problem solving instruction theory

Designing Instruction Through Cognitive Theory, Problem Solving, and Learning Transfer: An Ethnographic Case Study

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Adult Basic Education teachers indicate they need unique tools to help their students gain critical thinking and problem-solving skills. This ethnographic study provides a view into the process of designing such instruction by a team of graduate students whose expertise is in the area of learning and teaching rather than instructional design.

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Brookfield, S. (2005). Critical thinking: Unfinished business. New Directions for Community Colleges, 105 , 49–57.

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Hung, W. (2013). Problem-based learning: A learning environment for enhancing learning transfer. In L. M. R. Kaiser, K. Kaminski, & J. M. Foley (Eds.), Learning transfer in adult education. New directions for adult and continuing education (Vol. 137, pp. 49–60). New York: Wiley.

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Kaminski, K. (2015). Information and communication technologies and competencies in the 21st Century workforce. In J. Michael Spector (Ed.), Encyclopedia of educational technology . New York: Springer.

McGinty, J., Radin, J., & Kaminski, K. (2013). Brain friendly teaching supports learning transfer. In L. M. R. Kaiser, K. Kaminski, & J. M. Foley (Eds.), Learning transfer in adult education. New directions for adult and continuing education (Vol. 137, pp. 49–60). New York: Wiley.

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Kaminski, K. (2015). Designing Instruction Through Cognitive Theory, Problem Solving, and Learning Transfer: An Ethnographic Case Study. In: Hokanson, B., Clinton, G., Tracey, M. (eds) The Design of Learning Experience. Educational Communications and Technology: Issues and Innovations. Springer, Cham. https://doi.org/10.1007/978-3-319-16504-2_3

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Center for Teaching

Teaching problem solving.

Print Version

Tips and Techniques

Expert vs. novice problem solvers, communicate.

• Have students  identify specific problems, difficulties, or confusions . Don’t waste time working through problems that students already understand.
• If students are unable to articulate their concerns, determine where they are having trouble by  asking them to identify the specific concepts or principles associated with the problem.
• In a one-on-one tutoring session, ask the student to  work his/her problem out loud . This slows down the thinking process, making it more accurate and allowing you to access understanding.
• When working with larger groups you can ask students to provide a written “two-column solution.” Have students write up their solution to a problem by putting all their calculations in one column and all of their reasoning (in complete sentences) in the other column. This helps them to think critically about their own problem solving and helps you to more easily identify where they may be having problems. Two-Column Solution (Math) Two-Column Solution (Physics)

Encourage Independence

• Model the problem solving process rather than just giving students the answer. As you work through the problem, consider how a novice might struggle with the concepts and make your thinking clear
• Have students work through problems on their own. Ask directing questions or give helpful suggestions, but  provide only minimal assistance and only when needed to overcome obstacles.
• Don’t fear  group work ! Students can frequently help each other, and talking about a problem helps them think more critically about the steps needed to solve the problem. Additionally, group work helps students realize that problems often have multiple solution strategies, some that might be more effective than others

Be sensitive

• Frequently, when working problems, students are unsure of themselves. This lack of confidence may hamper their learning. It is important to recognize this when students come to us for help, and to give each student some feeling of mastery. Do this by providing  positive reinforcement to let students know when they have mastered a new concept or skill.

Encourage Thoroughness and Patience

• Try to communicate that  the process is more important than the answer so that the student learns that it is OK to not have an instant solution. This is learned through your acceptance of his/her pace of doing things, through your refusal to let anxiety pressure you into giving the right answer, and through your example of problem solving through a step-by step process.

Experts (teachers) in a particular field are often so fluent in solving problems from that field that they can find it difficult to articulate the problem solving principles and strategies they use to novices (students) in their field because these principles and strategies are second nature to the expert. To teach students problem solving skills,  a teacher should be aware of principles and strategies of good problem solving in his or her discipline .

The mathematician George Polya captured the problem solving principles and strategies he used in his discipline in the book  How to Solve It: A New Aspect of Mathematical Method (Princeton University Press, 1957). The book includes  a summary of Polya’s problem solving heuristic as well as advice on the teaching of problem solving.

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Jerome Bruner’s Theory Of Learning And Cognitive Development

Saul Mcleod, PhD

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Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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Jerome Bruner believed that children construct knowledge and meaning through active experience with the world around them. He emphasized the role of culture and language in cognitive development, which occurs in a spiral fashion with children revisiting basic concepts at increasing levels of complexity and abstraction.

