something with a property that is ruled out by the description, or without a property that is demanded by the description. This is exemplified by diagrams in the famous argument for the proposition that all triangles are isosceles: the meeting point of an angle bisector and the perpendicular bisector of the opposite side is represented as falling inside the triangle, when it has to be outside (Rouse Ball 1939; Maxwell 1959). Errors of this sort are comparatively rare, usually avoidable with a modicum of care, and not inherent in the nature of diagrams; so they do not warrant a general charge of unreliability.

The major sort of error is unwarranted generalisation. Typically diagrams (and other non-verbal visual representations) do not represent their objects as having a property that is actually ruled out by the intention or specification of the object to be represented. But diagrams very frequently do represent their objects as having properties that, though not ruled out by the specification, are not demanded by it. Verbal descriptions can be discrete, in that they supply no more information than is needed. But visual representations are typically indiscrete, in that they supply too much detail. This is often unavoidable, because for many properties or kinds \(F\), a visual representation cannot represent something as being \(F\) without representing it as being \(F\) in a particular way . Any diagram of a triangle, for instance, must represent it as having three acute angles or as having just two acute angles, even if neither property is required by the specification, as would be the case if the specification were “Let ABC be a triangle”. As a result there is a danger that in using a diagram to reason about an arbitrary instance of class \(K\), we will unwittingly rely on a feature represented in the diagram that is not common to all instances of the class \(K\). Thus the risk of unwarranted generalisation is a danger inherent in the use of many diagrams.

Indiscretion of diagrams is not confined to geometrical figures. The dot or pebble diagrams of ancient mathematics used to convince one of elementary truths of number theory necessarily display particular numbers of dots, though the truths are general. Here is an example, used to justify the formula for the \(n\) th triangular number, i.e., the sum of the first \(n\) positive integers.

The conclusion drawn is that the sum of integers from 1 to \(n\) is \((n \times n+1)/2\) for any positive integer \(n\), but the diagram presents the case for \(n = 6\). We can perhaps avoid representing a particular number of dots when we merely imagine a display of the relevant kind; or if a particular number is represented, our experience may not make us aware of the number—just as, when one imagines the sky on a starry night, for no particular number \(k\) are we aware that exactly \(k\) stars are represented. Even so, there is likely to be some extra specificity. For example, in imagining an array of dots of the form just illustrated, one is unlikely to imagine just two columns of three dots, the rectangular array for \(n = 2\). Typically the subject will be aware of imagining an array with more than two columns. This entails that an image is likely to have unintended exclusions. In this case it would exclude the three-by-two array. An image of a triangle representing all angles as acute would exclude triangles with an obtuse angle or a right angle. The danger is that the visual reasoning will not be valid for the cases that are unintentionally excluded by the visual representation, with the result that the step to the conclusion is an unwarranted generalisation.

What should we make of this? First, let us note that in a few cases the image or diagram will not be over-specific. When in geometry all instances of the relevant class are congruent to one another, for instance all circles or all squares, the image or diagram will not be over-specific for a generalisation about that class; so there will be no unintended exclusions and no danger of unwarranted generalisation. Here then are possibilities for reliable visual thinking in proving.

To get clear about the other cases, where there is a danger of over generalizing, it helps to look at generalisation in ordinary non-visual reasoning. Schematically put, in reasoning about things of kind \(K\), once we have shown that from certain premisses it follows that such-and-such a condition is true of arbitrary instance \(c\), we can validly infer from those same premisses that that condition is true of all \(K\)s, with the proviso that neither the condition nor any premiss mentions \(c\). The proviso is required, because if a premiss or the condition does mention \(c\), the reasoning may depend on a property of \(c\) that is not shared by all other \(K\)s and so the generalisation would be unsafe. For a trivial example consider a step from “\(x = c\)” to “\(\forall x [x = c]\)”.

A question we face is whether, in order to come to know the truth of a conclusion by following an argument involving generalisation on an arbitrary instance (a.k.a. universal generalisation, or universal quantifier introduction), the thinking must include a conscious, explicit check that the proviso is met. It is clearly not enough that the proviso is in fact met. For in that case it might just be the thinker’s good luck that the proviso is met; hence the thinker would not know that the generalisation is valid and so would not have genuinely thought through the proof at that step.

This leaves two options. The strict option is that without a conscious, explicit check one has not really thought through the proof. The relaxed option is that one can properly think through the proof without checking that the proviso is met, but only if one is sensitive to the potential error and would detect it in otherwise similar arguments. For then one is not just lucky that the proviso is met. Being sensitive in this context consists in being alert to dependence on features of the arbitrary instance not shared by all members of the class of generalisation, a state produced by a combination of past experience and current vigilance. Without compelling reason to prefer one of these options, decisions on what is to count as proving or following a proof must be conditional.

How does all this apply to generalizing from visual thinking about an arbitrary instance? Take the example of the visual route to the formula for triangular numbers using the diagram of Figure 2 . The diagram reveals that the formula holds for the 6 th triangular number. The generalisation to all triangular numbers is justified only if the visuo-spatial method used is applicable to the \(n\) th triangular number for all positive integers \(n\), that is, provided that the method used does not depend on a property not shared by all positive integers. A conscious, explicit check that this proviso is met requires making explicit the method exemplified for 6 and proving that the method is applicable for all positive integers in place of 6. (For a similar idea in the context of automating visual arguments, see Jamnik 2001). This is not done in practice when thinking visually, and so if we accept the strict option for thinking through a proof involving generalisation, we would have to accept that the visual route to the formula for triangular numbers does not amount to thinking through a proof of it; and the same would apply to the familiar visual routes to other general positive integer formulas, such as that \(n^2 =\) the sum of the first \(n\) odd numbers.

But what if the strict option for proving by generalisation on an arbitrary instance is too strict, and the relaxed option is right? When arriving at the formula in the visual way indicated, one does not pay attention to the fact that the visual display represents the situation for the 6 th triangular number; it is as if the mind had somehow extracted a general schema of visual reasoning from exposure to the particular case, and had then proceeded to reason schematically, converting a schematic result into a universal proposition. What is required, on the relaxed option, is sensitivity to the possibility that the schema is not applicable to all positive integers; one must be so alert to ways a schema of the given kind can fall short of universal applicability that if one had been presented with a schema that did fall short, one would have detected the failure.

In the example at hand, the schema of visual reasoning involves at the start taking a number \(k\) to be represented by a column of \(k\) dots, thence taking the triangular array of \(n\) columns to represent the sum of the first \(n\) positive integers, thence taking that array combined with an inverted copy to make a rectangular array of \(n\) columns of \(n+1\) dots. For a schema starting this way to be universally applicable, it must be possible, given any positive integer \(n\), for the sum of the first \(n\) positive integers to be represented in the form of a triangular array, so that combined with an inverted copy one gets a rectangular array. This actually fails at the extreme case: \(n = 1\). The formula \((n.(n + 1))/2\) holds for this case; but that is something we know by substituting “1” for the variable in the formula, not by the visual method indicated. That method cannot be applied to \(n = 1\), because a single dot does not form a triangular array, and combined with a copy it does not form a rectangular array. But we can check that the method works for all positive integers after the first, using visual reasoning to assure ourselves that it works for 2 and that if the method works for \(k\) it works for \(k+1\). Together with this reflective thinking, the visual thinking sketched earlier constitutes following a proof of the formula for the \(n\) th triangular number for all integers \(n > 1\), at least if the relaxed view of thinking through a proof is correct. Similar conclusions hold in the case of other “dot” arguments (Giaquinto 1993, 2007: ch. 8). So in some cases when the visual representation carries unwanted detail, the danger of over-generalisation in visual reasoning can be overcome.

But the fact that this is frequently missed by commentators suggests that the required sensitivity is often absent. Missing an untypical case is a common hazard in attempts at visual proving. A well-known example is the proof of Euler’s formula \(V - E + F = 2\) for polyhedra by “removing triangles” of a triangulated planar projection of a polyhedron. One is easily convinced by the thinking, but only because the polyhedra we normally think of are convex, while the exceptions are not convex. But it is also easy to miss a case which is not untypical or extreme when thinking visually. An example is Cauchy’s attempted proof (Cauchy 1813) of the claim that if a convex polygon is transformed into another polygon keeping all but one of the sides constant, then if some or all of the internal angles at the vertices increase, the remaining side increases, while if some or all of the internal angles at the vertices decrease, the remaining side decreases. The argument proceeds by considering what happens when one transforms a polygon by increasing (or decreasing) angles, angle by angle. But in a trapezoid, changing a single angle can turn a convex polygon into a concave polygon, and this invalidates the argument (Lyusternik 1963).

The frequency of such mistakes indicates that visual arguments (other than symbol manipulations) often lack the transparency required for proof. Even when a visual argument is in fact sound, its soundness may not be clear, in which case the argument is not a way of proving the truth of the conclusion, though it may be a way of discovering it. But this is consistent with the claim that visual non-symbolic thinking can be (and often is) part of a way of proving something.

An example from knot theory will substantiate the modal part of this claim. To present the example, we need some background information, which will be given with a minimum of technical detail.

A knot is a tame closed non-self-intersecting curve in Euclidean 3-space.

In other words, knots are just the tame curves in Euclidean 3-space which are homeomorphic to a circle. The word “tame” here stands for a property intended to rule out certain pathological cases, such as curves with infinitely nested knotting. There is more than one way of making this mathematically precise, but we have no need for these details. A knot has a specific geometric shape, size and axis-relative position. Now imagine it to be made of flexible yet unbreakable yarn that is stretchable and shrinkable, so that it can be smoothly transformed into other knots without cutting or gluing. Since our interest in a knot is the nature of its knottedness regardless of shape, size or axis-relative position, the real focus of interest is not just the knot but all its possible transforms. A way to think of this is to imagine a knot transforming continuously, so that every possible transform is realized at some time. Then the thing of central interest would be the object that persists over time in varying forms, with knots strictly so called being the things captured in each particular freeze frame. Mathematically, we represent the relevant entity as an equivalence class of knots.

Two knots are equivalent iff one can be smoothly deformed into the other by stretching, shrinking, twisting, flipping, repositioning or in any other way that does not involve cutting, gluing or passing one strand through another.

The relevant kind of deformation forbids eliminating a knotted part by shrinking it down to a point. Again there are mathematically precise definitions of knot-equivalence. Figure 3 gives diagrams of equivalent knots, instances of a trefoil.

Diagrams like these are not merely illustrations; they also have an operational role in knot theory. But not any picture of a knot will do for this purpose. We need to specify:

A knot diagram is a regular projection of a knot onto a plane which, when there is a crossing, tells us which strand passes over the other.

Regularity here is a combination of conditions. In particular, regularity entails that not more than two points of the strict knot project to the same point on the plane, and that two points of the strict knot project to the same point on the plane only where there is a crossing. For more on diagrams in knot theory see (De Toffoli and Giardino 2014).

A major task of knot theory is to find ways of telling whether two knot diagrams are diagrams of equivalent knots. In particular we will want to know if a given knot diagram represents a knot equivalent to an unknot , that is, a knot representable by a knot diagram without crossings.

One way of showing that a knot diagram represents a knot equivalent to an unknot is to show that the diagram can be transformed into one without crossings by a sequence of atomic moves, known as Reidemeister moves. The relevant background fact is Reidemeister’s theorem, which links the visualizable diagrammatic changes to the mathematically precise definition of knot equivalence: Two knots are equivalent if and only if there is a finite sequence of Reidemeister moves taking a knot diagram of one to a knot diagram of the other. Figure 4 illustrates. Each knot diagram is changed into the adjacent knot diagram by a Reidemeister move; hence the knot represented by the leftmost diagram is equivalent to the unknot.

In contrast to these, the knot presented by the left knot diagram of Figure 3 , a trefoil, may seem impossible to deform into an unknot. And in fact it is. To prove it, we can use a knot invariant known as colourability. An arc in a knot diagram is a maximal part between crossings (or the whole thing if there are no crossings). Colourability is this:

A knot diagram is colourable if and only if each of its arcs can be coloured one of three different colours so that (a) at least two colours are used and (b) at each crossing the three arcs are all coloured the same or all coloured differently.

The reference to colours here is inessential. Colourability is in fact a specific case of a kind of combinatorial property known as mod \(p\) labelling (for \(p\) an odd prime). Colourability is a knot invariant in the sense that if one diagram of a knot is colourable every diagram of that knot and of any equivalent knot is colourable. (By Reidemeister’s theorem this can be proved by showing that each Reidemeister move preserves colourability.) A standard diagram of an unknot, a diagram without crossings, is clearly not colourable because it has only one arc (the whole thing) and so two colours cannot be used. So in order to complete proving that the trefoil is not equivalent to an unknot, we only need prove that our trefoil diagram is colourable. This can be done visually. Colour each arc of the knot diagram one of the three colours red, green or blue so that no two arcs have the same colour (or visualize this). Then do a visual check of each crossing, to see that at each crossing the three meeting arcs are all coloured differently. That visual part of the proof is clearly non-superfluous and non-replaceable (without changing the proof). Moreover, the soundness of the argument is quite transparent. So here is a case of a non-formal, non-symbolic visual way of proving a mathematical truth.

Where notions involving the infinite are in play, such as many involving limits, the use of diagrams is famously risky. For this reason it has been widely thought that, beyond some very simple cases, arguments in real and complex analysis in which diagrams have a non-superfluous role are not genuine proofs. Bolzano [1817] expressed this attitude with regard to the intermediate value theorem for the real numbers (IVT) before giving a purely analytic proof, arguing that spatial thinking could not be used to help justify the IVT. James Robert Brown (1999) takes issue with Bolzano on this point. The IVT is this:

If \(f\) is a real-valued function of a real variable continuous on the closed interval \([a, b]\) and \(f(a) < c < f(b)\), then for some \(x\) in \((a, b), f(x) = c\).

Brown focuses on the special case when \(c = 0\). As the IVT can be deduced easily from this special case using the theorem that the difference of two continuous functions is continuous, there is no loss of generality here. Alluding to a diagram like Figure 5, Brown (1999) writes

We have a continuous line running from below to above the \(x\)-axis. Clearly, it must cross that axis in doing so. (1999: 26)

Later he claims:

Using the picture alone, we can be certain of this result—if we can be certain of anything. (1999: 28)

Bolzano’s diagram-free proof of the IVT is an argument from what later became known as the Dedekind completeness of the real numbers: every non-empty set of reals bounded above (below) has a least upper bound (greatest lower bound). The value of Bolzano’s deduction of the IVT from the Dedekind completeness of the reals, according to Brown, is not that it proves the IVT but that it gives us confirmation of Dedekind completeness, just as an empirical hypothesis in empirical science gets confirmed by deducing some consequence of the hypothesis and observing those consequence to be true. This view assumes that we already know the IVT to be true by observing a diagram relevantly like Figure 5 .

That assumption is challenged by Giaquinto (2011). Once we distinguish graphical concepts from associated analytic concepts, the underlying argument from the diagram is essentially this.

• 1. Any function \(f\) which is \(\varepsilon\textrm{-}\delta\) continuous on \([a, b]\) with \(f (a) < 0 < f (b)\) has a visually continuous graphical curve from below the horizontal line representing the \(x\)-axis to above.
• 2. Any visually continuous graphical curve from below a horizontal line to above it meets the line at a crossing point.
• 3. Any function whose graphical curve meets the line representing the \(x\)-axis at a crossing point has a zero value.
• 4. So, any \(\varepsilon\textrm{-}\delta\) continuous function \(f\) on \([a, b]\) with \(f (a) < 0< f (b)\) has a zero value.

What is inferred from the diagram is premiss 2. Premisses 1 and 3 are assumptions linking analytical with graphical conditions. These linking assumptions are disputed. With regard to premiss 1 Giaquinto (2011) argues that there are functions on the reals which meet the antecedent condition but do not have graphical curves, such as continuous but nowhere differentiable functions and functions which oscillate with unbounded frequency e.g., \(f(x) = x \cdot\sin(1/x)\) for non-zero \(x\) in \([-1, 1]\) and \(f(0) = 0\).

With regard to premiss 3 it is argued that, under the standard conventions of graphical representation of functions in a Cartesian co-ordinate frame, the graphical curve for \(x^2 - 2\) in the rationals is the same as the graphical curve for \(x^2- 2\) in the reals. This is because every real is a limit point of rationals; so for every point \(P\) with one or both co-ordinates irrational, there are points arbitrarily close to \(P\) with both co-ordinates rational; so no gaps would appear if irrational points were removed from the curve for \(x^2- 2\) in the reals. But for \(x\) in the rational interval [0, 2] the function \(x^2- 2\) has no zero value, even though it has a graphical curve which visually crosses the line representing the \(x\)-axis. So one cannot read off the existence of a zero of \(x^2- 2\) on the reals from the diagram; one needs to appeal to some property of the reals which the rationals lack, such as Dedekind completeness.

This raises some obvious questions. Do any theorems of analysis have proofs in which diagrams have a non-superfluous role? Littlewood (1953: 54–5) thought so and gives an example which is examined in Giaquinto (1994). If so, can we demarcate this class of theorems by some mathematical feature of their content? Another question is whether there is a significantly broad class of functions on the reals for which we could prove an intermediate value theorem (i.e., restricted to that class).

If there are theorems of analysis provable with diagrams we do not yet have a mathematical demarcation criterion for them. A natural place to look would be O-minimal structures on the reals—this was brought to the author’s attention by Ethan Galebach. This is because of some remarkable theorems about such structures which exclude all the pathological (hence vision-defying) functions on the reals (Van den Dries 1998), such as continuous nowhere differentiable functions and “space-filling” curves i.e., continuous surjections \(f:(0, 1)\rightarrow(0, 1)^2\). Is the IVT for functions in an O-minimal structure on the reals provable by visual means? Certainly one objection to the visual argument for the unrestricted IVT does not apply when the restriction is in place. This is the objection that continuous nowhere differentiable functions, having no graphical curve, provide counterexamples to the premiss that any \(\varepsilon\textrm{-}\delta\) continuous function \(f\) on \([a, b]\) with \(f (a) < c < f (b)\) has a visually continuous graphical curve from below the horizontal line representing \(y = c\) to above. But the existence of continuous functions with no graphical curve is not the only objection to the visual argument, contrary to a claim of Azzouni (2013: 327). There are also counterexamples to the premiss that any function that does have a graphical curve which visibly crosses the line representing \(y = c\) takes \(c\) as a value, e.g., the function \(x^2 - 2\) on the rationals with \(c = 0\). So the question of a visual proof of the IVT restricted to functions in an O-minimal structure on the reals is still open at the time of writing.

4. Visual thinking and discovery

Though philosophical discussion of visual thinking in mathematics has concentrated on its role in proof, visual thinking may be more valuable for discovery than proof. Three kinds of discovery important in mathematical practice are these:

• (1) propositional discovery (discovering, of a proposition, that it is true),
• (2) discovering a proof strategy (or more loosely, getting the idea for a proof of a proposition), and
• (3) discovering a property or kind of mathematical entity.

In the following subsections visual discovery of these kinds will be discussed and illustrated.

To discover a truth, as that expression is being used here, is to come to believe it by one’s own lights (as opposed to reading it or being told) in a way that is reliable and involves no violation of epistemic rationality (given one’s epistemic state). One can discover a truth without being the first to discover it (in this context); it is enough that one comes to believe it in an independent, reliable and rational way. The difference between merely discovering a truth and proving it is a matter of transparency: for proving or following a proof the subject must be aware of the way in which the conclusion is reached and the soundness of that way; this is not required for discovery.

Sometimes one discovers something by means of visual thinking using background knowledge, resulting in a cogent argument from which one could construct a proof. A nice example is a visual argument that any knot diagram with a finite number of crossings can be turned into a diagram of an unknot by interchanging the over-strand and under-strand of some of its crossings (Adams 2001: 58–90). That argument is a bit too long to present accessibly here. For a short example, here is a way of discovering that the geometric mean of two positive numbers is less than or equal to their arithmetic mean (Eddy 1985) using Figure 6.

Two circles (with diameters \(a\) and \(b\)) meet at a single point. A line is drawn between their centres through their common point; its length is \((a + b)/2\), the sum of the two radii. This line is the hypotenuse of a right angled triangle with one other side of length \((a - b)/2\), the difference of the radii. Pythagoras’s theorem is used to infer that the remaining side of the right-angled triangle has length \(\sqrt{(ab)}\).Then visualizing what happens to the triangle when the diameter of the smaller circle varies between 0 and the diameter of the larger circle, one infers that \(0 < \sqrt{(ab)} < (a + b)/2\); then verifying symbolically that \(\sqrt{(ab)} = (a + b)/2\) when \(a = b\), one concludes that for positive \(a\) and \(b\), \(\sqrt{(ab)} \le (a + b)/2\).

This thinking does not constitute a case of proving or following a proof of the conclusion, because it involves a step which we cannot clearly tell is valid. This is the step of attempting to visually imagine what would happen when the smaller circle varies in diameter between 0 and the diameter of the larger circle and inferring from the resulting experience that the line joining the centres of the circles will always be longer than the horizontal line from the centre of the smaller circle to the vertical diameter of the larger circle. This step seems sound (does not lead us into error) and may be sound; but its soundness is opaque. If in fact it is sound, the whole thinking process is a reliable way of reaching the conclusion; so in the absence of factors that would make it irrational to trust the thinking, it would be a way of discovering the conclusion to be true.

In some cases visual thinking inclines one to believe something on the basis of assumptions suggested by the visual representation that remain to be justified given the subject’s current knowledge. In such cases there is always the danger that the subject takes the visual representation to show the correctness of the assumptions and ends up with an unwarranted belief. In such a case, even if the belief is true, the subject has not made a discovery, as the means of belief-acquisition is unreliable. Here is an example using Figure 7 (Montuchi and Page 1988).

Using this diagram one can come to think the following about the real numbers. When for a constant \(k\) the positive values of \(x\) and \(y\) are constrained to satisfy the equation \(x \cdot y = k\), the positive values of \(x\) and \(y\) for which \(x + y\) is minimal are \(x = \sqrt{k} = y\). (Let “#” denote this claim.)

Suppose that one knows the conventions for representing functions by graphs in a Cartesian co-ordinate system, knows also that the diagonal represents the function \(y = x\), and that a line segment with gradient –1 from \((0, b)\) to \((b, 0)\) represents the function \(x + y = b\). Then looking at the diagram may incline one to think that for no positive value of \(x\) does the value of \(y\) in the function \(x\cdot y = k\) fall below the value of \(y\) in \(x + y = 2\sqrt{k}\), and that these functions coincide just at the diagonal. From these beliefs the subject may (correctly) infer the conclusion #. But mere attention to the diagram cannot warrant believing that, for a given positive \(x\)-value, the \(y\)-value of \(x\cdot y = k\) never falls below the \(y\)-value of \(x + y = 2\sqrt{k}\) and that the functions coincide just at the diagonal; for the conventions of representation do not rule out that the curve of \(x\cdot y = k\) meets the curve of \(x + y = 2\sqrt{k}\) at two points extremely close to the diagonal, and that the former curve falls under the latter in between those two points. So the visual thinking is not in this case a means of discovering proposition #.

But it is useful because it provides the idea for a proof of the conclusion—one of the major benefits of visual thinking in mathematics. In brief: for each equation \((x\cdot y = k\); \(x + y = 2\sqrt{k})\) if \(x = y\), their common value is \(\sqrt{k}\). So the functions expressed by those equations meet at the diagonal. To show that, for a fixed positive \(x\)-value, the \(y\)-values of \(x\cdot y = k\) never fall below the \(y\)-values of \(x + y = 2\sqrt{k}\), it suffices to show that \(2\sqrt{k} - x \le k/x\). As a geometric mean is less than or equal to the corresponding arithmetic mean, \(\sqrt{[x \cdot (k/x)]} \le [x + (k/x)]/2\). So \(2\sqrt{k} \le x + (k/x)\). So \(2\sqrt{k} - x \le k/x\).

