individual or group checklist in problem solving in mathematics

Module 2: General Problem Solving

Why it matters: general problem solving, why understand the basics of problem solving.

Thinking comes naturally. You don’t have to make it happen—it just does. But you can make it happen in different ways. For example, you can think positively or negatively. You can think with “heart” and you can think with rational judgment. You can also think strategically and analytically, and mathematically and scientifically. These are a few of multiple ways in which the mind can process thought.

What are some forms of thinking you use? When do you use them, and why?

As a college student, you are tasked with engaging and expanding your thinking skills. One of the most important of these skills is critical thinking. Critical thinking is important because it relates to nearly all tasks, situations, topics, careers, environments, challenges, and opportunities. It’s a “domain-general” thinking skill—not a thinking skill that’s reserved for a one subject alone or restricted to a particular subject area.

Great leaders have highly attuned critical thinking skills, and you can, too. In fact, you probably have a lot of these skills already. Of all your thinking skills, critical thinking may have the greatest value.

What Is Critical Thinking?

Critical thinking is clear, reasonable, reflective thinking focused on deciding what to believe or do. It means asking probing questions like, “How do we know?” or “Is this true in every case or just in this instance?” It involves being skeptical and challenging assumptions, rather than simply memorizing facts or blindly accepting what you hear or read.

Who are critical thinkers, and what characteristics do they have in common? Critical thinkers are usually curious and reflective people. They like to explore and probe new areas and seek knowledge, clarification, and new solutions. They ask pertinent questions, evaluate statements and arguments, and they distinguish between facts and opinion. They are also willing to examine their own beliefs, possessing a manner of humility that allows them to admit lack of knowledge or understanding when needed. They are open to changing their mind. Perhaps most of all, they actively enjoy learning, and seeking new knowledge is a lifelong pursuit.

This may well be you!

The following video, from Lawrence Bland, presents the major concepts and benefits of critical thinking.

Critical Thinking and Logic

Critical thinking is fundamentally a process of questioning information and data. You may question the information you read in a textbook, or you may question what a politician or a professor or a classmate says. You can also question a commonly-held belief or a new idea. With critical thinking, anything and everything is subject to question and examination for the purpose of logically constructing reasoned perspectives.

Questions of Logic in Critical Thinking

Let’s use a simple example of applying logic to a critical-thinking situation. In this hypothetical scenario, a man has a PhD in political science, and he works as a professor at a local college. His wife works at the college, too. They have three young children in the local school system, and their family is well known in the community. The man is now running for political office. Are his credentials and experience sufficient for entering public office? Will he be effective in the political office? Some voters might believe that his personal life and current job, on the surface, suggest he will do well in the position, and they will vote for him. In truth, the characteristics described don’t guarantee that the man will do a good job. The information is somewhat irrelevant. What else might you want to know? How about whether the man had already held a political office and done a good job? In this case, we want to ask, How much information is adequate in order to make a decision based on logic instead of assumptions?

The following questions are ones you may apply to formulating a logical, reasoned perspective in the above scenario or any other situation:

• What’s happening? Gather the basic information and begin to think of questions.
• Why is it important? Ask yourself why it’s significant and whether or not you agree.
• What don’t I see? Is there anything important missing?
• How do I know? Ask yourself where the information came from and how it was constructed.
• Who is saying it? What’s the position of the speaker and what is influencing them?
• What else? What if? What other ideas exist and are there other possibilities?

Problem-Solving with Critical Thinking

For most people, a typical day is filled with critical thinking and problem-solving challenges. In fact, critical thinking and problem-solving go hand-in-hand. They both refer to using knowledge, facts, and data to solve problems effectively. But with problem-solving, you are specifically identifying, selecting, and defending your solution.

Problem-Solving Action Checklist

Problem-solving can be an efficient and rewarding process, especially if you are organized and mindful of critical steps and strategies. Remember, too, to assume the attributes of a good critical thinker. If you are curious, reflective, knowledge-seeking, open to change, probing, organized, and ethical, your challenge or problem will be less of a hurdle, and you’ll be in a good position to find intelligent solutions.

Critical Thinking, Problem Solving, and Math

In previous math courses, you’ve no doubt run into the infamous “word problems.” Unfortunately, these problems rarely resemble the type of problems we actually encounter in everyday life. In math books, you usually are told exactly which formula or procedure to use, and are given exactly the information you need to answer the question. In real life, problem solving requires identifying an appropriate formula or procedure, and determining what information you will need (and won’t need) to answer the question.

