In part 1 of this series I recounted stumbling across Math Academy, thinking it might help me learn what an eigenvector is, and after doing some research signing up for it. In part 2 I took a pessimistic stance on whether or not Math Academy might be successful, whether with me or with the market in general.
In this post I turn the floor over to Math Academy itself, summarizing the main arguments of Part I of the book The Math Academy Way: Using the Power of Science to Supercharge Student Learning, by Justin Skycak (advised by Jason Roberts). (The book is still in draft form; I read, took notes on, and am referencing the draft as of December 27, 2024. However, at the time of writing there is a newer draft, dated January 28, 2025.)
The preface of the book lays out the questions the book aims to answer:
What techniques exist to maximize student learning and talent development, particularly in the context of math?
Why are these techniques so impactful, and if they are indeed so impactful, then why are they so often absent from traditional classrooms?
How does Math Academy leverage these techniques?
The book is aimed at pretty much anyone who might be interested in the Math Academy system, not excepting math hobbyists (like me).
The first set of chapters (Part I: Preliminaries) can be thought of as an extended “sales pitch” for Math Academy: identifying the problem and explaining how Math Academy is uniquely positioned to solve it. These and later chapters include copious quotations from and citations of the research literature relevant to Math Academy; as noted in the preface, “when faced with the decision to (a) build credibility by quoting the literature extensively, versus (b) streamline our communication, we have chosen to lean towards credibility.”
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 that . . .” or “The author claims that . . .” Terms in boldface are key concepts relevant to the Math Academy system.
Chapter 1. The Two-Sigma Solution
The first chapter riffs off the key claim of education researcher Benjamin Bloom: that one-on-one instruction can elevate a typical student to the 98th percentile, an up to two standard deviation improvement (hence the “two sigma solution”). [However, the book notes that this level of improvement is not seen by other researchers, and José Luis Ricón’s analysis echoes this skepticism.] Leaving aside the issue of exactly how effective it is, Bloom’s method would be extremely costly to do with human instructors even if we wanted to do it: more than $10,000 a year.
But in any case traditional schooling is at odds with the idea of talent development as it is practiced in other areas (e.g., sports and music). The key contrasts are grouping by ability vs. grouping by age, and short-term teacher involvement with a given group of students vs. long-term involvement (cross-sectional vs. longitudinal). Mixing these modes does not seem to be effective.
Talent development itself proceeds in three phases: an early phase focused on having fun learning, a middle phase focused on intensive skill development, and a final phase focused on application to new and original problems at the highest level. Acquiring knowledge and solving problems (the second phase) precedes creative endeavors (the third phase). Math Academy is focused on the second phase and assumes a willingness to learn on the part of the student.
The chapter concludes with a list of citations of relevant research papers, a practice repeated in subsequent chapters.
[To the extent that this book is a sales pitch—and I think in large part it is, whether the author acknowledges it or not—this chapter is in my opinion a missed opportunity. It’s a classic example of “feature-benefit” selling: “here is a feature, and here is the benefit it provides you, the prospective customer.” But this doesn’t address a critical point: why should I or anyone else care? Selling is much more effective when the customer has “pain” that the product can potentially relieve.]
[That’s why I led my previous post with the anecdote about me not knowing what an eigenvector is, goofy though it may be. It is a source of embarrassment and frustration to me that I still haven’t learned a key concept in an area of mathematics I’m interested in, and that embarrassment and frustration are intense enough that I was motivated to read this book and to consider spending $49 a month. So, Math Academy folks, please consider surfacing more pain in your prospects! And not general free-floating pain, but pain as it is specifically felt by students, parents, teachers, adults in the workforce, etc.]
Chapter 2. The Science of Learning
The key message of this chapter is that researchers have empirically demonstrated the key elements related to effective learning: active learning, deliberative practice, mastery learning, minimizing cognitive load, developing automaticity, layering, non-interference, spaced repetition (distributed practice), interleaving (mixed practice), the testing effect (retrieval practice), and gamification.
However, with minor exceptions [like the Math Academy], educators have not incorporated these finding into their teaching practice. The main reason seems to be that using them makes both teachers and students feel that students are not learning fast enough, and both teachers and students value the perception of learning over the actuality: illusion of comprehension prioritized over desirable difficulty.
[Although the book doesn’t mention it, this is true of parents as well: many value good grades more than actual learning.]
Technology can help here, by making it possible for teachers to implement techniques that are too time-consuming to do manually, e.g., creating personalized spaced repetition schedules for students. [But, technology is no panacea here, since effective use of spaced repetition requires a fair amount of discipline on the part of the student.]
The chapter concludes with the claim that by using these techniques the Math Academy system can accelerate learning by 4x.
Chapter 3. Core Science: How the Brain Works
This chapter begins by reviewing the distinction between sensory memory, short-term memory (working memory), and long-term memory. Proper learning techniques can compensate for lower short-term memory.