Bruner’s Ideas

• Like  Ausubel (and other cognitive psychologists), Bruner sees the learner as an active agent; emphasizing the importance of existing schemata in guiding learning.
• Bruner argues that students should discern for themselves the structure of subject content – discovering the links and relationships between different facts, concepts and theories (rather than the teacher simply telling them).
• Bruner (1966) hypothesized that the usual course of intellectual development moves through three stages: enactive, iconic, and symbolic, in that order. However, unlike Piaget’s stages, Bruner did not contend that these stages were necessarily age-dependent, or invariant.
• Piaget and, to an extent, Ausubel, contended that the child must be ready, or made ready, for the subject matter. But Bruner contends just the opposite. According to his theory, the fundamental principles of any subject can be taught at any age, provided the material is converted to a form (and stage) appropriate to the child.
• The notion of a “spiral curriculum” embodies Bruner’s ideas by “spiraling” through similar topics at every age, but consistent with the child’s stage of thought.
• His spiral curriculum revisits basic ideas repeatedly, building upon them into more complex, abstract concepts over time in a developmentally appropriate sequence.
• The aim of education should be to create autonomous learners (i.e., learning to learn).
• Cognitive growth involves an interaction between basic human capabilities and “culturally invented technologies that serve as amplifiers of these capabilities.”
• These culturally invented technologies include not just obvious things such as computers and television, but also more abstract notions such as the way a culture categorizes phenomena, and language itself.
• Bruner would likely agree with  Vygotsky  that language serves to mediate between environmental stimuli and the individual’s response.

Three Modes of Representation

Modes of representation are how information or knowledge is stored and encoded in memory.

Rather than neat age-related stages (like Piaget), the modes of representation are integrated and only loosely sequential as they “translate” into each other.

Bruner (1966) was concerned with how knowledge is represented and organized through different modes of thinking (or representation).

In his research on the cognitive development of children,  Jerome Bruner proposed three modes of representation:

• Enactive representation (action-based)
• Iconic representation (image-based)
• Symbolic representation (language-based)

Bruner’s constructivist theory suggests it is effective when faced with new material to follow a progression from enactive to iconic to symbolic representation; this holds true even for adult learners.

Bruner’s work also suggests that a learner even of a very young age is capable of learning any material so long as the instruction is organized appropriately, in sharp contrast to the beliefs of Piaget and other stage theorists.

Enactive Mode (0-1 year)

In the  enactive mode , knowledge is stored primarily in the form of motor responses. This mode is used within the first year of life (corresponding with Piaget’s sensorimotor stage ).

Thinking is based entirely on physical actions , and infants learn by doing, rather than by internal representation (or thinking).

It involves encoding physical action-based information and storing it in our memory. For example, in the form of movement as muscle memory, a baby might remember the action of shaking a rattle.

And this is not just limited to children. Many adults can perform a variety of motor tasks (typing, sewing a shirt, operating a lawn mower) that they would find difficult to describe in iconic (picture) or symbolic (word) form.

This mode continues later in many physical activities, such as learning to ride a bike.

Iconic Mode (1-6 years)

Information is stored as sensory images (icons), usually visual ones, like pictures in the mind. For some, this is conscious; others say they don’t experience it.

This may explain why, when we are learning a new subject, it is often helpful to have diagrams or illustrations to accompany the verbal information.

Thinking is also based on using other mental images (icons), such as hearing, smell or touch.

Symbolic Mode (7 years onwards)

This develops last. In the  symbolic stage , knowledge is stored primarily as language, mathematical symbols, or in other symbol systems.

This mode is acquired around six to seven years old (corresponding to Piaget’s concrete operational stage ).

In the symbolic stage, knowledge is stored primarily as words, mathematical symbols, or other symbol systems, such as music.

Symbols are flexible in that they can be manipulated, ordered, classified, etc., so the user isn’t constrained by actions or images (which have a fixed relation to that which they represent).

According to Bruner’s taxonomy, these differ from icons in that symbols are “arbitrary.” For example, the word “beauty” is an arbitrary designation for the idea of beauty in that the word itself is no more inherently beautiful than any other word.

The Importance of Language

Language is important for the increased ability to deal with abstract concepts.

Bruner argues that language can code stimuli and free an individual from the constraints of dealing only with appearances, to provide a more complex yet flexible cognition.

The use of words can aid the development of the concepts they represent and can remove the constraints of the “here & now” concept.

