Category Archives: Learning

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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