In this section, we will review several basic but powerful algebraic ideas: percents , rates , and proportions . We will then focus on the problem solving process, and explore how to use these ideas to solve problems where we don’t have perfect information.

• "Student Success-Thinking Critically In Class and Online."  Critical Thinking Gateway . St Petersburg College, n.d. Web. 16 Feb 2016. ↵
• Critical Thinking Skills. Authored by : Linda Bruce. Provided by : Lumen Learning. Located at : https://courses.lumenlearning.com/collegesuccess-lumen/chapter/critical-thinking-skills/ . Project : College Success. License : CC BY: Attribution
• Critical Thinking. Authored by : Critical and Creative Thinking Program. Located at : http://cct.wikispaces.umb.edu/Critical+Thinking . License : CC BY: Attribution
• Thinking Critically. Authored by : UBC Learning Commons. Provided by : The University of British Columbia, Vancouver Campus. Located at : http://www.oercommons.org/courses/learning-toolkit-critical-thinking/view . License : CC BY: Attribution
• Problem Solving. Authored by : David Lippman. Located at : http://www.opentextbookstore.com/mathinsociety/ . Project : Math in Society. License : CC BY-SA: Attribution-ShareAlike
• Critical Thinking.wmv. . Authored by : Lawrence Bland. Located at : https://youtu.be/WiSklIGUblo . License : All Rights Reserved . License Terms : Standard YouTube License

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Group-working Skills

In her article Group-worthy Tasks and Their Potential to Support Children to Develop Independent Problem-solving Skills , Jennie Pennant outlines the benefits of working on mathematics as a group, arguing that group work also supports children's individual problem-solving abilities.  Almost all NRICH tasks could be tackled by a group of learners, as opposed to an individual or pair, but the Developing Group-working Skills feature is a collection of activities which have been designed especially to develop children's group-working skills .  How can we help children get better at group work? We cannot expect learners to be able to work well in groups overnight.  We must help them develop the skills that are needed for successful group work and give them many opportunities to put these skills into practice.  In her article Developing Good Team-working Skills , Jenny Piggott offers the following list of skills related to working collaboratively, based on those found in Elizabeth Cohen's book 'Designing Groupwork':

• Asking questions - making sense of your own understanding
• Explaining by telling how and why
• Helping others - by responding to their needs
• Helping others - to do things for themselves
• Sharing knowledge and reasoning
• Finding out what others think - asking for, listening to and making sense of their ideas
• Reflecting on and making use of what has been said
• Being concise - communicating thinking
• Giving reasons for ideas - communicating reasoning
• Allowing everyone to contribute
• Pulling ideas together - sharing, listening, valuing all contributions
• Finding out if the group is ready to make a decision - consensus making.

Jenny suggests that these collaborative working skills can be developed through particular group activities and she offers six categories of team-building activities that can be used to focus on a range of the different skills.  Jenny's article links to several classroom activities within each category and we have chosen a subset of these in our Group Work feature which exemplifies the full range of skills. The featured activities For more details about the kinds of task and further example activities, please see Jenny's article . Number Match , a Stage 1 task, is only complete as an activity when every member of the group has completed their own part.  The task is undertaken in silence which helps group members respond to the needs of others.  Fraction Match , aimed at Stage 2 and above, is done in exactly the same way. In Counters in the Middle , a 'designer' makes an arrangement of counters without the team seeing it. The team has to agree on the final pattern by asking the minimum number of questions, which requires them to listen to each other, give reasons for their opinions and pull ideas together. In En-counters , each learner completes a picture themselves, based on the designer's instructions, but with support and advice from other members of the team.  This therefore encourages children to respond to the needs of others, help others do things for themselves and explain by telling how. Guess the Houses , a Stage 1 activity, depends on learners sharing reasoning, listening to opinions, reflecting and pulling ideas together. The team is required to guess the rule in the minimum number of questions. In What Shape? , one member of the group is trying to find out what is on their chosen card (the unknown) by asking as few questions as possible.  This task therefore depends on group members being concise, asking questions, listening and reflecting on what has been said. Arranging Cubes requires the group to recreate a 2D arrangement of cubes which matches all the information on their cards without showing each team member's information to anyone else.  Among other skills, learners must allow everyone to contribute, share knowledge and reasoning, reflect on (and make use of) what has been said and come to a consensus. Submitting solutions Due to the nature of these tasks, it will be a little difficult for children to submit solutions in the usual sense to the live problems in this feature.  However, we would love to hear about how the activity helped them work better as a group.  It could be that you as the teacher summarise your observations or it might be that the learners themselves can articulate their thoughts.  We would be delighted to hear from you.  You may find it helpful to use the list of skills above as an assessment 'checklist' (see Skills.doc or Skills.pdf ).  What next? Having tried these activities which aim to build learners' group-working skills, why not have a go at other NRICH activities using a group-work approach?  In May 2010, we created several group-worthy tasks, based on Jo Boaler's research on Complex Instruction, and in February 2010, all our problems were designed with collaborative mathematics in mind. Of course, as we mentioned at the start of this article, almost any NRICH activity could be worked on by a group, so you could encourage children to tackle any of our tasks using their group-working skills.

