The Math Academy curriculum and pedagogy leverage cutting-edge cognitive learning theory. Why? Because it’s been studied extensively, backed up and proven to work. We aren’t providing edu-tainment. This isn’t enrichment. We teach math as if we were training a professional athlete or musician, or anyone looking to acquire a skill to the highest degree possible. This is real work for a student who is serious about learning math. We expect every student using our system to actually master the material, and do it efficiently and effectively.

Math Academy harnesses technology as a method of instruction where the focus is on the role of feedback in learning and establishes a level of performance that all students must master before moving on. It has been shown to be one of the most powerful educational techniques ever discovered, however, to realize its potential required personal one-on-one tutoring, which is excessively resource-intensive - known as “Bloom’s 2-Sigma Problem”.

One of our main paradigms is mastery learning, also proposed by Bloom, in which students must demonstrate proficiency on prerequisite topics before moving on to more advanced topics. True mastery learning at a fully granular level requires fully personalized instruction, which is only attainable through one-on-one tutoring. However, through the use of a custom web-based learning platform, this and the following techniques can be combined into an exceptionally potent personalized learning system.

Further Reading:

An essential yet often-overlooked part of minimizing cognitive load is developing automaticity on basic skills – that is, the ability to execute low-level skills without having to devote conscious effort towards them. Automaticity is necessary because it frees up working memory to execute multiple lower-level skills in parallel and perform higher-level reasoning about the lower-level skills.

When you develop automaticity on a skill or piece of information you can use it without it occupying a slot in your working memory. Instead, the skill is stored in your long-term memory, where indefinitely many things can be held for indefinitely long without requiring cognitive effort.

To help students develop automaticity (and, consequently, expertise in mathematics), Math Academy requires students to practice each skill until they have reached a sufficient level of mastery. Students are not pushed forward along learning paths until they have mastered the prerequisite skills. Additionally, to help consolidate skills into long-term memory after mastery, skills are continually reviewed into the future through a systematic method called spaced repetition.

Further Reading:

- The relationship between working memory and mathematical problem solving in children at risk and not at risk for serious math difficulties.
- Investigating the predictive roles of working memory and IQ in academic attainment.
- The magical number seven, plus or minus two: Some limits on our capacity for processing information.
- The magical number 4 in short-term memory: A reconsideration of mental storage capacity.
- Systematic mathematical errors and cognitive load
- The expert mind.

Math Academy’s approach to math education is entirely centered around active learning – specifically, deliberate practice, which consists of individualized training activities specially chosen to improve specific aspects of a student’s performance through repetition and successive refinement. Students are solving problems (and receiving feedback) within minutes of starting a new lesson, after a minimum necessary dose of initial explanation. They spend the vast majority of their time engaged in active problem solving on new topics and topics most in need of review.

Further Reading:

Spaced repetition, also known as spaced retrieval or distributed practice, means that reviews should be spaced out or distributed over multiple sessions (as opposed to being crammed or massed into a single session) so that memory is not only restored, but also further consolidated into long-term storage, which slows its decay. The most effective means of constructing long-term memories is to re-engage with concepts at progressively greater intervals (the spacing effect). In contrast, doing a large number of a specific type of problem within a short-period, known as massed practice, has little to no long-term benefit but numerous negative side effects, most notably—boredom, burnout, and the inefficient use of time. Math Academy’s highly sophisticated algorithms track every single question a student completes and continually updates the student’s personal learning curve for optimal efficiency.