Bruner views the infant as an intelligent & active problem solver from birth, with intellectual abilities basically similar to those of the mature adult.

Educational Implications

Education should aim to create autonomous learners (i.e., learning to learn).

For Bruner (1961), the purpose of education is not to impart knowledge, but instead to facilitate a child’s thinking and problem-solving skills which can then be transferred to a range of situations. Specifically, education should also develop symbolic thinking in children.

In 1960 Bruner’s text, The Process of Education was published. The main premise of Bruner’s text was that students are active learners who construct their own knowledge.

Bruner (1960) opposed Piaget’s notion of readiness . He argued that schools waste time trying to match the complexity of subject material to a child’s cognitive stage of development.

This means students are held back by teachers as certain topics are deemed too difficult to understand and must be taught when the teacher believes the child has reached the appropriate stage of cognitive maturity .

The Spiral Curriculum

Bruner (1960) adopts a different view and believes a child (of any age) is capable of understanding complex information:

“We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development.” (p. 33)

Bruner (1960) explained how this was possible through the concept of the spiral curriculum. This involved information being structured so that complex ideas can be taught at a simplified level first, and then re-visited at more complex levels later on.

The underlying principle in this is that the student should review particular concepts at over and over again during their educative experience; each time building and their understanding and requiring more sophisticated cognitive strategies (and thus increase the sophistication of their understanding).

Therefore, subjects would be taught at levels of gradually increasing difficultly (hence the spiral analogy). Ideally, teaching his way should lead to children being able to solve problems by themselves.

Bruner argues that, as children age, they are capable of increasingly complex modes of representation (basically, ways of thinking) – and the spiral curriculum should be sensitive to this development;

• Initially, children learn better using an  enactive  mode of representation (i.e. they learn better through “doing things” such as physical and manual tasks) – for instance, the concept of addition might be first taught by asking the child to combine piles of beads and counting the results.
• As they grow older – and more familiar with subject content – pupils become more confident in using an  iconic  mode of representation; they are able to perform tasks by imagining concrete pictures in their heads. To continue the above example; as the child becomes more confident with addition, they should be able to imagine the beads in order to complete additions (without physically needing to manipulate the piles).
• Finally, students become capable of more abstract,  symbolic  modes of representation; without the need for either physical manipulation or mental imagery. Consequently, at this point, the student should have little problem with completing a series of written calculations; of numbers which are higher than is possible by “imagining beads”.

Discovery Learning Theory

Bruner (1960) developed the concept of Discovery Learning – arguing that students should “not be presented with the subject matter in its final form, but rather are required to organize it themselves…[requiring them] to discover for themselves relationships that exist among items of information”.

Bruner (1961) proposes that learners construct their own knowledge and do this by organizing and categorizing information using a coding system.

Bruner believed that the most effective way to develop a coding system is to discover it rather than being told by the teacher.

The concept of discovery learning implies that students construct their own knowledge for themselves (also known as a constructivist approach ).

The result is an extremely active form of learning, in which the students are always engaged in tasks, finding patterns or solving puzzles – and in which they constantly need to exercise their existing schemata , reorganizing and amending these concepts to address the challenges of the task.

The role of the teacher should not be to teach information by rote learning, but instead to facilitate the learning process. This means that a good teacher will design lessons that help students discover the relationship between bits of information.

To do this a teacher must give students the information they need, but without organizing for them. The use of the spiral curriculum can aid the process of discovery learning .

For example, in teaching a particular concept, the teacher should present the set of instances that will best help learners develop an appropriate model of the concept. The teacher should also model the inquiry process. Bruner would likely not contend that all learning should be through discovery.

For example, it seems pointless to have children “discover” the names of the U.S. Presidents, or important dates in history.

Bruner’s theory is probably clearest when illustrated with practical examples. The instinctive response of a teacher to the task of helping a primary-school child understand the concept of odd and even numbers, for instance, would be to explain the difference to them.

However, Bruner would argue that understanding of this concept would be much more genuine if the child discovered the difference for themselves; for instance, by playing a game in which they had to share various numbers of beads fairly between themselves and their friend.

Discovery is not just an instructional technique, but an important learning outcome in itself. Schools should help learners develop their own ability to find the “recurrent regularities” in their environment.

Bruner would likely not contend that all learning should be through discovery. For example, it seems pointless to have children “discover” the names of the U.S. Presidents, or important dates in history.

Scaffolding Theory

On the surface, Bruner’s emphasis on the learner discovering subject content for themselves seemingly absolves the teacher of a great deal of work.

