This article was converted by SimpRead; the original source is zhuanlan.zhihu.com
Math Academy’s Artificial Intelligence (AI) is an expert system that simulates the decision-making process of an outstanding tutor, recommending precisely which learning tasks a student should tackle at any given moment. This capability is made possible through the integration of several key technologies.
The Knowledge Graph stores all domain knowledge about the mathematical subject structure that an excellent tutor must possess. Specifically: Which knowledge topics exist? What are the simplest and most challenging problem variants within each topic? What prerequisite background knowledge must a student master before tackling each topic? And—if a student struggles with a particular problem type—which specific background concepts are most closely related to their current difficulty? Answers to all these questions (and many more) reside in our Knowledge Graph.
The Student Model receives students’ responses and compares them against the Knowledge Graph to determine which knowledge topics they have mastered—and to what degree. This forms the student’s “knowledge profile.” To construct this profile, our Student Model employs Spaced Repetition, a systematic method for determining when previously learned material needs review.
The Diagnostic Algorithm leverages the Knowledge Graph to assess the student’s knowledge profile using the fewest possible number of questions. It identifies both content the student already knows and gaps in foundational knowledge.
The Task Selection Algorithm selects optimal learning tasks—based on the student’s knowledge profile—to maximize learning progress. What new content should the student learn next? Which previously learned topics need review? In answering these questions, our Task Selection Algorithm always strives to maximize the amount of learning—the learning gain—a student achieves per unit of study time within the system.
Knowledge Graph
In graph theory—a branch of mathematics—the term “graph” refers to a diagram composed of objects (nodes) and arrows (edges) connecting those objects. In our Knowledge Graph, the objects represent individual math knowledge topics, and the arrows represent relationships among them—for example, one topic being a prerequisite for another. (In fact, many relationship types exist—including some involving fine-grained, “subatomic-level” components within topics—but for now, we focus exclusively on prerequisite relationships between topics.)
For instance, the small Knowledge Graph below shows that the top-level topic Adding Fractions with Unlike Denominators has two prerequisites:
- Adding Fractions with Unlike Denominators Using Models, and
- Adding Fractions and Whole Numbers (middle layer).
Both of these, in turn, share a common prerequisite: Adding Fractions and Whole Numbers Using Models (bottom layer).
The Knowledge Graph encodes vast amounts of complex information that would otherwise be difficult to describe or reason about logically. At a broader scale, the figure below shows the full Knowledge Graph for an entire course comprising approximately 300 knowledge topics.
At the highest level, Math Academy’s overall curriculum consists of thousands of interrelated knowledge topics spanning mathematics from Grade 4 through university-level coursework—all tightly interconnected within the Knowledge Graph. From this perspective, a course is simply a specific region of our Knowledge Graph. (In the visualization below, different colors represent different courses.)
Student Model
Our Student Model constructs a student’s knowledge profile based on their response history. Informally, a student’s knowledge profile reflects the developmental maturity of their mathematical thinking. Each time a student learns a new mathematical concept, it is like growing a new “neuron” and connecting it to existing ones. Initially, this new neuron is relatively “fragile” and requires frequent “nourishment”; over time, however, it becomes “stronger,” requiring less frequent “care.”
For example, the visualization below depicts the knowledge profile of a student studying second-semester calculus. Topics already studied are shaded with varying intensities (darker shades indicate more completed effective practice), and arrows between topics denote prerequisite relationships. (Note: The profile shown here represents only a subsystem of the student’s complete “mathematical brain”—though this calculus course contains hundreds of topics, Math Academy’s full K–university curriculum encompasses thousands.)
More precisely, a student’s knowledge profile quantifies how many times they have undergone “spaced repetition” practice for each knowledge topic. Spaced repetition—also known as distributed practice—is a systematic review method for revisiting previously learned material. It harnesses the spacing effect: when review sessions are distributed across multiple time intervals (rather than crammed into one session), memory not only recovers but strengthens over time, thereby slowing down forgetting. Thus, the more reviews a student completes—and the more optimally spaced those reviews are—the longer their memory retention lasts, and the longer the interval until the next required review. Accordingly, “one spaced repetition” can be understood as one productive review, delivered at the optimal moment, with a minimal effective dose.
Spaced Repetition Algorithm
The Spaced Repetition Algorithm determines when a student should review previously learned knowledge topics.
- If a review is delayed too long, excessive forgetting occurs, causing the student to regress within their spaced repetition trajectory.
- Conversely, if a review occurs too early, memory reinforcement is suboptimal, and learning progress slows. (This is inefficient: time could instead be spent learning new topics—or reviewing topics whose due date has genuinely arrived.)
Most existing spaced repetition algorithms operate in isolation, e.g., on flashcards—but such approaches fail for hierarchical, structured domains like mathematics. For example, when a student practices two-digit addition, they implicitly also practice one-digit addition! In general, repeated practice on higher-order topics should “trickle down” through the Knowledge Graph to update the review schedules of foundational topics that were implicitly practiced during the higher-order task.
