*(Quick announcement: I have a Patreon now!)*

I know Twitter’s reputation: it’s a cross between a skull, a mushroom cloud, and a swamp fire. But dang if I don’t love math teacher Twitter! So many friendly people. So many great questions. So many array photos.

I got a lasting reminder of this awesomeness in June, when I tweeted this:

I expected to hear two dominant kinds of memory: happy recollections of a teacher’s validation, and traumatic ones of a teacher’s censure. The actual flood of memories that filled my notifications were richer, weirder, and far more diverse.

Backseat revelations on car trips.

Games with parents and grandparents.

The endlessness of counting.

And—coolest of all—strong emotions excited not by a teacher, not by a peer, not by a parent, but by the mathematics itself. More than I’d have guessed, people remember early encounters with mathematical truth, with the deep persistence of pattern. It made them feel angry, joyous, fearful, awestruck.

I offer up these stories with a minimum of curation. Each is a vivid little window into one of the strangest symbioses on planet earth: the relationship of a human being to mathematical reality.

**“I Knew the Best Secret in the World, Because I Could Count Forever”:
The Endlessness of Number**

Several folks recalled glimpses of infinity, or (equivalently) the idea that there’s always a next number. I’m still reeling from this, and I am objectively growing old and dusty.

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**“We’ll Get There in One Episode of Sesame Street”:
Telling Time**

The quantification of time is one of the more peculiar achievements of civilization – and one with immediate importance to kids, who are always being made to wait for stuff.

(Interestingly, this “multiply by 10 then divide by 2” trick showed up in several people’s earliest memories!)

**“For a Six-Year-Old, It Was a Big Revelation”:
Making Discoveries**

It will surprise few of my math teacher colleagues that people remember the intellectual breakthroughs that felt like their own.

Your teacher tells you a thing? Fine.

You uncover a universal truth with *nothing but your mind*? More than fine!

**“Minus Three Blocks North”:
The Allure of Negatives**

Several folks remembered early encounters with negative numbers. Makes sense! They’re as weird as anything in Dr. Seuss or Lewis Carroll.

**“They Seemed Arbitrary to Me”:
Mathematical Symbols**

Those conversant in mathematics don’t think much about the symbols. They take a backseat to the thing symbolized.

But kids – just learning to shape symbols with their hands and to discern them with their eyes – put a bigger emphasis on the marks on paper. What shape is an 8? A 9?

**“I Still Think I Was Right”:
Conflict with a Teacher**

As a teacher, I half-dreaded and fully expected these answers. I know that disagreeing with a teacher burns a moment in memory far better than any mnemonic.

(As the follow-up tweet explains, Erick’s teacher couldn’t find one, and the vindicated Erick repeated the exercise the next day – this time with *five* colors.)

(I’m with the student on this one! 2-1 games at halftime rarely end 4-2.)

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**“Feeling Ecstatic That I Could Stack Two Odds”:
Odds and Evens
**

These recollections bolster my sense that “odd” and “even” are the best conceptual playground for the young mathematician. They’re abstract yet easy to visualize, rich yet elementary, not to mention a natural place to make the leap from specific (*what’s 6 + 7?*) to general (*what’s an odd plus an even?*).

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**“A Half of What?”:
Fractions**

Given how fractions dominate the upper-elementary curriculum (and the laments of secondary teachers), it’s no surprise that they surfaced often in folks’ memories.

These memories also point towards a meaningful context that stuck with people: coinage. “A quarter” = 1/4.

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**“Just Couldn’t Figure Out How My Dad Was Younger Than Me”:
Confusion & Clicking**

Two emotions predominated: the feeling of “WHAT THE HECK IS GOING ON” and its successor, the sweet relief of “oh THAT’S what the heck is going on.”

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**“We Were Now Properly Going to Start Mathematics”:
Visions of Things to Come**

Given the hierarchical structure of math education (year-by-year advancement, a sense of cumulative build-up, the sorting of topics as “beginning” or “advanced”), I’m not surprised that many folks remembered occasions when they got to glimpse around the corner, to see what lay ahead on their educational path.