In this example, visual attention to, and reasoning about, the diagram is not part of a way of discovering the conclusion. But if it gave one the idea for the argument just given, it would be part of what led to a way of discovering the conclusion, and that is important.

Can visual thinking lead to discovery of an idea for a proof in more advanced contexts? Yes. Carter (2010) gives an example from free probability theory. The case is about certain permutations (those denoted by “\(p\)” with a circumflex in Carter 2010) on a finite set of natural numbers. Using specific kinds of diagram, easily seen properties of the diagrams lead one naturally to certain properties of the permutations (crossing and non-crossing, having neighbouring pairs), and to a certain operation (cancellation of neighbouring pairs). All of these have algebraic definitions, but the ideas defined were noticed by thinking in terms of the diagrams. For the relevant permutations \(\sigma\), \(\sigma(\sigma(n)) = n\); so a permutation can be represented by a set of lines joining dots. The permutations represented on the left and right in Figure 8 are non-crossing and crossing respectively, the former with neighbouring pairs \(\{2, 3\}\) and \(\{6, 7\}\).

A permutation \(\sigma\) of \(\{1, 2, \ldots, 2p\}\) is defined to have a crossing just when there are \(a\), \(b\), \(c\), \(d\) in \(\{1, 2, \ldots, 2p\}\) such that \(a < b < c < d\) and \(\sigma(a) = c\) and \(\sigma(b) = d\). The focus is on the proof of a theorem which employs this notion. (The theorem is that when a permutation of \(\{1, 2, \ldots, 2p\}\) of the relevant kind is non-crossing, there will be exactly \(p+1\) R-equivalence classes, where \(R\) is a certain equivalence relation on \(\{1, 2, \ldots, 2p\}\) defined in terms of the permutation.) Carter says that the proofs of some lemmas “rely on a visualization of the setup”, in that to grasp the correctness of one or more of the steps one needs to visualize the situation. There is also a nice example of some reasoning in terms of a diagram which gives the idea for a proof (“suggests a proof strategy”) for the lemma that every non-crossing permutation has a neighbouring pair. Reflection on a diagram such as Figure 9 does the work.

The reasoning is this. Suppose that \(\pi\) has no neighbouring pair. Choose \(j\) such that \(\pi(j) - j = a\) is minimal, that is, for all \(k, \pi(j) - j \le \pi(k) - k\). As \(\pi\) has no neighbouring pair, \(\pi(j+1) \ne j\). So either \(\pi(j+1)\) is less than \(j\) and we have a crossing, or by minimality of \(\pi(j) - j\), \(\pi(j+1)\) is greater than \(j+a\) and again we have a crossing. Carter reports that this disjunction was initially believed by thinking in term of the diagram, and the proof of the lemma given in the published paper is a non-diagrammatic “version” of that reasoning. In this case study, visual thinking is shown to contribute to discovery in several ways; in particular, by leading the mathematicians to notice crucial properties—the “definitions are based on the diagrams”—and in giving them the ideas for parts of the overall proof.

In this section I will illustrate and then discuss the use of visual thinking in discovering kinds of mathematical entity, by going through a few of the main steps leading to geometric group theory, a subject which really took off in the 1980s through the work of Mikhail Gromov. The material is set out nicely in greater depth in Starikova (2012).

Sometimes it can be fruitful to think of non-spatial entities, such as algebraic structures, in terms of a spatial representation. An example is the representation of a finitely generated group by a Cayley graph. Let \((G, \cdot)\) be a group and \(S\) a finite subset of \(G\). Let \(S^{-1}\) be the set of inverses of members of \(S\). Then \((G, \cdot)\) is generated by \(S\) if and only if every member of \(G\) is the product (with respect to \(\cdot\)) of members of \(S\cup S^{-1}\). In that case \((G, \cdot, S)\) is said to be a finitely generated group. Here are a couple of examples.

First consider the group \(S_{3}\) of permutations of 3 elements under composition. Letting \(\{a, b, c\}\) be the elements, all six permutations can be generated by \(\rf\) and \(\rr\) where

\(\rf\) (for “flip”) fixes a and swaps \(b\) with \(c\), i.e., it takes to \(\langle a, b, c\rangle\) to \(\langle a, c, b\rangle\), and

\(\rr\) (for “rotate”) takes \(\langle a, b, c\rangle\) to \(\langle c, a, b\rangle\).

The Cayley graph for \((S_{3}, \cdot, \{\rf, \rr\})\) is a graph whose vertices represent the members of \(S_{3}\) and two “colours” of directed edges, representing composition with \(\rf\) and composition with \(\rr\). Figure 10 illustrates: red directed edges represent composition with \(\rr\) and black edges represent composition with \(\rf\). So a red edge from a vertex \(\rv\) representing \(\rs\) in \(S_{3}\) ends at a vertex representing \(\rs\rr\) and a black edge from \(\rv\) ends at a vertex representing \(\rs\rf\). (Notation: “\(\rs\rr\)” abbreviates “\(\rs \cdot \rr\)” which here denotes “\(\rs\) followed by \(\rr\)”; same for “\(\rf\)” in place of “\(\rr\)”.) A black edge has arrowheads both ways because \(\rf\) is its own inverse, that is, flipping and flipping again takes you back to where you started. (Sometimes a pair of edges with arrows in opposite directions is used instead.) The symbol “\(\re\)” denotes the identity.

An example of a finitely generated group of infinite order is \((\mathbb{Z}, +, \{1\})\). We can get any integer by successively adding 1 or its additive inverse \(-1\). Since 3 added to the inverse of 2 is 1, and 2 added to the inverse of 3 is \(-1\), we can get any integer by adding members of \(\{2, 3\}\) and their inverses. Thus both \(\{1\}\) and \(\{2, 3\}\) are generating sets for \((\mathbb{Z}, +)\). Figure 11 illustrates part of the Cayley graph for \((\mathbb{Z}, +, \{2, 3\})\). The horizontal directed edges represent +2. The directed edges ascending or descending obliquely represent \(+3\).

Another example of a generated group of infinite order is \(F_2\), the free group generated by a pair of members. The first few iterations of its Cayley graph are shown in Figure 12, where \(\{a, b\}\) is the set of generators and a right horizontal move between adjacent vertices represents composition with \(a\), an upward vertical move represents composition with \(b\), and leftward and downward moves represent composition with the inverse of \(a\) and the inverse of \(b\) respectively. The central vertex represents the identity.

Thinking of generated groups in terms of their Cayley graphs makes it very natural to view them as metric spaces. A path is a sequence of consecutively adjacent edges, regardless of direction. For example in the Cayley graph for \((\mathbb{Z}, +, \{2, 3\})\) the edges from \(-2\) to 1, from 1 to \(-1\), from \(-1\) to 2 (in that order) constitute a path, representing the action, starting from \(-2\), of adding 3, then adding \(-2\), then adding 3. Taking each edge to have unit length, the metric \(d_S\) for a group \(G\) generated by a finite subset \(S\) of \(G\) is defined: for any \(g\), \(h \in G\), \(d_{S}(g, h) =\) the length of a shortest path from \(g\) to \(h\) in the Caley graph of \((G, \cdot, S)\). This is the word metric for this generated group.

Viewing a finitely generated group as a metric space allows us to consider its growth function \(\gamma(n)\) which is the cardinality of the “ball” of radius \(\le n\) centred on the identity (the number of members of the group whose distance from the identity is not greater than \(n\)). A growth function for a given group depends on the set of generators chosen, but when the group is infinite the asymptotic behaviour as \(n \rightarrow \infty\) of the growth functions is independent of the set of generators.

Noticing the possibility of defining a metric on generated groups did not require first viewing diagrams of their Cayley graphs. This is because a word in the generators is just a finite sequence of symbols for the generators or their inverses (we omit the symbol for the group operation), and so has an obvious length visually suggested by the written form of the word, namely the number of symbols in the sequence; and then it is natural to define the distance between group members \(g\) and \(h\) to be the length of a shortest word that gets one from \(g\) to \(h\) by right multiplication, that is, \(\textrm{min}\{\textrm{length}(w): w = g^{-1}h\}\).

However, viewing generated groups by means of their Cayley graphs was the necessary starting point for geometric group theory, which enables us to view finitely generated groups of infinite order not merely as graphs or metric spaces but as geometric entities. The main steps on this route will be sketched briefly here; for more detail see Starikova (2012) and the references therein. The visual key is to start thinking in terms of the “coarse geometry” of the Cayley graph of the generated group, by zooming out in visual imagination so far that the discrete nature of the graph is transformed into a traditional geometrical object. For example, the Cayley graph of a generated group of finite order such as \((S_{3}, \cdot, \{f, r\})\) illustrated in Figure 11 becomes a dot; the Cayley graph for \((\mathbb{Z}, +, \{2, 3\})\) illustrated in Figure 12 becomes an uninterrupted line infinite in both directions.

The word metric of a generated group is discrete: the values are always in \(N\). How is this visuo-spatial association of a discrete metric space with a continuous geometrical object achieved mathematically? By quasi-isometry. While an isometry from one metric space to another is a distance preserving map, a quasi-isometry is a map which preserves distances to within fixed linear bounds. Precisely put, a map \(f\) from \((S, d)\) to \((S', d')\) is a quasi-isometry iff for some real constants \(L > 0\) and \(K \ge 0\) and all \(x\), \(y\) in \(S\) \[ d(x, y)/L - K \le d'(f(x), f(y)) \le L \cdot d(x, y) + K. \]

The spaces \((S, d)\) and \((S', d')\) are quasi - isometric spaces iff the quasi-isometry \(f\) is also quasi-surjective, in the sense that there is a real constant \(M \ge 0\) such that every point of \(S'\) is no further than \(M\) away from some point in the image of \(f\).

For example, \((\mathbb{Z}, d)\) is quasi-isometric to \((\mathbb{R}, d)\) where \(d(x, y) = |y - x|\), because the inclusion map \(\iota\) from \(\mathbb{Z}\) to \(\mathbb{R}\), \(\iota(n) = n\), is an isometry hence a quasi-isometry with \(L = 1\) and \(K = 0\), and each point in \(\mathbb{R}\) is no further than \(1/2\) away from an integer (in \(\mathbb{R}\)). Also, it is easy to see that for any real number \(x\), if \(g(x) =\) the nearest integer to \(x\) (or the greatest integer less than \(x\) if it is midway between integers) then \(g\) is a quasi-isometry from \(\mathbb{R}\) to \(\mathbb{Z}\) with \(L = 1\) and \(K =\frac{1}{2}\);.

The relation between metric spaces of being quasi-isometric is an equivalence relation. Also, if \(S\) and \(T\) are generating sets of a group \((G, \cdot)\), the Cayley graphs of \((G, \cdot, S)\) and \((G, \cdot, T)\) with their word metrics are quasi-isometric spaces. This means that properties of a generated group which are quasi-isometric invariants will be independent of the choice of generating set, and therefore informative about the group itself.

Moreover, it is easy to show that the Cayley graph of a generated group with word metric is quasi-isometric to a geodesic space. [ 1 ] A triangle with vertices \(x\), \(y\), \(z\) in this space is the union of three geodesic segments, between \(x\) and \(y\), between \(y\) and \(z\), and between \(z\) and \(x\). This is the gateway for the application of Gromov’s insights, some of which can be grasped with the help of visual geometric thinking.

Here are some indications. Recall the Poincaré open disc model of hyperbolic geometry: geodesics are diameters or arcs of circles orthogonal to the boundary, with unit distance represented by ever shorter Euclidean distances as one moves from the centre towards the boundary. (The boundary is not part of the model). All triangles have angle sum \(< \pi\) ( Figure 13, left ), and there is a global constant δ such that all triangles are δ-thin in the following sense:

A triangle \(T\) is δ- thin if and only if any point on one side of \(T\) lies within δ of some point on one of the other two sides.

This condition is equivalent to the condition that each side of \(T\) lies within the union of the δ-neighbourhoods of the other two sides, as illustrated in Figure 13 , right. There is no constant δ such that all triangles in a Euclidean plane are δ-thin, because for any δ there are triangles large enough that the midpoint of a longest side lies further than δ from all points on the other two sides.

Figure 13 [ 2 ]

The definition of thin triangles is sufficiently general to apply to any geodesic space and allows a generalisation of the concept of hyperbolicity beyond its original context:

• A geodesic space is hyperbolic iff for some δ all its triangles are δ-thin.
• A group is hyperbolic iff it has a Cayley graph quasi-isometric to a hyperbolic geodesic space.

The class of hyperbolic groups is large and includes important subkinds, such as finite groups, free groups and the fundamental groups of surfaces of genus \(\ge 2\). Some striking theorems have been proved for them. For example, for every hyperbolic group the word problem is solvable, and every hyperbolic group has a finite presentation. So we can reasonably conclude that the discovery of this mathematical kind, the hyperbolic groups, has been fruitful.

How important was visual thinking to the discoveries leading to geometric group theory? Visual thinking was needed to discover Cayley graphs as a means of representing finitely generated groups. This is not the triviality it might seem: Cayley graphs must be distinguished from the diagrams we use to present them visually. A Cayley graph is a mathematical representation of a generated group, not a visual representation. It consists of the following components: a set \(V\) (“vertices”), a set \(E\) of ordered pairs of members of \(V\) (“directed edges”) and a partition of \(E\) into distinguished subsets, (“colours”, each one for representing right multiplication by a particular generator). The Cayley graph of a generated group of infinite order cannot be fully represented by a diagram given the usual conventions of representation for diagrams of graphs, and distinct diagrams may visually represent the same Cayley graph: both diagrams in Figure 14 can be labelled so that under the usual conventions they represent the Cayley graph of \((S_{3}, \cdot, \{f, r\})\), already illustrated by Figure 10 . So the Cayley graph cannot be a diagram.

Diagrams of Cayley graphs were important in prompting mathematicians to think in terms of the coarse-grained geometry of the graphs, in that this idea arises just when one thinks in terms of “zooming out” visually. Gromov (1993) makes the point in a passage quoted in Starikova (2012:138)

This space [a Cayley graph with the word metric] may appear boring and uneventful to a geometer’s eye since it is discrete and the traditional (e.g., topological and infinitesimal) machinery does not run in [the group] Γ. To regain the geometric perspective one has to change one’s position and move the observation point far away from Γ. Then the metric in Γ seen from the distance \(d\) becomes the original distance divided by \(d\) and for \(d \rightarrow \infty\) the points in Γ coalesce into a connected continuous solid unity which occupies the visual horizon without any gaps and holes and fills our geometer’s heart with joy.

In saying that one has to move the observation point far away from Γ so that the points coalesce into a unity which occupies the visual horizon, he makes clear that visual imagination is involved in a crucial step on the road to geometric group theory. Visual thinking is again involved in discovering hyperbolicity as a property of general geodesic spaces from thinking about the Poincaré disk model of hyperbolic geometry. It is hard to see how this property would have been discovered without the use of visual resources.

While there is no reason to think that mental arithmetic (mental calculation in the integers and rational numbers) typically involves much visual thinking, there is strong evidence of substantial visual processing in the mental arithmetic of highly trained abacus users.

In earlier times an abacus would be a rectangular board or table surface marked with lines or grooves along which pebbles or counters could be moved. The oldest surviving abacus, the Salamis abacus, dated around 300 BCE, is a white marble slab, with markings designed for monetary calculation (Fernandes 2015, Other Internet Resources). These were superseded by rectangular frames within which wires or rods parallel to the short sides are fixed, with moveable holed beads on them. There are several kinds of modern abacus — the Chinese suanpan, the Russian schoty and the Japanese soroban for example — each kind with variations. Evidence for visual processing in mental arithmetic comes from studies with well trained users of the soroban, an example of which is shown in Figure 15.

Each column of beads represents a power of 10, increasing to the left. The horizontal bar, sometimes called the reckoning bar , separates the beads on each column into one bead of value 5 above and four beads of value 1 below. The number represented in a column is determined by the beads which are not separated from the reckoning bar. A column on which all beads are separated by a gap from the bar represents zero. For example, the number 6059 is represented on a portion of a schematic soroban in Figure 16.

On some sorobans there is a mark on the reckoning bar at every third column; if a user chooses one of these as a unit column, the marks will help the user keep track of which columns represent which powers of ten. Calculations are made by using forefinger and thumb to move beads according to procedures for the standard four numerical operations and for extraction of square and cube roots (Bernazzani 2005,Other Internet Resources). Despite the fact that the soroban has a decimal place representation of numbers, the soroban procedures are not ‘translations’ of the procedures normally taught for the standard operations using arabic numerals. For example, multidigit addition on a soroban starts by adding highest powers of ten and proceeds rightwards to lower powers, instead of starting with units thence proceeding leftwards to tens, hundreds and so on.

People trained to use a soroban often learn to do mental arithmetic by visualizing an abacus and imagining moving beads on it in accordance with the procedures learned for arithmetical calculations (Frank and Barner 2012). Mental abacus (MA), as this kind of mental arithmetic is known, compares favourably with other kinds of mental calculation for speed and accuracy (Kojima 1954) and MA users are often found among the medallists in the Mental Calculation World Cup.

Although visual and manual motor imagery is likely to occur, cognitive scientists have probed the question whether the actual processes of MA calculation consist in or involve imagining performing operations on a physical abacus. Brain imaging studies provide one source of evidence bearing on this question. Comparing well-trained abacus calculators with matched controls, evidence has been found that MA involves neural resources of visuospatial working memory with a form of abacus which does not depend on the modality (visual or auditory) of the numerical inputs (Chen et al. 2006). Another imaging study found that, compared to controls without abacus training, subjects with long term MA training from a young age had enhanced brain white matter related to motor and visuospatial processes (Hu et al. 2011).

Behavioural studies provide more evidence. Tests on expert and intermediate level abacus users strongly suggest that MA calculators mentally manipulate an abacus representation so that it passes through the same states that an actual abacus would pass through in solving an addition problem. Without using an actual abacus MA calculators were able to answer correctly questions about intermediates states unique to the abacus-based solution of a problem; moreover, their response times were a monotonic function of the position of the probed state in the sequence of states of the abacus process for solving the problem (Stigler 1984). On top of the ‘intermediate states’ evidence, there is ‘error type’ evidence. Mental addition tests comparing abacus users with American subjects revealed that abacus users made errors of a kind which the Americans did not make, but which were predictable from the distribution of errors in physical abacus addition (Stigler 1984).

Another study found evidence that when a sequence of numbers is presented auditorily (as a verbal whole “three thousand five hundred and forty seven” or as a digit sequence “Three, five, four, seven”) abacus experts encode it into an imaged abacus display, while non-experts encode it verbally (Hishitani 1990).

Further evidence comes from behavioural interference studies. In these studies subjects have to perform mental calculations, with and without a task of some other kind to be performed during the calculation, with the aim of seeing which kinds of task interfere with calculation as measured by differences of reaction time and error rate. An early study found that a linguistic task interfered weakly with MA performance (unless the linguistic task was to answer a mathematical question), while motor and visual tasks interfered relatively strongly. These findings suggested to the paper’s authors that MA representations are not linguistic in nature but rely on visual mechanisms and, for intermediate practitioners, on motor mechanisms as well (Hatano et al. 1977).

These studies provide impressive evidence that MA does involve mental manipulation of a visualized abacus. However, limits of the known capacities for perceiving or representing pluralities of objects seem to pose a problem. We have a parallel individuation system for keeping track of up to four objects simultaneously and an approximate number system (ANS) which allows us to gauge roughly the cardinality of a set of things, with an error which increases with the size of the set. The parallel individuation system has a limit of three or four objects and the ANS represents cardinalities greater than four only approximately. Yet mental abacus users would need to hold in mind with precision abacus representations involving a much larger number of beads than four (and the way in which those beads are distributed on the abacus). For example, the number 439 requires a precise distribution of twelve beads. Frank and Barner (2012) address this problem. In some circumstances we can perceive a plurality of objects as a single entity, a set, and simultaneously perceive those objects as individuals. There is evidence that we can keep track of up to three such sets in parallel and simultaneously make reliable estimates of the cardinalities of the sets (if not more than four). If the sets themselves can be easily perceived as (a) divided into disjoint subsets, e.g. columns of beads on an abacus, and (b) structured in a familiar way, e.g. as a distribution of four beads below a reckoning bar and one above, we have the resources for recognising a three-digit number from its abacus representation. The findings of (Frank and Barner 2012) suggest that this is what happens in MA: a mental abacus is represented in visuospatial working memory by splitting it into a series of columns each of which is stored as a unit with its own detailed substructure.

These cognitive investigations confirm the self-reports of mental abacus users that they calculate mentally by visualizing operating on an abacus as they would operate on a physical abacus. (See the 20-second movie Brief interview with mental abacus user , at the Stanford Language and Cognition Lab, for one such self-report.) There is good evidence that MA often involves processes linked to motor cognition in addition to active visual imagination. Intermediate abacus users often make hand movements, without necessarily attending to those movements during MA calculation, as shown in the second of the three short movies just mentioned. Experiments to test the possible role of motor processes in MA resulted in findings which led the authors to conclude that premotor processes involved in the planning of hand movements were involved in MA (Brooks et al. 2018).

6. A priori and a posteriori roles of visual experience

In coming to know a mathematical truth visual experience can play a merely “enabling” role. For example, visual experience may have been a factor in a person’s getting certain concepts involved in a mathematical proposition, thus enabling her to understand the proposition, without giving her reason to believe it. Or the visual experience of reading an argument in a text book may enable one to find out just what the argument is, without helping her tell that the argument is sound. In earlier sections visual experience has been presented as having roles in proof and propositional discovery that are not merely enabling. On the face of it this raises a puzzle: mathematics, as opposed to its application to natural phenomena, has traditionally been thought to be an a priori science; but if visual experience plays a role in acquiring mathematical knowledge which is not merely enabling, the result would surely be a posteriori knowledge, not a priori knowledge. Setting aside knowledge acquired by testimony (reading or hearing that such-&-such is the case), there remain plenty of cases where sensory experience seems to play an evidential role in coming to know some mathematical fact.

A plausible example of the evidential use of sensory experience is the case of a child coming to know that \(5 + 3 = 8\) by counting on her fingers. While there may be an important \(a\) priori element in the child’s appreciation that she can reliably generalise from the result of her counting experiment, getting that result by counting is an a posteriori route to it. For another example, consider the question: how many vertices does a cube have? With the background knowledge that cubes do not vary in shape and that material cubes do not differ from geometrical cubes in number of vertices (where a “vertex” of a material cube is a corner), one can find the answer by visually inspecting a material cube. Or if one does not have a material cube to hand, one can visually imagine a cube, and by attending to its top and bottom faces extract the information that the vertices of the cube are exactly the vertices of these two quadrangular faces. When one gets the answer by inspecting a material cube, the visual experience contributes to one’s grounds for believing the answer and that contribution is part of what makes the belief state knowledge. So the role of the visual experience is evidential; hence the resulting knowledge is not a priori . When one gets the answer by visually imagining a cube, one is drawing on the accumulated cognitive effects of past experiences of seeing material cubes to bring to mind what a cube looks like; so the experience of visual imagining has an indirectly evidential role in this case.

Do such examples show that mathematics is not an a priori science? Yes, if an a priori science is understood to be one whose knowable truths are all knowable only in an a priori way, without use of sense experience as evidence. No, if an a priori science is one whose knowable truths are all knowable in an a priori way, allowing that some may be knowable also in an a posteriori way.

Many cases of proving something (or following a proof of it) involve making, or imagining making, changes in a symbol array. A standard presentation of the proof of left-cancellation in group theory provides an example. “Left-cancellation” is the claim that for any members \(a\), \(b\), \(c\) of a group with operation \(\cdot\) and identity element \(\mathbf{e}\), if \(a \cdot b = a \cdot c\), then \(b = c\). Here is (the core of) a proof of it:

Suppose that one comes to know left-cancellation by following this sequence of steps. Is this an a priori way of getting this knowledge? Although following a mathematical proof is thought to be a paradigmatically a priori way of getting knowledge, attention to the role of visual experience here throws this into doubt. The case for claiming that the visual experience has an evidential role is as follows.