Trickling down repetition effects may sound straightforward—but its implementation involves numerous subtle complexities:
- Prerequisite topics are not always implicitly practiced. Sometimes, a student must first become familiar with a prerequisite to understand a new topic—even though that prerequisite itself receives insufficient explicit or implicit practice during the new topic’s study. Thus, repetition effects cannot simply trickle down to prerequisites; rather, they must trickle down to “included” topics—more fundamental constituent skills that are implicitly exercised.
- The effect of implicit repetition must be appropriately discounted: such practice often occurs too early to count as a full, effective review of the “included” topic—and therefore cannot directly determine the timing of the next scheduled review.
- Inclusion relationships are typically partial: constituent skills are often only partially practiced—not fully rehearsed.
Moreover, when a learning task contains multiple problems, the weight assigned to each toward spaced repetition must vary depending on the student’s performance. To comprehensively account for all these nuances, we developed an entirely new spaced repetition theory called Fractional Implicit Repetition (FIRe).
Diagnostic Algorithm
When students join Math Academy, they take an adaptive diagnostic assessment. Leveraging the Knowledge Graph, this test rapidly evaluates their knowledge profile—specifically identifying their “knowledge boundary”: the frontier separating what they already know from what they do not yet know. The knowledge boundary marks the set of topics a student is currently ready to learn.
Our diagnostic service is tailored to specific courses—but while assessing mastery of course content, it also evaluates the earlier foundational knowledge necessary to succeed in that course. New students often lack certain prerequisites. In traditional classrooms, this may cause serious difficulties—but our diagnostic system accurately detects even when a student’s knowledge boundary lies below their enrolled course level. After diagnosis, we help students fill all missing foundational gaps—while simultaneously allowing them to advance in course topics unrelated to those gaps.
Without sophisticated algorithms, inferring a student’s knowledge boundary would require administering a massive number of diagnostic questions. A typical course includes hundreds of topics students might already know—plus roughly twice as many foundational topics they might be missing—totaling 500–1,000 topics to assess. Yet, using a novel diagnostic question selection algorithm, we reduce the required number of questions by an order of magnitude.
First, the algorithm compresses the Knowledge Graph to identify the smallest core set of topics sufficient to fully “cover” the course and all its prerequisites at an appropriate granularity. Then, it iteratively selects topics whose assessment yields the most informative evidence for constructing the student’s knowledge profile. Each correct answer provides positive evidence that the student has mastered that topic—and its prerequisites and related topics. Conversely, each incorrect answer provides negative evidence that the student has not mastered that topic—and its “postconditions” (topics dependent upon it) and other related topics.
When contradictory evidence arises, the Diagnostic Algorithm carefully weighs positive and negative signals to form a highly nuanced, human-like diagnosis—and dynamically adjusts its conclusions as new observations arrive.
- For example, if a student submits a correct answer but takes significantly longer than typical for someone who has truly mastered that topic, the weight of that evidence is reduced (since the student likely hasn’t yet internalized the skill sufficiently to build upon it).
- Similarly, if cumulative evidence indicates the student falls just short of full mastery on certain topics, the system classifies them as “conditionally mastered.” Initially, it assumes mastery and assigns corresponding tasks—but if the student subsequently struggles, the system immediately initiates targeted “backward tracing” along the relevant learning path to reinforce understanding.
How the Task Selection Algorithm Chooses New Learning Topics
When deciding which topics a student should learn or review next, we aim to maximize learning gain per unit of study time. To achieve this, we integrate multiple cognitive learning strategies: mastery learning, incremental learning, spaced repetition, interleaved practice, and minimization of associative interference.
In mastery learning, students must first demonstrate proficiency in prerequisite knowledge before advancing to higher-level topics.
We elevate mastery learning to a higher level via incremental learning: once students prove mastery of prerequisites, they are immediately guided to new topics. As students continually layer new knowledge onto existing foundations—and actively apply prerequisite or constituent skills in doing so—their prior knowledge becomes more robust, well-organized, and deeply understood. This strengthens the structural integrity of their knowledge network and makes absorbing new material easier. Students do review past material regularly—but they are never unnecessarily “slowed down” by over-practicing already-mastered topics. Upon completing a lesson, the next one unlocks instantly.
Our Task Selection Algorithm also minimizes associative interference: a learning barrier that arises when highly similar concepts are learned simultaneously—or in rapid succession. Associative interference causes conceptual confusion, hinders memory retrieval, and severely limits the number of topics a student can effectively master concurrently—thus slowing overall progress. To mitigate it, we deliberately schedule dissimilar concepts for concurrent study—and space out highly related concepts over time. This reduces confusion, enhances retrieval strength, and enables students to efficiently learn multiple topics daily.
Finally, if diagnostics reveal foundational weaknesses, our algorithm optimizes the timing for initiating remediation. Generally, students feel greater motivation to study topics within their enrolled course than to revisit foundational gaps—and when students are intrinsically motivated, their learning efficiency and persistence increase. Therefore, we let students begin by mastering course topics independent of their weak foundations. This helps them build initial momentum, make tangible progress toward primary goals, and establish consistent study habits. Once they reach a point where advancing further in the course requires shoring up those foundations, the momentum they’ve built sustains high motivation throughout the remediation process.
