Americans Used to Be Smarter
why today's math teaching doesn't work
This one has a ton of photos. If your email client chokes, you can click the title above and a browser window should launch, taking you to read it on the Substack website or app.
A week ago, I published a story from my side hustle as a math tutor. It shows the appalling monstrosity called “chunking,” which is how kids are being taught to divide now.
My tutoring student, a little girl who’s about to turn eleven, still finds school math tiresome and emotionally painful. She spent three years in “math intervention” because, as far as I can tell, the school couldn’t admit that their way of implementing Common Core standards — almost exclusively through discovery-based methods — doesn’t work for very many kids.
Common Core did not mandate discovery learning. But its emphasis on conceptual understanding, reasoning, and multiple strategies was widely implemented through constructivist, discovery-based pedagogy. In practice, that often meant delaying explicit algorithms, discouraging memorization, and increasing cognitive load for beginners.
That’s why it took me months to convince my tutoring kid — a classic firstborn female perfectionist, the kind of junior adult whose parents regularly say things like, “Sweetheart, go read your book. Taking care of your brother and sister is our job, not yours” — that it was okay to memorize the multiplication tables.
The school had discouraged memorization so thoroughly that she thought she was cheating if she knew the answer too fast.
I have another tutoring story to tell. And then I’m going to show you something extraordinary: historical evidence that old-fashioned math instruction worked — and that abandoning it was a mistake.
Two sessions ago, we started short and long division. She fell in love with the algorithm.
Not because it was nostalgic.
Not because it was “adult math,” though that helped.
Because it was calm.
Clear.
Orderly.
Predictable.
She brought back the worksheet — designed on-the-fly, on a legal pad with a pencil — that I’d sent home, having correctly calculated the per-day cost of her parents’ house.
Last time, we moved on to division with decimals. And she had the kind of moment that discovery-based classrooms are constantly trying to manufacture — the spontaneous “light bulb” moment that’s supposed to arise from struggle and exploration.
It didn’t arise from struggle.
It arose from structure.
That distinction matters.
Discovery-based pedagogy assumes that insight emerges from wrestling. That if students explore long enough, the beauty will reveal itself.
It won’t.
It can’t — not reliably, and certainly not for beginners.
If a teacher is skilled, they can facilitate a few of those moments. But those moments come after the basics are stable, not before.
Math is a language.
Discovery-based instruction asks children to appreciate the poetry before they can read a sentence. To analyze the metaphor before they know the alphabet. To write free verse before they can ask for directions to the bathroom.
Structure is not the enemy of beauty.
It is the precondition for it.
After explicit instruction — the classic “I do, we do, you do” progression — I told her we were going to find the decimal equivalent of fractions.
First, I modeled one slowly, narrating every step.
Then we did one together.
Then she did one narrating.
Then she did one silently.
Only then did I say, “Okay. Now we’re going to do something cool.”
With 1/2 and 1/4 and 1/3 and 1/5, we’d used 1.00. For 1/7, I wrote:
1.00000000000000000000 ÷ 7
Some of you are grinning.
Some of you are confused.
The grinners remember that 1 ÷ 7 = 0.142857142857142857…
When she saw the pattern repeat for the third time, her face lit up.
“OH!!! It’s like how 1 divided by 3 goes on forever — but it’s a group!!”
How did that happen?
It happened after forty minutes of slow, careful, explicit instruction.
Not before.
Signal, Noise, and the Myth of “Discovery”
In the past week, Barbara Oakley joined Substack.
If you don’t know her name, you should. Oakley is the author of A Mind for Numbers (a book I cannot recommend highly enough; it was the most important study aid in my pursuit of my degree in mathematics) and co-creator of Learning How to Learn, the most popular online course in the world. She is not nostalgic, not reactionary, and not remotely anti-conceptual. She is a cognitive scientist who has spent years studying how beginners actually learn technical material.
Her most recent post is titled “The Teaching Method That Can’t Fail (and Why That’s the Problem).”
Her argument is devastatingly simple.
Constructivism — the philosophy behind discovery-based teaching — begins with a truism: learners construct knowledge in their own minds. Of course they do. No one thinks teachers beam facts into passive skulls.
But from that truism, a much larger claim gets smuggled in: because students construct knowledge, teachers should minimize explanation and maximize exploration.
Oakley points out the sleight of hand.
How learning works does not automatically tell us how teaching should look. That’s an empirical question.
And the empirical evidence is remarkably consistent: for novices, more guidance works better than less.
Explicit instruction.
Worked examples.
Clear explanations.
Structured practice.
When discovery-based classrooms underperform, constructivism never takes the blame. The teacher didn’t scaffold enough. The culture wasn’t ready. The test measured the wrong thing.
When explicit instruction is quietly reintroduced and scores improve, it gets relabeled “guided inquiry.”
Oakley calls this the “activity ratchet.” The prestige language remains student-centered. The effective components — explanation and practice — are smuggled back in and credited to the ideology that tried to replace them.
Her alternative is something she calls cognitive realism: the idea that brains have limits, working memory is finite, and instructional methods must respect those limits. Theories of teaching must be falsifiable. If no outcome can ever count against your philosophy, you’re not doing science.
Reading her post felt like watching someone describe, in clinical language, what I’ve been watching in living rooms and at kitchen tables for years.
