Yingzi Ye

Misunderstanding in Math Class: When Structure Fails Students

Students are often blamed for “not understanding,” but many of their errors are actually failures of representation, pacing, and information design.

As a math tutor, I have worked with students from middle school through college—algebra, geometry, precalculus, calculus, statistics. Over and over, I see the same pattern:

Students do not fail math. The structure around the math fails them.

The way information is presented—symbols, diagrams, pacing, and language—often makes concepts harder than they need to be, especially for neurodivergent students or those who never received careful foundational explanations.

1. The triangle that explains everything

One of my students, a tenth grader, was working on problems involving the area of triangles. She had memorized the formula:

area = 1/2 × base × height

But when she worked on problems, she would:

• pick any two sides • multiply them • divide by 2

It did not matter whether one of the sides was perpendicular or not. She treated all sides as interchangeable.

When I asked her which side was the base and which was the height, she looked at me with genuine confusion:

“What do you mean by height?”

No one had ever slowed down enough to explain that height is a specific, perpendicular segment, not “whichever side you feel like using.”

The issue was not intelligence, effort, or attitude. It was missing representation.

2. Symbols without anchors

Many students are introduced to symbolic notation without strong visual or conceptual anchors. They see:

• f(x) • Σ notation • exponents • limits • derivatives

But they rarely see:

• what these notations look like as pictures • how they change when inputs change • how to interpret them in everyday language

When I slow down and draw, narrate, or connect the idea to a story, students often say:

“Why didn’t my teacher just say that?”

The problem, again, is not that they “can’t do math.” It is that the information arrived in the wrong form.

3. Cognitive load and pacing

Neurodivergent students, especially, are often overwhelmed by:

• rapid switching between topics • multiple symbolic jumps in a single step • cluttered board work • verbal explanations with no visual support

I started designing flashcards originally for these students—clean layouts, limited text, one idea per card, visual anchors. Over time, I realized:

the same design principles that support neurodivergent learners help almost everyone.

Good information design is universal design.

4. Lessons for information systems

Tutoring has quietly shaped how I think about healthcare information:

• both involve specialized symbols • both assume background knowledge that many people do not have • both move too quickly over foundational concepts • both can easily lose people in representation choices

In math, this leads to anxiety, low confidence, and avoidance. In healthcare, it can lead to missed understanding, loss of agency, and worse outcomes.

5. Why this matters for my research identity

My work as a tutor informs my interest in:

• information clarity • visual and structural design • how people build mental models from complex systems • how small representational changes can unlock understanding

Whether I am designing a flashcard for a student or thinking about how symptoms are documented in an EHR, I am asking similar questions:

• What is the essential idea? • How is it represented? • Who gets left out of this representation? • How could we redesign it to be kinder and clearer?

Misunderstanding is rarely a personal failure. Much more often, it is a structural one.