Category Archives: Mathematics

Cognitive Science in Math Class: Implementing what we know

The rubber on the wheels of the learning bus meets the road in the classroom. How can knowing the following six scientific facts (distilled from the twelve listed in my previous post) inform and improve Mathematics learning and teaching?

  1. The brain can grow throughout your adult life; whether it grows depends on how you use it.
  2. Mistakes and errors stimulate brain growth; successful outcomes do not.
  3. Working memory is tiny, capable of holding only four or five information bits simultaneously.
  4. Knowing and internalizing items 1 and 2—the growth mindset—produces better learning outcomes than believing and internalizing the principle that the brain develops independently of its environment and for a fixed time period—the fixed mindset.
  5. Collaboration with others, including frequent self-testing, improves our understanding, retention, and recall of information.
  6. By organizing the information we receive according to deeper, binding principles and logics, we understand, retain, and recall it better than when we don’t.

First, we must stimulate the brain to shift to Reflective Mode from Automatic Mode.  We accomplish this shift by asking easy questions with hard answers.

Second, we must create the opportunity for students to relate and link the facts that they’ve learned to each other in ways that reveal deeper relationships in contexts that give them meaning and significance, i. e., to “chunk” the facts they’ve learned.  We achieve chunking by asking them why they believe a fact or statement, encouraging them to follow their path of reasoning, marking that path for others to follow, and stopping to check whether this path is leading them to their desired goal.

Third, we must offer students opportunities to discover or invent Mathematics.  We can offer them concrete situations that reflect actually occurring, non-mathematical events, challenge them to identify patterns in them and determine optimal outcomes.  Or, we can challenge them with mathematical events or circumstances and offer them the opportunity to respond to the challenge.

Fourth, we must encourage and require students to collaborate with their peers, parents, and teachers.  We achieve collaboration by creating circumstances that require them to collaborate to succeed.  Grouping them in groups of 3-6 and evaluating their work only as a group achieves collaboration.

Fifth, we must test their growth in ways that stimulate them to tackle hard problems and foster the growth mindset to accelerate brain growth and development.  Testing in this environment and under this philosophy occurs daily as an integral part of the previous four recommendations.  Teachers would test their students daily with hard problems and complex applications and evaluate their processes and results. Students would test their peers daily by asking for contributions to the group and evaluating such contributions.

In her secondary school, young Jacqui arrives in Math class, sits in her desk, sets her backpack on the floor, retrieves a pencil and a notebook from it, glances at the whiteboard on the front wall of the room and sees two or three problems to solve. The seating arrangement is proscenium.  Jacqui, in one ideal world, tackles the problems.  This exercise is called, “Bell Work.” Its purpose is to keep the students on learning tasks while the teacher takes attendance.  It usually consists of problems similar to problems in the previous night’s homework or to assignments from the past week.  After taking attendance, the teacher reviews the bell-work problems at the whiteboard and responds to students’ questions.  Fifteen to twenty minutes have elapsed.  Then, the lesson begins.  For the next thirty or thirty-five minutes, the teacher responds to questions about last night’s homework, presents a new topic according to the IWY (I-do, We-do, You-do) model of pedagogy.  The bell rings, Jacqui gathers her belongings and leaves for her next class or activity.  Jacqui was probably engaged in the bell-work and lesson for about fifteen minutes of the fifty minutes in class.  She repeats this process each of the non-test days of the year, approximately 130 days, and she repeats this process every non-test day of each year during each of the seven years in middle and high school.

During no interval in this scene has Jacqui engaged actively in learning mathematics or accelerating the growth of her brain. In terms of the six items listed above, she isn’t collaborating, trying, erring and trying alternatives, tackling hard problems, inquiring, discovering, constructing, inventing, or chunking.  Her brain grows less during math and while doing math homework than it grows during any other period in her day.  Her brain grows only because she hasn’t matured.  Why are we puzzled as to why she loathes her math classes and the math she encounters in them?