When beginners are overwhelmed with decisions — choose your strategy, explain your reasoning, justify your model — they don’t experience enlightenment.
They experience noise.
And when the noise clears, the light bulb moments come on their own.
Which brings me to the copybook.
Because none of this is new.
What An 1814 Schoolbook Knows About Cognitive Science
Below are photographs from an 1814–1850 Pennsylvania school copybook and farm ledger that I accidentally acquired while shopping on eBay for a used Civil War–era sketchbook. (If you want the full story of how a modest art purchase turned into custody of a bilingual Pennsylvania German manuscript signed “Christian Reist my hand, March the 2 day 1838,” you can read that here.)
For now, just look at the math. Because what you’re about to see is not nostalgia.
It is instructional design.
Numeracy Built a Nation
Let’s zoom out for a minute.
There were no calculators.
There were no spreadsheets. No accounting software.
No phones humming quietly in the background, ready to convert fractions to decimals or percentages to dollar amounts.
If you were a farmer in 1838, and you needed to calculate interest, rent, grain yield, labor cost, or transport time, you did it by hand.
In ink. On paper. Or in your head.
American capitalism did not grow into a world power because people were vibing their way through arithmetic.
It grew because ordinary people were numerate.
Not elites. Not specialists. Ordinary farmers, merchants, tradesmen.
They could calculate proportional labor.
They could compute interest.
They could convert between units.
They could track mixed-base currency.
They could do algebra — even if they didn’t call it that.
And they learned it the way you’re seeing in these pages: rule, example, repetition.
This was not ornamental schooling. It was, in a completely literal sense, our nation’s primary economic infrastructure.
The Tool In Your Hand
Josh Slocum gave me this abacus as a gift, because it’s a lovely green color that matches my aesthetic and he knows I love tactile objects that remind us how humans used to think.
Look at it.
It’s small. Beautiful. Completely manual. My tutoring kid has asked about it, and I used it to show her how to calculate 3x4 on it. (She proudly informed me that it could also just be 6x2, which was a great moment).
But the larger point is that it doesn’t hide the math.
It makes the structure visible, and you have to understand the problem to use it correctly.
Now compare that to what we hand children today.
We give them devices that compute instantly. We give them calculators before fluency.
And in Common Core classrooms, we give them “strategies” before structure.
And then we act surprised when a random adult chosen from America today struggles to compute a 15% tip without reaching for a phone.
A random adult chosen in 1850 — or even 1950 — would likely outperform that same modern adult on everyday arithmetic, even if that adult has a university degree.
Not because they were smarter. Because they were trained differently.
They were trained to execute.
They were trained to hold multi-step procedures in working memory.
They were trained to align units and track them carefully.
They were trained to reduce noise.
This kind of instruction — rule-based, structured, explicit — is why.
This is where Barbara Oakley’s brilliant, simple, devastating argument lands like a hammer. Because she’s not doing “new math bad” or longing for the past in some nostalgic way. She’s an actual scientist telling the truth about what we know empirically:
Working memory is finite. Novices overload very easily.
And structure — which kids only get because an adult authority who knows more than they do and isn’t afraid to act like it — reduces extraneous cognitive load.
When beginners are asked to choose strategies, justify reasoning, and explore before they are fluent, we increase noise.
When we give them clear procedures and guided practice, we reduce noise.
What you are seeing in this 19th-century copybook is not backwardness or, God help us all, “oppression.”
It is cognitive efficiency.
It is a society that understood — intuitively, long before fMRI machines — that before you can appreciate abstraction, you must be able to execute.
American economic expansion was not powered by vibes.
It was powered by numeracy.
And numeracy is built.
It is never, ever discovered.
Light Bulbs Can Only Illuminate Existing Structure
Somewhere in Pennsylvania in 1814, a young person sat at a desk with ink and copied:
Rule of Three.
They practiced until the layout felt natural.
Until proportion stopped feeling like juggling and started feeling like breathing.
Until multiplication facts surfaced without effort.
Until units stayed aligned without panic.
Years later, that same hand tracked wheat. Calculated rent. Computed interest.
Managed survival in the language of mathematics, the only way we have to reliably prove anything and to align ourselves with reality.
Two centuries later, a not-quite-eleven-year-old girl sat in my studio and, on a legal pad with a pencil, divided 1 by 7.
She did not discover the repeating pattern through exploration. She did not invent a strategy. She did not write a paragraph about her reasoning.
She followed a clear procedure, directly instructed in that procedure by an authority who knew more than she did.
And then — because the procedure was clear — she saw the pattern.
Her face lit up.
The light bulb moment came after structure, not instead of it.
Barbara Oakley calls this cognitive realism: the recognition that the brain has limits, that working memory is finite, and that instruction must respect those limits if insight is going to emerge at all.
The 19th-century copybook didn’t have the language of cognitive load theory.
It didn’t need it. It respected the limits of the novice mind.
It built fluency first. And it trusted that beauty would follow.
We didn’t become a numerate nation by accident.
We became one because children were taught, explicitly, how to compute.
If we want light bulbs instead of anxiety…
If we want numeracy instead of noise…
If we want adults who can think without reaching for a device…
We already know how to do that.
It’s written in ink.