In part 3 of this series I summarized the Math Academy “sales pitch” as embodied in Part I of the draft book The Math Academy Way: Using the Power of Science to Supercharge Student Learning. In this post I look at Part II of the book, “Addressing misconceptions,” which discusses the various objections that have been raised (or might be raised) against the Math Academy philosophy, pedagogy, and system.
As in my summary of Part I, the following sections are my paraphrases of the content of The Math Academy Way; my own comments are [enclosed in square brackets]. You should interpret statements not in brackets as being prefaced by “The book says . . .” or “The author claims that . . .” Terms in boldface are key concepts relevant to the Math Academy system.
Chapter 6. The persistence of neuromyths
This is a brief chapter that makes the point that laypeople and even experts believe things about the brain and learning that are demonstrably not true. The so-called “Mozart Effect” is given as an example. Belief in such neuromyths is common among those seeking to excuse education failures or looking for a “quick fix” that doesn’t require a lot of effort.
Chapter 7. Myths and Realities about Individual Differences
This chapter begins by pointing out the the idea of different ”learning styles” is a neuromyth: students may prefer receiving information in different ways (e.g., verbal vs. visual), but that does not affect the rate at which they actually learn.
A major factor in how people learn is working memory capacity (WMC): Larger WMC makes tasks easier (all other things being equal), improves the ability to do abstract thinking (which in turn affects the ability to apply learning to new contexts), and improves the speed at which people can learn.
It is not possible to improve WMC. [Although the author does not mention it by name, so-called ”dual n-back” training would seem to be an example of a technique that purports to do this. But see Gwern Branwen’s discussion of whether this actually affects IQ.] However WMC can be augmented by long-term memory that encodes domain-specific knowledge. [A trivial example: avoiding the need to multiple 4 times 16 because one has memorized the answer.] Since people are able to add more information to long-term memory (within limits), this is consistent with the ideas behind growth mindset.
In the context of mathematics, the field can be divided into roughly six levels, from basic arithmetic to Fields medal-level mathematical work. [I was probably at or close to level 4 upon graduating from college—capable of doing graduate-level work at least to some minimal level—and am now below level 3.]
People generally hit an abstraction ceiling, a point at which the time and effort required to learn math to a given level exponentially increases to a point where continuing is not a productive use of one’s time even given sufficient motivation. For people with lower WMC the exponential curve is steeper and they will hit the ceiling sooner, because it takes them more energy to reach a given level of proficiency.
However, if learning can be made more efficient, so that it requires less energy, the exponential curve flattens somewhat and students can reach levels of proficiency they might not otherwise be able to.
WMC and similar capacities have a genetic component, and we cannot assume everybody to be at the same level. Genetic-influenced ability interacts with environmental factors—access to instructors, amount of practice, etc.—to determine the overall level a person can reach in math. Instructors downplay this fact, partly to encourage students and partly for self-interested reasons, because they don’t want to lose students. [Does this same incentive affect Math Academy itself, and, if so, how?]
But people can generally learn more math than they do, and struggling in math is not really an indication of how much one can learn. For example, the struggle may be because a student didn’t master previous material and is not given the opportunity to remedy that. One of Math Academy’s claims is that they can detect and remediate such knowledge gaps with the aid of an ”adaptive, automated learning system.”
Math Academy also avoids such gaps in the first place via a combination of mastery learning, spaced repetition, and comprehensive coverage of all required topics. [See the chapter on the ”knowledge graph.”]
Struggle can also be caused by ineffective practice and insufficient practice. Math Academy can help here by providing suitable practice exercises to encourage active learning, but success is ultimately up to the student’s motivation to put in the work. Such motivation may be intrinsic (the student loves math) or extrinsic (the student needs math for a job, or is rewarded by competitive success or by parents).
Final discussion: If suitable instructional scaffolding, guidance, etc., can compensate for lower WMC and related issues, why couldn’t we employ this to educate all students at the same rate and to the same level?
But in practice some students can ”eat” bigger ”bites” of new material than others can, and this will in practice cause them to progress faster. This is consistent with the observation that math will become difficult for different students at different times in their study.
[This discussion reminds me of Nathan Robinson’s argument that we can’t conclude that hereditary differences in intelligence between individuals exist until/unless we spend many many years trying to teach someone something: “When we have given students a boundlessly kind, supportive engineering program, that lasts as long as they need and is structured around them with as many of society’s resources as possible put toward its perfection . . . then maybe we will know their ‘natural capacity’ for engineering.” The counter-argument in The Math Academy Way is that any given student’s “innate capacity” for mathematics is significantly higher than we suppose, but that students do vary in that capacity, such that some will “hit a wall” earlier than others even in an optimally-structured program.]
Chapter 8. Myths and Realities about Effective Practice
The first part of this chapter promotes direct instruction and criticizes constructivist instruction, discovery learning, and related approaches. One key point is that discovery learning is much more effective for experts who have pre-existing knowledge that they can use as a basis for further exploration, and also more effective in a work setting where no one person is expected to know everything and work output is often a group effort.
But these are not typically the case in traditional education: we want individuals to learn, not groups, and we don’t want student to have the illusion of learning when in reality they’re being carried by other people in the group.
The discussion then switches to the point that no learning is effortless, and learning requires a sustained effort at deliberative practice, i.e., ”individualized training activities specifically chosen to improve specific aspects of a student’s performance.” This can include practice testing and distributed practice (spaced repetition). “Learning is all about creating desirable difficulties.”
Finally, the discussion defends testing, repetition, competition, and (in the context of math) computation from those who feel they detract from learning. The main point here is that testing, repetition, and competition are key to talent development in other fields, and equally if not more so in math. As for computation, it is needed as a basic skill to help build further conceptual understanding.
Chapter 9. Myths and Realities About Mathematical Acceleration
This chapter discusses extensive research that shows that having qualified students take above-grade math courses has many positive benefits and no drawbacks. Students who have mastered prerequisites suffer no negative psychological effects, learn more material to the same depth (as they would without acceleration), don’t run out of math courses (many such courses beyond calculus), get better experience than with math competitions, can place out of college courses (possibly with special appeals to instructors), and better prepare themselves for advanced instruction in math or related fields (e.g., engineering).
Speculations on why schools don’t support acceleration include: It doesn’t fit the traditional one-grade-at-a-time model, especially at breakpoints between elementary and junior high school, and between junior high school and senior high school. Acceleration can also negatively impact school funding because students spend less time in school. [Another possible reason is apparent disparate impact, based on which students get admitted to accelerated classes and which don’t.]
This concludes my discussion of Part II of The Math Academy Way. In part 5 of this series I’ll discuss Part III of the book, which goes into more depth regarding the “features” of the Math Academy system, i.e., the learning strategies it implements.