Teaching Math Rightly

Now it’s personal…

I’ve taught math for over 10 years.

I’ve been to math conferences and done deep research into how to teach math well.

I’ve even done some writing about how to teach math and what’s broken with our approaches to teaching arithmetic.

My goddaughter is struggling with algebra. When my brother told me this during a recent visit, something shifted. Ten plus years of teaching math, attending conferences, and researching pedagogy suddenly felt intensely personal…

How kids (don’t) get math

As far as I can tell, there are four distinct ways students fail to understand math.

1. Kids misunderstand the vocabulary

This happens when a student doesn’t understand what a “numerator” or “denominator” is. The teacher uses the terms over and over and students do not have enough familiarity with them to immediately know what part of the fraction they are referring to.

2. Kids misunderstand the symbols

This happens when a student doesn’t understand what it means to write a number in smaller font as a superscript (an exponent) or what “f(x)” signifies.

3. Kids misunderstand the logic

This happens when a student knows what the exponent means and signifies, but has not thought through the logic of why this simplification is true (for x ≠ 0):

Following the exponent rule that “you subtract the exponent in the numerator from the exponent in the denominator” is not real understanding.

4. Kids (and teachers) misunderstand what it means to understand math

The first three are surface misunderstandings. The fourth is deeper and far more dangerous. Kids sometimes think that they understand math when they really do not. In lower grades, this looks like kids memorizing algorithms without understanding why they work. Or it looks like a kid who knows all his multiplication facts, but is unable to use them to solve new multiplication problems mentally.

In later grades, this usually manifests itself in bigger ways. Kids struggle with algebra or calculus after getting straight A’s in math until that point.

Learning math without truly understanding it is often referred to as “algorithmic” understanding. You know the algorithm, but you don’t know why or how it works.

What Even the Good Teachers Miss

Any good math teacher is familiar with the first of the three distinctions I made above. What I think even the best math teachers miss is the fourth. Many teachers think their students understand division once they can do the division algorithm. Many teachers think their students get “solving linear equations” once they can do all the problems of that type in the textbook.

An essential part of including this fourth type of understanding in the math classroom is increasing a student’s intellectual self-awareness. Early in my teaching career, during a one-year stint at Mathnasium, I encountered this approach and it reshaped my own thinking. After a student provides an answer to a mental math problem, tutors are encouraged to ask them “how did you get that?” even if the answer was correct.

This serves a double purpose. It allows the tutor to correct the exact mistake in the student’s logic (if there was one), but it also fosters in the student a habit of thinking about one’s thinking.

When a student can correctly explain his correct thinking about his correct solution to a problem, then he really understands it.

The Solution

The solution is not a simple textbook rewrite. The responsibility ultimately falls on teachers to fix a centuries-old problem that has spread because of the way teachers have learned and retaught mathematics. The problem goes back to the very foundations of mathematical thought.

The Ancient Greeks said that number included 2, 3, 4, 5… and so on. It was not a settled matter, but for the most part they did not consider that 1 was a number. And zero (nothing) was certainly not considered a number. Fractions, negatives, irrational numbers were far removed from consideration.

When my goddaughter struggles with negative numbers, she’s wrestling with the same conceptual leap that took mathematicians centuries to accept.

Over the centuries, mathematicians broadened their definition of number to include things like fractions and negatives, eventually even including irrational numbers like π or the square root of 2. Why? Generally speaking, mathematicians expanded these definitions to solve problems that were previously thought unsolvable.

Number, redefined by mathematicians over the centuries, is presented as one coherent and complete concept to the very young, when in fact it has been defined, redefined, and fought over by mathematicians of the highest caliber over the centuries. I hope to explore this in future articles.

Brilliant mathematicians over the course of history struggled or even outright refused to accept fractions, negatives, irrationals, algebra, calculus… into their mathematical schematic. If a 4th grader struggles with fractions or decimals, they’re in good company and we shouldn’t be surprised.

A New Enterprise

I believe that math education often fails because we teach the symbols and procedures of mathematics while forgetting the centuries of human struggle that gave rise to those symbols, and we teach the manipulation of symbols as if it were mathematics.

I am writing a series of essays called *Teaching Math Rightly* to expose these misunderstood concepts and present ways of teaching math that fosters true understanding and love for the subject.

This will go beyond my earlier post of a resources toolkit.

Until teachers re-learn math from the ground up themselves, no curriculum reform can fix math education.

When I think of my goddaughter working through her algebra, I don’t just see a struggling student. I see someone stepping into a centuries-long debate about what numbers really are.

Originally published at https://whatschoolsforget.substack.com.

Reclaiming the Beauty of Numbers: If you want to explore how to build a genuine culture of mathematics that goes beyond standardized rubrics, join our community. Over 1,000 readers receive my weekly articles directly via email. Subscribe at What Schools Forget.