In part 7 of this series I summarized the technical features of the Math Academy system. In this post I take a quick look at some of the questions people have had (or might have) regarding the Math Academy system, as presented in the “Frequently Asked Questions” section of The Math Academy Way. I also briefly discuss the “Notes for Future Additions” section.
As in previous posts, 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.
Frequently Asked Questions
Many of these are relevant primarily to people already taking Math Academy courses, but there are several exceptions.
First, the FAQ includes an explanation of how course lessons are divided up into instruction and active problem solving, interleaved with each other. More specifically, the lesson begins with an introduction (presented using slides), followed by a worked example covering the material just introduced and 2-3 practice problems similar to the worked example. The worked example plus practice problems is known as a knowledge point (KP), and there are typically 3-4 per lesson.
Failure to complete a KP results in “failing” the lesson, and the student is moved on to different lessons before coming back to the failed one. Math Academy claims an average pass rate of 95% on the first try, and 98% within two tries.
A subsequent question addresses the concern about the large amount of problem solving required of Math Academy students. The response is that only doing active problem solving as soon as possible after learning something will ensure that the information will in fact be committed to long-term memory. [This concern about Math Academy’s relentless focus on problem solving is also reminiscent of Dan Meyer’s complaint that Math Academy, like other proposed instructional metholodogies, is simply “[redefining] math to mean ‘becoming an absolute demon at math exercises.’” More on that later.]
Further questions reiterate core tenets of the Math Academy approach:
It’s not necessary to struggle in order to learn something—and in fact struggling is counterproductive. “The way to increase a student’s ability to make mental leaps is not by having them jump further, but by having them build bridges from which to jump.”
Automaticity is important, and something the Math Academy needs to check for. It is not necessary to achieve full automaticity to advance to higher-level topics, but a lack of automaticity will eventually catch up with students and impede further progress.
Consulting and leveraging worked examples is essential to make continued progress. “If you don’t have worked examples and instructional scaffolding to help carry you through once math becomes hard for you, then every problem basically blows up into a ‘research project’ for you.”
[This point is also relevant to Dan Meyer’s criticisms of the Math Academy approach. He dismisses the fact that Math Academy provides examples to teach concepts: “You don’t get Math Academy ‘experience points’ for reading conceptually rich explanations. You get them for completing exercises.” The counterpoint is that if you don’t read the explanations and examples, and as a consequence lack basic understanding of the concepts, then at some point you’ll no longer be able to successfully complete the exercises, and you’ll stop getting those sweet, sweet XP.]
[For example, Dan Meyer describes teaching his young children to mechanically calculate derivatives: “Easy! All I did was tell them to take the number above the x and write it next to the x, then to subtract one from the number above the x.” His kids could no doubt rack up some XP doing exercises of that type, but their XP acquisition would grind to a halt as soon as they came to a problem that asked them to calculate the slope of a tangent line to a curve.]
Math Academy courses are not structured like typical higher math textbooks because that style of instruction is demonstrably ineffective except for the most talented students. “Higher math textbooks and classes are typically not aligned with (and are often in direct opposition to) decades of research into the cognitive science of learning. Higher math is heavily g-loaded, which creates a cognitive barrier for many students. The goal of guided and scaffolded instruction is to help boost students over that barrier.”
Both interleaving of topics and the difficulty of reviews are designed to promote learning by making the task of retrieving information more effortful: It’s easier to remember something on a quiz or review if you’ve just covered the material, and it’s more satisfying to be able to ace every quiz or review, but that means that any learning may be shallow and the supposedly-learned information easily forgotten.
Learning with the Math Academy system requires that students put in a reasonable level of effort on a continuing basis. “Math Academy teaches math as though we were training a professional athlete or musician, or anyone looking to acquire a skill to the highest degree possible. . . . While it’s true that willingness to work hard is a bottleneck for many students, such students are not part of our target market.”
[This statement will no doubt be seized upon by critics who claim that any success Math Academy might have is due purely to selection effects: they are in effect “skimming off the cream” and leaving all the other students in the lurch. It’s certainly a fair criticism that Math Academy demands a high degree of self-motivation. One might hope that other people might be able to step in to help motivate students who are not as willing to put in the effort, but The Math Academy Way doesn’t spend much time on that issue—the “Coaching” chapter is one of the shortest ones in the book.]
[It’s easy to dismiss criticisms like this—for example, by questioning how much typical public school math teachers are able to motivate their own students who lack motivation. But I think it’s worth thinking about how Math Academy might be extended or supplemented to address this. One can blue-sky high-tech solutions, like an LLM designed to be an encouraging companion to an individual student, or low-tech solutions, like local Math Academy meet-ups where students could meet each other, share experiences, and possibly get special assistance from a volunteer or paid instructor. I’m not sure what might work or what might be feasible, but I think addressing this issue to at least some degree will be key to extending the addressable market for this method of online instruction.]
There are many other questions in the FAQ, but I’ll stop at this point.
Notes for Future Additions
This section contains material that is under consideration for incorporation in the main text of The Math Academy Way, and is interesting as a further look into the thinking of those creating the Math Academy system.
Apropos of my comments in the previous section, there’s a lot of suggestions for extending the “Coaching” chapter. For the most part the suggested additions continue the theme of Math Academy as an equivalent to intensive athletic or musical training, and the focus is almost exclusively on parents motivating their children. [Again, this reinforces the idea that Math Academy is not interested in—or has rationally chosen not to pursue—promoting its service to the traditional public education market.]
There is an interesting discussion of knowledge spaces, which are at least superficially comparable to the knowledge graphs of Math Academy, but proved to be too complex to implement in the context of a Math Academy-like system. [There is a commercial product, ALEKS, based on the knowledge spaces framework. It would be interesting to know more about how successful it has been, particularly since it’s sold by McGraw Hill and pitched to the K-12 education market, among others.]
One other interesting but brief mention is regarding “elaborative interrogration,” that is, asking students to elaborate on their understanding of particular concepts and procedures. [If Math Academy is going to incorporate LLMs in some contexts—and right now there’s no firm indication of this either way—this might be an area where they could be useful, if for no other reason than being able to better interpret free-form answers.]
The section includes with many more links to further reading. These may be of interest to anyone who’s gotten this far in the book and want to explore more material along the same lines.
This concludes my discussion of The Math Academy Way. In part 9 of this series I’ll discuss my own experience as a Math Academy student, starting with those aspects of the service unrelated to the actual learning of mathematics.