Math Academy

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How It Works

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Pedagogy

FAQ

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Math Academy's AI is an expert system that emulates the decisions of an expert tutor regarding what tasks a student should<br>work on at any given point in time. This is accomplished by combining the following pieces of technology.

The knowledge graph stores all the information that an expert tutor<br>would know about the structure of mathematics. What topics are there? What are the easiest and hardest variations of problems<br>within each topic? What background knowledge must a student have in order to learn each topic? If a student struggles with a<br>particular type of problem, what specific pieces of background knowledge are most relevant to their struggle? The answers to<br>all these questions (and many more) are stored within our knowledge graph.

The student model takes a student's answers, overlays them on the<br>knowledge graph, and figures out what topics the student knows (and how well they know it). This is called the student's<br>knowledge profile. To compute a student's knowledge profile, our student model uses spaced repetition,<br>a systematic method for determining when previously-learned material needs to be<br>reviewed.

The diagnostic algorithm leverages the knowledge graph to minimize<br>the number of questions needed to estimate a student's knowledge profile. It identifies what parts of the course the student<br>has already learned, and what gaps they have in their foundational knowledge.

The task-selection algorithm takes a student's knowledge profile and uses it to determine the optimal learning tasks that will<br>move the needle most on the student's learning. What should the student learn next? What do they need to<br>review? When answering these questions, our task selection algorithm is<br>always trying to maximize the amount of learning that<br>occurs per unit of time that the student spends on the system.

The Knowledge Graph

In the mathematical field of graph theory, the word “graph” refers to a diagram consisting of objects and arrows between them. In our<br>knowledge graph, the objects are mathematical topics and the arrows between them represent relationships, such as one topic being a<br>prerequisite for another topic. (There are lots of different kinds of relationships, some of which even refer to “sub-atomic”<br>components within topics - but for now, we'll just focus on prerequisite relationships between topics.)

For instance, the tiny knowledge graph below shows that the topic Adding Fractions With Unlike Denominators (top) has two<br>prerequisites: 1) Adding Fractions With Unlike Denominators Using Models, and 2) Adding Fractions and Whole Numbers (middle), and each<br>of those prerequisites itself has a prerequisite Adding Fractions and Whole Numbers Using Models (bottom).

Knowledge graphs can encode a lot of complicated information that would otherwise be hard to describe and reason about. Zooming out,<br>below is the knowledge graph for an entire course consisting of about 300 topics.

Fully zoomed out, Math Academy's entire curriculum consists of multiple thousands of interlinked topics spanning 4th Grade through<br>university-level math. All these topics are connected up together in the knowledge graph. In this view, a course is simply a section<br>of our knowledge graph. (In the visualization below, different colors represent different courses.)

The Student Model

Our student model uses a student's answer history to compute their knowledge profile. Loosely speaking, a student's knowledge<br>profile represents how developed their mathematical brain is. Every time they learn a new math topic, it's as if they grow a new brain<br>cell and connect it to existing brain cells. Initially, this new brain cell is weak and requires frequent nurturing, but over time it<br>becomes strong and requires less frequent care.

For instance, a knowledge profile for a second-semester calculus student is visualized below. Learned topics are shaded (with darker<br>shading indicating that more successful practice has been completed), and arrows between topics represent prerequisite relationships.<br>(Note that the knowledge profile below only shows a “subsystem” within the student's full mathematical brain - there are several<br>hundred topics in the calculus course below, but there are thousands of topics in our entire mathematical curriculum spanning<br>elementary school through university-level math.)

More precisely, a student's knowledge profile measures how many “spaced repetitions” they have accumulated on each topic. Spaced<br>repetition, also known as distributed practice, is a systematic method for reviewing previously-learned material that leverages the<br>spacing effect: when review is spaced out over multiple sessions (as opposed to being crammed into a single session), memory is not<br>only restored, but also further consolidated into long-term storage, which slows its decay. As a result, the more reviews...