To change the intellectual and emotional outcomes engendered by current mathematics pedagogy and content, we must change their daily practice and content.  What teachers, students, administrators, and parents do every day in and out of the classroom is critical to changing mathematics learning from hard labor—breaking rocks with sledge hammers—to art—painting or sculpting or playing with a rich palette and precision tools.

Assume a class of twenty-four students and one teacher.  Divide the students into four groups of six each or six groups of four each.  Exchange the usual one-piece student desks for eight or twelve large trapezoidal tables and twenty-four chairs. Assign to each group a pair of tables and to each student a chair. Each group would sit around each table most of the time.  At other times the tables can be configured differently, depending on the activities on a given day.

The teacher would:  1) pose problems or questions, 2) guide collaboration among group members, 3) respond to specific requests for help from groups of students (as a group) or from individual students, and 4) provide feedback to the groups and their members.  Each group would be required to record their daily progress:  the work they performed and the results of it.  The teacher would review it and comment on it before the next meeting.  Work outside class would not be assigned; independent inquiries by students or groups of students would be encouraged and supported.  The school or, at least, the teacher would eschew formal tests.

Consider some consequences of adopting this approach for teachers.  First, work load and asset allocation would shift from preparing lectures that duplicate the material in the written materials to preparing original materials or finding existing materials and preparing them for student study in group settings.  Second, work load would shift from preparing, giving and grading tests to reviewing and commenting on daily student work product, i. e., providing useful feedback to them.  Third, teachers would discover and invent maths with their students rather than tell them what we already know.  Fourth, teachers could allocate time and energy to giving and receiving meaningful feedback rather than checking to determine whether their students had attempted or completed assigned homework.  Fifth, such class administrative tasks as posting grades would be minimized or eliminated and teachers would have more time and energy to apply to learning, discovering and explaining mathematics with their students.

Consider, as well, some consequences of adopting this approach for students. First, they wouldn’t have to do maths at home; they would have more time for play and sleep.  Second, they wouldn’t have to worry about tests and grades.  Third, they wouldn’t have to learn how to cram for tests then cram for them.  Fourth, they would have the opportunity to demonstrate their growth to themselves and their peers daily by offering other students insights, hypotheses, suggestions, direct instructions, or explanations and by asking questions about new material or new questions about old material.  Fifth, they stop expecting simple, easy answers to simple questions and learn how to be patient with themselves and others.

By shifting the emphasis in the use of teacher and student classroom time and energy allocation from lecturing, assigning, testing, grading and recording to asking, discussing, commenting, collaborating and encouraging, we alter the nature of classroom learning.  The classroom shifts from one in which we adults determine performance expectations, instruct children in our expectations, show them how to achieve those expectations, and evaluate how well they’ve matched those expectations to a classroom in which adults ask students to learn how to learn independently in cooperation with their peers (not contradictory!) without regard to the subject matter.  Learning shifts from an environment in which we measure our children by how well they fit the glove we’ve designed to one in which we measure them by their designs of the gloves they choose to wear.  The result is that even should they fail to fulfill our hopes, their learning outcomes would exceed our wildest expectations.

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A Different Story for Mathematics

From our parents and teachers, we learn that, when performing a mathematical or arithmetical task, the result is a right answer or a wrong answer to a question posed by an adult. We learn that our answers result from a process and that there is only one process we can use to produce them. We learn that every question has an answer. We are pleased when our work produces a right answer and pleases the adult who asked the question and we are distressed when our work produces a wrong answer and displeases that adult. We conclude as young children that Arithmetic is rigorous, deductive, computational and useful. As our parents, teachers, and peers reinforce this conclusion throughout the first twelve years of school, we internalize and preserve it. It becomes bedrock to our view of the world and the places of things, events, and actions in it.

Neither History, Fiction, Poetry, Music, Physics, Chemistry, Biology nor, well, any subject is presented exclusively in this way. Sure, each field has its jargon to learn, its principles to apply and its recipes to follow. But, we learn in every field other than Math that authors embed their stories in context and tell them in several ways, teachers employ many devices and draw from many resources in their pedagogy. They tell the stories of their subjects as fields of inquiry that have evolved with civilization and like civilizations. They battle about the course of history, the nature and limits of fiction, music or poetry, or the proper use of its elements (character, plot, narrative voice, rhyme, meter, tone, key, scale, theme) the fundamental laws of matter and energy, the material composition of things or the building blocks of living things. Our teachers and parents demonstrate and encourage experimentation and discovery in every area—except Arithmetic and Mathematics.

