Making the cut

“You Don’t Get It”

The single biggest problem in America today is our lack of understanding. We are not going to solve the environment, the economy, the government, or anything else unless we can comprehend, communicate, analyze, evaluate, and act on big ideas.

A lot of ink has been spilled about empathy – there’s even a section about it in this very post – but the bigger issue is our inability to think. We are way too focused on feeling, and believing, and our capacity for abstraction is painfully underdeveloped.

There are more things in heaven and earth, Horatio,
Than are dreamt of in your philosophy. -William Shakespeare

There is a world of conceptual awareness beyond the stuff we can count or program that influences everything we can account for through direct experience. Learners would be much more successful in all fields if we addressed this directly and early.

We want to get our heads around ideas. Ignorance pisses us off. Toddlers get frustrated and angry because they don’t yet have the concepts or language to express themselves and feel understood – that’s the “terrible twos.” Adults get frustrated and angry because things happen they don’t understand, and they don’t trust information or logic – that’s the January 6 insurrection and protests against vaccinations and wearing masks.

Are You Bad at Math – Or Mad at Math?

A few years ago I was cleaning out my parents’ garage and I found a copy of my junior high school transcripts.

I always thought I was bad at math. Math always felt like a slog to me. I figured I just didn’t have a talent for it and I stopped taking math courses as soon as I had the chance.

Imagine my surprise when I read those transcripts. On every standardized test, I scored higher in math than I did in English. By a lot. Enough to make this author/ journalist/ blogger/ former English teacher wonder what else I might have done with my life.

Where did I get the idea that I was bad at math? It started with an impression – I didn’t like math. But why should that be? I’ve always pretty quick with arithmetic. My kids love it when I tell them “super speedster” problems like the ones my fifth grade teacher Mr. Friedman (who also worked as an auctioneer) used to rattle off at us for extra credit.

Then it hit me. I remembered the exact moment when I learned to hate math.

It was lunchtime on a bright, crisp winter day during my seventh grade year at Francisco Sepulveda Junior High School. I was the only student in my pre-algebra class who had to retake the chapter test. Mrs. Faught sat on her stool in front of the room next to the overhead projector, writing problems on a plastic sheet. The projector glass was covered by a white piece of paper so I couldn’t see the problems on the screen. She finished writing and looked up at me, still standing in the doorway.

Showing Our Work

“David, do you know why you’re here?”

(Yes, I thought. I am here because you are a jerk.)

“You didn’t show your work on the test problems, and you got them all right. So now you have to prove that you weren’t cheating.”

She made me put all my things down on the floor in the back of the room. She handed me a piece of paper and a pencil. Then, with dramatic flair, she removed the white piece of paper, revealing the problems she’d written just for me.

I finished in five minutes and I did not show my work. Mrs. Faught graded the paper and said, “Well, you got the right answers. But you didn’t show your work. You didn’t follow instructions.” She handed it back to me with a red “F” on top.

Math Is Not the Problem – It’s Us

Later in life, I would come to understand mathematics as a way to understand the world and  solve problems, a language that provides glimpses into the true nature and even splendor of physical reality. As Bertrand Russell put it, “Mathematics rightly viewed possesses not only truth but supreme beauty.”

But to many students, math is a bathroom pass of the mind, a tool that teachers use not only to control what students think about, but how they have to think about it. Demanding that you process information according to my steps – Think how I tell you to think – is oppression.

What is the point of teaching math? Are we helping each other discover truth and beauty as expressed in mathematical terms, or are we exerting our will and forcing people to think as we direct?

Our society can’t afford to graduate another generation of learners who believe – rightly or wrongly – that they are bad at math, or that learning is painful, or that abstract concepts are beyond our understanding.

We should start by understanding how we think about math.

The Count

Many young people do just fine with arithmetic because it’s concrete. If you see the symbol “1” or the word “one” you can pick up a physical object and assign the term: you are holding ONE of it. Put another thing next to that first thing – “add” it – and bathe in the sublime pleasure of “2.” Now you’re counting and operating. You have a foundation for the language you need to do arithmetical operations with real numbers.

