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:
- The brain can continue to grow and develop after it reaches physical maturity.
- Challenges and mistakes stimulate brain growth and formation of new connections;
- 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.
- 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.
- 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);
- 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.)
- 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.
- Total long-term memory and recall capacity is undefined and undefinable;
- We learn and retain more when we collaborate;
- We retain more when we test ourselves and each other frequently—take “test” in its broadest sense;
- 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);
- 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?