During our lives, some of us find by chance a few corners of Arithmetic that are fun: Rubik’s Cube, Sudoku, magic squares, and ciphers among them. Others among us discover that Mathematics is quite different from Arithmetic and attempt to cross the divide between them. But, crossing this divide is arduous; everyone who has attempted this crossing knows this. Many of us decide that it’s too arduous, reverse course, and return to the safety and certainty of the Arithmetic and Mathematics we know, to the Island of Conclusions. The remaining few travelers complete the crossing and seek roles for themselves in this new land. What do they find there? Why is this crossing so arduous?

They find a vast network of communities of people, each of whom inhabits some area in an open, limitless landscape that we can characterize as an evolving field of inquiry (a field of evolving inquiry, perhaps). Some communities inhabit developed areas; some inhabit territories on the frontiers of Mathematics and others inhabit undeveloped territories beyond such frontiers. Each generation of inhabitants, called Mathematicians, explores, settles and develops land beyond its frontiers and discovers new relationships, new networks, within the developed territories. This story is the story of Mathematics; Arithmetic and school Maths are chapters in it.

We make the crossing to Mathematics, the field of inquiry, from Arithmetic, the body of knowledge, arduous by failing to prepare our students for it. If they need to cross an ocean, they must build the boats, learn to sail them and learn to swim. If they need to cross a desert, they must build the wagons or find and train the camels, and learn to navigate by the night sky, the position of the Sun, or by compass. But, we the communities of educators and parents, have failed to provide our students and children the boats, wagons, camels, or compasses or to show them how to build, train or use them. We’ve given them boats; they need ships. We’ve taught them how to use a calculator for Arithmetic and to replicate algorithms for Algebra, Geometry, Trigonometry, and Calculus to find unique answers to manufactured problems. But, that is not enough. They need to learn how to recognize patterns and describe them with precision. They need to learn how to experiment with mathematical objects to discover patterns in their behavior. They need to learn to associate freely among concepts that appear similar or disparate, seeking connections they haven’t discovered, yet. For, although Mathematics is rigorous, deductive, computational and useful, it is, like any other subject, experimental, inductive and inferential. By framing it as the former alone, we hide the latter aspect of its nature.

Nearly all students learn to calculate, deduce and apply arithmetical and mathematical tools readily enough to get them through the days of their lives. But, the mathematics required for daily living is even more elementary than the reading required for it. Barcodes and computers total our groceries purchase and we use a plastic card to pay; no computation is required. Most daily math tasks are counting tasks. Calculators, hand-held or online, can calculate mortgage payments and their elements and other tasks that require more than counting. Students learn enough math for their daily living before they complete the seventh year of school. Demonstrations of and practice at applications of Algebra, Geometry, Trigonometry, Matrices, Vectors or Calculus convince our students that these tools are useful, indeed. But, such demonstrations and practice don’t convince them that they will ever use those tools. Mathematics’ usefulness in the world fails to motivate students to learn it after they’ve completed the sixth year. For the first three or four years, they needed only the approval of adults to motivate learning it; its usefulness didn’t matter to them. Once they’ve learned enough of the math they need to use daily, they lose interest in learning any further uses. Teaching them, in subsequent years, for example, how to use techniques in solving systems of linear equations in linear programming applications does not suffice to impel them to learn either how to solve systems of linear equations or how to apply those techniques in the world. The answer, “Because it’s useful,” to the question, “why do we have to learn this stuff?” is true, but it fails to stimulate students’ interest and impel them to learn such techniques and their applications. Only a few students will use them, and those students are unaware of their futures and don’t know, yet, that they will. WYSIATI (What You See Is What There Is) dominates their perceptions and their conceptions.