The visual experience reveals not only what the steps of the argument are but also that they are valid, thereby contributing to our grounds for accepting the argument and believing its conclusion. Consider, for example, the step from the second equation to the third. The relevant background knowledge, apart from the logic of identity, is that a group operation is associative. This fact is usually represented in the form of an equation that simply relocates brackets in an obvious way:

We see that relocating the brackets in accord with this format, the left-hand term of the second equation is transformed into the left-hand term of the third equation, and the same for the right-hand terms. So the visual experience plays an evidential role in our recognising as valid the step from the second equation to the third. Hence this quite standard route to knowledge of left-cancellation turns out to be a posteriori , even though it is a clear case of following a proof.

Against this, one may argue that the description just given of what is going on in following the proof is not strictly correct, as follows. Exactly the same proof can be expressed in natural language, using “the composition of \(x\) with \(y\)” for “\(x \cdot y\)”, but the result would be hard to take in. Or the proof can be presented using a different notational convention, one which forces a quite different expression of associativity. For example, we can use the Polish convention of putting the operation symbol before the operands: instead of “\(x \cdot y\)” we put “\(\cdot x y\)”. In that case associativity would be expressed in the following way, without brackets:

The equations of the proof would then need to be re-symbolised; but what is expressed by each equation after re-symbolisation and the steps from one to the next would be exactly as before. So we would be following the very same proof, step by step. But we would not be using visual experiences involved to notice the relocation of brackets this time. This suggests that the role of the different visual experiences involved in following the argument in its different guises is merely to give us access to the common reasoning: the role of the experience is merely enabling. On this account the visual experience does not strictly and literally enable us to see that any of the steps are valid; rather, recognition of (or sensitivity to) the validity of the steps results from cognitive processing at a more abstract level.

Which of these rival views is correct? Does our visual experience in following the argument presented with brackets (1) reveal to us the validity of some of the steps, given the relevant background knowledge ? Or (2) merely give us access to the argument? The core of the argument against view (1) is this:

Seeing the relocation of brackets is not essential to following the argument.

So seeing merely gives access to the argument; it does not reveal any step to be valid.

The step to this conclusion is faulty. How one follows a proof may, and in this case does, depend on how it is presented, and different ways of following a proof may be different ways of coming to know its conclusion. While seeing the relocation of brackets is not essential to all ways of following this argument, it is essential to the normal way of following the argument when it is symbolically presented with brackets in the way given above.

Associativity, expressed without symbols, is this: When the binary group operation is applied twice in succession on an ordered triple of operands \(\langle a, b, c\rangle\), it makes no difference whether the first application is to the initial two operands or the final two operands. While this is the content of associativity, for ease of processing associativity is almost always expressed as a symbol-manipulation rule. Visual perception is used to tell in particular cases whether the rule thus expressed is correctly implemented, in the context of prior knowledge that the rule is correct. What is going on here is a familiar division of labour in mathematical thinking. We first establish the soundness of a rule of symbol-manipulation (in terms of the governing semantic conventions—in this case the matter is trivial); then we check visually that the rule is correctly implemented. Processing at a more abstract, semantic level is often harder than processing at a purely syntactic level; it is for this reason that we often resort to symbol-manipulation techniques as proxy for reasoning directly with meanings to solve a problem. (What is six eighths divided by three fifths, without using any symbolic technique?) When we do use symbol-manipulation in proving or following a proof, visual experience is required to discern that the moves conform to permitted patterns and thus contributes to our grounds for accepting the argument. Then the way of coming to know the conclusion has an a posteriori element.

Must a use of visual experience in knowledge acquisition be evidential , if the visual experience is not merely enabling? Here is an example which supports a negative answer. Imagine a square or look at a drawing of one. Each of its four sides has a midpoint. Now visualize the “inner” square whose sides run between the midpoints of adjacent sides of the original square (Figure 17, left). By visualizing this figure, it should be clear that the original square is composed precisely of the inner square plus four corner triangles, each side of the inner square being the base of a corner triangle. One can now visualize the corner triangles folding over, with creases along the sides of the inner square. The starting and end states of the imagery transformation can be represented by the left and right diagrams of Figure 17.

Visualizing the folding-over within the remembered frame of the original square results in an image of the original square divided into square quarters, its quadrants, and the sides of the inner square seem to be diagonals of the quadrants. Many people conclude that the corner triangles can be arranged to cover the inner square exactly, without any gap or overlap. Thence they infer that the area of the original square is twice the size of the inner square. Let us assume that the propositions concerned are about Euclidean figures. Our concern is with the visual route to the following:

The parts of a square beyond its inner square (formed by joining midpoints of adjacent sides of the original square) can be arranged to fit the inner square exactly, without overlap or gap, without change of size or shape.

The experience of visualizing the corner triangles folding over can lead one to this belief. But it cannot provide good evidence for it. This is because visual experience (of sight or imagination) has limited acuity and so does not enable us to discriminate between a situation in which the outer triangles fit the inner square exactly and a situation in which they fit inexactly but well enough for the mismatch to escape visual detection. (This contrasts with the case of discovering the number of vertices of a cube by seeing or visualizing one.) Even though visualizing the square, the inner square and then visualizing the corner triangles folding over is constrained by the results of earlier perceptual experience of scenes with relevant similarities, we cannot draw from it reliable information about exact equality of areas, because perception itself is not reliable about exact equalities (or exact proportions) of continuous magnitudes.

Though the visual experience could not provide good evidence for the belief, it is possible that we erroneously use the experience evidentially in reaching the belief. But it is also possible, when reaching the belief in the way described, that we do not take the experience to provide evidence. A non-evidential use is more likely, if when one arrives at the belief in this way one feels fairly certain of it, while aware that visual perception and imagination have limited acuity and so cannot provide evidence for a claim of exact fit.

But what could the role of the visualizing experience possibly be, if it were neither merely enabling nor evidential? One suggestion is that we already have relevant beliefs and belief-forming dispositions, and the visualizing experience could serve to bring to mind the beliefs and to activate the belief-forming dispositions (Giaquinto 2007). These beliefs and dispositions will have resulted from prior possession of cognitive resources, some subject-specific such as concepts of geometrical figures, some subject-general such as symmetry perception about perceptually salient vertical and horizontal axes. A relevant prior belief in this case might be that a square is symmetric about a diagonal. A relevant disposition might be the disposition to believe that the quadrants of a square are congruent squares upon seeing or visualizing a square with a horizontal base plus the vertical and horizontal line segments joining midpoints of its opposite sides. (These dispositions differ from ordinary perceptual dispositions to believe what we see in that they are not cancelled when we mistrust the accuracy of the visual experience.)

The question whether the resulting belief would be knowledge depends on whether the belief-forming dispositions are reliable (truth-conducive) and the pre-existing belief states are states of knowledge. As these conditions can be met without any violation of epistemic rationality, the visualizing route described incompletely here can be a route to knowledge. In that case we would have an example of a use of visual experience which is integral to a way of knowing a truth, which is not merely enabling and yet not evidential. A fuller account and discussion is given in chapters 3 and 4 of Giaquinto (2007).

There are other significant uses of visual representations in mathematics. This final section briefly presents a couple of them.

Although the use of diagrams in arguments in analysis faces special dangers (as noted in 3.3 ), the use of diagrams to illustrate symbolically presented operations can be very helpful. Consider, for example, this pair of operations \(\{ f(x) + k, f(x + k) \}\). Grasping them and the difference between them can be aided by a visual illustration; similarly for the sets \(\{ f(x + k), f(x - k) \}\), \(\{ |f(x)|, f(|x|) \}\), \(\{ f(x)^{-1}, f^{-1}(x), f(x^{-1}) \}\). While generalization on the basis of a visual illustration is unreliable, we can use them as checks against calculation errors and overgeneralization. The same holds for properties. Consider for example, functions for which \(f(-x) = f(x)\), known as even functions, and functions for which \(f(-x) = -f(x)\), the odd functions: it can be helpful to have in mind the images of graphs of \(y = x^2\) and \(y = x^{3}\) as instances of evenness and oddness, to remind one that even functions are symmetrical about the \(y\)-axis and odd functions have rotation symmetry by \(\pi\) about the origin. They can serve as a reminder and check against over-generalisation: any general claim true of all odd functions, for example, must be true of \(y = x^{3}\) in particular.

The utility of visual representations in real and complex analysis is not confined to such simple cases. Visual representations can help us grasp what motivates certain definitions and arguments, and thereby deepen our understanding. Abundant confirmation of this claim can be gathered from working through the text Visual Complex Analysis (Needham 1997). Some mathematical subjects have natural visual representations, which then give rise to a domain of mathematical entities in their own right. This is true of geometry but is also true of subjects which become algebraic in nature very quickly, such as graph theory, knot theory and braid theory. Techniques of computer graphics now enable us to use moving images. For an example of the power of kinematic visual representations to provide and increase understanding of a subject, see the first two “chapters” of the online introduction to braid theory by Ester Dalvit (2012, Other Internet Resources).

With regard to proofs, a minimal kind of understanding consists in understanding each line (proposition or formula) and grasping the validity of each step to a new line from earlier lines. But we can have that stepwise grasp of proof without any idea of why it proceeds by those steps. One has a more advanced (or deeper) kind of understanding when one has the minimal understanding and a grasp of the motivating idea(s) and strategy of the proof. The point is sharply expressed by Weyl (1995 [1932]: 453), quoted in (Tappenden 2005:150)

We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and the road; we want to understand the idea of the proof, the deeper context.

Occasionally the author of a proof gives readers the desired understanding by adding commentary. But this is not always needed, as the idea of a proof is sometimes revealed in the presentation of the proof itself. Often this is done by using visual representations. An example is Fisk’s proof of Chvátal’s “art gallery” theorem. This theorem is the answer to a combinatorial problem in geometry. Put concretely, the problem is this. Let the \(n\) walls of a single-floored gallery make a polygon. What is the smallest number of stationary guards needed to ensure that every point of the gallery wall can be seen by a guard? If the polygon is convex (all interior angles < 180°), one guard will suffice, as guards may rotate. But if the polygon is not convex, as in Figure 18, one guard may not be enough.

Chvátal’s theorem gives the answer: for a gallery with \(n\) walls, \(\llcorner n/3\lrcorner\) guards suffice, where \(\llcorner n/3\lrcorner\) is the greatest integer \(\le n/3\). (If this does not sound to you sufficiently like a mathematical theorem, it can be restated as follows: Let \(S\) be a subset of the Euclidean plane. For a subset \(B\) of \(S\) let us say that \(B\) supervises \(S\) iff for each \(x \in S\) there is a \(y \in B\) such that the segment \(xy\) lies within \(S\). Then the smallest number \(f(n)\) such that every set bounded by a simple \(n\)-gon is supervised by a set of \(f(n)\) points is at most \(\llcorner n/3.\lrcorner\)

Here is Steve Fisk’s proof. A short induction shows that every polygon can be triangulated, i.e., non-crossing edges between non-adjacent vertices (“diagonals”) can be added so that the polygon is entirely composed of non-overlapping triangles. So take any \(n\)-sided polygon with a fixed triangulation. Think of it as a graph, a set of vertices and connected edges, as in Figure 19.

The first part of the proof shows that the graph is 3-colourable, i.e., every vertex can be coloured with one of just three colours (red, white and blue, say) so that no edge connects vertices of the same colour.

The argument proceeds by induction on \(n \ge 3\), the number of vertices.

For \(n = 3\) it is trivial. Assume it holds for all \(k\), where \(3 \le k < n\).

Let triangulated polygon \(G\) have \(n\) vertices. Let \(u\) and \(v\) be any two vertices connected by diagonal edge \(uv\). The diagonal \(uv\) splits \(G\) into two smaller graphs, both containing \(uv\). Give \(u\) and \(v\) different colours, say red and white, as in Figure 20.

By the inductive assumption, we may colour each of the smaller graphs with the three colours so that no edge joins vertices of the same colour, keeping fixed the colours of \(u\) and \(v\). Pasting together the two smaller graphs as coloured gives us a 3-colouring of the whole graph.

What remains is to show that \(\llcorner n/3\lrcorner\) or fewer guards can be placed on vertices so that every triangle is in the view of a guard. Let \(b\), \(r\) and \(w\) be the number of vertices coloured blue, red and white respectively. Let \(b\) be minimal in \(\{b, r, w\}\). Then \(b \le r\) and \(b \le w\). Then \(2b \le r + w\). So \(3b \le b + r + w = n\). So \(b \le n/3\) and so \(b \le \llcorner n/3\lrcorner\). Place a guard on each blue vertex. Done.

The central idea of this proof, or the proof strategy, is clear. While the actual diagrams produced here are superfluous to the proof, some visualizing enables us to grasp the central idea.

Thinking which involves the use of seen or visualized images, which may be static or moving, is widespread in mathematical practice. Such visual thinking may constitute a non-superfluous and non-replaceable part of thinking through a specific proof. But there is a real danger of over-generalisation when using images, which we need to guard against, and in some contexts, such as real and complex analysis, the apparent soundness of a diagrammatic inference is liable to be illusory.

Even when visual thinking does not contribute to proving a mathematical truth, it may enable one to discover a truth, where to discover a truth is to come to believe it in an independent, reliable and rational way. Visual thinking can also play a large role in discovering a central idea for a proof or a proof-strategy; and in discovering a kind of mathematical entity or a mathematical property.

The (non-superfluous) use of visual thinking in coming to know a mathematical truth does in some cases introduce an a posteriori element into the way one comes to know it, resulting in a posteriori mathematical knowledge. This is not as revolutionary as it may sound as a truth knowable a posteriori may also be knowable a priori . More interesting is the possibility that one can acquire some mathematical knowledge in a way in which visual thinking is essential but does not contribute evidence; in this case the role of the visual thinking may be to activate one’s prior cognitive resources. This opens the possibility that non-superfluous visual thinking may result in a priori knowledge of a mathematical truth.

Visual thinking may contribute to understanding in more than one way. Visual illustrations may be extremely useful in providing examples and non-examples of analytic concepts, thus helping to sharpen our grasp of those concepts. Also, visual thinking accompanying a proof may deepen our understanding of the proof, giving us an awareness of the direction of the proof so that, as Hermann Weyl put it, we are not forced to traverse the steps blindly, link by link, feeling our way by touch.

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How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Bernazzani, D., 2005, Soroban Abacus Handbook .
• Dalvit, E., 2012, A Journey through the Mathematical Theory of Braids .
• Fernandes, L. 2015 The Abacus: A brief history. .
• Lauda, A., 2005, Frobenius algebras and planar open string topological field theories , at arXiv.org.

a priori justification and knowledge | Bolzano, Bernard | Dedekind, Richard: contributions to the foundations of mathematics | diagrams | mathematical: explanation | proof theory | quantifiers and quantification | Weyl, Hermann

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18 Types of Diagrams You Can Use to Visualize Data (Templates Included)

Have you ever found yourself stuck while trying to explain a complex concept to someone? Or struggling to put your idea into words?

This is where diagrams come in.

While simple text is best for highlighting figures or information, diagrams are handy for conveying complex ideas and loads of information without overwhelming your audience. They can visualize almost anything, from numerical data to qualitative relationships, making them versatile tools in numerous fields.

Whether you’re in the academe or enterprise setting, this guide is for you. We’ll explore the different types of diagrams with a brief explanation for each type, the best time to use a diagram type, and how you can use them to be a better visual storyteller and communicator. You’ll also find examples and templates for each type of diagram.

Let’s get on with it.

You can also follow along by creating a free account . Select a template to get started.

What exactly is a diagram? 

A diagram is a visual snapshot of information. Think of diagrams as visual representations of data or information that communicate a concept, idea, or process in a simplified and easily understandable way. You can also use them to illustrate relationships, hierarchies, cycles, or workflows. 

Diagrams aren’t just used to show quantitative data, such as sales earnings or satisfaction ratings with a diagram. They’re equally helpful if you want to share qualitative data. For example, a diagram could be used to illustrate the life cycle of a butterfly, showcasing each transformation stage. 

Now, let’s jump into the various types of diagrams, ranging from simple flow charts to the more complex Unified Modeling Language (UML) diagrams.

18 diagram types and when to use each type 

Whether you’re doing data analysis or need a simple visual representation of data, there is a wide array of diagrams at your fingertips. If you’re having a hard time choosing the right diagram for your data visualization needs, use the list below as a quick guide. 

1. Flowchart 

A flowchart is a type of diagram that acts as a roadmap for a process or workflow. It uses shapes and arrows to guide you through each step, making complex procedures simple to understand.

Flowcharts are best for : Simplifying complex processes into understandable stages, making it easier for your readers to follow along and see the ‘big picture”. 

2. Line graph

Line graphs , sometimes called line charts, visualizes numerical data points connected by straight lines. In a line graph or line chart, data points representing different time periods are plotted and connected by a line. This helps with easy visualization of trends and patterns.

Line graphs are best for: Representing the change of one or more quantities over time, making them excellent for tracking the progression of data points.

3. Bar chart 

A bar chart , often interchangeable with bar graphs, is a type of diagram used primarily to display and compare data. For this diagram type, rectangular bars of varying lengths represent data of different categories or groups. Each bar represents a category, and the length or height of the bar corresponds to the numeric data or quantity.

Variations of bar charts include stacked bar charts, grouped bar charts, and horizontal bar charts. 

Bar charts are best for : Comparing the frequency, count, or other measures (such as average) for different categories or groups. A bar chart is particularly useful if you want to display data sets that can be grouped into categories.

4. Circle diagram or pie chart

A pie chart is a circular diagram that represents data in slices. Each slice of the pie chart represents a different category and its proportion to the whole.

Pie charts are best for: Displaying categorical data where you want to highlight each category’s percentage of the total.

5.Venn diagrams

A Venn diagram compares the differences and similarities of groups of things. As a diagram based on overlapping circles, each circle in a Venn diagram represents a different set, and their overlap represents the intersection of the data sets. 

Venn diagrams are best for : Visualizing the relationships between different groups of things. They are helpful when you want to show areas of overlap between elements. A good example is if you want to compare the features of different products or two overlapping concepts, like in the Ikigai Venn diagram template below. Easily create your Venn diagram with Piktochart’s online Venn diagram maker .

6. Tree diagrams

A tree diagram is a diagram that starts with one central idea and expands with branching lines to show multiple paths, all possible outcomes, decisions, or steps. Each ‘branch’ represents a possible outcome or decision in a tree diagram, moving from left to right. Tree diagrams are best for : Representing hierarchy like organizational roles, evolutionary relationships, or possible outcomes of events like when a company launches a product. 

7. Organizational chart 

Organizational charts are diagrams used to display the structure of an organization. In an organizational chart, each box or node represents a different role or department, and lines connecting the boxes illustrate the lines of authority, communication, and responsibility. The chart typically starts with the highest-ranking individual or body (like a CEO or Board of Directors) at the top and branches downwards to various levels of management and individual employees.

Organizational charts are best for : Showing relationships between different members and departments in a company or organization. 

8. Gantt charts 

Gantt charts are typically used in project management to represent the timeline of a project. They consist of horizontal bars, with each bar representing a task or activity.

For this type of diagram, each chart is represented by a horizontal bar spanning from its start date to its end date. The length of the bar corresponds to the duration of the task. Tasks are listed vertically, often in the order they need to be completed. In some projects, tasks are grouped under larger, overarching activities or phases.

Gantt charts are best for : Projects where you need to manage multiple tasks that occur over time, often in a specific sequence, and may depend on each other.

9. Unified Modeling Language (UML) diagram

Software engineers use Unified Modeling Language (UML) diagrams to create standardized diagrams that illustrate the building blocks of a software system.

UML diagrams, such as class diagrams, sequence diagrams, and state diagrams, provide different perspectives on complex systems. Class diagrams depict a system’s static structure, displaying classes, attributes, and relationships. Meanwhile, sequence diagrams illustrate interactions and communication between system entities, providing insight into system functionality. 

UML diagrams are best for : Visualizing a software system’s architecture in software engineering.

10. SWOT analysis diagrams 

A SWOT analysis diagram is used in business strategy for evaluating internal and external factors affecting the organization. The acronym stands for Strengths, Weaknesses, Opportunities, and Threats. Each category is represented in a quadrant chart, providing a comprehensive view of the business landscape.

SWOT diagrams are best for : Strategic planning and decision-making. They represent data that can help identify areas of competitive advantage and inform strategy development.

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11. Fishbone diagram 

Fishbone diagrams, sometimes called cause-and-effect diagrams,  are used to represent the causes of a problem. They consist of a central idea, with different diagrams or branches representing the factors contributing to the problem.

Fishbone diagrams are best for : Brainstorming and problem-solving sessions.

12. Funnel chart

A funnel chart is a type of diagram used to represent stages or progress. In a funnel chart, each stage is represented by a horizontal bar, and the length of the bar corresponds to the quantity or value at that stage. The chart is widest at the top, where the quantity or value is greatest, and narrows down to represent the decrease at each subsequent stage.

Funnel charts are best for: Visual representation of the sales pipeline or data visualization of how a broad market is narrowed down into potential leads and a select group of customers.

13. SIPOC diagrams

A SIPOC diagram is used in process improvement to represent the different components of a process. The acronym stands for Suppliers, Inputs, Process, Outputs, and Customers.

SIPOC diagrams are best for: Providing a high-level view of a process which helps visualize the sequence of events and their interconnections.

14. Swimlane diagrams

Swimlane diagrams are best for mapping out complex processes that involve multiple participants or groups.

Keep in mind that each lane (which can be either horizontal or vertical) in a swimlane diagram represents a different participant or group involved in the process. The steps or activities carried out by each participant are plotted within their respective lanes. This helps clarify roles and responsibilities as well as the sequence of events and points of interaction.

Swimlane diagrams are best for : Visualizing how different roles or departments interact and collaborate throughout a workflow or process.

15. Mind maps

A mind map starts with a central idea and expands outward to include supporting ideas, related subtopics, concepts, or tasks, which can be further subdivided as needed. The branches radiating out from the central idea represent hierarchical relationships and connections between the different pieces of information in a mind map.

Mind maps are best for : Brainstorming, taking notes, organizing information, and visualizing complex concepts in a digestible format.

16. Scatter Plots

Scatter plots are used to compare data and represent the relationship between two variables. In a scatter plot, each dot represents a data point with its position along the x and y axes representing the values of two variables.

Scatter plots are best for : Observing relationships and trends between the two variables. These scatter plots are useful for regression analysis, hypothesis testing, and data exploration in various fields such as statistics, economics, and natural sciences.

17. PERT chart

PERT (Project Evaluation Review Technique) charts are project management tools used to schedule tasks. Each node or arrow represents each task, while lines represent dependencies between tasks. The chart includes task duration and earliest/latest start/end times.

Construction project managers often use PERT charts to schedule tasks like design, site prep, construction, and inspection. Identifying the critical path helps focus resources on tasks that impact the project timeline.

PERT charts are best for : Visualizing the sequence of tasks, the time required for each task, and project timelines.

18. Network diagrams

A network diagram visually represents the relationships between elements in a system or project. In network diagrams, each node represents an element, such as a device in a computer network or a task in a project. The lines or arrows connecting the nodes represent the relationships or interactions between these elements.

Network diagrams are best for: Visually representing the relationships or connections between different elements in a system or a project. They are often used in telecommunications, computer networking, project management, and organization planning.

Choosing the right diagram starts with a good understanding of your audience

Understanding your audience’s needs, expectations, and context is necessary before designing diagrams. The best diagram is not the one that looks the most impressive but the one that communicates complex information most clearly and effectively to your intended audience.