After you master addition, subtraction, multiplication, and division, you’ll be ready for fractions and percentages. Apart from using the same terms as basic arithmetic, these operations have practical referents and applications in popular culture: sale prices, sports statistics, and pizza slices.

X Marks the Spot

Algebra is a whole ‘nother ballgame. We go from nice, whole numbers to variables. {x} stands for numbers, or sets of numbers, or possibilities.

Equations can be intimidating, especially when they’re used in applications such as engineering or philosophy. How is it that so many people, decade after decade, can count stuff in the world and then completely fall apart when the math becomes conceptual?

In their paper “Solving Equations: The Transition From Arithmetic to Algebra,” researchers Teresa Rojano and Eugenio Filloy describe “Conceptual and/or symbolic changes which mark a difference between arithmetical and algabraic thought in the individual.”

This is a big deal.

The Didactic Cut

“One of these cuts,” write Rojano and Filloy, “is particularly interesting for the theme of problem solving…”

It turns out that arithmetical skill isn’t the right mental foundation or frame for algebra. It’s not even the right language. Telling students to practice more arithmetic to prepare for algebra is like telling them to read more Spanish so they can understand Russian.

We literally have to learn a different language to understand algebra. According to Rojano and Filloy, “It is necessary to construct, or acquire, some elements of an algebraic syntax… The construction of these syntactic elements is based on arithmetical knowledge which has worked well up to a certain point, but it must also break with certain arithmetical notions – hence the presence of a cut.”

It’s easy to confuse algebra with arithmetic, so consider the following two equations. First:

Ax+ B = C

That’s easy to solve using arithmetic; if you know C, you can use the concept of equality to “unwind” the operations and deduce the value of x.

However, things get trickier in an equation like this:

Ax+ B = Cx + D

Now we need operations drawn from outside the domain of arithmetic – what Rojano and Filloy call “operations on the unknown.”

Making this sort of operation meaningful to the learner requires redefining the concepts of equation and equality in numbers. According to Rojano and Filloy, “the learner must at least understand that the expressions on both sides of the equals sign are of the same nature (or structure), and that there are actions which give meaning to the equality of the expressions (for instance, the action of substituting a numerical value for the unknown).”


What would have happened if Mrs. Faught would have simply asked me to explain my thought process in solving problems? It would at least have given us the chance to better understand each other. I imagine that I would have had to slow down, and – in essence – show my work. I like to imagine that she would have learned something too.

Learners take many different paths to understanding. It is presumptuous to assume that any one methodology will suit them all. Especially when we’re actually introducing a new subject, and not merely a higher level of the same subject.

The Larger Point

As Lakoff and Johnson wrote in Metaphors We Live By, “Most concepts are partially understood in terms of other concepts.”

That word: partially. For decades, educators have relied on Skinner and Hunter to “scaffold” content-based learning on previously acquired skills and/or lived experiences. But developing different ways of thinking is just as important.

Knowing arithmetic helps us learn the arithmetical elements of algebra. That’s effective as far as it goes – and that’s as far as it goes.

The same is true for any topic or field that expands into abstraction when it transcends what we can know with our senses:

  • Biology –> How to treat a cold –> Epidemiology/ Covid-19 policy
  • Money –> Sale price –> Macrofinance/ Cryptocurrency
  • Big –> Ocean big –> Astronomy
  • Small –> Molecule small –> Quantum mechanics
  • Voting –> Representative democracy –> Today’s zeitgeist

Operant conditioning isn’t enough when there is nothing in our immediate environment to operate on. We have to build confidence in our capacity for reason. We have to be able to distinguish logic and the scientific method from nonsense.

“See the North Star? Yes. You’re pointing at the right dot. But that’s actually not happening right now. The light you’re looking at is three or four hundred years old. I know! Weird. But also cool, but also weird… right?” And now we’re learning about light years.

Understanding abstraction takes training that involves discipline, yes, but also kindness, patience, and communication about schema and language.

We’re going to have to think our way through this.


As we begin 2022, let’s keep the conversation going. What’s your least favorite subject? When did you realize that learning and school were two different things? Contact me and share your story.