Mathematics is a body of knowledge, a compendium of results (truths) developed over at least seven millennia. Its method of verification is most rigorous of the methods of the sciences and its facts live longest of them. Logical deduction plays a unique role in its practice. Yet, Mathematics is more than a static body of knowledge. It’s an open, evolving field of inquiry into the nature of numbers and their relationship to the world. Its practitioners, Mathematicians, pursue answers to questions that arise in their endeavor to locate and understand the fundamental elements of the mathematical universe and explain their existence and behavior because such questions interest and challenge them and because they feel themselves continue to grow and learn, which they like to do. A Mathematician enjoys feelings similar to the feeling an author feels when he’s written a satisfying paragraph or an effective stanza or to a musician when he’s executed a passage “perfectly” or his performance jells with an ensemble.

Importance and applicability are reasons necessary to the pursuit of mathematical knowledge, and they are reasons necessary to the pursuit of any and all other knowledge. But, they are insufficient. In this regard, too, Mathematics is like all other fields of inquiry. We pursue our inquiries into them because we enjoy the process, and our success in discovering and answering questions amplifies this enjoyment a thousand times. The pursuit of mathematical knowledge is interesting, gratifying and creative. These factors impel practitioners to practice. These factors will impel students to learn to practice it, too.

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Three Myths of Mathematics and Mathematics Education

To understand current Math teaching practice, IWY (I-do, We-do, You-do), we need to recognize at least three myths about Mathematics and Mathematics education—stories about what it is, whether and how it differs from other fields of inquiry, why we need to learn it, and how we learn it:

  • Myth 1: Mathematics is unique among all fields of inquiry in its structure and practice. Mathematics is deductive and rigorous—and nothing else;
  • Myth 2: Logical rigor and deductive method—and nothing else—must frame Mathematics pedagogy;
  • Myth 3: Mathematics is a tool for use in the construction of other things such as buildings, rockets, theories and submarines—and is nothing more.

Math is different, right? There’s always a unique, right answer to every problem. Math is rigorous: you can always verify whether your answer is correct or incorrect with absolute certainty. When you read a topic section in a Math textbook or an academic paper, you encounter definitions, first, then theorems and their proofs. The theorems use the definitions to assert some claims about the world. These theorems follow by strict rules of formal logic from other theorems that were proven previously. A machine could, in principle, prove the theorems in your textbook. New research results are always published in essays in academic journals and rarely included (as new results) in textbooks. Although there is little or no exposition in an academic paper, there is almost always some exposition in a textbook, usually pictures, diagrams or graphs of a situation, or examples of algorithms, or all of them. The pictures, diagrams or graphs are heuristic devices; they motivate and depict the concept you’re supposed to learn to help you retain them for use. The examples are heuristics, too. They show you how to use the algorithms you learned to solve problems posed in the book, that is, how to do the homework problems. By telling the story of Mathematics only in this way in books, classrooms, and lecture halls, textbook authors, publishers, professors, and teachers present an incomplete picture of it. They create the myths that: Mathematics consists of definitions, theorems, proofs, and problems; theorems are deduced rigorously from other theorems by strict, formal rules of logic; and these theorems are deployed as algorithms to solve problems—and nothing else.

This story of Mathematics became the dominant narrative of Mathematics as the result of cognitive processes developed to promote our survival as a species over millions of years of evolution by trial and error. WYSIATI combines with representation, cognitive ease, and repetition to embed this myth in our memories and enable its instant, subconscious recall. WYSIATI is an acronym for What You See is All There Is. It is a slogan for the fact that the associative machine—our minds operating in “current awareness only” mode (which is 99% of the time)—dominates our awareness at any and all given moments; we are “hardwired” by evolution to jump to conclusions based on limited information. If the only story about Mathematics that you’ve been told is that it is rigorous, deductive, and computational, you will jump to the conclusion that it is only rigorous, deductive, and computational. This jump occurs early in school, in the first grade when we learn to add and before we’ve learned to reflect on and filter our immediate experience. Jumping to the Island of Conclusions is easy. But, you can’t jump back and there are no bridges or boats; you must swim back to the mainland, which is hard, so you are unlikely to try. Soon, we develop a stereotype of Mathematics that we use to represent it in all of our thoughts and emotions about it. This stereotype is reinforced by the repetition of IWY pedagogy over the first twelve years of school. It becomes the easy representation that we invoke subconsciously whenever we think about Mathematics or perform Mathematical tasks. By the end of first grade, we use this stereotype to represent Math throughout the school years and for the remainder of our lives, unless and until we learn another story and stereotype with which to replace it. Go on, try it! Try to think of Mathematics in some other way. How did you do?