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• Our Mission

The Power of Visualization in Math

Creating visual representations for math students can open up understanding. We have resources you can use in class tomorrow.

When do you know it’s time to try something different in your math lesson?

For me, I knew the moment I read this word problem to my fifth-grade summer school students: “On average, the sun’s energy density reaching Earth’s upper atmosphere is 1,350 watts per square meter. Assume the incident, monochromatic light has a wavelength of 800 nanometers (each photon has an energy of 2.48 × 10 -19 joules at this wavelength). How many photons are incident on the Earth’s upper atmosphere in one second?”

My students couldn’t get past the language, the sizes of the different numbers, or the science concepts addressed in the question. In short, I had effectively shut them down, and I needed a new approach to bring them back to their learning. So I started drawing on the whiteboard and created something with a little whimsy, a cartoon photon asking how much energy a photon has.

Immediately, students started yelling out, “2.48 × 10 -19 joules,” and they could even cite the text where they had learned the information. I knew I was on to something, so the next thing I drew was a series of boxes with our friend the photon.

If all of the photons in the image below were to hit in one second, how much energy is represented in the drawing?

Students realized that we were just adding up all the individual energy from each photon and then quickly realized that this was multiplication. And then they knew that the question we were trying to answer was just figuring out the number of photons, and since we knew the total energy in one second, we could compute the number of photons by division.

The point being, we reached a place where my students were able to process the learning. The power of the visual representation made all the difference for these students, and being able to sequence through the problem using the visual supports completely changed the interactions they were having with the problem.

If you’re like me, you’re thinking, “So the visual representations worked with this problem, but what about other types of problems? Surely there isn’t a visual model for every problem!”

The power of this moment, the change in the learning environment, and the excitement of my fifth graders as they could not only understand but explain to others what the problem was about convinced me it was worth the effort to pursue visualization and try to answer these questions: Is there a process to unlock visualizations in math? And are there resources already available to help make mathematics visual?

I realized that the first step in unlocking visualization as a scaffold for students was to change the kind of question I was asking myself. A powerful question to start with is: “How might I represent this learning target in a visual way?” This reframing opens a world of possible representations that we might not otherwise have considered. Thinking about many possible visual representations is the first step in creating a good one for students.

The Progressions published in tandem with the Common Core State Standards for mathematics are one resource for finding specific visual models based on grade level and standard. In my fifth-grade example, what I constructed was a sequenced process to develop a tape diagram—a type of visual model that uses rectangles to represent the parts of a ratio. I didn’t realize it, but to unlock my thinking I had to commit to finding a way to represent the problem in a visual way. Asking yourself a very simple series of questions leads you down a variety of learning paths, and primes you for the next step in the sequence—finding the right resources to complete your visualization journey.

Posing the question of visualization readies your brain to identify the right tool for the desired learning target and your students. That is, you’ll more readily know when you’ve identified the right tool for the job for your students. There are many, many resources available to help make this process even easier, and I’ve created a matrix of clickable tools, articles, and resources .

The process to visualize your math instruction is summarized at the top of my Visualizing Math graphic; below that is a mix of visualization strategies and resources you can use tomorrow in your classroom.

Our job as educators is to set a stage that maximizes the amount of learning done by our students, and teaching students mathematics in this visual way provides a powerful pathway for us to do our job well. The process of visualizing mathematics tests your abilities at first, and you’ll find that it makes both you and your students learn.

Initial Thoughts

Perspectives & resources, what is high-quality mathematics instruction and why is it important.

• Page 1: The Importance of High-Quality Mathematics Instruction
• Page 2: A Standards-Based Mathematics Curriculum
• Page 3: Evidence-Based Mathematics Practices

What evidence-based mathematics practices can teachers employ?

• Page 4: Explicit, Systematic Instruction

Page 5: Visual Representations

• Page 6: Schema Instruction
• Page 7: Metacognitive Strategies
• Page 8: Effective Classroom Practices
• Page 9: References & Additional Resources
• Page 10: Credits

Research Shows

• Students who use accurate visual representations are six times more likely to correctly solve mathematics problems than are students who do not use them. However, students who use inaccurate visual representations are less likely to correctly solve mathematics problems than those who do not use visual representations at all. (Boonen, van Wesel, Jolles, & van der Schoot, 2014)
• Students with a learning disability (LD) often do not create accurate visual representations or use them strategically to solve problems. Teaching students to systematically use a visual representation to solve word problems has led to substantial improvements in math achievement for students with learning disabilities. (van Garderen, Scheuermann, & Jackson, 2012; van Garderen, Scheuermann, & Poch, 2014)
• Students who use visual representations to solve word problems are more likely to solve the problems accurately. This was equally true for students who had LD, were low-achieving, or were average-achieving. (Krawec, 2014)

Visual representations are flexible; they can be used across grade levels and types of math problems. They can be used by teachers to teach mathematics facts and by students to learn mathematics content. Visual representations can take a number of forms. Click on the links below to view some of the visual representations most commonly used by teachers and students.

How does this practice align?

High-leverage practice (hlp).

• HLP15 : Provide scaffolded supports

CCSSM: Standards for Mathematical Practice

• MP1 : Make sense of problems and persevere in solving them.

Number Lines

Definition : A straight line that shows the order of and the relation between numbers.

Common Uses : addition, subtraction, counting

Strip Diagrams

Definition : A bar divided into rectangles that accurately represent quantities noted in the problem.

Common Uses : addition, fractions, proportions, ratios

Definition : Simple drawings of concrete or real items (e.g., marbles, trucks).

Common Uses : counting, addition, subtraction, multiplication, division

Graphs/Charts

Definition : Drawings that depict information using lines, shapes, and colors.

Common Uses : comparing numbers, statistics, ratios, algebra

Graphic Organizers

Definition : Visual that assists students in remembering and organizing information, as well as depicting the relationships between ideas (e.g., word webs, tables, Venn diagrams).

Common Uses : algebra, geometry

Before they can solve problems, however, students must first know what type of visual representation to create and use for a given mathematics problem. Some students—specifically, high-achieving students, gifted students—do this automatically, whereas others need to be explicitly taught how. This is especially the case for students who struggle with mathematics and those with mathematics learning disabilities. Without explicit, systematic instruction on how to create and use visual representations, these students often create visual representations that are disorganized or contain incorrect or partial information. Consider the examples below.

Elementary Example

Mrs. Aldridge ask her first-grade students to add 2 + 4 by drawing dots.

Notice that Talia gets the correct answer. However, because Colby draws his dots in haphazard fashion, he fails to count all of them and consequently arrives at the wrong solution.

High School Example

Mr. Huang asks his students to solve the following word problem:

The flagpole needs to be replaced. The school would like to replace it with the same size pole. When Juan stands 11 feet from the base of the pole, the angle of elevation from Juan’s feet to the top of the pole is 70 degrees. How tall is the pole?

Compare the drawings below created by Brody and Zoe to represent this problem. Notice that Brody drew an accurate representation and applied the correct strategy. In contrast, Zoe drew a picture with partially correct information. The 11 is in the correct place, but the 70° is not. As a result of her inaccurate representation, Zoe is unable to move forward and solve the problem. However, given an accurate representation developed by someone else, Zoe is more likely to solve the problem correctly.

Manipulatives

Some students will not be able to grasp mathematics skills and concepts using only the types of visual representations noted in the table above. Very young children and students who struggle with mathematics often require different types of visual representations known as manipulatives. These concrete, hands-on materials and objects—for example, an abacus or coins—help students to represent the mathematical idea they are trying to learn or the problem they are attempting to solve. Manipulatives can help students develop a conceptual understanding of mathematical topics. (For the purpose of this module, the term concrete objects refers to manipulatives and the term visual representations refers to schematic diagrams.)

It is important that the teacher make explicit the connection between the concrete object and the abstract concept being taught. The goal is for the student to eventually understand the concepts and procedures without the use of manipulatives. For secondary students who struggle with mathematics, teachers should show the abstract along with the concrete or visual representation and explicitly make the connection between them.

A move from concrete objects or visual representations to using abstract equations can be difficult for some students. One strategy teachers can use to help students systematically transition among concrete objects, visual representations, and abstract equations is the Concrete-Representational-Abstract (CRA) framework.

If you would like to learn more about this framework, click here.

Concrete-Representational-Abstract Framework

• Concrete —Students interact and manipulate three-dimensional objects, for example algebra tiles or other algebra manipulatives with representations of variables and units.
• Representational — Students use two-dimensional drawings to represent problems. These pictures may be presented to them by the teacher, or through the curriculum used in the class, or students may draw their own representation of the problem.
• Abstract — Students solve problems with numbers, symbols, and words without any concrete or representational assistance.

CRA is effective across all age levels and can assist students in learning concepts, procedures, and applications. When implementing each component, teachers should use explicit, systematic instruction and continually monitor student work to assess their understanding, asking them questions about their thinking and providing clarification as needed. Concrete and representational activities must reflect the actual process of solving the problem so that students are able to generalize the process to solve an abstract equation. The illustration below highlights each of these components.

For Your Information

One promising practice for moving secondary students with mathematics difficulties or disabilities from the use of manipulatives and visual representations to the abstract equation quickly is the CRA-I strategy . In this modified version of CRA, the teacher simultaneously presents the content using concrete objects, visual representations of the concrete objects, and the abstract equation. Studies have shown that this framework is effective for teaching algebra to this population of students (Strickland & Maccini, 2012; Strickland & Maccini, 2013; Strickland, 2017).

Kim Paulsen discusses the benefits of manipulatives and a number of things to keep in mind when using them (time: 2:35).

Kim Paulsen, EdD Associate Professor, Special Education Vanderbilt University

View Transcript

Transcript: Kim Paulsen, EdD

Manipulatives are a great way of helping kids understand conceptually. The use of manipulatives really helps students see that conceptually, and it clicks a little more with them. Some of the things, though, that we need to remember when we’re using manipulatives is that it is important to give students a little bit of free time when you’re using a new manipulative so that they can just explore with them. We need to have specific rules for how to use manipulatives, that they aren’t toys, that they really are learning materials, and how students pick them up, how they put them away, the right time to use them, and making sure that they’re not distracters while we’re actually doing the presentation part of the lesson. One of the important things is that we don’t want students to memorize the algorithm or the procedures while they’re using the manipulatives. It really is just to help them understand conceptually. That doesn’t mean that kids are automatically going to understand conceptually or be able to make that bridge between using the concrete manipulatives into them being able to solve the problems. For some kids, it is difficult to use the manipulatives. That’s not how they learn, and so we don’t want to force kids to have to use manipulatives if it’s not something that is helpful for them. So we have to remember that manipulatives are one way to think about teaching math.

I think part of the reason that some teachers don’t use them is because it takes a lot of time, it takes a lot of organization, and they also feel that students get too reliant on using manipulatives. One way to think about using manipulatives is that you do it a couple of lessons when you’re teaching a new concept, and then take those away so that students are able to do just the computation part of it. It is true we can’t walk around life with manipulatives in our hands. And I think one of the other reasons that a lot of schools or teachers don’t use manipulatives is because they’re very expensive. And so it’s very helpful if all of the teachers in the school can pool resources and have a manipulative room where teachers can go check out manipulatives so that it’s not so expensive. Teachers have to know how to use them, and that takes a lot of practice.

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Creating visual explanations improves learning

Eliza bobek.

1 University of Massachusetts Lowell, Lowell, MA USA

Barbara Tversky

2 Stanford University, Columbia University Teachers College, New York, NY USA

Associated Data

Many topics in science are notoriously difficult for students to learn. Mechanisms and processes outside student experience present particular challenges. While instruction typically involves visualizations, students usually explain in words. Because visual explanations can show parts and processes of complex systems directly, creating them should have benefits beyond creating verbal explanations. We compared learning from creating visual or verbal explanations for two STEM domains, a mechanical system (bicycle pump) and a chemical system (bonding). Both kinds of explanations were analyzed for content and learning assess by a post-test. For the mechanical system, creating a visual explanation increased understanding particularly for participants of low spatial ability. For the chemical system, creating both visual and verbal explanations improved learning without new teaching. Creating a visual explanation was superior and benefitted participants of both high and low spatial ability. Visual explanations often included crucial yet invisible features. The greater effectiveness of visual explanations appears attributable to the checks they provide for completeness and coherence as well as to their roles as platforms for inference. The benefits should generalize to other domains like the social sciences, history, and archeology where important information can be visualized. Together, the findings provide support for the use of learner-generated visual explanations as a powerful learning tool.

Electronic supplementary material

The online version of this article (doi:10.1186/s41235-016-0031-6) contains supplementary material, which is available to authorized users.

Significance

Uncovering cognitive principles for effective teaching and learning is a central application of cognitive psychology. Here we show: (1) creating explanations of STEM phenomena improves learning without additional teaching; and (2) creating visual explanations is superior to creating verbal ones. There are several notable differences between visual and verbal explanations; visual explanations map thought more directly than words and provide checks for completeness and coherence as well as a platform for inference, notably from structure to process. Extensions of the technique to other domains should be possible. Creating visual explanations is likely to enhance students’ spatial thinking skills, skills that are increasingly needed in the contemporary and future world.

Dynamic systems such as those in science and engineering, but also in history, politics, and other domains, are notoriously difficult to learn (e.g. Chi, DeLeeuw, Chiu, & Lavancher, 1994 ; Hmelo-Silver & Pfeffer, 2004 ; Johnstone, 1991 ; Perkins & Grotzer, 2005 ). Mechanisms, processes, and behavior of complex systems present particular challenges. Learners must master not only the individual components of the system or process (structure) but also the interactions and mechanisms (function), which may be complex and frequently invisible. If the phenomena are macroscopic, sub-microscopic, or abstract, there is an additional level of difficulty. Although the teaching of STEM phenomena typically relies on visualizations, such as pictures, graphs, and diagrams, learning is typically revealed in words, both spoken and written. Visualizations have many advantages over verbal explanations for teaching; can creating visual explanations promote learning?

Learning from visual representations in STEM

Given the inherent challenges in teaching and learning complex or invisible processes in science, educators have developed ways of representing these processes to enable and enhance student understanding. External visual representations, including diagrams, photographs, illustrations, flow charts, and graphs, are often used in science to both illustrate and explain concepts (e.g., Hegarty, Carpenter, & Just, 1990 ; Mayer, 1989 ). Visualizations can directly represent many structural and behavioral properties. They also help to draw inferences (Larkin & Simon, 1987 ), find routes in maps (Levine, 1982 ), spot trends in graphs (Kessell & Tversky, 2011 ; Zacks & Tversky, 1999 ), imagine traffic flow or seasonal changes in light from architectural sketches (e.g. Tversky & Suwa, 2009 ), and determine the consequences of movements of gears and pulleys in mechanical systems (e.g. Hegarty & Just, 1993 ; Hegarty, Kriz, & Cate, 2003 ). The use of visual elements such as arrows is another benefit to learning with visualizations. Arrows are widely produced and comprehended as representing a range of kinds of forces as well as changes over time (e.g. Heiser & Tversky, 2002 ; Tversky, Heiser, MacKenzie, Lozano, & Morrison, 2007 ). Visualizations are thus readily able to depict the parts and configurations of systems; presenting the same content via language may be more difficult. Although words can describe spatial properties, because the correspondences of meaning to language are purely symbolic, comprehension and construction of mental representations from descriptions is far more effortful and error prone (e.g. Glenberg & Langston, 1992 ; Hegarty & Just, 1993 ; Larkin & Simon, 1987 ; Mayer, 1989 ). Given the differences in how visual and verbal information is processed, how learners draw inferences and construct understanding in these two modes warrants further investigation.

Benefits of generating explanations

Learner-generated explanations of scientific phenomena may be an important learning strategy to consider beyond the utility of learning from a provided external visualization. Explanations convey information about concepts or processes with the goal of making clear and comprehensible an idea or set of ideas. Explanations may involve a variety of elements, such as the use of examples and analogies (Roscoe & Chi, 2007 ). When explaining something new, learners may have to think carefully about the relationships between elements in the process and prioritize the multitude of information available to them. Generating explanations may require learners to reorganize their mental models by allowing them to make and refine connections between and among elements and concepts. Explaining may also help learners metacognitively address their own knowledge gaps and misconceptions.

Many studies have shown that learning is enhanced when students are actively engaged in creative, generative activities (e.g. Chi, 2009 ; Hall, Bailey, & Tillman, 1997 ). Generative activities have been shown to benefit comprehension of domains involving invisible components, including electric circuits (Johnson & Mayer, 2010 ) and the chemistry of detergents (Schwamborn, Mayer, Thillmann, Leopold, & Leutner, 2010 ). Wittrock’s ( 1990 ) generative theory stresses the importance of learners actively constructing and developing relationships. Generative activities require learners to select information and choose how to integrate and represent the information in a unified way. When learners make connections between pieces of information, knowledge, and experience, by generating headings, summaries, pictures, and analogies, deeper understanding develops.

The information learners draw upon to construct their explanations is likely important. For example, Ainsworth and Loizou ( 2003 ) found that asking participants to self-explain with a diagram resulted in greater learning than self-explaining from text. How might learners explain with physical mechanisms or materials with multi-modal information?

Generating visual explanations

Learner-generated visualizations have been explored in several domains. Gobert and Clement ( 1999 ) investigated the effectiveness of student-generated diagrams versus student-generated summaries on understanding plate tectonics after reading an expository text. Students who generated diagrams scored significantly higher on a post-test measuring spatial and causal/dynamic content, even though the diagrams contained less domain-related information. Hall et al. ( 1997 ) showed that learners who generated their own illustrations from text performed equally as well as learners provided with text and illustrations. Both groups outperformed learners only provided with text. In a study concerning the law of conservation of energy, participants who generated drawings scored higher on a post-test than participants who wrote their own narrative of the process (Edens & Potter, 2003 ). In addition, the quality and number of concept units present in the drawing/science log correlated with performance on the post-test. Van Meter ( 2001 ) found that drawing while reading a text about Newton’s Laws was more effective than answering prompts in writing.

One aspect to explore is whether visual and verbal productions contain different types of information. Learning advantages for the generation of visualizations could be attributed to learners’ translating across modalities, from a verbal format into a visual format. Translating verbal information from the text into a visual explanation may promote deeper processing of the material and more complete and comprehensive mental models (Craik & Lockhart, 1972 ). Ainsworth and Iacovides ( 2005 ) addressed this issue by asking two groups of learners to self-explain while learning about the circulatory system of the human body. Learners given diagrams were asked to self-explain in writing and learners given text were asked to explain using a diagram. The results showed no overall differences in learning outcomes, however the learners provided text included significantly more information in their diagrams than the other group. Aleven and Koedinger ( 2002 ) argue that explanations are most helpful if they can integrate visual and verbal information. Translating across modalities may serve this purpose, although translating is not necessarily an easy task (Ainsworth, Bibby, & Wood, 2002 ).

It is important to remember that not all studies have found advantages to generating explanations. Wilkin ( 1997 ) found that directions to self-explain using a diagram hindered understanding in examples in physical motion when students were presented with text and instructed to draw a diagram. She argues that the diagrams encouraged learners to connect familiar but unrelated knowledge. In particular, “low benefit learners” in her study inappropriately used spatial adjacency and location to connect parts of diagrams, instead of the particular properties of those parts. Wilkin argues that these learners are novices and that experts may not make the same mistake since they have the skills to analyze features of a diagram according to their relevant properties. She also argues that the benefits of self-explaining are highest when the learning activity is constrained so that learners are limited in their possible interpretations. Other studies that have not found a learning advantage from generating drawings have in common an absence of support for the learner (Alesandrini, 1981 ; Leutner, Leopold, & Sumfleth, 2009 ). Another mediating factor may be the learner’s spatial ability.

The role of spatial ability

Spatial thinking involves objects, their size, location, shape, their relation to one another, and how and where they move through space. How then, might learners with different levels of spatial ability gain structural and functional understanding in science and how might this ability affect the utility of learner-generated visual explanations? Several lines of research have sought to explore the role of spatial ability in learning science. Kozhevnikov, Hegarty, and Mayer ( 2002 ) found that low spatial ability participants interpreted graphs as pictures, whereas high spatial ability participants were able to construct more schematic images and manipulate them spatially. Hegarty and Just ( 1993 ) found that the ability to mentally animate mechanical systems correlated with spatial ability, but not verbal ability. In their study, low spatial ability participants made more errors in movement verification tasks. Leutner et al. ( 2009 ) found no effect of spatial ability on the effectiveness of drawing compared to mentally imagining text content. Mayer and Sims ( 1994 ) found that spatial ability played a role in participants’ ability to integrate visual and verbal information presented in an animation. The authors argue that their results can be interpreted within the context of dual-coding theory. They suggest that low spatial ability participants must devote large amounts of cognitive effort into building a visual representation of the system. High spatial ability participants, on the other hand, are more able to allocate sufficient cognitive resources to building referential connections between visual and verbal information.

Benefits of testing

Although not presented that way, creating an explanation could be regarded as a form of testing. Considerable research has documented positive effects of testing on learning. Presumably taking a test requires retrieving and sometimes integrating the learned material and those processes can augment learning without additional teaching or study (e.g. Roediger & Karpicke, 2006 ; Roediger, Putnam, & Smith, 2011 ; Wheeler & Roediger, 1992 ). Hausmann and Vanlehn ( 2007 ) addressed the possibility that generating explanations is beneficial because learners merely spend more time with the content material than learners who are not required to generate an explanation. In their study, they compared the effects of using instructions to self-explain with instructions to merely paraphrase physics (electrodynamics) material. Attending to provided explanations by paraphrasing was not as effective as generating explanations as evidenced by retention scores on an exam 29 days after the experiment and transfer scores within and across domains. Their study concludes, “the important variable for learning was the process of producing an explanation” (p. 423). Thus, we expect benefits from creating either kind of explanation but for the reasons outlined previously, we expect larger benefits from creating visual explanations.

Present experiments

This study set out to answer a number of related questions about the role of learner-generated explanations in learning and understanding of invisible processes. (1) Do students learn more when they generate visual or verbal explanations? We anticipate that learning will be greater with the creation of visual explanations, as they encourage completeness and the integration of structure and function. (2) Does the inclusion of structural and functional information correlate with learning as measured by a post-test? We predict that including greater counts of information, particularly invisible and functional information, will positively correlate with higher post-test scores. (3) Does spatial ability predict the inclusion of structural and functional information in explanations, and does spatial ability predict post-test scores? We predict that high spatial ability participants will include more information in their explanations, and will score higher on post-tests.

Experiment 1

The first experiment examines the effects of creating visual or verbal explanations on the comprehension of a bicycle tire pump’s operation in participants with low and high spatial ability. Although the pump itself is not invisible, the components crucial to its function, notably the inlet and outlet valves, and the movement of air, are located inside the pump. It was predicted that visual explanations would include more information than verbal explanations, particularly structural information, since their construction encourages completeness and the production of a whole mechanical system. It was also predicted that functional information would be biased towards a verbal format, since much of the function of the pump is hidden and difficult to express in pictures. Finally, it was predicted that high spatial ability participants would be able to produce more complete explanations and would thus also demonstrate better performance on the post-test. Explanations were coded for structural and functional content, essential features, invisible features, arrows, and multiple steps.

Participants

Participants were 127 (59 female) seventh and eighth grade students, aged 12–14 years, enrolled in an independent school in New York City. The school’s student body is 70% white, 30% other ethnicities. Approximately 25% of the student body receives financial aid. The sample consisted of three class sections of seventh grade students and three class sections of eighth grade students. Both seventh and eighth grade classes were integrated science (earth, life, and physical sciences) and students were not grouped according to ability in any section. Written parental consent was obtained by means of signed informed consent forms. Each participant was randomly assigned to one of two conditions within each class. There were 64 participants in the visual condition explained the bicycle pump’s function by drawing and 63 participants explained the pump’s function by writing.

