How MathMap Works
A diagnostic engine built on structural mathematics — not surface-level practice.
Why Not Just More Practice?
- ✗Adds practice without identifying structural gaps
- ✗Treats symptoms, not root causes
- ✗Increases quantity without building understanding
- ✗Doesn't show the student where they actually are
- ✗Cannot predict which topics will break next
- ✓Diagnosable: every node has a clear L0-L4 level
- ✓Structural: shows how ideas connect and build
- ✓Targeted: repair exactly what is broken
- ✓Predictive: reveals what gaps will block future topics
- ✓Bilingual: parents understand the diagnosis too
A student who “doesn't understand polynomials” might have mastered Expressions (L3) and Equations (L2) but completely missing the root-factor-graph connection. Without this map, both the student and tutor are guessing. With MathMap, the gap is visible — and so is the precise repair path.
Structure Diagnosis
Every node in the map is assessed on 3 dimensions:
How deeply does the student understand this concept? From Not Diagnosed (L0) to Transfer (L4).
Is knowledge stable, fragile, or exposed? Stability predicts whether the student will hold up under pressure.
Specific observed errors and misconceptions — the raw material for targeted repair.
Ability Levels: L0 → L4
Five precisely defined levels of mathematical understanding — each with a Chinese parent explanation.
Student has not been assessed on this node.
我们还没有评估孩子在这个环节的水平。
Can identify the concept or feature when presented.
孩子见过这个概念，能认出来，但还不能独立操作。
Can perform standard operations or solve familiar problems.
孩子会做标准题，但换个题型或换个问法可能就不稳了。
Can explain why, connect representations, and justify reasoning.
孩子不仅会做，还能解释为什么，能把代数、图像、表格等不同方式联系起来。
Can generalize, model, and apply the idea in unfamiliar contexts.
孩子能举一反三，遇到没见过的情境也能灵活运用，这是真正扎实的标志。
10 Repair Methods
Proven teaching interventions for building structural understanding
Student labels every component of a mathematical object with its proper name
Given 3x⁴ - 2x² + 7: name the degree, leading coefficient, constant term, number of terms
Present the same mathematical idea in algebraic, graphical, tabular, and verbal forms
Show y = x² - 4 as: equation, graph, table of values, and verbal description
Without looking at the original graph, rebuild it from algebraic information alone
Given f(x) = (x+1)(x-2)²: sketch the graph justifying every feature
Given the answer or result, construct a problem that would produce it
Write a polynomial that has roots at x = -1 and x = 3 (touching)
Analyze an incorrect solution and categorize the type of error
A student wrote x² - 9 = (x-3)(x-3). What kind of error is this?
Place two similar but different objects next to each other and articulate differences
Compare: x² - 4 = 0 vs x² - 4. One is an equation, one is an expression. What's different?
Student must explain their solution process out loud, in complete sentences
Solve and then explain: why does setting each factor equal to zero give us the roots?
5-10 carefully designed questions, each testing exactly one structural element
10-question polynomial diagnostic: one question per node in the structure
Apply the same structural idea in a completely new context
You understand polynomial roots. Now: how does the same idea work for rational functions?
Fit multiple models to the same data and argue which is most appropriate
This data could be linear or exponential. Fit both. Which is better? Why?
School Pathways
Three U.S. high school math sequences — MathMap works with all of them.
The most common U.S. high school math sequence
For students who take Algebra 1 in 8th grade
Combines algebra, geometry, and statistics each year
Structural Math Education for the Next Generation
LIGHT HOPE combines rigorous structural math diagnosis with bilingual parent communication to help Chinese-American students build the mathematical depth needed for college and beyond. MathMap V5.3 is our core diagnostic infrastructure.