The myth that Mathematics consists only of definitions, theorems, proofs, and problems (applications) frames our pedagogy and thereby limits our practice to stocking our student’s inventory of facts and algorithms to use them. We employ IWY as the default pedagogy because we believe this myth. We learned this convenient, easy representation of Mathematics from our professors, teachers, and textbooks, and we use it in our classrooms and lecture halls. It “worked” for us; it should work for everyone, shouldn’t it? We want our children, our students, to succeed, don’t we? After all, success is good; it reinforces what they’ve learned. Practice makes perfect. Homework is the opportunity to practice and, thereby, perfect and make permanent what they’ve learned. Solve the problems, achieve success, feel good about what you’ve done and who you are. Of course, we use heuristics to motivate topics. Thus, we use the rectangle to illustrate the commutative property: A rectangle’s area is the product of its length and width; the order of multiplication doesn’t matter (32 = 8 x 4 = 4 x 8). Or, we use a pair of scissors to illustrate the Hinge Theorem: the length of the side of a triangle is proportional to the measure of the angle opposite it; in a triangle, the side opposite a 60º angle is longer than the side opposite a 30º angle (√3 times longer). The hinge of the scissors represents the vertex of the relevant angle and the opening between the tips of the blades represents the side opposite the angle. As you open the scissors, the angle between the blades increases and the distance between their tips increases; as you close them, the distance between the tips shrinks as the angle between the blades decreases. We intend such heuristics to point to the path to understanding and retention, but they don’t.

Perhaps the most surprising and wonderful fact about Mathematics is its fruitful application to nearly every aspect of the world we inhabit. Its applicability is integral to the prevailing story of Mathematics. Applicability would,  of course, be integral to any story of Mathematics. But, its role in this story is to play the sole reason to learn and do Mathematics. We learn Mathematics because it’s useful. Its value lies in its instrumentality. In at least one widely used high school textbook, at the beginning of each section there is a short list: what the section is about, what the student is going to learn how to do, and why the student should learn it. The third item in this list is, invariably, “It’s useful.” And, naturally, the section includes examples of its use in the world. Thus, we learn algorithms to perform such counting tasks as totaling the cost of a basket of groceries or the addition to your house or the size of your farm, or to perform such engineering tasks as determining the thrust required for a rocket of a given size to achieve escape velocity. We also learn algorithms to help us tell which algorithm(s) to apply to which circumstances. We call it “cookbook math” or “engineering math”. Our students—our children—do not meet “real” Mathematics until they’ve completed a twelve-to-fourteen-year apprenticeship that fails utterly to prepare them for this meeting.

So, Mathematics is, according to these three myths, in fact, and indeed, a body of knowledge consisting of definitions, theorems, their proofs and the algorithms that we use to apply those proofs to solve problems. Its value and, therefore, our motivation to learn it lie in is instrumentality. Together, these three myths comprise the meta-myth of Mathematics.

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Our Brains and Learning: What we’ve learned that has revolutionized our beliefs about the brain and learning

 

School teachers in all types of schools, private tutors, tutoring companies, military instructors, coaches, performance arts teachers, and all other types of teacher/instructor use the teaching method known as “I-do, We-do, You-do” (“IWY”): I show you how to do it; we do it together; you do it without my help, whatever “it” might be. We check the outcome of you doing it without my help. If the outcome is acceptable, we consider ourselves successful—you at learning, me at teaching. Martial arts, performance art, trade-school, factory, business and professional-school instructors employ this method as the prevailing method. Students and parents expect their private tutors to use this method almost exclusively; they expect the tutor to show the student how to apply the theory he or she learned in class that day or week.