**“This Makes the Patterns in My House Better”:
**

**Sorting and Arranging**

I have a conjecture for why there weren’t *more* memories like this: because people don’t necessarily think of this work as mathematical!

It is, of course – pattern and classification are deep, essential mathematical tasks. But I wouldn’t be surprised if some folks’ minds’ skipped past similar memories, and latched onto something arithmetical, i.e., something more quintessentially “mathematical.”

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**“I Could Break the Ice and Scrape the Fragments into Triangles”:
Shape & Geometry**

I was surprised to find so little geometry among these memories. The few that did surface were notably beautiful.

**“What Did That Little x Do?”:
From Addition to Multiplication**

Most folks have a sense of autobiographical memory that kicks into high gear around age 7 – right when they’re starting to encounter multiplication.

Apparently, meditating on the meaning of “2 x 3” is a very common activity for mathematicians at this age!

**“I Am Still So Satisfied and Delighted By This”:
Memorable Tricks**

Whether self-discovered or not, clever numerical tricks and patterns have a way of sticking in the brain.

**“By the Teens I Was Struggling”:
Place Value**

Some memories concerned not just the endlessness of number, but the peculiar nature of our numeral system itself, with place value and positional notation.

This is, after all, a pretty fancy technology, one that took centuries to develop, and which then swept the globe. The early experience of students often recapitulates historical development, in one fashion or another.

**“I Am Five! How Should I Know?”:
Memorization and Drill**

These early memories of drill and memorization aren’t all negative. But it’s striking, I think, how *few* of the memories overall concern drill!

Perhaps it’s because drill isn’t as common as we think.

Perhaps it’s because repeated events, by nature, are hard to pluck out as a “first” memory.

Or perhaps it’s because drill doesn’t stick in the mind quite the way other experiences do.

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**“Using This Little Guy”:
Technology**

Only three of the memories dealt with calculators, and none with computers. A generational thing? Or – as I suspect – do technological experiences (e.g., playing with a calculator) stick in memory less than social (e.g., conversing with a parent) or interior experiences (e.g., sitting in the back of a car, dreaming mathematical thoughts) do?

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**“A Puzzle To Be Solved:
Learning Through Games**

Again, given how much kids love games, I was surprised how few adults remembered playing them!

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**“I Always Wanted the Book with the Counting Monkey”:
Falling in Love with Math**

No further comment necessary.

My sister- if you have one cookie. And I take your one cookie. How many do you have left

I do not remember much about early childhood math other than I received good grades in school. Then came geometry and algebra and I lost any love I had for math class. To my credit, however, I can count change without the aid of a calculating device and managing budget is one of my duties at work.

I have warm memories of my daughter driving her matchbox car along a number line (we homeschooled, there was a number line in the kitchen, at kid-level) with a sing-song voice chanting “1 is odd, 2 is even, 3 is odd, 4 is even,…”

My earliest mathematical memory that I can remember, funnily enough, was reading More Murderous Maths by Kjartan Poskitt in first grade, and being absolutely enamored by the Mutilated Chessboard problem mentioned within. Such a seemingly-impenetrable problem, felled by the simple observation that the squares on a chessboard are black and white!

My mother was a math teacher. We had an alarm clock that was quite a ways off. I recall sitting under the table where she was correcting papers at about age 5, knowing what time Mom had correctly set the clock and what time it was now vs. what the clock said. I figured out, by laborious guess and check with serial addition, that it was losing about 4 minutes per hour over several hours. Didn’t have a clue I was doing division. Also, about a year later at about age 6, to keep me amused while we waited in the car for the elderly lady we took shopping weekly, she taught me to solve 2 equations in 2 unknowns writing with my finger in the condensation on the inside of the car windows. I thought balancing equations was a lovely game!

First – after a long day, your math posts focus my mind so I can let the busy thoughts go. Helps me fall asleep.

Second – my first memory of math is that my grandfather always carried me up and down stairs BUT he always counted every step. I’m sure there’s a more mathematical memory in there… But nothing more significant than that counting stairs before I could walk.

Not the earliest memory but a profound early memory.

I was not getting this borrowing concept with subtraction. But I found an algorithm that worked. So I started using it. But then we went up to the blackboard, and I couldn’t explain the logic of my process.