The materials consisted of a 12-inch Spalding bicycle pump, a blank 8.5 × 11 in. sheet of paper, and a post-test (Additional file 1 ). The pump’s chamber and hose were made of clear plastic; the handle and piston were black plastic. The parts of the pump (e.g. inlet valve, piston) were labeled.

Spatial ability was assessed using the Vandenberg and Kuse ( 1978 ) mental rotation test (MRT). The MRT is a 20-item test in which two-dimensional drawings of three-dimensional objects are compared. Each item consists of one “target” drawing and four drawings that are to be compared to the target. Two of the four drawings are rotated versions of the target drawing and the other two are not. The task is to identify the two rotated versions of the target. A score was determined by assigning one point to each question if both of the correct rotated versions were chosen. The maximum score was 20 points.

The post-test consisted of 16 true/false questions printed on a single sheet of paper measuring 8.5 × 11 in. Half of the questions related to the structure of the pump and the other half related to its function. The questions were adapted from Heiser and Tversky ( 2002 ) in order to be clear and comprehensible for this age group.

The experiment was conducted over the course of two non-consecutive days during the normal school day and during regularly scheduled class time. On the first day, participants completed the MRT as a whole-class activity. After completing an untimed practice test, they were given 3 min for each of the two parts of the MRT. On the second day, occurring between two and four days after completing the MRT, participants were individually asked to study an actual bicycle tire pump and were then asked to generate explanations of its function. The participants were tested individually in a quiet room away from the rest of the class. In addition to the pump, each participant was one instruction sheet and one blank sheet of paper for their explanations. The post-test was given upon completion of the explanation. The instruction sheet was read aloud to participants and they were instructed to read along. The first set of instructions was as follows: “A bicycle pump is a mechanical device that pumps air into bicycle tires. First, take this bicycle pump and try to understand how it works. Spend as much time as you need to understand the pump.” The next set of instructions differed for participants in each condition. The instructions for the visual condition were as follows: “Then, we would like you to draw your own diagram or set of diagrams that explain how the bike pump works. Draw your explanation so that someone else who has not seen the pump could understand the bike pump from your explanation. Don’t worry about the artistic quality of the diagrams; in fact, if something is hard for you to draw, you can explain what you would draw. What’s important is that the explanation should be primarily visual, in a diagram or diagrams.” The instructions for the verbal condition were as follows: “Then, we would like you to write an explanation of how the bike pump works. Write your explanation so that someone else who has not seen the pump could understand the bike pump from your explanation.” All participants then received these instructions: “You may not use the pump while you create your explanations. Please return it to me when you are ready to begin your explanation. When you are finished with the explanation, you will hand in your explanation to me and I will then give you 16 true/false questions about the bike pump. You will not be able to look at your explanation while you complete the questions.” Study and test were untimed. All students finished within the 45-min class period.

Spatial ability

The mean score on the MRT was 10.56, with a median of 11. Boys scored significantly higher (M = 13.5, SD = 4.4) than girls (M = 8.8, SD = 4.5), F(1, 126) = 19.07, p  < 0.01, a typical finding (Voyer, Voyer, & Bryden, 1995 ). Participants were split into high or low spatial ability by the median. Low and high spatial ability participants were equally distributed in the visual and verbal groups.

Learning outcomes

It was predicted that high spatial ability participants would be better able to mentally animate the bicycle pump system and therefore score higher on the post-test and that post-test scores would be higher for those who created visual explanations. Table  1 shows the scores on the post-test by condition and spatial ability. A two-way factorial ANOVA revealed marginally significant main effect of spatial ability F(1, 124) = 3.680, p  = 0.06, with high spatial ability participants scoring higher on the post-test. There was also a significant interaction between spatial ability and explanation type F(1, 124) = 4.094, p  < 0.01, see Fig.  1 . Creating a visual explanation of the bicycle pump selectively helped low spatial participants.

Post-test scores, by explanation type and spatial ability

Scores on the post-test by condition and spatial ability

Coding explanations

Explanations (see Fig.  2 ) were coded for structural and functional content, essential features, invisible features, arrows, and multiple steps. A subset of the explanations (20%) was coded by the first author and another researcher using the same coding system as a guide. The agreement between scores was above 90% for all measures. Disagreements were resolved through discussion. The first author then scored the remaining explanations.

Examples of visual and verbal explanations of the bicycle pump

Coding for structure and function

A maximum score of 12 points was awarded for the inclusion and labeling of six structural components: chamber, piston, inlet valve, outlet valve, handle, and hose. For the visual explanations, 1 point was given for a component drawn correctly and 1 additional point if the component was labeled correctly. For verbal explanations, sentences were divided into propositions, the smallest unit of meaning in a sentence. Descriptions of structural location e.g. “at the end of the piston is the inlet valve,” or of features of the components, e.g. the shape of a part, counted as structural components. Information was coded as functional if it depicted (typically with an arrow) or described the function/movement of an individual part, or the way multiple parts interact. No explanation contained more than ten functional units.

Visual explanations contained significantly more structural components (M = 6.05, SD = 2.76) than verbal explanations (M = 4.27, SD = 1.54), F(1, 126) = 20.53, p  < 0.05. The number of functional components did not differ between visual and verbal explanations as displayed in Figs.  3 and ​ and4. 4 . Many visual explanations (67%) contained verbal components; the structural and functional information in explanations was coded as depictive or descriptive. Structural and functional information were equally likely to be expressed in words or pictures in visual explanations. It was predicted that explanations created by high spatial participants would include more functional information. However, there were no significant differences found between low spatial (M = 5.15, SD = 2.21) and high spatial (M = 4.62, SD = 2.16) participants in the number of structural units or between low spatial (M = 3.83, SD = 2.51) and high spatial (M = 4.10, SD = 2.13) participants in the number of functional units.

Average number of structural and functional components in visual and verbal explanations

Visual and verbal explanations of chemical bonding

Coding of essential features

To further establish a relationship between the explanations generated and outcomes on the post-test, explanations were also coded for the inclusion of information essential to its function according to a 4-point scale (adapted from Hall et al., 1997 ). One point was given if both the inlet and the outlet valve were clearly present in the drawing or described in writing, 1 point was given if the piston inserted into the chamber was shown or described to be airtight, and 1 point was given for each of the two valves if they were shown or described to be opening/closing in the correct direction.

Visual explanations contained significantly more essential information (M = 1.78, SD = 1.0) than verbal explanations (M = 1.20, SD = 1.21), F(1, 126) = 7.63, p  < 0.05. Inclusion of essential features correlated positively with post-test scores, r = 0.197, p  < 0.05).

Coding arrows and multiple steps

For the visual explanations, three uses of arrows were coded and tallied: labeling a part or action, showing motion, or indicating sequence. Analysis of visual explanations revealed that 87% contained arrows. No significant differences were found between low and high spatial participants’ use of arrows to label and no signification correlations were found between the use of arrows and learning outcomes measured on the post-test.

The explanations were coded for the number of discrete steps used to explain the process of using the bike pump. The number of steps used by participants ranged from one to six. Participants whose explanations, whether verbal or visual, contained multiple steps scored significantly higher (M = 0.76, SD = 0.18) on the post-test than participants whose explanations consisted of a single step (M = 0.67, SD = 0.19), F(1, 126) = 5.02, p  < 0.05.

Coding invisible features

The bicycle tire pump, like many mechanical devices, contains several structural features that are hidden or invisible and must be inferred from the function of the pump. For the bicycle pump the invisible features are the inlet and outlet valves and the three phases of movement of air, entering the pump, moving through the pump, exiting the pump. Each feature received 1 point for a total of 5 possible points.

The mean score for the inclusion of invisible features was 3.26, SD = 1.25. The data were analyzed using linear regression and revealed that the total score for invisible parts significantly predicted scores on the post-test, F(1, 118) = 3.80, p  = 0.05.

In the first experiment, students learned the workings of a bicycle pump from interacting with an actual pump and creating a visual or verbal explanation of its function. Understanding the functionality of a bike pump depends on the actions and consequences of parts that are not visible. Overall, the results provide support for the use of learner-generated visual explanations in developing understanding of a new scientific system. The results show that low spatial ability participants were able to learn as successfully as high spatial ability participants when they first generated an explanation in a visual format.

Visual explanations may have led to greater understanding for a number of reasons. As discussed previously, visual explanations encourage completeness. They force learners to decide on the size, shape, and location of parts/objects. Understanding the “hidden” function of the invisible parts is key to understanding the function of the entire system and requires an understanding of how both the visible and invisible parts interact. The visual format may have been able to elicit components and concepts that are invisible and difficult to integrate into the formation of a mental model. The results show that including more of the essential features and showing multiple steps correlated with superior test performance. Understanding the bicycle pump requires understanding how all of these components are connected through movement, force, and function. Many (67%) of the visual explanations also contained written components to accompany their explanation. Arguably, some types of information may be difficult to depict visually and verbal language has many possibilities that allow for specificity. The inclusion of text as a complement to visual explanations may be key to the success of learner-generated explanations and the development of understanding.

A limitation of this experiment is that participants were not provided with detailed instructions for completing their explanations. In addition, this experiment does not fully clarify the role of spatial ability, since high spatial participants in the visual and verbal groups demonstrated equivalent knowledge of the pump on the post-test. One possibility is that the interaction with the bicycle pump prior to generating explanations was a sufficient learning experience for the high spatial participants. Other researchers (e.g. Flick, 1993 ) have shown that hands-on interactive experiences can be effective learning situations. High spatial ability participants may be better able to imagine the movement and function of a system (e.g. Hegarty, 1992 ).

Experiment 1 examined learning a mechanical system with invisible (hidden) parts. Participants were introduced to the system by being able to interact with an actual bicycle pump. While we did not assess participants’ prior knowledge of the pump with a pre-test, participants were randomly assigned to each condition. The findings have promising implications for teaching. Creating visual explanations should be an effective way to improve performance, especially in low spatial students. Instructors can guide the creation of visual explanations toward the features that augment learning. For example, students can be encouraged to show every step and action and to focus on the essential parts, even if invisible. The coding system shows that visual explanations can be objectively evaluated to provide feedback on students’ understanding. The utility of visual explanations may differ for scientific phenomena that are more abstract, or contain elements that are invisible due to their scale. Experiment 2 addresses this possibility by examining a sub-microscopic area of science: chemical bonding.

Experiment 2

In this experiment, we examine visual and verbal explanations in an area of chemistry: ionic and covalent bonding. Chemistry is often regarded as a difficult subject; one of the essential or inherent features of chemistry which presents difficulty is the interplay between the macroscopic, sub-microscopic, and representational levels (e.g. Bradley & Brand, 1985 ; Johnstone, 1991 ; Taber, 1997 ). In chemical bonding, invisible components engage in complex processes whose scale makes them impossible to observe. Chemists routinely use visual representations to investigate relationships and move between the observable, physical level and the invisible particulate level (Kozma, Chin, Russell, & Marx, 2002 ). Generating explanations in a visual format may be a particularly useful learning tool for this domain.

For this topic, we expect that creating a visual rather than verbal explanation will aid students of both high and low spatial abilities. Visual explanations demand completeness; they were predicted to include more information than verbal explanations, particularly structural information. The inclusion of functional information should lead to better performance on the post-test since understanding how and why atoms bond is crucial to understanding the process. Participants with high spatial ability may be better able to explain function since the sub-microscopic nature of bonding requires mentally imagining invisible particles and how they interact. This experiment also asks whether creating an explanation per se can increase learning in the absence of additional teaching by administering two post-tests of knowledge, one immediately following instruction but before creating an explanation and one after creating an explanation. The scores on this immediate post-test were used to confirm that the visual and verbal groups were equivalent prior to the generation of explanations. Explanations were coded for structural and functional information, arrows, specific examples, and multiple representations. Do the acts of selecting, integrating, and explaining knowledge serve learning even in the absence of further study or teaching?

Participants were 126 (58 female) eighth grade students, aged 13–14 years, with written parental consent and enrolled in the same independent school described in Experiment 1. None of the students previously participated in Experiment 1. As in Experiment 1, randomization occurred within-class, with participants assigned to either the visual or verbal explanation condition.

The materials consisted of the MRT (same as Experiment 1), a video lesson on chemical bonding, two versions of the instructions, the immediate post-test, the delayed post-test, and a blank page for the explanations. All paper materials were typed on 8.5 × 11 in. sheets of paper. Both immediate and delayed post-tests consisted of seven multiple-choice items and three free-response items. The video lesson on chemical bonding consisted of a video that was 13 min 22 s. The video began with a brief review of atoms and their structure and introduced the idea that atoms combine to form molecules. Next, the lesson showed that location in the periodic table reveals the behavior and reactivity of atoms, in particular the gain, loss, or sharing of electrons. Examples of atoms, their valence shell structure, stability, charges, transfer and sharing of electrons, and the formation of ionic, covalent, and polar covalent bonds were discussed. The example of NaCl (table salt) was used to illustrate ionic bonding and the examples of O 2 and H 2 O (water) were used to illustrate covalent bonding. Information was presented verbally, accompanied by drawings, written notes of keywords and terms, and a color-coded periodic table.

On the first of three non-consecutive school days, participants completed the MRT as a whole-class activity. On the second day (occurring between two and three days after completing the MRT), participants viewed the recorded lesson on chemical bonding. They were instructed to pay close attention to the material but were not allowed to take notes. Immediately following the video, participants had 20 min to complete the immediate post-test; all finished within this time frame. On the third day (occurring on the next school day after viewing the video and completing the immediate post-test), the participants were randomly assigned to either the visual or verbal explanation condition. The typed instructions were given to participants along with a blank 8.5 × 11 in. sheet of paper for their explanations. The instructions differed for each condition. For the visual condition, the instructions were as follows: “You have just finished learning about chemical bonding. On the next piece of paper, draw an explanation of how atoms bond and how ionic and covalent bonds differ. Draw your explanation so that another student your age who has never studied this topic will be able to understand it. Be as clear and complete as possible, and remember to use pictures/diagrams only. After you complete your explanation, you will be asked to answer a series of questions about bonding.”

For the verbal condition the instructions were: “You have just finished learning about chemical bonding. On the next piece of paper, write an explanation of how atoms bond and how ionic and covalent bonds differ. Write your explanation so that another student your age who has never studied this topic will be able to understand it. Be as clear and complete as possible. After you complete your explanation, you will be asked to answer a series of questions about bonding.”

Participants were instructed to read the instructions carefully before beginning the task. The participants completed their explanations as a whole-class activity. Participants were given unlimited time to complete their explanations. Upon completion of their explanations, participants were asked to complete the ten-question delayed post-test (comparable to but different from the first) and were given a maximum of 20 min to do so. All participants completed their explanations as well as the post-test during the 45-min class period.

The mean score on the MRT was 10.39, with a median of 11. Boys (M = 12.5, SD = 4.8) scored significantly higher than girls (M = 8.0, SD = 4.0), F(1, 125) = 24.49, p  < 0.01. Participants were split into low and high spatial ability based on the median.

The maximum score for both the immediate and delayed post-test was 10 points. A repeated measures ANOVA showed that the difference between the immediate post-test scores (M = 4.63, SD = 0.469) and delayed post-test scores (M = 7.04, SD = 0.299) was statistically significant F(1, 125) = 18.501, p  < 0.05). Without any further instruction, scores increased following the generation of a visual or verbal explanation. Both groups improved significantly; those who created visual explanations (M = 8.22, SD = 0.208), F(1, 125) = 51.24, p  < 0.01, Cohen’s d  = 1.27 as well as those who created verbal explanations (M = 6.31, SD = 0.273), F(1,125) = 15.796, p  < 0.05, Cohen’s d  = 0.71. As seen in Fig.  5 , participants who generated visual explanations (M = 0.822, SD = 0.208) scored considerably higher on the delayed post-test than participants who generated verbal explanations (M = 0.631, SD = 0.273), F(1, 125) = 19.707, p  < 0.01, Cohen’s d  = 0.88. In addition, high spatial participants (M = 0.824, SD = 0.273) scored significantly higher than low spatial participants (M = 0.636, SD = 0.207), F(1, 125) = 19.94, p  < 0.01, Cohen’s d  = 0.87. The results of the test of the interaction between group and spatial ability was not significant.

Scores on the post-tests by explanation type and spatial ability

Explanations were coded for structural and functional content, arrows, specific examples, and multiple representations. A subset of the explanations (20%) was coded by both the first author and a middle school science teacher with expertise in Chemistry. Both scorers used the same coding system as a guide. The percentage of agreement between scores was above 90 for all measures. The first author then scored the remainder of the explanations. As evident from Fig.  4 , the visual explanations were individual inventions; they neither resembled each other nor those used in teaching. Most contained language, especially labels and symbolic language such as NaCl.

Structure, function, and modality

Visual and verbal explanations were coded for depicting or describing structural and functional components. The structural components included the following: the correct number of valence electrons, the correct charges of atoms, the bonds between non-metals for covalent molecules and between a metal and non-metal for ionic molecules, the crystalline structure of ionic molecules, and that covalent bonds were individual molecules. The functional components included the following: transfer of electrons in ionic bonds, sharing of electrons in covalent bonds, attraction between ions of opposite charge, bonding resulting in atoms with neutral charge and stable electron shell configurations, and outcome of bonding shows molecules with overall neutral charge. The presence of each component was awarded 1 point; the maximum possible points was 5 for structural and 5 for functional information. The modality, visual or verbal, of each component was also coded; if the information was given in both formats, both were coded.

As displayed in Fig.  6 , visual explanations contained a significantly greater number of structural components (M = 2.81, SD = 1.56) than verbal explanations (M = 1.30, SD = 1.54), F(1, 125) = 13.69, p  < 0.05. There were no differences between verbal and visual explanations in the number of functional components. Structural information was more likely to be depicted (M = 3.38, SD = 1.49) than described (M = 0.429, SD = 1.03), F(1, 62) = 21.49, p  < 0.05, but functional information was equally likely to be depicted (M = 1.86, SD = 1.10) or described (M = 1.71, SD = 1.87).

Functional information expressed verbally in the visual explanations significantly predicted scores on the post-test, F(1, 62) = 21.603, p  < 0.01, while functional information in verbal explanations did not. The inclusion of structural information did not significantly predict test scores. As seen Fig.  7 , explanations created by high spatial participants contained significantly more functional components, F(1, 125) = 7.13, p  < 0.05, but there were no ability differences in the amount of structural information created by high spatial participants in either visual or verbal explanations.

Average number of structural and functional components created by low and high spatial ability learners

Ninety-two percent of visual explanations contained arrows. Arrows were used to indicate motion as well as to label. The use of arrows was positively correlated with scores on the post-test, r = 0.293, p  < 0.05. There were no significant differences in the use of arrows between low and high spatial participants.

Specific examples

Explanations were coded for the use of specific examples, such as NaCl, to illustrate ionic bonding and CO 2 and O 2 to illustrate covalent bonding. High spatial participants (M = 1.6, SD = 0.69) used specific examples in their verbal and visual explanations more often than low spatial participants (M = 1.07, SD = 0.79), a marginally significant effect F(1, 125) = 3.65, p  = 0.06. Visual and verbal explanations did not differ in the presence of specific examples. The inclusion of a specific example was positively correlated with delayed test scores, r = 0.555, p  < 0.05.

Use of multiple representations

Many of the explanations (65%) contained multiple representations of bonding. For example, ionic bonding and its properties can be represented at the level of individual atoms or at the level of many atoms bonded together in a crystalline compound. The representations that were coded were as follows: symbolic (e.g. NaCl), atomic (showing structure of atom(s), and macroscopic (visible). Participants who created visual explanations generated significantly more (M =1.79, SD = 1.20) than those who created verbal explanations (M = 1.33, SD = 0.48), F (125) = 6.03, p  < 0.05. However, the use of multiple representations did not significantly correlate with delayed post-test scores on the delayed post-test.

Metaphoric explanations

Although there were too few examples to be included in the statistical analyses, some participants in the visual group created explanations that used metaphors and/or analogies to illustrate the differences between the types of bonding. Figure  4 shows examples of metaphoric explanations. In one example, two stick figures are used to show “transfer” and “sharing” of an object between people. In another, two sharks are used to represent sodium and chlorine, and the transfer of fish instead of electrons.

In the second experiment, students were introduced to chemical bonding, a more abstract and complex set of phenomena than the bicycle pump used in the first experiment. Students were tested immediately after instruction. The following day, half the students created visual explanations and half created verbal explanations. Following creation of the explanations, students were tested again, with different questions. Performance was considerably higher as a consequence of creating either explanation despite the absence of new teaching. Generating an explanation in this way could be regarded as a test of learning. Seen this way, the results echo and amplify previous research showing the advantages of testing over study (e.g. Roediger et al., 2011 ; Roediger & Karpicke, 2006 ; Wheeler & Roediger, 1992 ). Specifically, creating an explanation requires selecting the crucial information, integrating it temporally and causally, and expressing it clearly, processes that seem to augment learning and understanding without additional teaching. Importantly, creating a visual explanation gave an extra boost to learning outcomes over and above the gains provided by creating a verbal explanation. This is most likely due to the directness of mapping complex systems to a visual-spatial format, a format that can also provide a natural check for completeness and coherence as well as a platform for inference. In the case of this more abstract and complex material, generating a visual explanation benefited both low spatial and high spatial participants even if it did not bring low spatial participants up to the level of high spatial participants as for the bicycle pump.

Participants high in spatial ability not only scored better, they also generated better explanations, including more of the information that predicted learning. Their explanations contained more functional information and more specific examples. Their visual explanations also contained more functional information.

As in Experiment 1, qualities of the explanations predicted learning outcomes. Including more arrows, typically used to indicate function, predicted delayed test scores as did articulating more functional information in words in visual explanations. Including more specific examples in both types of explanation also improved learning outcomes. These are all indications of deeper understanding of the processes, primarily expressed in the visual explanations. As before, these findings provide ways that educators can guide students to craft better visual explanations and augment learning.

General discussion

Two experiments examined how learner-generated explanations, particularly visual explanations, can be used to increase understanding in scientific domains, notably those that contain “invisible” components. It was proposed that visual explanations would be more effective than verbal explanations because they encourage completeness and coherence, are more explicit, and are typically multimodal. These two experiments differ meaningfully from previous studies in that the information selected for drawing was not taken from a written text, but from a physical object (bicycle pump) and a class lesson with multiple representations (chemical bonding).

The results show that creating an explanation of a STEM phenomenon benefits learning, even when the explanations are created after learning and in the absence of new instruction. These gains in performance in the absence of teaching bear similarities to recent research showing gains in learning from testing in the absence of new instruction (e.g. Roediger et al., 2011 ; Roediger & Karpicke, 2006 ; Wheeler & Roediger, 1992 ). Many researchers have argued that the retrieval of information required during testing strengthens or enhances the retrieval process itself. Formulating explanations may be an especially effective form of testing for post-instruction learning. Creating an explanation of a complex system requires the retrieval of critical information and then the integration of that information into a coherent and plausible account. Other factors, such as the timing of the creation of the explanations, and whether feedback is provided to students, should help clarify the benefits of generating explanations and how they may be seen as a form of testing. There may even be additional benefits to learners, including increasing their engagement and motivation in school, and increasing their communication and reasoning skills (Ainsworth, Prain, & Tytler, 2011 ). Formulating a visual explanation draws upon students’ creativity and imagination as they actively create their own product.

As in previous research, students with high spatial ability both produced better explanations and performed better on tests of learning (e.g. Uttal et al., 2013 ). The visual explanations of high spatial students contained more information and more of the information that predicts learning outcomes. For the workings of a bicycle pump, creating a visual as opposed to verbal explanation had little impact on students of high spatial ability but brought students of lower spatial ability up to the level of students with high spatial abilities. For the more difficult set of concepts, chemical bonding, creating a visual explanation led to much larger gains than creating a verbal one for students both high and low in spatial ability. It is likely a mistake to assume that how and high spatial learners will remain that way; there is evidence that spatial ability develops with experience (Baenninger & Newcombe, 1989 ). It is possible that low spatial learners need more support in constructing explanations that require imagining the movement and manipulation of objects in space. Students learned the function of the bike pump by examining an actual pump and learned bonding through a video presentation. Future work to investigate methods of presenting material to students may also help to clarify the utility of generating explanations.