IWY is endemic because it’s successful. It has been successful among sentient species since their origination and will continue to be successful. Its hallmark features are a demonstration, guided or supervised practice, and independent practice and application. It is the “tried-and-true” method to teach the rudiments of how to do nearly anything: to hunt an antelope, grow corn, dig a well, assemble a bicycle, dance the samba, play the clarinet, build a chair or frame a house, wire a kitchen, wire a lamp, fly an aircraft, sail a boat, drive a car, assemble a rifle, learn kung fu, or assemble a computer. Through experience—trial and error—we enhance and perfect the techniques we learn through training. Practice makes it permanent; experience, if we survive the trials (think of flying a small aircraft in bad weather, or climbing your first 100-foot rock wall without gear), perfects it. The ancient paradigm of this process is the guru instructing his or her pupil.

The invention of new technologies, of economical methods to produce them, and their subsequent, broad deployment in the physical, biological and social sciences have enabled researchers in the cognitive, brain, language and learning sciences to investigate the links between physical processes and behavioral outcomes more deeply and more broadly than was possible before their invention and deployment. In particular, “MRI”—Magnetic Resonance Imaging; digital signal processing; digital audio and visual recording; Wi-fi and the Internet; the capacity to consume, digest and restructure vast quantities of data; and machine learning capabilities have enabled insightful research into crucial causal connections and correlations between brain processes and their outcomes in human behavior.  Since the year 2000, we’ve learned a lot about the relationship between our minds and our brains.

The “London Black-Cab Driver” study exemplifies this point. In the early 2000s, scientists chose to study three hundred London black-cab drivers for brain changes as the drivers took years of complex spatial training. MRI and related technologies enabled these researchers to measure changes to the brain without killing the subjects or drilling through their skulls. To qualify as a black-cab driver, applicants must learn 25,000 street intersections and 20,000 landmarks. They are tested for their knowledge of the city. Applicants take two to four years to complete the course (and acquire The Knowledge, as it’s called) and fail the test an average of four times. The drivers in this study were (and still are) mature adults, whose brains were believed to have completed their development prior to undertaking the course. Researchers found that at the end of the course, the hippocampus in the drivers’ brains had grown significantly (as much as 20%). Additional, independent studies confirmed this result. This result and its confirmation revolutionized learning science. Before the publication of the first black-cab studies in 2006, scientists believed that we are born with a capacity to learn that was fixed at birth to reach a genetically determined maximum. This capacity, they believed, is distributed among individuals according to the normal probability distribution; its measure is the “IQ”—Intelligence Quotient score on either of two specific tests. Since the Black-Cab study result, they know that one’s learning capacity is not fixed at birth, that the brain remains plastic throughout our lifespan, and its plasticity is a function of the extent and nature of its use. Our capacity to learn throughout our life is, if not unlimited, undefined and undefinable.

Neuroscientists and psychologists have learned from other studies and experiments that:

  1. The brain can continue to grow and develop after it reaches physical maturity.
  2. Challenges and mistakes stimulate brain growth and formation of new connections;
  3. Repetition and success do not stimulate growth and new connections;
    To confirm this hypothesis, the “black-cab” research team performed the same study on London bus drivers, and the team found no significant brain growth among the bus drivers. They attributed this stasis to the lack of challenges for the drivers: they drove the same, assigned route every day. Once they had learned the route, there was no more stimulation of the kind that compels the brain to grow: no more mistakes or failures.
  4. The brain will return to its original, mature state if growth activities cease;To test the permanence of this growth, the same researchers studied cab drivers as they retired. They found that among those who stayed active intellectually, the brain growth they experienced as drivers did not change; among drivers who did not stay active, the brain shrank to its size and complexity before they took the driving course.
  5. Specific behavioral and operational capabilities are located in specific regions of the brain, i. e., that a cartography of the brain is possible (and underway);
  6. The development of the frontal cortex lags development of the cortex by several years;
    (This phenomenon, by the way, goes a long way toward explaining the “strange” behavior of your adolescent children, if you are parents.)
  7. Working memory is tiny: it is capable of holding 4, perhaps, 5 chunks of information at once:
    Try to repeat any spoken, random, nine-digit number sequence backward; then, try to repeat one forward. Repeat this experiment with the sequence 123456789 (or any sequence of consecutive integers) . Compare the result.  You will have completed the third task accurately and failed at the first two.