In practice, however, his model requires the teacher to be actively involved in lessons; providing cognitive scaffolding which will facilitate learning on the part of the student.

On the one hand, this involves the selection and design of appropriate stimulus materials and activities which the student can understand and complete – however Bruner also advocates that the teacher should circulate the classroom and work with individual students, performing six core “functions” (Wood, Bruner and Ross: 1976):

• Recruitment : ensuring that the student is interested in the task, and understands what is required of them.
• Reducing degrees of freedom : helping the student make sense of the material by eliminating irrelevant directions and thus reducing the “trial and error” aspect of learning.
• Direction Maintenance : ensuring that the learner is on-task and interest is maintained – often by breaking the ultimate aim of the task into “sub-aims” which are more readily understood and achieved.
• Marking critical features : highlighting relevant concepts or processes and pointing out errors.
• Frustration Control : stopping students from “giving up” on the task.
• Demonstration : providing models for imitation or possible (partial solution).

In this context, Bruner’s model might be better described as guided discovery learning; as the teacher is vital in ensuring that the acquisition of new concepts and processes is successful.

Bruner and Vygotsky

Both Bruner and Vygotsky emphasize a child’s environment, especially the social environment, more than Piaget did. Both agree that adults should play an active role in assisting the child’s learning.

Bruner, like Vygotsky, emphasized the social nature of learning, citing that other people should help a child develop skills through the process of scaffolding.

“[Scaffolding] refers to the steps taken to reduce the degrees of freedom in carrying out some task so that the child can concentrate on the difficult skill she is in the process of acquiring” (Bruner, 1978, p. 19).

He was especially interested in the characteristics of people whom he considered to have achieved their potential as individuals.

The term scaffolding first appeared in the literature when Wood, Bruner, and Ross described how tutors” interacted with a preschooler to help them solve a block reconstruction problem (Wood et al., 1976).

The concept of scaffolding is very similar to Vygotsky’s notion of the zone of proximal development , and it’s not uncommon for the terms to be used interchangeably.

Scaffolding involves helpful, structured interaction between an adult and a child with the aim of helping the child achieve a specific goal.

The purpose of the support is to allow the child to achieve higher levels of development by:

• Simplifying the task or idea.
• Motivating and encouraging the child.
• Highlighting important task elements or errors.
• Giving models that can be imitated.

Bruner and Piaget

There are similarities between Piaget and Bruner, but a significant difference is that Bruner’s modes are not related in terms of which presuppose the one that precedes it. While sometimes one mode may dominate in usage, they coexist.

Bruner states that the level of intellectual development determines the extent to which the child has been given appropriate instruction together with practice or experience.

So – the right way of presentation and explanation will enable a child to grasp a concept usually only understood by an adult. His theory stresses the role of education and the adult.

Although Bruner proposes stages of cognitive development, he doesn’t see them as representing different separate modes of thought at different points of development (like Piaget).

Instead, he sees a gradual development of cognitive skills and techniques into more integrated “adult” cognitive techniques.

Bruner views symbolic representation as crucial for cognitive development, and since language is our primary means of symbolizing the world, he attaches great importance to language in determining cognitive development.

• Children are innately PRE-ADAPTED to learning
• Children have a NATURAL CURIOSITY
• Children’s COGNITIVE STRUCTURES develop over time
• Children are ACTIVE participants in the learning process
• Cognitive development entails the acquisition of SYMBOLS
• Social factors, particularly language, were important for cognitive growth. These underpin the concept of ‘scaffolding’.
• The development of LANGUAGE is a cause not a consequence of cognitive development
• You can SPEED-UP cognitive development. You don’t have to wait for the child to be ready
• The involvement of ADULTS and MORE KNOWLEDGEABLE PEERS makes a big difference

Bruner, J. S. (1957). Going beyond the information given. New York: Norton.

Bruner, J. S. (1960). The Process of education. Cambridge, Mass.: Harvard University Press.

Bruner, J. S. (1961). The act of discovery. Harvard Educational Review , 31, 21-32.

Bruner, J. S. (1966). Toward a theory of instruction , Cambridge, Mass.: Belkapp Press.

Bruner, J. S. (1973). The relevance of education . New York: Norton.

Bruner, J. S. (1978). The role of dialogue in language acquisition. In A. Sinclair, R., J. Jarvelle, and W. J.M. Levelt (eds.) The Child’s Concept of Language. New York: Springer-Verlag.

