In part 4 of this series I discussed Math Academy’s responses to the various objections lodged against the system. I now look at more in-depth explanations of the Math Academy system, as described in Part III (“Cognitive Learning Strategies”) of the book The Math Academy Way.

As in my 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.

Chapter 10. Active Learning

This chapter reemphasizes that active learning (doing exercises, etc.) is much more effective than passive learning (watching videos and lectures, reading and re-reading textbooks,etc.). In classroom settings this requires all students to be actively learning individually, not just doing exercises as a group.

Multiple examples show the need for active learning. In the first (hypothetical) example, a personal tennis coach talks about tennis and demonstrates moves, but does not have the student practice them. The obvious result would be no learning. In a second (real-life) example, a system was tested that provided instructors improved feedback for lectures, but it was discovered most students were not paying attention anyway.

A third example is MIT physics courses incorporating active learning and reducing the number of students failing by almost 2/3. Finally, it was discovered that elite skaters spend 6x more practice time vs. rest time on jumps, etc., that they’re trying to master. (The Math Academy system has students spending 7x time doing exercises vs. reading worked examples.)

The neuroscience behind active learning: active learning leads to more brain activity, both during active learning and during subsequent passive learning.

Why are there misconceptions around active learning? Passive learning is more convenient for teachers and less stressful for students—and thus less stressful for teachers in turn. Students think they are learning something when they actually aren’t.

Chapter 11. Direct Instruction (in-progress)

[This is a fragmentary chapter with minimal material thus far.]

We should not reject direct instruction on the basis that active learning with passive guidance is more effective than passive learning plus direct instruction. This tells us nothing about the effectiveness of direct instruction on its own, but rather just demonstrates the superiority of active learning over passive learning: Active + Direct > Active + Unguided > Passive + Direct.

Chapter 12. Deliberate Practice

Deliberate practice is “mindful repetition at the edge of one’s ability,” contrasted with mindless repetition of things one already knows. The chapter includes much discussion about the effectiveness of deliberate practice vs. non-deliberative, as well as the fact that deliberate practice is hard for students, since one must continually repeat things “at the edge of one’s ability.” One can supplement deliberate practice with other more fun things to help motivate students, but these are not an effective substitute for it.

Chapter 13. Mastery Learning

The basic idea here is to require the student to demonstrate proficiency in whatever areas are prerequisites for their next learning challenge. In the limit this requires one-on-one instruction, either by a tutor or by a system like Math Academy.

The idea of mastery learning is resisted by traditional educators because it plays havoc with the traditional grade by grade progression of students.

Math Academy has implemented mastery learning at a very granular level through its “knowledge graph,” discussed above. The knowledge graph dictates what a student can next learn. This is compared to the zone of proximal development, i.e., that set of problems which a student can solve with support but not without it. This corresponds with the knowledge frontier or edge of mastery. The goal is for the student to continually expand that frontier outward.

The knowledge profile is the set of topics in the knowledge graph that the student has already mastered. Placement diagnostics determine what that profile is for a new student, so that they can be presented with material at their knowledge frontier.

Chapter 14, Minimizing Cognitive Load

Math Academy instruction is very fine-grained, with a given area having about 10x the number of steps (e.g., worked example followed by exercise) than typical mathematics curricula. The goal is to minimize cognitive load, i.e., the amount of working memory required to complete a task. This helps prevent students from getting stuck on a particular point in the progression of the curriculum.

Each topic is divided into several knowledge points, each consisting of a worked example plus practice exercises. The example given is adding two digit whole numbers. The first knowledge point might be adding two digit numbers where carrying is not required, e.g., 63 + 12. The second knowledge point might be adding two digit numbers with carrying, e.g., 63 + 18. The third knowledge point might be adding two digit numbers with carrying into the hundreds place, e.g., 63 + 38. And so on. Each knowledge point has a subgoal label, e.g., “adding two digit numbers with carrying.”

Knowledge points also can contain diagrams for visualization in addition to verbal explanations, to leverage dual coding and distribute mental processing between visual processing (visuo-spatial sketchpad) and verbal/audio processing (phonological loop). For higher-level topics this can include flow charts.