Creating visual explanations had greater benefits than those accruing from creating verbal ones. Surely some of the effectiveness of visual explanations is because they represent and communicate more directly than language. Elements of a complex system can be depicted and arrayed spatially to reflect actual or metaphoric spatial configurations of the system parts. They also allow, indeed, encourage, the use of well-honed spatial inferences to substitute for and support abstract inferences (e.g. Larkin & Simon, 1987 ; Tversky, 2011 ). As noted, visual explanations provide checks for completeness and coherence, that is, verification that all the necessary elements of the system are represented and that they work together properly to produce the outcomes of the processes. Visual explanations also provide a concrete reference for making and checking inferences about the behavior, causality, and function of the system. Thus, creating a visual explanation facilitates the selection and integration of information underlying learning even more than creating a verbal explanation.

Creating visual explanations appears to be an underused method of supporting and evaluating students’ understanding of dynamic processes. Two obstacles to using visual explanations in classrooms seem to be developing guidelines for creating visual explanations and developing objective scoring systems for evaluating them. The present findings give insights into both. Creating a complete and coherent visual explanation entails selecting the essential components and linking them by behavior, process, or causality. This structure and organization is familiar from recipes or construction sets: first the ingredients or parts, then the sequence of actions. It is also the ingredients of theater or stories: the players and their actions. In fact, the creation of visual explanations can be practiced on these more familiar cases and then applied to new ones in other domains. Deconstructing and reconstructing knowledge and information in these ways has more generality than visual explanations: these techniques of analysis serve thought and provide skills and tools that underlie creative thought. Next, we have shown that objective scoring systems can be devised, beginning with separating the information into structure and function, then further decomposing the structure into the central parts or actors and the function into the qualities of the sequence of actions and their consequences. Assessing students’ prior knowledge and misconceptions can also easily be accomplished by having students create explanations at different times in a unit of study. Teachers can see how their students’ ideas change and if students can apply their understanding by analyzing visual explanations as a culminating activity.

Creating visual explanations of a range of phenomena should be an effective way to augment students’ spatial thinking skills, thereby increasing the effectiveness of these explanations as spatial ability increases. The proverbial reading, writing, and arithmetic are routinely regarded as the basic curriculum of school learning and teaching. Spatial skills are not typically taught in schools, but should be: these skills can be learned and are essential to functioning in the contemporary and future world (see Uttal et al., 2013 ). In our lives, both daily and professional, we need to understand the maps, charts, diagrams, and graphs that appear in the media and public places, with our apps and appliances, in forms we complete, in equipment we operate. In particular, spatial thinking underlies the skills needed for professional and amateur understanding in STEM fields and knowledge and understanding STEM concepts is increasingly required in what have not been regarded as STEM fields, notably the largest employers, business, and service.

This research has shown that creating visual explanations has clear benefits to students, both specific and potentially general. There are also benefits to teachers, specifically, revealing misunderstandings and gaps in knowledge. Visualizations could be used by teachers as a formative assessment tool to guide further instructional activities and scoring rubrics could allow for the identification of specific misconceptions. The bottom line is clear. Creating a visual explanation is an excellent way to learn and master complex systems.

Additional file

Post-tests. (DOC 44 kb)

Acknowledgments

The authors are indebted to the Varieties of Understanding Project at Fordham University and The John Templeton Foundation and to the following National Science Foundation grants for facilitating the research and/or preparing the manuscript: National Science Foundation NSF CHS-1513841, HHC 0905417, IIS-0725223, IIS-0855995, and REC 0440103. We are grateful to James E. Corter for his helpful suggestions and to Felice Frankel for her inspiration. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the funders. Please address correspondence to Barbara Tversky at the Columbia Teachers College, 525 W. 120th St., New York, NY 10025, USA. Email: [email protected].

Authors’ contributions

This research was part of EB’s doctoral dissertation under the advisement of BT. Both authors contributed to the design, analysis, and drafting of the manuscript. Both authors read and approved the final manuscript.

Competing interests

The author declares that they have no competing interests.

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5. Visual Representation

How can you design computer displays that are as meaningful as possible to human viewers? Answering this question requires understanding of visual representation - the principles by which markings on a surface are made and interpreted. The analysis in this article addresses the most important principles of visual representation for screen design, introduced with examples from the early history of graphical user interfaces . In most cases, these principles have been developed and elaborated within whole fields of study and professional skill - typography , cartography, engineering and architectural draughting, art criticism and semiotics . Improving on the current conventions requires serious skill and understanding. Nevertheless, interaction designers should be able, when necessary, to invent new visual representations.

Introduction to Visual Representation by Alan Blackwell

Alan Blackwell on applying theories of Visual Representation

• 5.1 Typography and text

For many years, computer displays resembled paper documents. This does not mean that they were simplistic or unreasonably constrained. On the contrary, most aspects of modern industrial society have been successfully achieved using the representational conventions of paper, so those conventions seem to be powerful ones. Information on paper can be structured using tabulated columns, alignment, indentation and emphasis , borders and shading. All of those were incorporated into computer text displays. Interaction conventions, however, were restricted to operations of the typewriter rather than the pencil. Each character typed would appear at a specific location. Locations could be constrained, like filling boxes on a paper form. And shortcut command keys could be defined using onscreen labels or paper overlays. It is not text itself, but keyboard interaction with text that is limited and frustrating compared to what we can do with paper (Sellen and Harper 2001).

But despite the constraints on keyboard interaction, most information on computer screens is still represented as text. Conventions of typography and graphic design help us to interpret that text as if it were on a page, and human readers benefit from many centuries of refinement in text document design. Text itself, including many writing systems as well as specialised notations such as algebra, is a visual representation that has its own research and educational literature. Documents that contain a mix of bordered or coloured regions containing pictures, text and diagrammatic elements can be interpreted according to the conventions of magazine design, poster advertising, form design, textbooks and encyclopaedias. Designers of screen representations should take care to properly apply the specialist knowledge of those graphic and typographic professions. Position on the page, use of typographic grids, and genre-specific illustrative conventions should all be taken into account.

Author/Copyright holder: Unknown (pending investigation). Copyright terms and licence: Unknown (pending investigation). See section "Exceptions" in the copyright terms below.

Figure 5.1 : Contemporary example from the grid system website

Figure 5.2 : Example of a symbolic algebra expression (the single particle solution to Schrodinger's equation)

Figure 5.3 : Table layout of funerals from the plague in London in 1665

Figure 5.4 : Tabular layout of the first page of the Gutenberg Bible: Volume 1, Old Testament, Epistle of St. Jerome. The Gutenberg Bible was printed by Johannes Gutenberg, in Mainz, Germany in the 1450s

• 5.1.1 Summary

Most screen-based information is interpreted according to textual and typographic conventions, in which graphical elements are arranged within a visual grid, occasionally divided or contained with ruled and coloured borders. Where to learn more:

thegridsystem.org

Resnick , Elizabeth (2003): Design for Communication: Conceptual Graphic Design Basics. Wiley

• 5.2 Maps and graphs

The computer has, however, also acquired a specialised visual vocabulary and conventions. Before the text-based computer terminal (or 'glass teletype') became ubiquitous, cathode ray tube displays were already used to display oscilloscope waves and radar echoes. Both could be easily interpreted because of their correspondence to existing paper conventions. An oscilloscope uses a horizontal time axis to trace variation of a quantity over time, as pioneered by William Playfair in his 1786 charts of the British economy. A radar screen shows direction and distance of objects from a central reference point, just as the Hereford Mappa Mundi of 1300 organised places according to their approximate direction and distance from Jerusalem. Many visual displays on computers continue to use these ancient but powerful inventions - the map and the graph. In particular, the first truly large software project, the SAGE air defense system, set out to present data in the form of an augmented radar screen - an abstract map, on which symbols and text could be overlaid. The first graphics computer, the Lincoln Laboratory Whirlwind, was created to show maps, not text.

Figure 5.5 : The technique invented by William Playfair, for visual representation of time series data.

Author/Copyright holder: Courtesy of Premek. V. Copyright terms and licence: pd (Public Domain (information that is common property and contains no original authorship)).

Figure 5.6 : Time series data as shown on an oscilloscope screen

Author/Copyright holder: Courtesy of Remi Kaupp. Copyright terms and licence: CC-Att-SA (Creative Commons Attribution-ShareAlike 3.0 Unported)

Figure 5.7 : Early radar screen from HMS Belfast built in 1936

Author/Copyright holder: Courtesy of NOAA's National Weather Service. Copyright terms and licence: pd (Public Domain (information that is common property and contains no original authorship)).

Figure 5.8 : Early weather radar - Hurricane Abby approaching the coast of British Honduras in 1960

Figure 5.9 : The Hereford Mappa Mundi of 1300 organised places according to their approximate direction and distance from Jerusalem

Author/Copyright holder: Courtesy of Wikipedia. Copyright terms and licence: Unknown (pending investigation). See section "Exceptions" in the copyright terms below.

Figure 5.10 : The SAGE system in use. The SAGE system used light guns as interaction devices.

Author/Copyright holder: The MITRE Corporation. Copyright terms and licence: All Rights Reserved. Reproduced with permission. See section "Exceptions" in the copyright terms below.

Figure 5.11 : The Whirlwind computer at the MIT Lincoln Laboratory

• 5.2.1 Summary

Basic diagrammatic conventions rely on quantitative correspondence between a direction on the surface and a continuous quantity such as time or distance. These should follow established conventions of maps and graphs.

Where to learn more:

MacEachren , Alan M. (2004): How Maps Work: Representation, Visualization, and Design. The Guilford Press

• 5.3 Schematic drawings

Ivan Sutherland's groundbreaking PhD research with Whirlwind's successor TX-2 introduced several more sophisticated alternatives (Sutherland 1963). The use of a light pen allowed users to draw arbitrary lines, rather than relying on control keys to select predefined options. An obvious application, in the engineering context of Massachusetts Institute of Technology (MIT) where Sutherland worked, was to make engineering drawings such as the girder bridge in Figure 13. Lines on the screen are scaled versions of the actual girders, and text information can be overlaid to give details of force calculations. Plans of this kind, as a visual representation, are closely related to maps. However, where the plane of a map corresponds to a continuous surface, engineering drawings need not be continuous. Each set of connected components must share the same scale, but white space indicates an interpretive break, so that independent representations can potentially share the same divided surface - a convention introduced in Diderot's encyclopedia of 1772, which showed pictures of multiple objects on a page, but cut them loose from any shared pictorial context.

Author/Copyright holder: Courtesy of Ivan Sutherland. Copyright terms and licence: CC-Att-SA-3 (Creative Commons Attribution-ShareAlike 3.0).

Figure 5.12 : The TX-2 graphics computer, running Ivan Sutherland's Sketchpad software

Figure 5.13 : An example of a force diagram created using Sutherland's Sketchpad

Figure 5.14 : A page from the Encyclopédie of Diderot and d'Alembert, combining pictorial elements with diagrammatic lines and categorical use of white space.

• 5.3.1 Summary

Engineering drawing conventions allow schematic views of connected components to be shown in relative scale, and with text annotations labelling the parts. White space in the representation plane can be used to help the reader distinguish elements from each other rather than directly representing physical space. Where to learn more:

Engineering draughting textbooks

Ferguson , Eugene S. (1994): Engineering and the Mind's Eye. MIT Press

• 5.4 Pictures

The examples so far may seem rather abstract. Isn't the most 'natural' visual representation simply a picture of the thing you are trying to represent? In that case, what is so hard about design? Just point a camera, and take the picture. It seems like pictures are natural and intuitive, and anyone should be able to understand what they mean. Of course, you might want the picture to be more or less artistic, but that isn't a technical concern, is it? Well, Ivan Sutherland also suggested the potential value that computer screens might offer as artistic tools. His Sketchpad system was used to create a simple animated cartoon of a winking girl. We can use this example to ask whether pictures are necessarily 'natural', and what design factors are relevant to the selection or creation of pictures in an interaction design context.

We would not describe Sutherland's girl as 'realistic', but it is an effective representation of a girl. In fact, it is an unusually good representation of a winking girl, because all the other elements of the picture are completely abstract and generic. It uses a conventional graphic vocabulary of lines and shapes that are understood in our culture to represent eyes, mouths and so on - these elements do not draw attention to themselves, and therefore highlight the winking eye. If a realistic picture of an actual person was used instead, other aspects of the image (the particular person) might distract the viewer from this message.

Figure 5.15 : Sutherland's 'Winking Girl' drawing, created with the Sketchpad system

It is important, when considering the design options for pictures, to avoid the 'resemblance fallacy', i.e. that drawings are able to depict real object or scenes because the viewer's perception of the flat image simulates the visual perception of a real scene. In practice, all pictures rely on conventions of visual representation, and are relatively poor simulations of natural engagement with physical objects, scenes and people. We are in the habit of speaking approvingly of some pictures as more 'realistic' than others (photographs, photorealistic ray-traced renderings, 'old master' oil paintings), but this simply means that they follow more rigorously a particular set of conventions. The informed designer is aware of a wide range of pictorial conventions and options.

As an example of different pictorial conventions, consider the ways that scenes can be rendered using different forms of artistic perspective. The invention of linear perspective introduced a particular convention in which the viewer is encouraged to think of the scene as perceived through a lens or frame while holding his head still, so that nearby objects occupy a disproportionate amount of the visual field. Previously, pictorial representations more often varied the relative size of objects according to their importance - a kind of 'semantic' perspective. Modern viewers tend to think of the perspective of a camera lens as being most natural, due to the ubiquity of photography, but we still understand and respect alternative perspectives, such as the isometric perspective of the pixel art group eBoy, which has been highly influential on video game style.

Author/Copyright holder: Courtesy of Masaccio (1401-1428). Copyright terms and licence: pd (Public Domain (information that is common property and contains no original authorship))

Figure 5.16 : Example of an early work by Masaccio, demonstrating a 'perspective' in which relative size shows symbolic importance

Author/Copyright holder: eBoy.com. Copyright terms and licence: All Rights Reserved. Reproduced with permission. See section "Exceptions" in the copyright terms below.

Figure 5.17 : Example of the strict isometric perspective used by the eBoy group

Author/Copyright holder: Courtesy of Masaccio (1401-1428). Copyright terms and licence: pd (Public Domain (information that is common property and contains no original authorship)).

Figure 5.18 : Masaccio's mature work The Tribute Money, demonstrating linear perspective

As with most conventions of pictorial representation, new perspective rendering conventions are invented and esteemed for their accuracy by critical consensus, and only more slowly adopted by untrained readers. The consensus on preferred perspective shifts across cultures and historical periods. It would be naïve to assume that the conventions of today are the final and perfect product of technical evolution. As with text, we become so accustomed to interpreting these representations that we are blind to the artifice. But professional artists are fully aware of the conventions they use, even where they might have mechanical elements - the way that a photograph is framed changes its meaning, and a skilled pencil drawing is completely unlike visual edge-detection thresholds. A good pictorial representation need not simulate visual experience any more than a good painting of a unicorn need resemble an actual unicorn. When designing user interfaces, all of these techniques are available for use, and new styles of pictorial rendering are constantly being introduced.

• 5.4.1 Summary

Pictorial representations, including line drawings, paintings, perspective renderings and photographs rely on shared interpretive conventions for their meaning. It is naïve to treat screen representations as though they were simulations of experience in the physical world. Where to learn more:

Micklewright , Keith (2005): Drawing: Mastering the Language of Visual Expression. Harry N. Abrams

Stroebel , Leslie, Todd , Hollis and Zakia , Richard (1979): Visual Concepts for Photographers. Focal Press

• 5.5 Node-and-link diagrams

The first impulse of a computer scientist, when given a pencil, seems to be to draw boxes and connect them with lines. These node and link diagrams can be analysed in terms of the graph structures that are fundamental to the study of algorithms (but unrelated to the visual representations known as graphs or charts). A predecessor of these connectivity diagrams can be found in electrical circuit schematics, where the exact location of components, and the lengths of the wires, can be arranged anywhere, because they are irrelevant to the circuit function. Another early program created for the TX-2, this time by Ivan Sutherland's brother Bert, allowed users to create circuit diagrams of this kind. The distinctive feature of a node-and-link connectivity diagram is that, since the position of each node is irrelevant to the operation of the circuit, it can be used to carry other information. Marian Petre's research into the work of electronics engineers (Petre 1995) catalogued the ways in which they positioned components in ways that were meaningful to human readers, but not to the computer - like the blank space between Diderot's objects this is a form of 'secondary notation' - use of the plane to assist the reader in ways not related to the technical content.

Circuit connectivity diagrams have been most widely popularised through the London Underground diagram, an invention of electrical engineer Henry Beck. The diagram clarified earlier maps by exploiting the fact that most underground travellers are only interested in order and connectivity, not location, of the stations on the line. (Sadly, the widespread belief that a 'diagram' will be technical and hard to understand means that most people describe this as the London Undergound 'map', despite Beck's insistence on his original term).

Author/Copyright holder: Courtesy of Harry C. Beck and possibly F. H. Stingemore, born 1890, died 1954. Stingmore designed posters for the Underground Group and London Transport 1914-1942. Copyright terms and licence: Unknown (pending investigation). See section "Exceptions" in the copyright terms below.

Figure 5.19 : Henry Beck's London Underground Diagram (1933)

Author/Copyright holder: Computer History Museum, Mountain View, CA, USA. Copyright terms and licence: All Rights Reserved. Reproduced with permission. See section "Exceptions" in the copyright terms below.

Figure 5.20 : Node and link diagram of the kind often drawn by computing professionals

Figure 5.21 : Map of the London Underground network, as it was printed before the design of Beck's diagram (1932)

• 5.5.1 Summary

Node and link diagrams are still widely perceived as being too technical for broad acceptance. Nevertheless, they can present information about ordering and relationships clearly, especially if consideration is given to the value of allowing human users to specify positions. Where to learn more:

Diagrammatic representation books

Lowe , Ric (1992): Successful Instructional Diagram.

• 5.6 Icons and symbols

Maps frequently use symbols to indicate specific kinds of landmark. Sometimes these are recognisably pictorial (the standard symbols for tree and church), but others are fairly arbitrary conventions (the symbol for a railway station). As the resolution of computer displays increased in the 1970s, a greater variety of symbols could be differentiated, by making them more detailed, as in the MIT SDMS (Spatial Data Management System) that mapped a naval battle scenario with symbols for different kinds of ship. However, the dividing line between pictures and symbols is ambiguous. Children's drawings of houses often use conventional symbols (door, four windows, triangle roof and chimney) whether or not their own house has two storeys, or a fireplace. Letters of the Latin alphabet are shapes with completely arbitrary relationship to their phonetic meaning, but the Korean phonetic alphabet is easier to learn because the forms mimic the shape of the mouth when pronouncing those sounds. The field of semiotics offers sophisticated ways of analysing the basis on which marks correspond to meanings. In most cases, the best approach for an interaction designer is simply to adopt familiar conventions. When these do not exist, the design task is more challenging.

It is unclear which of the designers working on the Xerox Star coined the term 'icon' for the small pictures symbolising different kinds of system object. David Canfield Smith winningly described them as being like religious icons, which he said were pictures standing for (abstract) spiritual concepts. But 'icon' is also used as a technical term in semiotics. Unfortunately, few of the Xerox team had a sophisticated understanding of semiotics. It was fine art PhD Susan Kare's design work on the Apple Macintosh that established a visual vocabulary which has informed the genre ever since. Some general advice principles are offered by authors such as Horton (1994), but the successful design of icons is still sporadic. Many software publishers simply opt for a memorable brand logo, while others seriously misjudge the kinds of correspondence that are appropriate (my favourite blooper was a software engineering tool in which a pile of coins was used to access the 'change' command).

It has been suggested that icons, being pictorial, are easier to understand than text, and that pre-literate children, or speakers of different languages, might thereby be able to use computers without being able to read. In practice, most icons simply add decoration to text labels, and those that are intended to be self-explanatory must be supported with textual tooltips. The early Macintosh icons, despite their elegance, were surprisingly open to misinterpretation. One PhD graduate of my acquaintance believed that the Macintosh folder symbol was a briefcase (the folder tag looked like a handle), which allowed her to carry her files from place to place when placed inside it. Although mistaken, this belief never caused her any trouble - any correspondence can work, so long as it is applied consistently.

Copyright terms and licence: pd (Public Domain (information that is common property and contains no original authorship)).

Figure 5.22 : In art, the term Icon (from Greek, eikon, "image") commonly refers to religious paintings in Eastern Orthodox, Oriental Orthodox, and Eastern-rite Catholic jurisdictions. Here a 6th-century encaustic icon from Saint Catherine's Monastery, Mount Sinai

Author/Copyright holder: Apple Computer, Inc. Copyright terms and licence: All Rights Reserved. Reproduced with permission. See section "Exceptions" in the copyright terms below.

Figure 5.23 : In computing, David Canfield Smith described computer icons as being like religious icons, which he said were pictures standing for (abstract) spiritual concepts.

• 5.6.1 Summary

The design of simple and memorable visual symbols is a sophisticated graphic design skill. Following established conventions is the easiest option, but new symbols must be designed with an awareness of what sort of correspondence is intended - pictorial, symbolic, metonymic (e.g. a key to represent locking), bizarrely mnemonic, but probably not monolingual puns. Where to learn more:

Napoles , Veronica (1987): Corporate Identity Design.

• 5.7 Visual metaphor

The ambitious graphic designs of the Xerox Star/Alto and Apple Lisa/Macintosh were the first mass-market visual interfaces. They were marketed to office professionals, making the 'cover story' that they resembled an office desktop a convenient explanatory device. Of course, as was frequently noted at the time, these interfaces behaved nothing like a real desktop. The mnemonic symbol for file deletion (a wastebasket) was ridiculous if interpreted as an object placed on a desk. And nobody could explain why the desk had windows in it (the name was derived from the 'clipping window' of the graphics architecture used to implement them - it was at some later point that they began to be explained as resembling sheets of paper on a desk). There were immediate complaints from luminaries such as Alan Kay and Ted Nelson that strict analogical correspondence to physical objects would become obstructive rather than instructive. Nevertheless, for many years the marketing story behind the desktop metaphor was taken seriously, despite the fact that all attempts to improve the Macintosh design with more elaborate visual analogies , as in General Magic and Microsoft Bob, subsequently failed.

The 'desktop' can be far more profitably analysed (and extended) by understanding the representational conventions that it uses. The size and position of icons and windows on the desktop has no meaning, they are not connected, and there is no visual perspective, so it is neither a map, graph nor picture. The real value is the extent to which it allows secondary notation, with the user creating her own meaning by arranging items as she wishes. Window borders separate areas of the screen into different pictorial, text or symbolic contexts as in the typographic page design of a textbook or magazine. Icons use a large variety of conventions to indicate symbolic correspondence to software operations and/or company brands, but they are only occasionally or incidentally organised into more complex semiotic structures.

Author/Copyright holder:Apple Computer, Inc and Computer History Museum, Mountain View, CA. Copyright terms and licence: All Rights Reserved. Reproduced with permission. See section "Exceptions" in the copyright terms below.

Figure 5.24 : Apple marketed the visual metaphor in 1983 as a key benefit of the Lisa computer. This advertisement said 'You can work with Lisa the same familiar way you work at your desk'. However a controlled study by Carroll and Mazur (1986) found that the claim for immediately familiar operation may have been exaggerated.