    In the first experiment, each number was a chunk of information unrelated to the other numbers (except by the fact that they were numbers). In the second experiment, the principle, “n + 1” linked them, thereby, creating a single chunk of information. You could have used 2, 4, 6, 8, 10, 12, 14, 16, 18. Given the pattern, principle or rule, “2n”, you could repeat any sequence of nine such numbers easily; it’s one chunk instead of nine.

  8. Total long-term memory and recall capacity is undefined and undefinable;
  9. We learn and retain more when we collaborate;
  10. We retain more when we test ourselves and each other frequently—take “test” in its broadest sense;
  11. Practice makes permanent—use it or lose it (this principle holds for bad and good behaviors, sound and unsound beliefs, sound technique and unsound technique);
  12. Mindset matters.  People with growth mindsets—people who “know” that their capacity to learn and grow is not defined and delimited at birth—learn more than people with fixed mindsets—people who believe they are born “smart,” “average,” or “stupid.” People with growth mindsets feel better about themselves and are more optimistic about their prospects than people with fixed mindsets.

None of these facts are specific to learning Mathematics, Arithmetic or any other subject, learning domain, or field of inquiry. This generality implies that learning Mathematics and Arithmetic is no different from learning English, Geography, Physics, Chemistry, Music, Dance or Carpentry. It suggests, too, that, considered as a field of inquiry, Mathematics is similar to other fields in some ways essential to our  capacity to learn it.

Chemistry, Physics, Biology, Psychology and their related interdisciplinary subjects are experimental areas of inquiry. Progress and learning in them result from trial and error guided by hypotheses. The literary and performing arts are experimental, too. Progress and learning in those areas result from trial and error guided by hypotheses. Progress and learning in Mathematics result from experimentation—trial and error—guided by hypotheses. Mathematics is an experimental science, too. In the sciences, humanities and arts, we anchor the ladder of abstraction on the apparent world of mid-size objects. In Mathematics, we anchor this ladder on the Natural Numbers—the positive whole numbers. Like practitioners in other fields, mathematicians observe the behavior of numbers, attempt to codify it, and when they succeed in understanding and codifying the behavior they observe, they build new structures on those numbers and observe their behavior as well. Physicists develop theories to describe, predict and explain the behavior of matter and energy; psychologists develop theories to describe, predict and explain the behavior of individual humans; sociologists develop theories to describe, predict and explain the behavior of people in large groups. Mathematicians develop theories to describe, predict and explain the behavior of numbers.

Why, then, is the experience of learning and teaching Mathematics distinct from the experience of learning and teaching other subjects? Why is Mathematics detested by nearly every student in the schools and avoided like the plague at college or university? Is there something about numbers that makes them less accessible or more intimidating to us than planetary motion, light, people, cars, or bridges, than words, paragraphs, and stories, or than sounds, music notation, and sonatas? How can knowing these twelve facts about the science of learning inform our experiences of learning and teaching Mathematics?

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I-do, You-do, We-do: An essay on teaching mathematics – 1

 

 

In a typical mathematics classroom, the student desks are arranged in proscenium theater fashion. From the front of the room, the teacher presents definitions of new terms, new theorems, proofs of these new theorems, and demonstrations of how to apply them to example situations. The teacher writes a mathematics problem on the board, solves it, and asks for questions from the class. After this interlude for questions, the teacher writes a new problem on the board, then leads the students step-by-step through the algorithm used commonly to solve it. Teachers vary this process according to their preferences, but it is essentially the same for all who use it. Next, the teacher writes another problem or two on the board and instructs the students to solve them without guidance. After a short time, the teacher gives the solution and asks for questions. If time permits, the teacher assigns homework, the students begin working, and the teacher strolls around the room responding to student requests for help. This method is called, “I-do, we-do, you-do.” It is the prevailing method of teaching mathematics in the secondary schools. Most teachers vary their methods, but, even among those who do, most of them use it most of the time. Although the community of math education researchers with which I’m familiar recommends different methods, many, probably most, math teachers still lecture, demonstrate solutions to homework problems and test student recall in the same way we did in most math classes for the past century.