As students learn the material, scaffolding is removed: a review question may call upon knowledge from one or two of many different worked examples, and timed quizzes further test knowledge in a context where it is not possible to “look at the book.”

Chapter 15. Developing Automaticity

“Automaticity is the ability to perform low-level skills without conscious effort. [emphasis added]” An analogy is to athletes who can perform low-level skills (dribbling a basketball) while thinking consciously about higher-level game strategies. [A similar analogy would be musicians who can perform low-level tasks of playing an instrument while thinking about higher-level tasks like playing expressively or in a certain style.]

One can achieve automaticity (e.g., in mathematics) by leveraging long-term memory to relieve pressure on the limited capacity of short-term memory. Long-term memory thus becomes an extension of short-term memory that a person can draw upon at will.

Automaticity goes beyond familiarity, and requires accessing learned knowledge quickly and accurately. This is required as a necessary foundation before learning more advanced topics that depend on that knowledge already being learned.

Examples of automaticity and the lack thereof: Students are being taught how to compute cubes as the number multiplied by itself and then multiplied by itself again, for example, 43 = 4 x 4 x 4. The first student knows 4 x 4 = 16 (from having the multiplication table in long-term memory) and then can apply the learned procedure for multiplying a two digit number by a one digit number. The second student doesn’t know the multiplication table, so needs to compute 4 * 4 as 4 + 4 + 4 + 4 = 16. The third student doesn’t even know the addition fact 4 + 4 = 8, but must count up from 4 by 1 four times: 4 + 1 = 5, 5 + 1 = 6, 6 + 1 = 7, 7 + 1 = 8. This results in the second two students making mistakes in their calculations, requiring additional teacher time to correct their understanding and causing student frustration that they’re not “good at math.”

“Automaticity is a necessary component of creativity.” An example is writing: if a person has difficulty with basic issues of spelling and grammar, they will have difficulty in expressing themselves in a creative way. [This is especially true with skills like writing and doing mathematics that—unlike oral language learning—do not come naturally to students based on their having inborn capabilities.]

Automaticity is also necessary for higher-level thinking, and automaticity in knowing and recalling standard mathematical facts is critical to achieving mathematical literacy and academic success in mathematics.

Finally, the chapter discusses the neuroscience underlying automaticity, the idea that it prevents disruptions to background thinking (the “default mode network”), disruptions that reduce the amount of attention and thought a person can devote to a higher-level task.

Chapter 16. Layering

“Layering is the act of building on top of existing knowledge. [emphasis added]” Layering promotes retroactive facilitation (solving a problem using existing knowledge reinforces memory of that knowledge) and proactive facilitation (knowledge acquired in solving previous problems improving knowledge acquisition needed in solving new problems).

Layering also improves the structural integrity of a person’s acquired knowledge, i.e., having that knowledge not have holes where understanding is lacking.

Math Academy promotes layering by having mastery of one topic lead directly into a new topic, and by leveraging a complete and comprehensive knowledge graph in which all new topics depend and build on previous topics. It also uses additional techniques to promote connections between topics, like presenting multi-part problems requiring knowledge of many previous topics to solve.

A key principle: “Any lesson should cover all types of problems that a student could reasonably be expected to solve if they truly know the topic.” Some other approaches violate this by, for example, presenting calculus in a way that does not require algebra.

Chapter 17. Non-Interference

Learning two related topics at the same time (or close together) can inhibit learning of both (associative interference). The Math Academy system avoids this by spacing related topics out in time, and presenting students with a choice of unrelated next topics, thus achieving non-interference. In addition to promoting learning, this also keeps students interested by increasing variety and reducing unnecessary repetition.

Chapter 18. Spaced Repetition (Distributed Practice)

This chapter reviews conventional information about spaced repetition: that by spacing review out in time, students can mitigate the effect of memory decay (the forgetting curve) and (ideally) retain information indefinitely.

The book criticizes traditional educational practices for neglecting the effectiveness of spaced repetition, and thereby leading students to forget information once they have been tested on it, reducing the amount of information retained by them.

