A Musician's Nightmare
"Imagine you're in music class, but..."
You never get to hear music. No singing, no instruments, no melodies.
Instead, for twelve years, you spend music class learning musical notation. You memorize what a quarter note looks like. You practice drawing treble clefs. You fill in worksheets converting eighth notes to sixteenth notes.
You never understand why someone would write a symphony. You never feel the joy of music. You just memorize rules about notation.
After twelve years of this, someone asks: "Do you like music?"
And you say: "I hate music. It's so boring."
That nightmare is from Paul Lockhart's essay "A Mathematician's Lament" (2002). He's a mathematician who taught at universities and then chose to teach at a K-12 school because he was horrified by how mathematics is taught.
His provocative claim: The way we teach math is like teaching music notation without music. We've stripped away everything beautiful and meaningful, leaving only the mechanics.
The Argument: What We're Doing Wrong
Lockhart argues that real mathematics is:
Mathematicians explore patterns, pose problems, and discover beautiful relationships. They create proofs the way artists create paintings—through vision, inspiration, and craft.
Mathematics at its best is play. "What if we try this? What patterns emerge? What happens if we change this assumption?" It's driven by curiosity and delight.
The joy comes from figuring things out yourself, from having insights, from the "aha!" moment when something clicks.
Real mathematicians work on problems they care about because the problems are interesting, beautiful, or important—not because they're on page 47 of the textbook.
But instead, Lockhart argues, we teach:
"Here's the formula for the area of a triangle. Memorize it. Use it on these 20 problems. Don't ask why it works."
Success means getting the right answer quickly. Understanding doesn't matter. Insight doesn't matter. Just get the answer and move on.
Students never discover anything. They're told formulas that mathematicians proved centuries ago and told to apply them. No ownership, no creativity.
"Do these problems because I said so. They'll be on the test. You'll need this for next year." But why does anyone care about factoring polynomials?
An Example: The Area of a Triangle
Lockhart gives a powerful example. Most students are taught:
"The area of a triangle is ½ × base × height. Memorize this. Use it on the test."
But imagine instead:
Teacher: "Here's a triangle. How could we figure out how much space it takes up?"
Students experiment: Drawing rectangles around it. Cutting it up. Comparing it to squares.
Someone notices: "Hey, if you put two identical triangles together, they make a rectangle!"
Another student: "So the triangle is half of that rectangle!"
Class discovers together: Area = ½ × base × height. Not because someone told them, but because they figured it out.
Which approach makes you care about triangles? Which one feels like real thinking?
Is He Right?
Lockhart's essay is controversial. Some teachers say he's being unrealistic—there's too much to cover, and not every student will become a mathematician. Others say he's exactly right and that's why students hate math.
Your Investigation
Part 1: Read the Essay (45 minutes)
Find and read "A Mathematician's Lament" by Paul Lockhart. It's available free online (search for "Lockhart Mathematician's Lament PDF").
As you read, mark:
- Three points where you completely agree with him
- Three points where you disagree or think he's exaggerating
- One idea that completely changes how you think about math
Part 2: Examine Your Own Experience (30 minutes)
Reflect on your math education honestly:
1. Moments of Discovery: Can you remember a time in math class when you actually discovered something yourself rather than being told it? Describe that experience. How did it feel different from typical math class?
2. The Formula Problem: Pick one formula you've been taught (area of a circle, Pythagorean theorem, slope formula, whatever). Do you understand why it's true, or did you just memorize it? If you only memorized it, does that bother you?
3. Beauty vs. Boredom: Have you ever found something in math genuinely beautiful or fascinating? Or has math class trained you to see math as purely procedural?
4. The "Why" Question: How often do you ask "why?" in math class versus "how do I solve this?" Would your teacher welcome "why" questions or see them as distractions?
Part 3: Essay Response (60 minutes)
Write a 2-3 page essay addressing:
- Your Verdict: Is Lockhart right? Is the way we typically teach math killing the beauty and joy of mathematics? Or is he being unfair? Use specific examples from your own experience.
- The Practical Problem: Even if Lockhart is right, is his vision practical? Can teachers really let students discover everything themselves? Should they? What would you keep from traditional math education and what would you change?
- Personal Impact: How has this essay changed (or not changed) how you think about math? Will you approach your math homework differently now? Why or why not?
- The Path Forward: If you could redesign your math class to be more like Lockhart envisions, what would it look like? Be specific. What would a typical class period involve?
Part 4: Try It Yourself (Optional, 30 minutes)
Pick a mathematical topic you've been taught. Instead of reviewing the formula, try to discover it from scratch.
Examples:
- Why does (a + b)² = a² + 2ab + b²? Draw pictures until you see it.
- Why is the sum of angles in a triangle always 180°? Can you prove it?
- Why does the Pythagorean theorem work? Can you discover a proof?
Write about the experience. Was it more satisfying than just being told? More frustrating? Both?
The Big Picture
What This Means for Your Math Class
Here's the uncomfortable truth: Lockhart is probably at least partially right. Much of how mathematics is taught does strip away the beauty. The formulas you memorize were discoveries that thrilled the people who found them—but you're just given them as facts to apply.
But here's the thing: You don't have to wait for your teacher or your school to change.
When you're given a formula, you can ask "Why is this true?" and investigate it yourself. When you're solving problems, you can treat them as puzzles to explore rather than boxes to check. You can choose to engage with mathematics as a creative art rather than a mechanical procedure.
The system might be flawed, but you can still find the beauty. Yes, you need to learn the formulas. Yes, you need to practice the procedures. But you can do that while also asking deeper questions, making connections, and seeking understanding.
Think of it this way: Even if music class focused too much on notation, you could still listen to music at home and learn to appreciate it. Even if math class focuses too much on procedures, you can still explore the ideas, wonder about the patterns, and find the art in mathematics.
Lockhart's essay is a reminder: Mathematics is beautiful. It's creative. It's profound. If your experience of math class hasn't shown you that, it's not because math isn't beautiful—it's because beauty was hidden from you. But now you know it's there. And you can go looking for it.
The next time you're working through math homework, pause and ask: "What's actually happening here? Why does this work? Is there a pattern I'm seeing?" Those questions transform math from a chore into an exploration. That's the difference between music notation and music itself.