Further Reading:

- Want to Remember Everything You’ll Ever Learn? Surrender to This Algorithm
- The Effects of Spacing and Mixing Practice Problems
- Memory: A contribution to experimental psychology.
- Maintenance of foreign language vocabulary and the spacing effect.
- Individual differences in associative learning and forgetting.
- Forgetting Curve

Presenting problems out of context is the most effective means of improving a learner's ability to match a problem with the appropriate concept or procedure. This strategy is also known as interleaving. Interleaving promotes vastly superior retention and generalization of an acquired skill, which makes it an effective review strategy. Together with distributed practice, the two perform an especially powerful mix, however, few textbooks have ever been written with any awareness of these techniques, therefore they are all but absent from the classroom - much to the chagrin of researchers in cognitive learning theory. Math Academy spreads out review problems, or interleaves problems, over multiple review assignments that each cover a broad mix of previously-learned topics. In addition to being more efficient, this also helps students match problems with the appropriate solution techniques.

Further Reading:

- The Interleaving Effect: Mixing It Up Boosts Learning
- Interleaving helps students distinguish among similar concepts.
- Research commentary: The effects of spacing and mixing practice problems.
- Learning concepts and categories: Is spacing the “enemy of induction”?
- People’s mind wander more during massed than spaced inductive learning

When students learn progressively more advanced material, they reinforce and deepen their foundational knowledge. And as the number of neural connections increase, the more ingrained a concept or skill becomes. Acquiring new skills and concepts that exercise prerequisite or component concepts, is an especially efficient method of increasing the number of connections to existing knowledge. The more connections (neural, cognitive, social, and experiential) there are to a piece of knowledge, the more ingrained, organized, and deeply understood it is, and the easier it is to recall via spreading activation through connections. The most efficient way to increase the number of connections to existing knowledge is to continue layering on top of it But that's not all. It has the added, and even greater, benefit of increasing the organization and comprehension of existing knowledge. Math Academy’s knowledge graph makes connections between all prerequisites and encompassing topics so that earlier topics mastered are applied and reinforced in higher level topics.

To reap the benefits of layering, Math Academy employs two key features:

- Moving students forward to new topics immediately after they demonstrate mastery of prerequisites.
- A highly-connected curriculum where new topics exercise and build on earlier topics.

Further Reading:

Learning highly similar or related items simultaneously, or in close succession, causes encoding interference and impedes recall. The typical math curriculum is divided into units of related material and taught in subsequent lessons, which has the adverse effect of causing both proactive and retroactive interference. New concepts should be taught alongside or following dissimilar material so as to avoid this problem. When this method is utilized, multiple topics can be successfully taught simultaneously, thereby enabling students to make smooth, fast progress through their courses. Our students have the ability to choose any unlocked topic that becomes available to them as they advance through the system. As each task is completed, the student's knowledge graph is updated and the system chooses new topics to guide them most efficiently through the course. Just like a personal trainer packs a wide variety of exercises into each workout to maintain motivation, Math Academy packs a wide variety of topics into each learning session to keep things interesting.

Further Reading:

Cognitive load refers to the amount of working memory that is required to complete a task. Different students have different working memory capacities, and if the cognitive load of a learning task exceeds a student’s working memory capacity, then the student will not be able to complete the task due to cognitive overload. By minimizing the cognitive load and avoiding cognitive overload, we make learning accessible to all students regardless of their working memory capacity.

Learning is like climbing a staircase. Each step is a learning task – the higher the step, the more advanced the topic is. Math Academy’s solution is to split individual stairs into even smaller stairs so that all students can climb them. The smaller we make the individual stairs, the more students can climb all the way to the top. For instance, a normal calculus textbook might consist of 100 steps (10 chapters x 10 sections in each chapter). But in our calculus course, we have almost 1000 steps (~300 topics x 3 knowledge points or stages of increasing difficulty per topic). In other words, our content is about 10x more finely scaffolded than what you’d find elsewhere.

Even within individual knowledge points, we take additional measures to minimize cognitive load. To take advantage of the worked-example effect, each knowledge point starts with a worked example. In the explanation, we leverage subgoal labeling by grouping steps into meaningful units that help students grasp the structure of the problem, and we also leverage dual-coding theory by including visualizations and diagrams when possible to help students develop mental images. After the worked example, students solve problems that are similar to the worked example, only progressing onto the next knowledge point once they have demonstrated mastery of the previous one.