By this process, the teacher programs students to execute algorithms. Which algorithm to execute depends on the data fed into the students. A teacher’s lesson plan consists of four stages:

  1. Present the material—load the program for students to compile;
  2. Test the program—run it with sample input data to test the students’ compilation of it by working through a problem or two with them;
  3. Debug the students’ compilers—walk through the program step by step to find compilation errors;
  4. Input new data for processing—assign classwork and homework. The students run their new program(s) in class, first, then, at home to check their compilations for efficacy.

The next day, everyone checks their results. For those students with no results or with error messages and given enough time the teacher checks the program for errors, tries to fix them and the students run the debugged program to check its efficacy: Repeat debugging their compilers and programs until all or “enough” students produce acceptable output and receive no more error messages. No one knows what the students whose programs don’t need to be debugged do during this period.

Consider the process known as “solving for the unknown” or “solving for x.” To present this method, the teacher writes a specific example on the board at the front (or side or rear) of the room and determines the value of the variable. If her example is “x + 3 = 11,” we subtract 3 from both sides of the equal sign and produce the statement, “x = 8.” To enable this algorithm to handle more complex data, we call another subroutine. Given, “2x + 3 = 11,” we call two subroutines, the subtraction and division subroutines: 1) “subtract 3 from 2x + 3 and from 11 to obtain, “2x = 8”; 2) divide 2x and 8 by 2 to obtain, “x = 4.” For some students, the latter algorithm becomes the algorithm for solving problems involving subtraction and division; the former algorithm becomes a special case of the latter (i. e., when the coefficient of x is “1”). They have one program (composed of two routines) to call; the students who don’t realize that the two simple algorithms are special cases of a single algorithm still have two routines to call.

Because everyone’s working memory is tiny, students acquire and store these algorithms in their long-term memories. These algorithms become more complex and more numerous during the school year. To use them, students must first locate them in long-term memory, then choose an appropriate algorithm to apply to a particular problem, third, call it into working memory, and finally apply the algorithm correctly. As they learn more—acquire more algorithms—this process of applying their knowledge becomes more difficult exponentially. To mitigate or cope with this accumulation of algorithms, teachers have three options: do nothing, letting their students figure out how to cope with it; repeat IWY often with small variations, thereby imitating the practice and rehearsal used (successfully) in sports and performance arts; or, introduce new algorithms to automate the choice process, such as, “If ‘2x’ appears as a line in the derivation”, “then, ‘Get ‘division algorithm’ appears as a line in the derivation.” This second level of programming can be as complex and sophisticated as we wish. The question in the classroom is who programs whom? Mathematicians and logicians learned in the twentieth century that most of mathematics can be produced or reproduced, depending on your philosophical perspective, by algorithms from a tiny number of symbols, sentences, and rules. As students journey through their various mathematics classes, subjects and topics in their schools (by “school” I mean grades K-12), some students are more proficient at assembling the routines and subroutines they learn into programs they can apply to a wide spectrum of problems; some students are proficient at collecting these algorithms and identifying appropriate situations for their use; some students are proficient at both processes and some are proficient at neither process. At which of these a given student is proficient appears to be serendipitous or a combination of factors that are difficult to control. Eventually, the education system rewards those students who learn to simulate the behavior and produce the output of computing machines best. The “best” students are identified and their rewards are distributed solely according to the results of such tests. The tests don’t differentiate the processes the students use. The best students continue to accumulate and apply more of the algorithms they are fed, and commercial and academic societies esteem their accomplishment. The best of them become, in effect, carbon-based, biological computers. Or, they become mathematicians, physicists, or artists.

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