After 5,000 Homework Assignments, I Know Why Math Fails Students (And How We Can All Do Better)

Picture this: You're sitting in differential equations class, staring at a second-order linear equation on your exam. You know dozens of formulas by heart. You've memorized the steps for Laplace transforms, variation of parameters, characteristic equations. But as you look at this problem, you freeze.

Which method should you use? The problem doesn't say "use the Laplace transform"—you're supposed to figure that out yourself. You know how to execute each technique perfectly, but you have no idea when to apply which one.

Over the past four years as a teaching assistant, I've graded over 5,000 homework assignments and 2,000 exams across courses ranging from differential equations to machine learning to mathematical finance. And here's what I've discovered: the math struggle isn't really about math being inherently difficult. It's about how we're approaching it from both sides of the classroom.

The students I see aren't lazy or incapable. The educators aren't incompetent or uncaring. But we've created a system that turns one of humanity's most beautiful intellectual achievements into a frustrating game of memorization and pattern matching. And both students and teachers have the power to change this.

The Real Problem: When Math Becomes a Foreign Language

Here's what I see repeatedly in those thousands of assignments: students who can execute complex procedures flawlessly when told which one to use, but completely freeze when they have to choose the method themselves. They'll solve a differential equation perfectly using Laplace transforms if the problem says "use Laplace transforms," but give them the same equation without that hint and they're lost. They've memorized the recipe, but they can't recognize when to cook which dish.

This isn't stupidity—it's the predictable result of treating math like a collection of formulas to memorize rather than a toolkit of problem-solving strategies to recognize and apply.

The fundamental issue is that we've focused on formula memorization instead of pattern recognition. Students can recite dozens of mathematical procedures, but they can't look at a problem and identify which tool from their toolkit actually fits. It's like memorizing every single chess move but never learning to recognize when you're in an opening, middlegame, or endgame situation.

When I grade exams, I see two types of errors consistently. The first type is calculation mistakes—dropped negative signs, arithmetic errors, algebraic slip-ups. These bother me less because they're usually just carelessness under time pressure. The second type is strategic confusion—students who can execute individual techniques perfectly but choose completely the wrong approach for the problem type. These errors tell me we're teaching procedures without teaching recognition.

What Students Can Do: Three Strategies That Actually Work

Based on what I've observed in thousands of student submissions, here are the three most effective strategies for students who want to actually understand math instead of just surviving it:

1. Always Ask "Why Am I Doing This?"

Before you dive into any new mathematical concept, spend time understanding the bigger picture. What problem is this chapter trying to solve? Why did we move from the previous topic to this one? What real-world situation motivated mathematicians to develop these tools?

Most textbooks do this backwards—they throw the algorithm at you first, then maybe mention applications later. Fight this tendency. When you encounter a new concept, immediately ask: "What is this for?"

If your instructor doesn't make this clear (and unfortunately, many don't), don't give up—we'll address how to get these answers in the next section.

2. Use AI as Your Patient Tutor

This might be controversial, but AI has revolutionized how students can learn math. Unlike office hours, where you might feel rushed or embarrassed to ask "basic" questions, ChatGPT will patiently explain concepts as many times as you need.

Here's how to use it effectively: When you're stuck on a concept, don't just ask for the answer. Ask for explanations. "Why did we choose this method over that one?" "Can you walk me through this step more slowly?" "What would happen if we tried a different approach?"

GPT recently released a study mode specifically for this kind of interactive learning. The key is to be conversational—treat it like you're talking to a knowledgeable friend who has infinite patience and no judgment.

3. Practice Strategically, Not Blindly

Here's something I've learned from grading: the students who improve fastest aren't necessarily the ones who do the most practice problems. They're the ones who practice most strategically.

When you get a set of practice problems, don't just work through them mechanically. First, scan through them and identify which ones test concepts you already understand versus which ones challenge you to recognize new problem types. If you can immediately see which method to apply to a problem, you might not need to actually work through all the arithmetic—just verify that you know the approach.

Spend your time on problems where the solution strategy isn't immediately obvious to you. These are the problems that will actually train your pattern recognition skills and expand your ability to choose the right tool for each situation.