If we had 23-15. I would subtract 3 from 5 giving 2. Then I would “reverse it.” Most simply you could say I subtracted 2 from 10 to give 8. But, that isn’t how I was thinking about it. I was thinking about an involution (not that I knew what an involution was) ….all the numbers in [0, 9] have have their pair. 1 pairs with 9 and 2 pairs with 8, and 5 pairs with itself.

The teacher thought I might be retarded.

Okay, that puts me in mind of another. I am at UCSF being assessed, They have decided that I am not retarded, but there is definitely something different going on in my brain. Test has cards with colors and shapes, and arrows that suggest some sort of logic to their arrangement, and I am supposed to guess what are the properties of the missing object on the card. After doing this for an hour, we get to the last card, and I say “black square” and the expert says “red circle” and I explain my logic as to why I think it is black square, and the tester agrees with me that black square makes more sense than red circle.

Certainly, you have reviewed this test and know the answers and understand where the answers come from!

And the test administer tells me that he has given this test to lots of children, but we stop when we get so many wrong answers, and that he has have never gotten to the end before.

Doug, I don’t know whether you have ever heard Tom Lehrer’s song “New Math,” but in his intro, he explains how people “used to” do subtraction, which I never understood at all. But it starts out more or less the way you did it… Worth listening to in any case: https://www.youtube.com/watch?v=W6OaYPVueW4 (as is all of the rest of Tom Lehrer)

It is not my memory exactly. It was my grandmother’s birthday that was 90 years old, so the great-grandchildren around her began to count how many more years she would have 91, 92, 93, 94, 95, 96, 97, 98, 99, … and a great silence . They did not know what came after 99. Then one of the great-grandchildren said, “Ah, it’s already many years of life, is not it great-grandmother?” General laughter of adults. Detail, my grandmother lived up to 104!

I remember learning how to calculate the area of a triangle – base times height, divided by two. I could do it, but I couldn’t understand *why* it works to calculate the area! But then my mom (I think, but she does not remember this) showed me visually how it makes sense – if you build a rectangle with the same base and height, its area is base times height (I knew this already). The triangle’s height divides the rectangle into 2 smaller rectangles, and the sides of the triangle *obviously* divide each mini-rectangle in half. So if this part of the triangle is half of this mini-rectangle, and that part of the triangle is half of that mini-rectangle, than both parts of the triangle together are half of both mini-rectangles – and so, the area of the triangle is half of the area of the rectangle!

I remember talking with my father about sixty coming after fifty, seventy coming after sixty, eighty after seventy, ninety after eighty. What comes after ninety? “I know!” I said, “ten-dee.”

I was five and I wasn’t new to counting but I was thinking about the number two and wondering how anything could ever be so alike something else that you could lump them together into one category and say there were two of them. I thought about how even if you got two apples that were as alike as you could get them they would still have slight differences in colour, in weight, in shape, one might a little bit sweeter or tangier, and one of them would be eaten sooner than the other one in time. But you could also say two apples to mean a green one that had grown in one country and a red one that had grown in a different country. I was thinking about how strange it was that all these differences could be overlooked, how could there ever be more than one item in any category?

I tried to explain this to an adult but I didn’t have the words and they didn’t have the patience, “come on, you know what two means” and I did know, but it just seemed so odd.

Mine is from when my mom mentioned that she was teaching her pre-algebra class negative numbers. I was in 1st or 2nd grade, so I still conceptualized numbers as always being numbers *of things.* Three apples, five pennies, four pencils–there had to be a concrete object there for there to be a number *of* in my mind. So I wondered if negative numbers meant there was just part of a thing.

I was very disappointed when I learned that that was called “fractions.” “But then what’s a negative?” Fortunately, by the time I got to pre-algebra class myself and learned what negatives really were, I’d reached the point where I could understand numbers beyond “this many physical objects.”

I’m also told that I used to get detention a lot in 1st grade for altering math tests to make them harder, because “a test is to show what you know.” But I don’t remember that as clearly as I remember picturing half a teddy bear and thinking “Is that negative one?”