Taking the tasks home

We have rewritten some of these tasks so that they are suitable for just one child to do together with an adult. We've collected these here . References Cohen, E. G. (1994) Designing Groupwork - Strategies for the Heterogeneous Classroom. Second Edition, Teachers College Press. Here is a PDF version of this article.

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Module 1: Problem Solving Strategies

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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

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Group Problem Solving

Students are given a problem to solve. This problem can be just about anything from esoteric philosophical issues to mathematics. The goal is to come up with a solution to the problem. In some cases, the solution will be tested and reported on.

Stern, M. (1995).  Visions for a sustainable city: Owings Mills, MD . http://www3.iath.virginia.edu/stern/

Goals and Objectives

The goals of group problem solving are to develop the students ability to solve problems in the given context. Goals can also include building team skills, experience testing solutions, evaluative skills among alternatives, etc. Sample objective statements include:

During and after performing the Group Problem Solving activity, students will…

• develop group problem solving skills,
• interpret facts and propose solutions,
• recognize key facets of a problem situation,
• identify motives creating a problem situation,
• outline a course of action,

…as determined by successfully attending to 80% of rubric items.

Materials and Resources

The instructor must present the students with the problem. This may include access to numerous background materials. The instructor should also provide a group discussion area for each group and clear guidelines on their activities.

Guiding Questions for this Lesson

As with most activities, the guiding question will depend on the purpose. For example, if group skills is the primary purpose, the guiding questions might be how well can students work together to resolve issue x? Another guiding question might be, how effectively can a student group address the given problem?

Lesson Outline and Procedure

• Either pre-select of allow self selection of students into groups. Groups of 3-5 are usually effective in online education activities.
• Present students with the problem to be solved and any background materials needed to begin solving the problem.
• Provide time for group formation. Follow the groups communications very closely at the beginning to make sure that everything starts off well.
• If it is a long process, have the groups submit progress reports.
• Final group solutions may be submitted to the instructor or the class for peer review and comments.
• In some cases, students may test out their solution and report on the results as a stage 2 process to the group problem solving activity.
• Assess both individual and group performance.

Teaching Strategies

• Group work is a fickle fiend in online education. Resources are provided below regarding effectively implementing online group work.
• If groups have already been used in the course, the same groups may be maintained to reduce group formation time.
• Do not present the students with too much information at the beginning. In other words, give them enough to guide them to the solution without actually telling them the solution.
• Use problems that have multiple or complex solutions so as to more effectively make use of group synergy relationships. Easy problems do not require group work.

Accommodations

What accommodations may be needed for students with disabilities or other special needs? The primary accommodation will involve communication changes as needed for those with disabilities to actively and effectively participate in the group activity. Other accommodations may be needed depending on the nature of the problem to be solved.

The time required to complete a group problem solving exercise can very widely. Small groups could be formed in a synchronous session and given 10 minutes to come up with a problem solution. In most cases, for a class activity involving group work, at least 3 days should be given for group formation in an online course. An additional week may be needed for the group to meet and address the problem. 3-5 more days would be needed for the group to compose and submit a problem response. These would be average times that will vary by the needs of the exercise.

Ideas for Lesson Evaluation and Teacher Reflection

How did the students like the lesson? End of semester evaluations should ask about the usefulness and learning accomplished through such activities. Also, the conversation that occurs during the activity will help guage how the students are enjoying various aspects and whether they are learning and/or participating.

Questions the instructor should ask when evaluating the lesson include: Were the students engaged in efficiently working together? Does the solution actually address the problem? Is there creativity in the solution? Is the solution feasible? How quickly did they arrive at a solution?