Figure 5.25 : The Xerox Alto and Apple Lisa, early products in which bitmapped displays allowed pictorial icons to be used as mnemonic cues within the 'desktop metaphor'

Author/Copyright holder: Courtesy of Mschlindwein. Copyright terms and licence: CC-Att-SA (Creative Commons Attribution-ShareAlike 3.0 Unported).

Figure 5.26 : Apple Lisa

• 5.7.1 Summary

Theories of visual representation, rather than theories of visual metaphor, are the best approach to explaining the conventional Macintosh/Windows 'desktop'. There is huge room for improvement. Where to learn more:

Blackwell , Alan (2006): The reification of metaphor as a design tool . In ACM Transactions on Computer-Human Interaction , 13 (4) pp. 490-530

• 5.8 Unified theories of visual representation

The analysis in this article has addressed the most important principles of visual representation for screen design, introduced with examples from the early history of graphical user interfaces. In most cases, these principles have been developed and elaborated within whole fields of study and professional skill - typography, cartography, engineering and architectural draughting, art criticism and semiotics. Improving on the current conventions requires serious skill and understanding. Nevertheless, interaction designers should be able, when necessary, to invent new visual representations.

One approach is to take a holistic perspective on visual language, information design, notations, or diagrams. Specialist research communities in these fields address many relevant factors from low-level visual perception to critique of visual culture. Across all of them, it can be necessary to ignore (or not be distracted by) technical and marketing claims, and to remember that all visual representations simply comprise marks on a surface that are intended to correspond to things understood by the reader. The two dimensions of the surface can be made to correspond to physical space (in a map), to dimensions of an object, to a pictorial perspective, or to continuous abstract scales (time or quantity). The surface can also be partitioned into regions that should be interpreted differently. Within any region, elements can be aligned, grouped, connected or contained in order to express their relationships. In each case, the correspondence between that arrangement, and the intended interpretation, must be understood by convention, explained, or derived from the structural and perceptual properties of marks on the plane. Finally, any individual element might be assigned meaning according to many different semiotic principles of correspondence.

The following table summarises holistic views, as introduced above, drawing principally on the work of Bertin, Richards, MacEachren, Blackwell & Engelhardt and Engelhardt. Where to learn more:

Engelhardt , Yuri (2002). The Language of Graphics. A framework for the analysis of syntax and meaning in maps, charts and diagrams (PhD Thesis) . University of Amsterdam

Table 5.1 : Summary of the ways in which graphical representations can be applied in design, via different systems of correspondence

Table 5.2 : Screenshot from the site gapminder.org, illustrating a variety of correspondence conventions used in different parts of the page

As an example of how one might analyse (or working backwards, design) a complex visual representation, consider the case of musical scores. These consist of marks on a paper surface, bound into a multi-page book, that is placed on a stand at arms length in front of a performer. Each page is vertically divided into a number of regions, visually separated by white space and grid alignment cues. The regions are ordered, with that at the top of the page coming first. Each region contains two quantitative axes, with the horizontal axis representing time duration, and the vertical axis pitch. The vertical axis is segmented by lines to categorise pitch class. Symbols placed at a given x-y location indicate a specific pitched sound to be initiated at a specific time. A conventional symbol set indicates the duration of the sound. None of the elements use any variation in colour, saturation or texture. A wide variety of text labels and annotation symbols are used to elaborate these basic elements. Music can be, and is, also expressed using many other visual representations (see e.g. Duignan for a survey of representations used in digital music processing).

• 5.9 Where to learn more

The historical examples of early computer representations used in this article are mainly drawn from Sutherland (Ed. Blackwell and Rodden 2003), Garland (1994), and Blackwell (2006). Historical reviews of visual representation in other fields include Ferguson (1992), Pérez-Gómez and Pelletier (1997), McCloud (1993), Tufte (1983). Reviews of human perceptual principles can be found in Gregory (1970), Ittelson (1996), Ware (2004), Blackwell (2002). Advice on principles of interaction with visual representation is distributed throughout the HCI literature, but classics include Norman (1988), Horton (1994), Shneiderman ( Shneiderman and Plaisant 2009, Card et al 1999, Bederson and Shneiderman 2003) and Spence (2001). Green's Cognitive Dimensions of Notations framework has for many years provided a systematic classification of the design parameters in interactive visual representations. A brief introduction is provided in Blackwell and Green (2003).

Research on visual representation topics is regularly presented at the Diagrams conference series (which has a particular emphasis on cognitive science), the InfoDesign and Vision Plus conferences (which emphasise graphic and typographic information design), the Visual Languages and Human-Centric Computing symposia (emphasising software tools and development), and the InfoVis and Information Visualisation conferences (emphasising quantitative and scientific data visualisation ).

• 5.9.0.1 IV - International Conference on Information Visualization

2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998

• 5.9.0.2 DIAGRAMS - International Conference on the Theory and Application of Diagrams

2008 2006 2004 2002 2000

• 5.9.0.3 VL-HCC - Symposium on Visual Languages and Human Centric Computing

2008 2007 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 1990

• 5.9.0.4 InfoVis - IEEE Symposium on Information Visualization

2005 2004 2003 2002 2001 2000 1999 1998 1997 1995

• 5.10 References

Anderson , Michael, Meyer , Bernd and Olivier , Patrick (2002): Diagrammatic Representation and Reasoning. London, UK,

Bederson , Benjamin B. and Shneiderman , Ben (2003): The Craft of Information Visualization : Readings and Reflections. Morgan Kaufman Publishers

Bertin , Jacques (1967): Semiology of Graphics: Diagrams, Networks, Maps (Sémiologie graphique: Les diagrammes - Les réseaux - Les cartes). English translation by W. J. Berg. Madison, WI, USA, University of Wisconsin Press

Blackwell , Alan (2002): Psychological perspectives on diagrams and their users. In: Anderson , Michael, Meyer , Bernd and Olivier , Patrick (eds.). "Diagrammatic Representation and Reasoning". London, UK: pp. 109-123

Blackwell , Alan and Engelhardt , Yuri (2002): A Meta-Taxonomy for Diagram Research. In: Anderson , Michael, Meyer , Bernd and Olivier , Patrick (eds.). "Diagrammatic Representation and Reasoning". London, UK: pp. 47-64

Blackwell , Alan and Green , T. R. G. (2003): Notational Systems - The Cognitive Dimensions of Notations Framework. In: Carroll , John M. (ed.). "HCI Models, Theories, and Frameworks". San Francisco: Morgan Kaufman Publisherspp. 103-133

Carroll , John M. and Mazur , Sandra A. (1986): LisaLearning . In Computer , 19 (11) pp. 35-49

Garland , Ken (1994): Mr . Beck's Underground Map. Capital Transport Publishing

Goodman , Nelson (1976): Languages of Art. Hackett Publishing Company

Gregory , Richard L. (1970): The Intelligent Eye. London, Weidenfeld and Nicolson

Horton , William (1994): The Icon Book: Visual Symbols for Computer Systems and Documentation. John Wiley and Sons

Ittelson , W. H. (1996): Visual perception of markings . In Psychonomic Bulletin & Review , 3 (2) pp. 171-187

Mccloud , Scott (1994): Understanding Comics: The Invisible Art. Harper Paperbacks

Norman , Donald A. (1988): The Design of Everyday Things. New York, Doubleday

Petre , Marian (1995): Why Looking Isn't Always Seeing: Readership Skills and Graphical Programming . In Communications of the ACM , 38 (6) pp. 33-44

Pérez-Gómez , Alberto and Pelletier , Louise (1997): Architectural Representation and the Perspective Hinge. MIT Press

Richards , Clive (1984). Diagrammatics: an investigation aimed at providing a theoretical framework for studying diagrams and for establishing a taxonomy of their fundamental modes of graphic organization. Unpublished Phd Thesis . Royal College of Art, London, UK

Sellen , Abigail and Harper , Richard H. R. (2001): The Myth of the Paperless Office. MIT Press

Shneiderman , Ben and Plaisant , Catherine (2009): Designing the User Interface : Strategies for Effective Human-Computer Interaction (5th ed.). Addison-Wesley

Spence , Robert (2001): Information Visualization. Addison Wesley

Sutherland , Ivan E. (1963). Sketchpad, A Man-Machine Graphical Communication System. PhD Thesis at Massachusetts Institute of Technology, online version and editors' introduction by Alan Blackwell & K. Rodden. Technical Report 574 . Cambridge University Computer Laboratory

Tufte , Edward R. (1983): The Visual Display of Quantitative Information. Cheshire, CT , Graphics Press

Ware , Colin (2004): Information Visualization: Perception for Design, 2nd Ed. San Francisco, Morgan Kaufman

• 5 Visual Representation

Human-Computer Interaction: The Foundations of UX Design

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5.10 commentary by ben shneiderman.

Since computer displays are such powerful visual appliances, careful designers devote extensive effort to getting the visual representation right. They have to balance the demands of many tasks, diverse users, and challenging requirements, such as short learning time, rapid performance, low error rates, and good retention over time. Designing esthetic interfaces that please and even delight users is a further expectation that designers must meet to be successful. For playful and discretionary tasks esthetic concerns may dominate, but for life critical tasks, rapid performance with low error rates are essential. Alan Blackwell's competent description of many visual representation issues is a great start for newcomers with helpful reminders even for experienced designers. The videos make for a pleasant personal accompaniment that bridges visual representation for interface design with thoughtful analyses of representational art. Blackwell's approach might be enriched by more discussion of visual representations in functional product design tied to meaningful tasks. Learning from paintings of Paris is fine, but aren't there other lessons to learn from visual representations in airport kiosks, automobile dashboards, or intensive care units? These devices as well as most graphical user interfaces and mobile devices raise additional questions of changing state visualization and interaction dynamics. Modern designers need to do more than show the right phone icon, they need to show ringing, busy, inactive, no network, conference mode, etc., which may include color changes (highlighted, grayed out), animations, and accompanying sounds. These designers also need to deal with interactive visual representations that happen with a click, double-click, right-click, drag, drag-and-drop, hover, multi-select, region-select, brushing-linking, and more. The world of mobile devices such as phones, cameras, music players, or medical sensors is the new frontier for design, where visual representations are dynamic and tightly integrated with sound, haptics, and novel actions such as shaking, twisting, or body movements. Even more challenging is the expectation that goes beyond the solitary viewer to the collaboration in which multiple users embedded in a changing physical environment produce new visual representations. These changing and interactive demands on designers invite creative expressions that are very different from designs for static signs, printed diagrams, or interpretive art. The adventure for visual representation designers is to create a new language of interaction that engages users, accelerates learning, provides comprehensible feedback, and offers appropriate warnings when dangers emerge. Blackwell touches on some of these issues in the closing Gapminder example, but I was thirsty for more.

5.11 Commentary by Clive Richards

If I may be permitted a graphically inspired metaphor Alan Blackwell provides us with a neat pen sketch of that extensive scene called 'visual representation' (Blackwell 2011).

"Visualisation has a lot more to offer than most people are aware of today" we are told by Robert Kosara at the end of his commentary (Kosara 2010) on Stephen Few's related article on ' Data visualisation for human perception ' (Few 2010). Korsara is right, and Blackwell maps out the broad territory in which many of these visualisation offerings may be located. In this commentary I offer a few observations on some prominent features in that landscape: dynamics, picturing, semiotics and metaphor.

Ben Shneiderman's critique of Blackwell's piece points to a lack of attention to "... additional questions of changing state visualisations and interaction dynamics" (Shneiderman 2010). Indeed the possibilities offered by these additional questions present some exciting challenges for interaction designers - opportunities to create novel and effective combinations of visual with other sensory and motor experiences in dynamic operational contexts. Shneiderman suggests that: "These changing and interactive demands on designers invite creative expressions that are very different from design for static signs, printed diagrams, or interpretive art". This may be so up to a point, but here Shneinderman and I part company a little. The focus of Blackwell's essay is properly on the visual representation side of facilities available to interaction designers, and in that context he is quite right to give prominence to highly successful but static visual representation precedents, and also to point out the various specialist fields of endeavour in which they have been developed. Some of these representational approaches have histories reaching back thousands of years and are deeply embedded within our culture. It would be foolhardy to disregard conventions established in, say, the print domain, and to try to re-invent everything afresh for the screen, even if this were a practical proposition. Others have made arguments to support looking to historical precedents. For example Michael Twyman has pointed out that when considering typographic cueing and "... the problems of the electronic age ... we have much to learn from the manuscript age" (Twyman 1987, p5). He proposes that studying the early scribes' use of colour, spacing and other graphical devices can usefully inform the design of today's screen-based texts. And as Blackwell points out in his opening section on 'Typography and text' "most information on computer screen is still presented as text".

It is also sometimes assumed that the pictorial representation of a dynamic process is best presented dynamically. However it can be argued that the comic book convention of using a sequence of static frames is sometimes superior for focusing the viewer's attention on the critical events in a process, rather than using an animated sequence in which key moments may be missed. This is of course not to deny the immense value of the moving and interactive visual image in the right context. The Gapminder charts are a case in point (http://www.gapminder.org). Blackwell usefully includes one of these, but as a static presentation. These diagrams come to life and really tell their story through the clustering of balloons that inflate or deflate as they move about the screen when driven through simulated periods of time.

While designing a tool for engineers to learn about the operation and maintenance of an oil system for an aircraft jet engine, Detlev Fischer devised a series of interactive animations, called 'Cinegrams' to display in diagrammatic form various operating procedures (Fischer and Richards 1995). He used the cinematic techniques of time compression and expansion in one animated sequence to show how the slow accumulation of debris in an oil filter, over an extended period of time, would eventually create a blockage to the oil flow and trigger the opening of a by-pass device in split seconds. Notwithstanding my earlier comment about the potential superiority of the comic strip genre for displaying some time dependant processes this particular Cinegram proved very instructive for the targeted users. There are many other examples one could cite where dynamic picturing of this sort has been deployed to similarly good effect in interactive environments.

Shneinderman also comments that: "Blackwell's approach might be enriched by more discussion of visual representation in functional product design tied to meaningful tasks". An area I have worked in is the pictorial representation of engineering assemblies to show that which is normally hidden from view. Techniques to do this on the printed page include 'ghosting' (making occluding parts appear as if transparent), 'exploding' (showing components separately, set out in dis-assembly order along an axis) and cutting away (taking a slice out of an outer shell to reveal mechanisms beneath). All these three-dimensional picturing techniques were used by, if not actually invented by, Leonardo Da Vinci (Richards 2006). All could be enhanced by interactive viewer control - an area of further fruitful exploration for picturing purposes in technical documentation contexts.

Blackwell's section on 'Pictures' warns us that when considering picturing options to avoid the "resemblance fallacy" pointing out the role that convention plays, even in so called photo-realistic images. He also points out that viewers can be distracted from the message by incidental information in 'realistic' pictures. From my own work in the field I know that technical illustrators' synoptic black and white outline depictions are regarded as best for drawing the viewer's attention to the key features of a pictorial representation. Research in this area has shown that when using linear perspective type drawings the appropriate deployment of lines of varying 'weight', rather than of a single thickness, can have a significant effect on viewers' levels of understanding about what is depicted (Richards, Bussard and Newman 2007). This work was done specifically to determine an 'easy to read' visual representational style when manipulating on the screen images of CAD objects. The most effective convention was shown to be: thin lines for edges where both planes forming the edge are visible and thicker lines for edges where only one plane is visible - that is where an outline edge forms a kind of horizon to the object.

These line thickness conventions appear on the face of it to have little to do with how we normally perceive the world, and Blackwell tells us that: "A good pictorial representation need not simulate visual experience any more than a good painting of a unicorn need resemble an actual unicorn". And some particular representations of unicorns can aid our understanding of how to use semiotic theory to figure out how pictures may be interpreted and, importantly, sometimes misunderstood - as I shall describe in the following.

Blackwell mentions semiotics, almost in passing, however it can help unravel some of the complexities of visual representation. Evelyn Goldsmith uses a Charles Addams cartoon to explain the relevance of the 'syntactic', 'semantic' and 'pragmatic' levels of semiotic analysis when applied to pictures (Goldsmith 1978). The cartoon in question, like many of those by Charles Addams, has no caption. It shows two unicorns standing on a small island in the pouring rain forlornly watching the Ark sailing away into the distance. Goldsmith suggests that most viewers will have little trouble in interpreting the overlapping elements in the scene, for example that one unicorn is standing behind the other, nor any difficulty understanding that the texture gradient of the sea stands for a receding horizontal plane. These represent the syntactic level of interpretation. Most adults will correctly identify the various components of the picture at the semantic level, however Goldsmith proposes that a young child might mistake the unicorns for horses and be happy with 'boat' for the Ark. But at the pragmatic level of interpretation, unless a viewer of the picture is aware of the story of Noah's Ark, the joke will be lost  - the connection will not be made between the scene depicted in the drawing and the scarcity of unicorns. This reinforces the point that one should not assume that the understanding of pictures is straightforward. There is much more to it than a simple matter or recognition. This is especially the case when metaphor is involved in visual representation.

Blackwell's section on 'Visual metaphor' is essentially a critique of the use of "theories of visual metaphor" as an "approach to explaining the conventional Mackintosh/Windows 'desktop' ". His is a convincing argument but there is much more which may be said about the use of visual metaphor - especially to show that which otherwise cannot be pictured. In fact most diagrams employ a kind of spatial metaphor when not depicting physical arrangements, for example when using the branches of a tree to represent relations within a family (Richards 2002). The capability to represent the invisible is the great strength of the visual metaphor, but there are dangers, and here I refer back to semiotics and particularly the pragmatic level of analysis. One needs to know the story to get the picture.

In our parental home, one of the many books much loved by my two brothers and me, was The Practical Encyclopaedia for Children (Odhams circa 1948). In it a double page spread illustration shows the possible evolutionary phases of the elephant. These are depicted as a procession of animals in a primordial swamp cum jungle setting. Starting with a tiny fish and passing to a small aquatic creature climbing out of the water onto the bank the procession progresses on through eight phases of transformation, including the Moeritherium and the Paleomatodon, finishing up with the land-based giant of today's African Elephant. Recently one of my brothers confessed to me that through studying this graphical diorama he had believed as a child that the elephant had a life cycle akin to that of a frog. He had understood that the procession was a metaphor for time. He had just got the duration wrong - by several orders of magnitude. He also hadn't understood that each separate depiction was of a different animal. He had used the arguably more sophisticated concept that it was the same animal at different times and stages in its individual development.

Please forgive the cliché if I say that this anecdote clearly illustrates that there can be more to looking at a picture than meets the eye? Blackwell's essay provides some useful pointers for exploring the possibilities of this fascinating territory of picturing and visual representation in general.   

• Blackwell A 2011 'Visual representation' Interaction-Design.org
• Few S 2010 ' Data visualisation for human perception ' Interaction-Design.org
• Fischer D and Richards CJ 1995 'The presentation of time in interactive animated systems diagrams' In: Earnshaw RA and Vince JA (eds) Multimedia Systems and Applications London: Academic Press Ltd (pp141 - 159). ISBN 0-12-227740-6
• Goldsmith E 1978 An analysis of the elements affecting comprehensibility of illustrations intended as supportive of text PhD thesis (CNAA) Brighton Polytechnic
• Korsa R 2010 ' Commentary on Stephen Few's article : Data visualisation for human perception' Interaction-Design.org Odhams c. 1949 The practical encyclopaedia for children (pp 194 - 195)
• Richards CJ 2002 'The fundamental design variables of diagramming' In: Oliver P, Anderson M and Meyer B (eds) Diagrammatic representation and reasoning London: Springer Verlag (pp 85 - 102) ISBN 1-85233-242-5
• Richards CJ 2006 'Drawing out information - lines of communication in technical illustration' Information Design Journal 14 (2) 93 - 107
• Richards CJ, Bussard N, Newman R 2007 'Weighing up line weights: the value of differing line thicknesses in technical illustrations' Information Design Journal 15 (2) 171 - 181
• Shneiderman B 2011 'Commentary on Alan Blackwell's article: Visual representation' Interaction-Design.org
• Twyman M 1982 'The graphic representation of language' Information Design Journal 3 (1) 2 - 22

5.12 Commentary by Peter C-H. Cheng

Alan Blackwell has provided us with a fine introduction to the design of visual representations. The article does a great job in motivating the novice designer of visual representations to explore some of the fundamental issues that lurk just beneath the surface of creating effective representations.  Furthermore, he gives us all quite a challenge:

Alan, quite rightly, claims that we must consider the fundamental principles of symbolic correspondence, if we are to design new genres of visual representations beyond the common forms of displays and interfaces.  The report begins to equip the novice visual representation designer with an understanding of the nature of symbolic correspondence between the components of visual representations and the things they represent, whether objects, actions or ideas.  In particular, it gives a useful survey of how correspondence works in a range of representations and provides a systematic framework of how systems of correspondence can be applied to design. The interactive screen shot is an exemplary visual representation that vividly reveals the correspondence techniques used in each part of the example diagram.

However, suppose you really wished to rise to the challenge of creating novel visual representations, how far will a knowledge of the fundamentals of symbolic correspondence take you? Drawing on my studies of the role of diagrams in the history of science, experience of inventing novel visual representations and research on problem solving and learning with diagrams, from the perspective of Cognitive Science, my view is that such knowledge will be necessary but not sufficient for your endeavours.  So, what else should the budding visual representation designer consider? From the perspective of cognitive science there are at least three aspects that we may profitably target.

First, there is the knowledge of how human process information; specifically the nature of the human cognitive architecture. By this, I mean more than visual perception, but an understanding of how we mentally receive, store, retrieve, transform and transmit information. The way the mind deals with each of these basic types of information processing provides relevant constrains for the design of visual representations. For instance, humans often, perhaps even typically, encode concepts in the form of hierarchies of schemas, which are information structures that coordinate attributes that describe and differentiate classes of concepts. These hierarchies of schemas underpin our ability to efficiently generalize or specialize concepts. Hence, we can use this knowledge to consider whether particular forms of symbolic correspondence will assist or hinder the forms of inference that we hope the user of the representation may make. For example, are the main symbolic correspondences in a visual representation consistent with the key attributes of the schemas for the concepts being considered?

Second, it may be useful for the designer to consider the broader nature of the tasks that the user may wish to do with the designed representation.  Resource allocation, optimization, calculating quantities, inferences about of possible outcomes, classification, reasoning about extreme or special cases, and debugging: these are just a few of the many possibilities. These tasks are more generic than the information-oriented options considered in the 'design uses' column of Figure 27 in the article. They are worth addressing, because they provide constraints for the initial stages of representation design, by narrowing the search for what are likely to be effective correspondences to adopt. For example, if taxonomic classification is important, then separation and layering will be important correspondences; whereas magnitude calculations may demand scale mapping, Euclidian and metrical correspondences.

The third aspect concerns situations in which the visual representation must support not just a single task, but many diverse tasks. For example, a visual representation to help students learn about electricity will be used to explain the topology of circuits, make computations with electrical quantities, provide explanations of circuit behaviour (in terms of formal algebraic models and as qualitative causal models), facilitate fault finding or trouble shooting, among other activities. The creation of novel representations in such circumstances is perhaps one of the most challenging for designers. So, what knowledge can help? In this case, I advocate attempting to design representations on the basis of an analysis of the underlying conceptual structure of the knowledge of the target domain. Why? Because the nature of the knowledge is invariant across different classes of task. For example, for problem solving and learning of electricity, all the tasks depend upon the common fundamental conceptual structures of the domain that knit together the laws governing the physical properties of electricity and circuit topology. Hence, a representation that makes these concepts readily available through effective representation designed will probably be effective for a wide range of tasks.

In summary, it is desirable for the aspiring visual representation designer to consider symbolic correspondence, but I recommend they cast their net more widely for inspiration by learning about the human cognitive architecture, focusing on the nature of the task for which they are designing, and most critically thinking about the underlying conceptual structure of the knowledge of the target domain.