Math Academy has found a way to improve on traditional spaced repetition methods based on flashcards, using fractional implicit repetition (FIRe). In a hierarchical body of knowledge like mathematics, by reviewing a given topic the student is implicitly also reviewing those topics on which the original topic depends; this must be taken into account when constructing a review schedule for a student. The techniques by which this is done have been refined over many years by Math Academy [and therefore form part of its proprietary advantage].

Spaced repetition also promotes generalization: that by reviewing material on a suitable schedule, the student can discover new connections between the topic being reviewed and other topics, and therefore can better transfer their knowledge to related but different topics.

What about the objection that spaced repetition requires reviewing a very large number of items during each review session? Because mathematics is a hierarchical body of knowledge (see above), more advanced skills encompass many more basic skills. Thus the number of reviews can be reduced (repetition compression) by reviewing the advanced skill, which also serves as a review of the basic skills. The example given is that of multiplying a 2-digit number by a single-digit number: reviewing this also reviews multiplying a single-digit number by another single-digit number, as well as adding a single-digit number to a 2-digit number.

However, this cannot always be done. If a student’s learning speed is below average, the Math Academy system will not do implicit reviews but will drop back to doing explicit reviews of more basic material. In this case a review counts as a fraction of a spaced repetition. Conversely, if a student’s learning speed is above average, each review will count as more than one spaced repetition. The Math Academy system computes student-topic learning speeds for each individua student in order to do this effectively.

Spaced repetition can be contrasted with the spiral approach, where an instructor periodically revisits material previously covered. Spiraling amounts to spaced repetition with a fixed schedule, and is less effective than actual spaced repetition. However, it is easier for instructors to implement; true spaced repetition with individualized schedules requires supporting technology like that found in the Math Academy system.

Chapter 19. Interleaving (Mixed Practice)

Interleaving or mixed practice—mixing up exercises on different topics in a single practice session—is contrasted with blocking or blocked practice—doing a bunch of similar exercises on the same topic. [“Blocking” and “blocked” are here used in the sense of doing a homogeneous “block” of exercises. This was what I was doing by systematically doing linear algebra exercises one by one in the order presented in the textbook I was using.]

Interleaving is more effective for a variety of reasons. First, it is more efficient: blocking leads to diminishing returns as the number of similar exercises increases. Second, it helps students better match problem solving techniques to problems, especially when the technique needed is not obvious from the statement of the problem. With blocking, students end up reusing the same technique from problem to problem and can get lost when a different type of problem is posed.

However, blocking can appear to be more effective, and to some degree can be more effective when first learning a skill. This makes it attractive to both students and teachers (who are motivated by the appearance of rapid learning). However, interleaving is more effective for long-term retention of material, as has been experimentally demonstrated. It involves desirable difficulties, i.e., difficulties that promote learning.

Interleaving can occur at two levels, and the Math Academy system features both:

Macro-interleaving is done at the level of topics: the student will be presented with a variety of different topics, as opposed to working on the same topic for an extended period of time. Micro-interleaving is done at the level of review, mixing up problems from different topics.

However, there is a trade-off here, in that it would take an excessively long time to fully interleave all review exercises for a topic (i.e., with exercises for several other topics) before featuring them on quizzes on that topic. So the Math Academy system compromises by blocking exercises during lessons and counting them toward spaced repetition credit.

Chapter 20. The Testing Effect (Retrieval Practice)

The best way to review material is not to re-read it (or re-watch it, or re-listen to it), it’s to be quizzed on it. (This is referred to as the testing effect or retrieval practice effect.) The act of retrieving material helps fix it in long-term memory. This is especially effective when quizzes are combined with spaced repetition.

Frequent quizzes are not used in traditional educational settings as much as they might be; a more typical pattern is to have one mid-term exam and one final exam. However, the Math Academy system does quick quizzes very frequently (continuous assessment), and also does evaluation as part of spaced repetition review.