How was student learning verified? Participation can be assessed in discussion sessions. A rubric can also be set up to help guage the quality of final work.

Sample rubrics for group work available online:

Useful Online References on Group Work in General

• Austin, D., & Mescia, M. D. (n.d.). Strategies to incorporate active learning into online teaching . from http://www.icte.org/T01_Library/T01_245.pdf
• Lopez-Ortiz, B. I., & Lin, L. (2005, February). What makes an online group project work? Students’ perceptions before and after an online collaborative problem/project-based learning (PBL) experience.  International Journal of Instructional Technology and Distance Learning, 2 (2). from  http://www.itdl.org/Journal/Feb_05/article04.htm

A Few Books with Information on Group Work Online

• Harasim, L. (2007). Assessing online collaborative learning: A theory, methodology, and toolset. In B. H. Khan (Ed.). Flexible learning in an information society . pp. 282-293. Hershey, PA: Information Science Publishing.
• Horton, W. (2000). Designing Web-based training . New York: John Wiley & Sons, Inc.
• Ko, S., & Rossen, S. (2001). Teaching online: A practical guide . Boston: Houghton Mifflin Co.
• Nicolay, J. A. (2002). Group assessment in the on-line learning environment. In R. S. Anderson, J. F. Bauer, & B. W. Speck (Eds.).  Assessment strategies for the on-line class: From theory to practice . pp. 43-52. San Francisco: Jossey-Bass Publishers.
• Palloff, R. M., & Pratt, K. (1999). Building learning communities in cyberspace: Effective strategies for the online classroom . San Francisco: Jossey-Bass Publishers.
• Palloff, R. M., & Pratt, K. (2001). Lessons from the cyberspace classroom: The realities fo online teaching . San Francisco: Jossey-Bass Publishers.
• Salmon, G. (2002).  e-tivities: The key to active online learning . London: Kogan Page Limited.

Meaningful Assessments of Students Who Struggle to Learn Mathematics

• First Online: 12 July 2022

Cite this chapter

• Dake Zhang 6 ,
• Carolyn A. Maher 6 &
• Louise C. Wilkinson 7  

Part of the book series: Research in Mathematics Education ((RME))

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This chapter presents the results of collaboration among special education, mathematics education, and psycholinguistic researchers, with a focus on meaningful assessments for students who have difficulties with learning mathematics. Gaining knowledge of the mathematical abilities of students who struggle or who have particular learning disabilities is often limited in traditional assessments. Access to struggling students’ ability to reason and solve problems may be obscured because of inadequate approaches and limitations in assessing the knowledge of these students using traditional test items. Our goal is to address these concerns and other obstacles to assessment and suggest additional and alternate ways of measuring students’ knowledge. It is noteworthy for researchers to attend to what are termed “learning disabilities” in mathematics and the characteristics of goal requirements in students’ individualized education programs (IEPs). IEP requirements are understood in relation to assessment approaches that are used to identify students who struggle to learn mathematics. In particular, it is essential to consider former, existing, and emerging theoretical views of mathematics and special education educators regarding what might be considered the “ability” of students to learn mathematics.

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1.1 A Summary of Assessment Categories

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Zhang, D., Maher, C.A., Wilkinson, L.C. (2022). Meaningful Assessments of Students Who Struggle to Learn Mathematics. In: Xin, Y.P., Tzur, R., Thouless, H. (eds) Enabling Mathematics Learning of Struggling Students. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-95216-7_6

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5 Steps for Setting Up Small Group Math Instruction in the Elementary Grades

Small group instruction is a key strategy for meeting the diverse needs of students, but figuring out how to manage it can be tricky.

Every teacher knows that small group instruction is an effective way to help students reach mastery and get the differentiated instruction they need. Although that sounds easy, it can take years for teachers to master this skill in the mathematics classroom.

My first year teaching, I spent my time making sure that my whole group lessons were up to par. I thought that was enough. I didn’t have any training on small group instruction; it was just expected that I would pull small groups to meet the needs of my students.

When I tried to figure out small group instruction, I didn’t know where to start. What data should I use? What would I do in small groups with them? How often did I need to meet with each student? Carrying out differentiated small groups that grow mathematicians is a master teacher skill that takes years to develop. Or does it? I worked with hundreds of teachers to come up with five concrete and easy-to-understand steps to bridge this learning gap.