5.13 Commentary by Brad A. Myers

I have been teaching human-computer interaction to students with a wide range of backgrounds for many years. One of the most difficult areas for them to learn seems to be visual design. Students seem to quickly pick up rules like Nielsen's Heuristics for interaction (Nielsen & Molich, 1990), whereas the guidelines for visual design are much more subtle. Alan Blackwell's article presents many useful points, but a designer needs to know so much more! Whereas students can achieve competence at achieving Nielsen's "consistency and standards," for example, they struggle with selecting an appropriate representation for their information. And only a trained graphic designer is likely to be able to create an attractive and effective icon. Some people have a much better aesthetic sense, and can create much more beautiful and appropriate representations. A key goal of my introductory course, therefore, is to try to impart to the students how difficult it is to do visual design, and how wide the set of choices is. Studying the examples that Blackwell provides will give the reader a small start towards effective visual representations, but the path requires talent, study, and then iterative design and testing to evaluate and improve a design's success.

• Nielsen, J., & Molich, R. (1990). Heuristic evaluation of user interfaces. Paper presented at the Proc. ACM CHI'90 Conf, Seattle, WA, 249-256.
• See also: http://www.useit.com/papers/heuristic/heuristic_list.html

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Understanding Without Words: Visual Representations in Math, Science and Art

• First Online: 02 November 2021

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• Kathleen Coessens 5 ,
• Karen François 6 &
• Jean Paul Van Bendegem 7  

Part of the book series: Educational Research ((EDRE,volume 11))

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As knowledge can be condensed in different non-verbal ways of representation, the integration of graphic and visual representations and design in research output helps to expand insight and understanding. Layers of visual charts, maps, diagrams not only aim at synergizing the complexity of a topic with visual simplicity, but also to guide a personal search for and insights into knowledge. However, from research over graphic representation to interpretation and understanding implies a move that is scientific, epistemic, artistic and, last but not least, ethical. This article will consider these four aspects from both the side of the researcher and the receiver/interpreter from three different perspectives. The first perspective will consider the importance of visual representations in science and its recent developments. As a second perspective, we will analyse the discussion concerning the use of diagrams in the philosophy of mathematics. A third perspective will be from an artistic perspective on diagrams, where the visual tells us (sometimes) more than the verbal.

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Visual Reasoning in Science and Mathematics

Diagrams in Mathematics: On Visual Experience in Peirce

Why Do Mathematicians Need Diagrams? Peirce’s Existential Graphs and the Idea of Immanent Visuality

This is the school typically associated with the mathematician David Hilbert. Although he himself saw formalism as a particular strategy to solve certain specific mathematical questions such as the consistency of arithmetic, nevertheless in the hands mainly of the French Bourbaki group it became an overall philosophy and the famous expression that mathematics is a game of meaningless signs was born. See (Detlefsen, 2005 ).

This seemingly simple graph consisting of 10 vertices and 15 edges is nevertheless of supreme importance in graph theory because of the impressive list of properties it possesses. Starikova ( 2017 ) presents a nice and thorough analysis of the graph (in order to discuss its aesthetic qualities). We just mention that the graph has 120 symmetries.

To be found at http://mathworld.wolfram.com/PetersenGraph.html , consulted Sunday, 17 September 2017.

A famous example is a proof of Augustin Cauchy wherein he made the mistake of inverting the quantifiers. A statement of the form ‘For all x, there is a y such that …’ was interpreted as ‘There is a y, such that for all x …’, which is a stronger statement. It is interesting to mention that this case was already (partially) studied by Imre Lakatos, see (Lakatos, 1976 , Appendix 1), who is often seen as the founding father of the study of mathematical practices.

That being said, the interest in the topic is growing. We just mention (Giaquinto, ), (Manders, ), (Giardino, ) and (Carter, 2010 ) as initiators. Of special interest is the connection that is being made between the philosophical approach and the opportunities offered by cognitive science to study the multiple ways that diagrams can be used an interpreted, see (Mumma & Hamami, 2013 ).

It is interesting that, under the same topic, David Bridges (this volume) develops a similar point of view on arts-based research for education. While Bridges questions the ambiguity of the potential and use of artistic means and expressions as research, we rather consider artistic expressions as enriching methods for knowledge construction, opening new insights by their complexity and layeredness.

Alsina, C., & Nelsen, R. B. (2006). Math made visual . The Mathematical Association of America.

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Acknowledgements

Thanks to Joachim Frans (2017) who directed my attention to the work of Nelsen (1993, 2000) in his inspiring Ph.D. thesis on ‘Mathematical explanation’.

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About this chapter

Coessens, K., François, K., Van Bendegem, J.P. (2021). Understanding Without Words: Visual Representations in Math, Science and Art. In: Smeyers, P., Depaepe, M. (eds) Production, Presentation, and Acceleration of Educational Research: Could Less be More?. Educational Research, vol 11. Springer, Singapore. https://doi.org/10.1007/978-981-16-3017-0_9

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Blog Graphic Design 15 Effective Visual Presentation Tips To Wow Your Audience

15 Effective Visual Presentation Tips To Wow Your Audience

Written by: Krystle Wong Sep 28, 2023

So, you’re gearing up for that big presentation and you want it to be more than just another snooze-fest with slides. You want it to be engaging, memorable and downright impressive. 

Well, you’ve come to the right place — I’ve got some slick tips on how to create a visual presentation that’ll take your presentation game up a notch. 

Packed with presentation templates that are easily customizable, keep reading this blog post to learn the secret sauce behind crafting presentations that captivate, inform and remain etched in the memory of your audience.

Click to jump ahead:

What is a visual presentation & why is it important?

15 effective tips to make your visual presentations more engaging, 6 major types of visual presentation you should know , what are some common mistakes to avoid in visual presentations, visual presentation faqs, 5 steps to create a visual presentation with venngage.

A visual presentation is a communication method that utilizes visual elements such as images, graphics, charts, slides and other visual aids to convey information, ideas or messages to an audience. 

Visual presentations aim to enhance comprehension engagement and the overall impact of the message through the strategic use of visuals. People remember what they see, making your point last longer in their heads. 

Without further ado, let’s jump right into some great visual presentation examples that would do a great job in keeping your audience interested and getting your point across.

In today’s fast-paced world, where information is constantly bombarding our senses, creating engaging visual presentations has never been more crucial. To help you design a presentation that’ll leave a lasting impression, I’ve compiled these examples of visual presentations that will elevate your game.

1. Use the rule of thirds for layout

Ever heard of the rule of thirds? It’s a presentation layout trick that can instantly up your slide game. Imagine dividing your slide into a 3×3 grid and then placing your text and visuals at the intersection points or along the lines. This simple tweak creates a balanced and seriously pleasing layout that’ll draw everyone’s eyes.

2. Get creative with visual metaphors

Got a complex idea to explain? Skip the jargon and use visual metaphors. Throw in images that symbolize your point – for example, using a road map to show your journey towards a goal or using metaphors to represent answer choices or progress indicators in an interactive quiz or poll.

3. Visualize your data with charts and graphs

The right data visualization tools not only make content more appealing but also aid comprehension and retention. Choosing the right visual presentation for your data is all about finding a good match. 

For ordinal data, where things have a clear order, consider using ordered bar charts or dot plots. When it comes to nominal data, where categories are on an equal footing, stick with the classics like bar charts, pie charts or simple frequency tables. And for interval-ratio data, where there’s a meaningful order, go for histograms, line graphs, scatterplots or box plots to help your data shine.

In an increasingly visual world, effective visual communication is a valuable skill for conveying messages. Here’s a guide on how to use visual communication to engage your audience while avoiding information overload.

4. Employ the power of contrast

Want your important stuff to pop? That’s where contrast comes in. Mix things up with contrasting colors, fonts or shapes. It’s like highlighting your key points with a neon marker – an instant attention grabber.

5. Tell a visual story

Structure your slides like a storybook and create a visual narrative by arranging your slides in a way that tells a story. Each slide should flow into the next, creating a visual narrative that keeps your audience hooked till the very end.

Icons and images are essential for adding visual appeal and clarity to your presentation. Venngage provides a vast library of icons and images, allowing you to choose visuals that resonate with your audience and complement your message. 

6. Show the “before and after” magic

Want to drive home the impact of your message or solution? Whip out the “before and after” technique. Show the current state (before) and the desired state (after) in a visual way. It’s like showing a makeover transformation, but for your ideas.

7. Add fun with visual quizzes and polls

To break the monotony and see if your audience is still with you, throw in some quick quizzes or polls. It’s like a mini-game break in your presentation — your audience gets involved and it makes your presentation way more dynamic and memorable.

8. End with a powerful visual punch

Your presentation closing should be a showstopper. Think a stunning clip art that wraps up your message with a visual bow, a killer quote that lingers in minds or a call to action that gets hearts racing.

9. Engage with storytelling through data

Use storytelling magic to bring your data to life. Don’t just throw numbers at your audience—explain what they mean, why they matter and add a bit of human touch. Turn those stats into relatable tales and watch your audience’s eyes light up with understanding.

10. Use visuals wisely

Your visuals are the secret sauce of a great presentation. Cherry-pick high-quality images, graphics, charts and videos that not only look good but also align with your message’s vibe. Each visual should have a purpose – they’re not just there for decoration. 

11. Utilize visual hierarchy

Employ design principles like contrast, alignment and proximity to make your key info stand out. Play around with fonts, colors and placement to make sure your audience can’t miss the important stuff.

12. Engage with multimedia

Static slides are so last year. Give your presentation some sizzle by tossing in multimedia elements. Think short video clips, animations, or a touch of sound when it makes sense, including an animated logo . But remember, these are sidekicks, not the main act, so use them smartly.

13. Interact with your audience

Turn your presentation into a two-way street. Start your presentation by encouraging your audience to join in with thought-provoking questions, quick polls or using interactive tools. Get them chatting and watch your presentation come alive.

When it comes to delivering a group presentation, it’s important to have everyone on the team on the same page. Venngage’s real-time collaboration tools enable you and your team to work together seamlessly, regardless of geographical locations. Collaborators can provide input, make edits and offer suggestions in real time. 

14. Incorporate stories and examples

Weave in relatable stories, personal anecdotes or real-life examples to illustrate your points. It’s like adding a dash of spice to your content – it becomes more memorable and relatable.

15. Nail that delivery

Don’t just stand there and recite facts like a robot — be a confident and engaging presenter. Lock eyes with your audience, mix up your tone and pace and use some gestures to drive your points home. Practice and brush up your presentation skills until you’ve got it down pat for a persuasive presentation that flows like a pro.

Venngage offers a wide selection of professionally designed presentation templates, each tailored for different purposes and styles. By choosing a template that aligns with your content and goals, you can create a visually cohesive and polished presentation that captivates your audience.

Looking for more presentation ideas ? Why not try using a presentation software that will take your presentations to the next level with a combination of user-friendly interfaces, stunning visuals, collaboration features and innovative functionalities that will take your presentations to the next level. 

Visual presentations come in various formats, each uniquely suited to convey information and engage audiences effectively. Here are six major types of visual presentations that you should be familiar with:

1. Slideshows or PowerPoint presentations

Slideshows are one of the most common forms of visual presentations. They typically consist of a series of slides containing text, images, charts, graphs and other visual elements. Slideshows are used for various purposes, including business presentations, educational lectures and conference talks.

2. Infographics

Infographics are visual representations of information, data or knowledge. They combine text, images and graphics to convey complex concepts or data in a concise and visually appealing manner. Infographics are often used in marketing, reporting and educational materials.

Don’t worry, they are also super easy to create thanks to Venngage’s fully customizable infographics templates that are professionally designed to bring your information to life. Be sure to try it out for your next visual presentation!

3. Video presentation

Videos are your dynamic storytellers. Whether it’s pre-recorded or happening in real-time, videos are the showstoppers. You can have interviews, demos, animations or even your own mini-documentary. Video presentations are highly engaging and can be shared in both in-person and virtual presentations .

4. Charts and graphs

Charts and graphs are visual representations of data that make it easier to understand and analyze numerical information. Common types include bar charts, line graphs, pie charts and scatterplots. They are commonly used in scientific research, business reports and academic presentations.

Effective data visualizations are crucial for simplifying complex information and Venngage has got you covered. Venngage’s tools enable you to create engaging charts, graphs,and infographics that enhance audience understanding and retention, leaving a lasting impression in your presentation.

5. Interactive presentations

Interactive presentations involve audience participation and engagement. These can include interactive polls, quizzes, games and multimedia elements that allow the audience to actively participate in the presentation. Interactive presentations are often used in workshops, training sessions and webinars.

Venngage’s interactive presentation tools enable you to create immersive experiences that leave a lasting impact and enhance audience retention. By incorporating features like clickable elements, quizzes and embedded multimedia, you can captivate your audience’s attention and encourage active participation.

6. Poster presentations

Poster presentations are the stars of the academic and research scene. They consist of a large poster that includes text, images and graphics to communicate research findings or project details and are usually used at conferences and exhibitions. For more poster ideas, browse through Venngage’s gallery of poster templates to inspire your next presentation.

Different visual presentations aside, different presentation methods also serve a unique purpose, tailored to specific objectives and audiences. Find out which type of presentation works best for the message you are sending across to better capture attention, maintain interest and leave a lasting impression. 

To make a good presentation , it’s crucial to be aware of common mistakes and how to avoid them. Without further ado, let’s explore some of these pitfalls along with valuable insights on how to sidestep them.

Overloading slides with text

Text heavy slides can be like trying to swallow a whole sandwich in one bite – overwhelming and unappetizing. Instead, opt for concise sentences and bullet points to keep your slides simple. Visuals can help convey your message in a more engaging way.

Using low-quality visuals

Grainy images and pixelated charts are the equivalent of a scratchy vinyl record at a DJ party. High-resolution visuals are your ticket to professionalism. Ensure that the images, charts and graphics you use are clear, relevant and sharp.

Choosing the right visuals for presentations is important. To find great visuals for your visual presentation, Browse Venngage’s extensive library of high-quality stock photos. These images can help you convey your message effectively, evoke emotions and create a visually pleasing narrative. 

Ignoring design consistency

Imagine a book with every chapter in a different font and color – it’s a visual mess. Consistency in fonts, colors and formatting throughout your presentation is key to a polished and professional look.

Reading directly from slides

Reading your slides word-for-word is like inviting your audience to a one-person audiobook session. Slides should complement your speech, not replace it. Use them as visual aids, offering key points and visuals to support your narrative.

Lack of visual hierarchy

Neglecting visual hierarchy is like trying to find Waldo in a crowd of clones. Use size, color and positioning to emphasize what’s most important. Guide your audience’s attention to key points so they don’t miss the forest for the trees.

Ignoring accessibility

Accessibility isn’t an option these days; it’s a must. Forgetting alt text for images, color contrast and closed captions for videos can exclude individuals with disabilities from understanding your presentation. 

Relying too heavily on animation

While animations can add pizzazz and draw attention, overdoing it can overshadow your message. Use animations sparingly and with purpose to enhance, not detract from your content.

Using jargon and complex language

Keep it simple. Use plain language and explain terms when needed. You want your message to resonate, not leave people scratching their heads.

Not testing interactive elements

Interactive elements can be the life of your whole presentation, but not testing them beforehand is like jumping into a pool without checking if there’s water. Ensure that all interactive features, from live polls to multimedia content, work seamlessly. A smooth experience keeps your audience engaged and avoids those awkward technical hiccups.

Presenting complex data and information in a clear and visually appealing way has never been easier with Venngage. Build professional-looking designs with our free visual chart slide templates for your next presentation.

What software or tools can I use to create visual presentations?

You can use various software and tools to create visual presentations, including Microsoft PowerPoint, Google Slides, Adobe Illustrator, Canva, Prezi and Venngage, among others.

What is the difference between a visual presentation and a written report?

The main difference between a visual presentation and a written report is the medium of communication. Visual presentations rely on visuals, such as slides, charts and images to convey information quickly, while written reports use text to provide detailed information in a linear format.

How do I effectively communicate data through visual presentations?

To effectively communicate data through visual presentations, simplify complex data into easily digestible charts and graphs, use clear labels and titles and ensure that your visuals support the key messages you want to convey.

Are there any accessibility considerations for visual presentations?

Accessibility considerations for visual presentations include providing alt text for images, ensuring good color contrast, using readable fonts and providing transcripts or captions for multimedia content to make the presentation inclusive.

Most design tools today make accessibility hard but Venngage’s Accessibility Design Tool comes with accessibility features baked in, including accessible-friendly and inclusive icons.

How do I choose the right visuals for my presentation?

Choose visuals that align with your content and message. Use charts for data, images for illustrating concepts, icons for emphasis and color to evoke emotions or convey themes.

What is the role of storytelling in visual presentations?

Storytelling plays a crucial role in visual presentations by providing a narrative structure that engages the audience, helps them relate to the content and makes the information more memorable.

How can I adapt my visual presentations for online or virtual audiences?

To adapt visual presentations for online or virtual audiences, focus on concise content, use engaging visuals, ensure clear audio, encourage audience interaction through chat or polls and rehearse for a smooth online delivery.

What is the role of data visualization in visual presentations?

Data visualization in visual presentations simplifies complex data by using charts, graphs and diagrams, making it easier for the audience to understand and interpret information.

How do I choose the right color scheme and fonts for my visual presentation?

Choose a color scheme that aligns with your content and brand and select fonts that are readable and appropriate for the message you want to convey.

How can I measure the effectiveness of my visual presentation?

Measure the effectiveness of your visual presentation by collecting feedback from the audience, tracking engagement metrics (e.g., click-through rates for online presentations) and evaluating whether the presentation achieved its intended objectives.

Ultimately, creating a memorable visual presentation isn’t just about throwing together pretty slides. It’s about mastering the art of making your message stick, captivating your audience and leaving a mark.

Lucky for you, Venngage simplifies the process of creating great presentations, empowering you to concentrate on delivering a compelling message. Follow the 5 simple steps below to make your entire presentation visually appealing and impactful:

1. Sign up and log In: Log in to your Venngage account or sign up for free and gain access to Venngage’s templates and design tools.

2. Choose a template: Browse through Venngage’s presentation template library and select one that best suits your presentation’s purpose and style. Venngage offers a variety of pre-designed templates for different types of visual presentations, including infographics, reports, posters and more.

3. Edit and customize your template: Replace the placeholder text, image and graphics with your own content and customize the colors, fonts and visual elements to align with your presentation’s theme or your organization’s branding.

4. Add visual elements: Venngage offers a wide range of visual elements, such as icons, illustrations, charts, graphs and images, that you can easily add to your presentation with the user-friendly drag-and-drop editor.

5. Save and export your presentation: Export your presentation in a format that suits your needs and then share it with your audience via email, social media or by embedding it on your website or blog .

So, as you gear up for your next presentation, whether it’s for business, education or pure creative expression, don’t forget to keep these visual presentation ideas in your back pocket.

Feel free to experiment and fine-tune your approach and let your passion and expertise shine through in your presentation. With practice, you’ll not only build presentations but also leave a lasting impact on your audience – one slide at a time.

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Using Visual Representations in Mathematics

On this page:, drawing on technology tools, in the classroom, online resources for visual representations, introduction.

All students can benefit from using visual representations, although struggling students may require additional, focused support and practice. Visual representations are a powerful way for students to access abstract mathematical ideas.  To be college and career ready, students need to be able to draw a situation, graph lists of data, or place numbers on a number line. Developing this strategy early during the elementary grades gives students tools for engaging with—and ways of thinking about—increasingly abstract concepts. Over time, they will work toward developing Common Core Standards for Mathematical Practice:

• CCSS.Math.Practice.MP2 (opens in a new window) Reason abstractly and quantitatively.
• CCSS.Math.Practice.MP4 (opens in a new window) Model with mathematics.
• CCSS.Math.Practice.MP5 (opens in a new window) Use appropriate tools strategically.

WAYS TO SUPPORT STUDENTS

Helping students choose the “right” visual representation often depends on content and context. In some contexts, there are multiple ways to represent the same idea. Show your students a variety of examples in order to demonstrate when (and why) they should choose each one (see UDL Checkpoint 2.5: Illustrate through multiple media (opens in a new window) ). Consider how you could use the following strategies to support your students:

• Check for understanding to determine a starting point. For example, you could ask the following questions: Why do you think that? How do you know that is correct? How does that picture represent the problem? Can you explain your answer? Is there another way you could do that?
• Ask students about features of the visual representation (including labels and scales, when appropriate).
• As students create visual representations, ask questions to ensure that they understand all the features of the representations. Prompt students to focus on the information the visual representations provide.
• When possible, include alternative visual representations and discuss the similarities and differences between them.
• Vary the shapes and orientations of representations so that students focus only on the important features as they learn about the objects and situations represented.
• Show your students a specific representation—a graph or a table—that is missing an important feature. Ask them to identify the missing feature.

New technologies are constantly expanding our ability to visualize data and explain mathematical concepts. For teachers looking to incorporate technology into the classroom, using virtual manipulatives (instead of physical ones) can be a good start. Students can begin with simple graphical representations of mathematical concepts and then work toward more complex modules that require them to create the data or work within a system of rules, like a game. Infographics (opens in a new window) —visualizations that are designed to communicate complex information effectively—have become increasingly popular. They can be used to “tell a story” with numbers, such as international democracy rankings (opens in a new window) or climate change impacts (opens in a new window) . Learning to create infographics gives students additional tools to communicate data and other quantitative information.

3D printing is a technology that, until recently, has been too expensive to make use of in a classroom. However, thanks to falling prices, they have now started to appear in high schools and it may not be long before elementary schools and middle schools also embrace this technology. 3D printing allows you to create solid, three-dimensional models from a digital design. You can explore what others have created (opens in a new window) to get a sense of what is possible. Imagine having students design and create their own mathematical models and manipulatives!

For more ideas on using technology to create visual representations, visit the Tech Matters blog (opens in a new window) or PowerUp’s Pinterest page (opens in a new window) . You can also check out the “ Virtual Manipulatives (opens in a new window) ” video, which supports students’ use of visual representations.

Geometry lends itself naturally to teaching with visual representations, as can be seen in Ms. Richardson’s Grade 6 class. So far, students have learned how to classify different quadrilaterals and triangles, and they are beginning to decompose polygons. They have also started using software (e.g., GeoGebra (opens in a new window) ) that can support their understanding by emphasizing the connections between mathematical language and visualization.

Ms. Richardson’s lesson objective is to have students decompose polygons into triangles, rectangles, and trapezoids. She will address two s Common Core State in this lesson:

• CCSS Math 6.G.1 (opens in a new window) Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
• CCSS Math MP4 (opens in a new window) Model with mathematics.

Ms. Richardson has students work on these standards within the context of a real-world example—a painting by the artist Sol LeWitt.

Sol LeWitt. Wall Drawing #1113. On a wall, a triangle within a rectangle, each with broken bands of color, 2003. Hirshhorn Museum and Sculpture Garden, Smithsonian Institution.

Students will build on their existing technology skills and create a model of this work, decomposing polygons and creating their own virtual LeWitt in the process. Ms. Richardson’s lesson plan is organized into three sections: a warm-up exercise to review concepts, the main learning task, and a closing discussion and assessment.

Lesson plan

This article draws from the PowerUp WHAT WORKS (opens in a new window) website, particularly the Visual Representations Instructional Strategy Guide (opens in a new window) . PowerUp is a free, teacher-friendly website that requires no log-in or registration. The Instructional Strategy Guide on visual representations includes a brief overview with an accompanying slide show; a list of the relevant mathematics Common Core State Standards; evidence-based teaching strategies to differentiate instruction using technology; short videos; and links to resources that will help you use technology to support mathematics instruction. If you want to dig deeper into the research foundation behind best practices in the use of virtual manipulatives, take a look at our Tech Research Brief (opens in a new window) on the topic. If you are responsible for professional development, the PD Support Materials (opens in a new window) provide helpful ideas and materials for using the resources. Want more information? See PowerUp WHAT WORKS (opens in a new window) .

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