The Math Academy systems tries to mitigate test-induced anxiety by doing quick quizzes on material the student is already ready to be tested on. Timed tests should not be introduced too early and tests in general should be matched to the student’s current level of proficiency, thus preventing desirable difficulties from turning into undesirable difficulties. Lower proficiency can lead to math anxiety as students find themselves unprepared for tests and do poorly on them.

Doing frequent quizzes helps build self-confidence and prepares the student for more extensive timed testing. Topics show up on low-stakes non-timed quizzes first, with opportunities to retake quizzes and go back to review material. Timed tests on the same topics are done only after the student has demonstrated proficiency on those topics.

Chapter 21. Targeted Remediation

The Math Academy system helps students struggling with certain topics (or certain components within a given topic) not by trying to lower the difficulty of the student’s tasks, e.g., by providing extra feedback and hints (“adaptive feedback”), but rather by giving them additional time and practice on exactly those topics (or components within topics) that are causing them the most difficulty (targeted remediation).

“Targeted remediation at Math Academy’s level of granularity (individual students on individual topics) and integrity (maintaining the bar for success) has not been studied in academic literature. [emphasis in the original]”

Corrective remedial support is tailored to the specific circumstances: providing more questions if a student is struggling with a task, switching them to unrelated topics if they fail a topic before coming back to the original topic, and providing remedial reviews if they appear to be stuck at a particular place. Note that remedial reviews may be for a topic that’s some distance back in the knowledge graph hierarchy, but which is a prerequisite for the current topic. For example, in the topic of cubing a number, a student may have problems with calculating (-4)3 = (-4) x (-4) x (-4) because they have unremediated problems with multiplying negative numbers.

Preventative remediation occurs when the student’s learning speed for a topic is predicted to be low based on their learning speed for other related topics. In this case the Math Academy system can prevent problems by scheduling additional reviews.

Foundational remediation occurs when students start a Math Academy course with holes in their knowledge of the foundational topics for the course. For example, they may not have mastered some topics in arithmetic needed for algebra. In this case the Math Academy system can let them proceed with topics that don’t depend on the unmastered foundational topics, and go back and remediate those unmastered topics when needed.

Finally, the Math Academy developers track student learning to see if there are any topics that students are having particular struggles with. They can then do content remediation, for example by providing additional worked examples or review points within the topic, or by splitting it up into multiple separate topics.

Chapter 22. Gamification

Gamification can improve both student learning and enjoyment if it is properly aligned with education objectives and student motivations and is designed to prevent students gaming the system.

The Math Academy system uses eXperience Points (XP) tied to completion of tasks, each XP representing a minute of sustained effort by an average student. (Optional) competitive weekly leaderboards keep track of students’ activity versus other students of comparable ability. Students accumulating lots of XP get promoted into higher leagues, students not doing so much get relegated to lower divisions [as in the English Premier League].

Students can earn extra XP with perfect performances on tasks, earn little or no XP for nearly passable or poor performance, and get penalized with negative XP if they are perceived to be blowing off tasks.

Student progress is measured separately from XP, based on the percentage of a course that they’ve completed. Progress slows down near the end of a course due to the need to review material from earlier in the course—but the system will never let lesson time (vs. review time) fall below 25% of the total time. There is no leaderboard or other gamification for course progress.

Chapter 23. Leveraging Cognitive Learning Strategies Requires Technology

Teachers are reluctant to implement educational strategies like those embodied in the Math Academy system, but not through any fault of their own. They are working under structural constraints that make it difficult to adopt such strategies, for example, the system of grade-to-grade progression. Even if they could adopt some or all of the strategies then it would be physically impossible to implement them beyond a 1-on-1 tutoring scenario, because students have differing knowledge profiles and learn at different speeds. Thus implementing these strategies via technology is the only possible solution.

[This raises a question: How did this problem play out in the original Math Academy program in Pasadena, the one from which the Math Academy learning system arose? I think this chapter would benefit from a more in-depth treatment of that story, including an account of how it motivated creation of the Math Academy system.]

This concludes my discussion of Part III of The Math Academy Way. In part 6 of this series I’ll discuss Part IV of the book, “Coaching,” a relatively short and incomplete section that discusses how parents can best help children using the Math Academy system, as well as how students can help themselves.