Solving a math problem (like calculating the value of 43 ) is a coordinated effort between sensory, short-term, and long-term memory: Sensory memory is used for initial understanding of the problem (calculate 43 ) and for concrete storage of intermediate results (e.g., writing down “4 * 16” after having done the initial multiplication of 4 * 4). Long-term memory is used for retrieval of memorized facts (e.g., 4 * 4 = 16) and memorized procedures (e.g., how to multiple a two-digit number). Working memory retrieves information from sensory memory and long-term memory, does calculations as needed, and stores the results in sensory memory (as intermediate results) or long-term memory (as final results).
The author notes that items stored in long-term memory can reduce the work done by short-term memory, e.g., memorizing the values of 23 , 33 , 43 , 53 , etc.
[An alternate—though less general—possibility along the same lines is being familar with “computer arithmetic” and recalling that 16 * 4 = 64. Or, a variant: recasting 43 as (22 )3 = 26 based on rules involving addition of exponents, and then using the memorized values of powers of two to produce 26 = 64.]
Chapter 4. Core Technology: The Knowledge Graph
This chapter introduces the idea of a knowledge graph, a [directed acyclic] graph showing which (detailed) topics are prerequisites for other (detailed) topics. Some topics may be prerequisites for more than one topic, and some topics may have multiple prerequisites.
[The knowledge graph is analogous to a “tech tree” in Sid Meier’s Civilization and similar games, in which certain technologies must be discovered first in order to enable discovery of others. It’s a directed graph because the edges go in a single direction from a more basic prerequisite topic to a more advanced topic that depends on that prerequisite. It’s an acyclic graph because there are no circular dependencies, in which one topic is a prequisite for another topic, which in turn is a prerequisite for the first.]
The full knowledge graph for Math Academy consists of thousands of topics [nodes], covering mathematics from elementary school to college. Courses are simply subsets of the overall knowledge graph, typically containing a few hundred topics; these topics can then be combined into a course graph.
[The construction of the overall knowledge graph must have taken a lot of work. The graph is thus a possible source of competitive advantage for Math Academy. However, since it’s exposed in the course pages an unscrupulous competitor could presumably copy the entire graph, either directly or—more sneakily—by enrolling an LLM as a student and training it on the course material.]
Mastery learning: Students must demonstrate proficiency in all prerequisites for a topic before being allowed to move on to that topic. Mastery of a topic thus unlocks new portions of the knowledge graph into which students can advance.
A topic can encompass other topics that are prerequisites to it. This simplifies review of material: a student reviewing a given topic does not have to do detailed review of that topic’s prerequisites.
[The book discusses this point in much more detail later. For now, note that this encompassing is made possible by the hierarchical nature of mathematics, in which more advanced areas subsume basic topics. It would not necessarily be possible with other subjects. Also, many people like me who’ve tried learning topics using spaced repetition have felt burdened by the sheer number of items to be reviewed each day, especially when learning a new set of facts. Greatly reducing the number of review items can thus be a major advantage for Math Academy.]
Finally, a new Math Academy student takes a diagnostic exam to determine which topics in the knowledge graph they have mastered and which they have not. This enables them to skip topics they already know in favor of addressing foundational topics that they have not yet mastered.
Chapter 5. Accountability and Incentives
Maximizing learning is difficult and at odds with other possible goals: enjoyment, ease of practice, etc. Doing it successfully requires making decisions on pedagogy, etc., based on how those affect measurable learning. However students and teachers (as well as parents and sometimes employers) resist this approach (which is admittedly not easy to implement).
Accountability is lacking in traditional education because it is diffuse, spread out over multiple teachers: a teacher may be ineffective, and will leave it to the teacher of the next grade to remedy any student deficiencies. [This is of course a consequence of the way education is traditionally structured.]
Grade inflation is rampant. This means that grades as a measure of learning cannot be trusted and can set students up for failure in future courses. Experience during COVID-19 demonstrates this, as math grades inflated and remained inflated after the main part of the epidemic was over. But COVID-19 just accelerated an existing trend.
One response to criticism of grade inflation is to deny the existence of objective learning (radical constructivism), making the student the judge of their own learning. [Although the author does not mention it, this can seen as an example of the general postmodernist approach.] This denial also can be seen as a response to concerns over disparate impact, both during COVID-19 and otherwise. [It’s interesting to speculate whether radical constructivism would have gained a following if such concerns were absent.]
In contrast, Math Academy is held accountable for learning by those who pay for it and [it is implied] exert the discipline to complete the course, and must therefore ensure it employs effective learning strategies. This is in contrast to free or “freemium” offerings that cannot afford to turn away students unwilling to put in serious work.
[Note that charging in and of itself is necessary but not sufficient: There are lots of education offerings that charge a lot of money and purport to deliver actual learning. The key element has to be independent validation of learning by a third party mechanism, e.g., standardized tests or class grades.]
This concludes my discussion of Part I of The Math Academy Way. In part 4 of this series I’ll discuss Part II of the book, which can be thought of as addressing prospects’ objections to the sales pitch of Part I.