That said, if you struggle with time management and calculation speed, then grinding through more problems can help build fluency. Know your own weaknesses and tailor your practice accordingly.

What Educators Can Do: Three Changes That Transform Learning

Now, having seen the educator side through my TA experience, here's what I believe teachers need to change to help students actually learn math:The Engagement Crisis

This brings me to what I think might be our biggest challenge: we're inadvertently turning students away from mathematics by making it seem distant and irrelevant.

I've watched too many bright students reach differential equations and think, "This is it? This is what all that preparation led to?" They get the impression that calculus represents everything mathematics has to offer, and when it doesn't connect to their interests or goals, they dismiss the entire field.

But imagine if these same students had been exposed to cryptography through discrete math, or seen how linear algebra powers the recommendation algorithms they use every day, or learned how game theory explains everything from auction design to international relations. We might see a very different relationship with mathematical literacy in this country.

1. Start with Intuition, Always

This is the most important change we can make. Every new concept should begin with motivation and intuition before diving into technical details. Don't start with the algorithm—start with the problem the algorithm solves.

When introducing Laplace transforms, begin by explaining that we're looking for a way to turn difficult differential equations into easier algebraic equations. When teaching matrix multiplication, start by explaining that we're trying to capture how linear transformations compose. When introducing derivatives, begin with the intuitive idea of instantaneous rate of change.

Too many textbooks (and lectures) do this backwards: they present the formal definition first, then maybe get around to motivation later. This is pedagogically backwards. Students need the "why" before they can meaningfully engage with the "how."

2. Create More Opportunities for Questions and Slower Explanations

Students need more time and space to ask questions when concepts aren't clear. The traditional lecture-homework-exam cycle doesn't provide enough opportunities for this kind of interactive learning.

Recitation sections are incredibly valuable, but we need to use them better. Instead of just working through more practice problems (which assumes students already understand the concepts), use these sessions to re-explain core ideas at a slower pace and simpler level.

Think of it as providing a "baseline understanding" that students can build on. When they go back to the lecture notes with this foundation, suddenly everything clicks together. This could be done through student-led sessions, pre-recorded videos, or more interactive office hours.

3. Stop Testing Arithmetic When You're Teaching Concepts

Here's my most specific piece of advice, drawn from hundreds of hours of grading: stop giving problems where 80% of the work is algebraic manipulation and only 20% is applying the actual mathematical concept you're trying to teach.

When I grade exams, I'm very lenient on calculation errors. If a student demonstrates that they understand the concept and choose the right approach, I'm not going to heavily penalize them for dropping a negative sign or making an arithmetic mistake in an intermediate step. These kinds of errors tell me nothing about their mathematical understanding.

We live in an age where calculators and computer algebra systems can handle complex arithmetic faster and more accurately than humans. Our job as educators should be to teach students what mathematical tools exist, how to recognize which problems require which approaches, and how to set up solutions correctly.

The goal of mathematical education should be developing pattern recognition and strategic thinking, not testing who can memorize the most formulas or execute procedures fastest.

The Bigger Picture: What Math Education Should Be

Here's what I've learned from grading thousands of student assignments: math struggles aren't usually about students lacking ability or effort. They're about a mismatch between how we teach math and how humans actually learn complex ideas.

Mathematics is fundamentally about problem-solving patterns and logical reasoning. It's about learning to translate real-world situations into mathematical language, then using established tools and techniques to find solutions. It's not about memorizing formulas or executing algorithmic procedures without understanding.

When students struggle with math, it's often because we've asked them to memorize the tools without explaining what they're for, or to execute procedures without understanding their logic. We've turned one of humanity's greatest intellectual achievements into a series of arbitrary rules to be followed.

But here's the encouraging news: both students and educators have the power to change this. Students can take responsibility for seeking deeper understanding and using available tools more strategically. Educators can restructure how they present concepts to prioritize intuition and application over rote procedure.

The math itself isn't the enemy. The way we've been approaching it is.

After grading all those assignments and seeing the same patterns of struggle repeated semester after semester, I'm convinced that math education can be so much better than it is. We just need both sides of the classroom to commit to understanding over memorization, intuition over procedure, and patience over speed.

Because in the end, math isn't hard. We've just been making it harder than it needs to be.