Getting started

1. Create a short assessment anticipating what you want to see from your students. When you’re creating the assessment, be strategic about getting data on the exact teaching points you want to target. Here are some things to consider:

• Embed the targeted math concept into rigorous problem-solving to get a deep look into processing behaviors, organization strategies, and the math strategies used to solve the problem.
• Keep assessments for small group instruction focused on the most essential math skills that students need to master.
• Keep it short. Use only one or two questions so that students can go deeper in the work. 
• Use a checklist or embedded representations so that you can check in on specific behaviors and strategies.

I’ve put together a lower grade example and an upper grade example .

2. Use an If… Then… chart to drill down to specific behaviors that need support. One of the biggest misconceptions I witness in creating small groups is grouping students by the problem they missed. There could be so many reasons why a number of students missed the same problem. You could end up with a student who only made a simple mistake in a group with another student who didn’t even know where to start and yet another student who needs help with number lines.

When students are grouped this way, the time together isn’t meaningful to all students. Instead, approach the creation of small groups by behavior to create more impactful learning experiences for students. Using a simple If… Then... chart is the key to staying targeted, which keeps small groups intentional and short.

Take one piece of student work and create a T-chart. Practice pinpointing student misconceptions and think of a concrete action that students could take to master that skill.

For example, if a student answers only the first part of the problem, then I can teach them to write an answer statement in order to ensure that they focus on the question and answer the appropriate part. If a student subtracts across zeros incorrectly, then I can teach them to subtract across zeros using a place value chart with base 10 blocks.

The T-chart also helps to break problem-solving into steps of comprehension. This continuum can help you get started with naming teaching points for a progression of problem-solving. As you look at each piece of student work, figure out what teaching points they’ll need and plug them into this chart . Then list what each student in the group needs. You may need many chart pages, depending on how many learning targets you end up with. Continue this process as you look at each student’s piece of work. 

3. Choose appropriate skills for learners who have a lot to work on or the ones who got the correct answer. These students can be the trickiest to figure out. When looking at a student’s work that shows there’s much to work on, choose the next step they’re ready for. That might mean you start with figuring out where they are on the problem-solving continuum or an essential math concept that will help with other concepts throughout the year.

For students who have the correct answer, it’s still important to ensure that they’re flexible in their thinking and they can fully communicate their justification. Another option is to do inquiry work with these students. Give them a rich math task and tell them what you want to see in their work as they collaborate with their peers in the group. Check in on them throughout with facilitating questions and support as necessary while you pull other groups.

4. Use a predictable lesson structure and reuse your assessment. You might be wondering what you should do with students once you get them into groups. One of the biggest favors you can do for yourself is to have a predictable lesson structure to provide an avenue to quick and effective lessons. There are many different structures you can choose from. Here’s one you can start with.

Launch: Begin the lesson with one of the options below. (This should take one minute.) 

• Pose a question to strike up curiosity and excitement.
• Start with a compliment based on a common strength of the group. 
• Present a real-world example that links to the math concept.

Teach: State the learning target several times while using one of the approaches    listed. (This could take four to five minutes.)  

• Show an exemplar (student work or teacher created) of the learning target, and point out what students should notice and how it will help them. 
• You can do a direct teach if it’s the first time a student has been exposed to a strategy. Use this method sparingly, as it promotes a low level of thinking in students.

Try it: Let students try it out with a new problem in pairs or independently.

• Offer learning targets that don’t call for students to solve. For example, if the lesson is about visualizing and drawing a quick sketch to match the problem, that will be the only thing to work on during this time. 
• Instead of telling students what to do, refer them back to the anchor chart or example to try to think it out. Another option is a simple prompt to remind them to reread. 

Empower: Dismiss each student as they finish, and send them off with one of the options below.

• Compliment them on a strength you noticed as they tried out the new skill.
• Give them an artifact, such as a mini anchor chart or written note, to remind them of what they learned and to use as a guide as they try it out in their independent work. 

Use the problem from the assessment you gave in the “Teach” or in the “Try it” to save you some time.

5. Make small group preparation a part of what you’re already doing. It doesn’t have to be an extra task. Make this meaningful work part of your grading routine. As you’re grading or giving feedback, put students into groups. Then you can start your week with fresh data each time you dive into student work.

Don’t be afraid to just try it out. The only wrong way to do small groups is to not do them. Meeting with students this way builds strong relationships, helps you gain more insight into student thinking, and focuses on specific skills of your mathematicians to close gaps. Being so intentional is empowering. You’ll see students make gains right before your eyes.