Wood, D. J., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychiatry and Psychology , 17(2), 89-100.

InstructionalDesign.org

Home » Learning Theories » Structural Learning Theory (Joseph Scandura)

Structural Learning Theory (Joseph Scandura)

According to structural learning theory, what is learned are rules which consist of a domain, range, and procedure. There may be alternative rule sets for any given class of tasks. Problem solving may be facilitated when higher order rules are used, i.e., rules that generate new rules. Higher order rules account for creative behavior (unanticipated outcomes) as well as the ability to solve complex problems by making it possible to generate (learn) new rules.

Unlike information processing theories which often assume more complex control mechanisms and production rules, structural learning theory postulates a single, goal-switching control mechanism with minimal assumptions about the processor and allows more complex rule structures. Structural learning theory also assumes that “working memory” holds both rules and data (i.e., rules which do not act on other rules); the memory load associated with a task depends upon the rule(s) used for the task at hand.

Structural analysis is a methodology for identifying the rules to be learned for a given topic or class of tasks and breaking them done into their atomic components. The major steps in structural analysis are: (1) select a representative sample of problems, (2) identify a solution rule for each problem, (3) convert each solution rule into a higher order problem whose solutions is that rule, (4) identify a higher order solution rule for solving the new problems, (5) eliminate redundant solution rules from the rule set (i.e., those which can be derived from other rules), and (6) notice that steps 3 and 4 are essentially the same as steps 1 and 2, and continue the process iteratively with each newly-identified set of solution rules. The result of repeatedly identifying higher order rules, and eliminating redundant rules, is a succession of rule sets, each consisting of rules which are simpler individually but collectively more powerful than the ones before.

Structural learning prescribes teaching the simplest solution path for a problem and then teaching more complex paths until the entire rule has been mastered. The theory proposes that we should teach as many higher-order rules as possible as replacements for lower order rules. The theory also suggests a strategy for individualizing instruction by analyzing which rules a student has/has not mastered and teaching only the rules, or portions thereof, that have not been mastered.

Application

Structural learning theory has been applied extensively to mathematics and also provides an interpretation of  Piagetian theory  (Sandura & Scandura, 1980). The primary focus of the theory is  problem solving  instruction (Scandura, 1977). Scandura has applied the theoretical framework to the development of authoring tools and software engineering.

Here is an example of structural learning theory in the context of subtraction provided by Scandura (1977):

• The first step involves selecting a representative sample of problems such as 9-5, 248-13, or 801-302.
• The second step is to identify the rules for solving each of the selected problems. To achieve this step, it is necessary to determine the minimal capabilities of the students (e.g., can recognize the digits 0-9, minus sign, column and rows). Then the detailed operations involved in solving each of the representative problems must be worked out in terms of the minimum capabilities of the students. For example, one subtraction rule students might learn is the “borrowing” procedure that specifies if the top number is less than the bottom number in a column, the top number in the column to the right must be made smaller by 1.
• The next step is to identify any higher order rules and eliminate any lower order rules they subsume. In the case of subtraction , we could replace a number of partial rules with a single rule for borrowing that covers all cases.
• The last step is to test and refine the resulting rule(s) using new problems and extend the rule set if necessary so that it accounts for all problems in the domain. In the case of subtraction, we would use problems with varying combinations of columns and perhaps different bases.
• Whenever possible, teach higher order rules that can be used to derive lower order rules.
• Teach the simplest solution path first and then teach more complex paths or rule sets.
• Rules must be composed of the minimum capabilities possessed by the learners.
• Scandura, J.M. (1970). The role of rules in behavior: Toward an operational definition of what (rule) is learned. Psychological Review , 77, 516-533.
• Scandura, J.M. (1973). Structural Learning I: Theory and Research. London: Gordon & Breach.
• Scandura, J.M. (1976). Structural Learning II: Issues and Approaches. London: Gordon & Breach.
• Scandura, J.M. (1977). Problem Solving: A Structural/Process Approach with Instructional Applications. NY: Academic Press.
• Scandura, J.M. & Scandura, A. (1980). Structural Learning and Concrete Operations: An Approach to Piagetian Conservation. NY: Praeger.
• Scandura, J.M. (1984). Structural (cognitive task) analysis: A method for analyzing content. Part II: Precision, objectivity, and systematization. Journal of Structural Learning , 8, 1-28.
• Scandura, J.M. (2004). Structural Learning Theory: Current Status and New Perspectives.

Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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