How the Task Selection Algorithm Selects Topics for Review
As noted, review activities follow spaced repetition schedules to optimize long-term memory retention. However, when selecting learning tasks, we perform deeper optimization to minimize the total number of reviews needed. Often, we can select tasks whose implicit repetition effects are sufficient to “eliminate” overdue reviews for other topics—and even postpone upcoming ones.
For example, suppose a student has overdue reviews for One-Digit Multiplication, One- and Two-Digit Addition, and Two-Digit × One-Digit Multiplication. Performing just one review on Two-Digit × One-Digit Multiplication suffices to cover all three—because its practice effect “trickles down” to the two implicitly exercised topics.
In summary, whenever a student has overdue reviews, we compress and consolidate them into a streamlined set of learning tasks—ensuring those tasks implicitly cover (via implicit practice) all overdue items. Where possible, we use new lesson assignments to “eliminate” overdue reviews—allowing students to fulfill necessary review requirements without actually slowing down their progression in new content. We call this process repetition compression, because it packs as many spaced repetitions as possible into as few tasks as possible—like triggering a full domino cascade with a single, minimal push.Spaced repetition naturally gives rise to “interleaved practice” (also known as “mixed practice”)—a highly effective review strategy whose core principle is distributing minimal effective doses of practice across a variety of different skills. Interleaved practice stands in sharp contrast to “blocked practice,” an inefficient approach involving prolonged, repetitive practice of a single skill. Blocked practice often fosters an illusion of mastery and fluency, because it trains students to mechanically and thoughtlessly apply the same problem-solving method to the same type of problem. By contrast, interleaved practice introduces a form of “desirable difficulty”: this productive challenge significantly enhances long-term retention and transferability of knowledge, making it a far more efficient review strategy.
How Algorithms Adapt Dynamically to Individual Learning Pace
All the algorithms discussed above adapt dynamically to each student’s learning pace.
Our spaced repetition algorithm adjusts individually for every student—and for every distinct knowledge topic they study. Each student possesses a “student-topic learning speed” metric, quantifying how easy or difficult that specific topic is for them. This speed metric reflects how effectively each review session extends the spacing interval for that topic.
- For instance, a speed of 2× indicates the topic is relatively easy for the student: they progress through spaced repetition for this topic at twice the standard rate, since they require less frequent practice on this particular subject.
- Conversely, a speed of 0.5× means the topic is more challenging: they progress at half the standard rate, requiring more frequent practice on this specific topic.
The task selection algorithm performs an even more granular level of adaptation: when the system judges a topic to be particularly difficult for a given student, it forces explicit review of that topic (rather than relying solely on implicit practice). We provide this support specifically for students who need additional practice to internalize and consolidate harder concepts learned implicitly—and who benefit from extra scaffolding to understand and generalize the idea that “what was previously learned is a special case (or component) of what is being learned now.”
Even within a single learning task, instructional pacing dynamically adjusts to the student’s actual ability—speeding up or slowing down accordingly. The higher the student’s accuracy rate within a task, the fewer questions they must answer to complete it. When a student answers incorrectly, we increase the required number of questions to provide more practice opportunities, more learning moments, and more chances to demonstrate mastery.
If a student accumulates too many incorrect answers, we pause the course entirely—because persisting in struggle without progress is an inefficient use of their time. During the pause, we guide the student to learn other unrelated topics and make tangible progress elsewhere before returning to re-attempt the paused course. Often, a brief break and a change of perspective are enough for the student to overcome the obstacle.
However, if the student again stalls upon re-attempting the paused course—and makes no further progress—we slow the pace still further and deliver targeted remedial review focused precisely on the foundational knowledge domains most closely linked to the core conceptual hurdle they’re facing. Our knowledge graph tracks the key prerequisite knowledge applied at each section of every course, enabling us to pinpoint exactly which topics require remediation—even if those topics reside much earlier in the mathematical hierarchy.
A corresponding remediation protocol also applies during quizzes: whenever a student answers a quiz question incorrectly, we slow the pace and immediately assign a remedial review session focused on the associated topic. Each quiz draws a diverse, randomized set of topics from the student’s prior learning history, prioritizing topics from their currently enrolled course while deprioritizing topics that have recently appeared in past quizzes or that are already subsumed by other assessable topics. The goal is comprehensive assessment—uncovering any unexpected gaps—and enabling immediate consolidation of those areas.
To further align with individual learning pace, quiz item selection also adapts question difficulty based on the student’s demonstrated ability—aiming for an average score near 80%. Students scoring above 80% will encounter slightly more challenging variants on their next quiz; those scoring below 80% will receive slightly easier variants. Empirical evidence confirms that quizzes maximize learning value when they are challenging—but never overwhelming.
Thoughts Memo Translation Team
Translated by gemini-2.5-pro-exp; reviewed by Jarrett Ye
Original article: Math Academy