#teaching math

Modern College Algebra Textbook

A couple days ago, I wrote:

My math education thought I could manage strings of confusing symbols in my head in two directions at once, but that I couldn't do any problem that required me to identify the unknown in relationship to the known. ?????????

Because of my Quest to Be Friendlier With Numbers, my take on my K-12 math education changes almost daily. The bottom-line take, "My K-12 math education was bad," stays the same. But every day, I learn more about why and how it was bad.

If you asked me today, I'd say the primary failing with my K-12 math education was not that it utterly missed my dyscalculia or how poorly it prepared anyone to understand numbers. Instead, I'd say its primary failing was its failure to allow us to experience ourselves as people who could confront an unknown, consider the tools available to us, and move the unknown into a known.

Here's an illustration:

In 2024, I took over a high school library that had been in storage for seven years.

The upside to unpacking an entire library is that the unpacker gets to decide how the books will be shelved. If you want to move from Dewey Decimal to, say, a genre-based system, there is no better time to do it.

After wrestling with that choice for three weeks (the time it took to unpack 12,000 books), I decided we would abandon Dewey in favor of a topic heading-based call number system. When I presented this to my principal, I said:

"Do the kids need to know poetry is shelved under 811? No. Do they need to experience themselves as people who can walk into a library and find the poetry? Yes."

As a teacher, I see my job less as "impart knowledge" than as "create the conditions under which you can experience yourself as a builder and user of knowledge."

Sometimes that does require me to impart some basic knowledge. More often, and especially at the high school level, it means I need to set up situations in which students can use their knowledge. Where they see an unknown and navigate to it using their available knowns.

I'm finally getting the hang of long division. But I'm still doing problems on the tens unit blocks, even though I can also do them on paper. The blocks give me that sense of "see an unknown, use your tools and strategies to navigate to it, turn it into a known" that I firmly believe is the foundation of all real learning.

A friend sent me a soroban for Christmas.

(Pictured: a soroban, reading "5" all the way across because I forgot to reset it before taking the picture.)

The soroban, or "Japanese abacus," is used in several countries (not just Japan) for rapid, accurate calculations. While a lot of schools have abandoned the abacus except in early childhood education, Japan still teaches entire classes on using it.

Watching people speed-math on this thing is truly impressive. Here's a seven year old girl adding 12-digit numbers as fast as they can be read to her. Poke around YouTube, and you'll find other videos of people mathing even faster.

When I told my friend about being obsessed with the abacus (but not allowed to touch it) in first grade, I expected to get a primary school abacus. Instead, I got this. And I'm not sorry, because this is much cooler.

Reasons my dyscalculic brain likes the soroban:

I don't actually have to do mental math.

"Not doing mental math" includes never having to read a number one way but math it the other way. All numbers are inserted and read on the soroban from left to right.

If the beads are moved correctly, the answer is correct.

It's easy to move beads correctly because I can both feel and see whether they are right or not.

Moving beads correctly requires one to have memorized number bonds from 1-5 and 1-10 - especially in subtraction. But even here, I don't have to deal with the symbols for numbers as long as I know what the beads are supposed to look like.

For instance, my brain doesn't have to deal with "10 = 6 - 4." To add 4, I can just see that I don't have enough beads in my ones column, move a 10, and subtract 6 from the ones. I know I'm right without counting if I move the top bead and one bottom bead. Done.

For me, the soroban solves the problem I have with calculators. Calculators still require number symbols (like "4") to make sense - which they do not always do. Sometimes my brain just rejects that input. ("Ensign 9 is not on the Enterprise." "Where did he go?" "Please restate the question.")

The soroban doesn't. I have to know what the beads look and feel like, not what arcane picture humanity's ancestors drew to represent it. That helps me calculate. It helps a lot.

I have no plans to become a speed math wizard on this thing. But it'd be cool if I could reliably do my own taxes with it.

Further Reading:

Math With Soroban: A Japanese Abacus by the SAI Speed Math Academy. While there are several other resources online for learning soroban, Math With Soroban is my favorite so far for the clarity of its examples and the pages' worth of example problems available.

Soroban: The Japanese Abacus. Intro lessons from the Japan Society in the UK. (See also the League for Soroban Education in America.)

Soroban Abacus Handbook by David Bernazzani

Things That Are Changing

I am less anxious about basic math.

I have more tools for doing basic arithmetic problems.

I'm better at identifying which will be the fastest or easiest tool for any given problem.

I can more quickly and easily ID when an exact answer is needed or when an estimate will suffice.

I'm marginally better at noticing when an answer can't be correct.

Things That Are Not Changing

I still transpose numbers frequently.

I still transpose operations frequently (adding when I should subtract, dividing when I should multiply, etc.)

I still have initial anxiety when looking at a math problem, before the "oh yeah, I have more tools for addressing this now" kicks in.

I still frequently mix up my right and my left.

My sense of direction is still bad.

I cracked Maths - No Problem! Textbook 4A today, putting me halfway through the series. I'm making this list for future reference, because I suspect the things that aren't changing will continue to not change.

Better math education won't change the fact that I have dyscalculia. I didn't expect it to, but I also didn't know what it would or wouldn't change. When I started this, I didn't know where my dyscalculia ended and my poor math education or math anxiety began.

My education in long division took a BIG leap. We went from "reciting multiplication tables backwards" to "long division" with absolutely no steps in between - and absolutely no attempts to explain why long division works the way it does.

So when I got to the first division chapter in the Maths - No Problem series, there was yelling. Especially at these pages:

Left: A page I yelled at because no one ever explained to me that division uses regrouping just like addition, subtraction, and multiplication do.

Again, was I supposed to figure this out? Do ya'll nerds who don't have dyscalculia figure this out on your own? Brag about your lack of dyscalculia in the comments.

Right: The first page that introduces division the way I was taught to do it.

The right side, however, comes with an explanation we never got: why on earth you put the numbers where they go and why you do what you do to them when they get there.

We were walked through the steps of this sort of division like we were little computers who required programming. Which I guess we were. A lot of 1980s US education seemed to think kids were empty vessels who needed to be programmed for maximum usefulness to our elders and betters (aka the Me Generation). And I don't just mean classroom education.

A couple times now, I've talked about "check your work." Specifically, I've talked about meaningful or material work checking.

What I'm trying to find words for is "a way to check the work that demonstrates that the answer reached is in fact correct, per the basics of how numbers work." *

This is why I was dissatisfied earlier with "10 - 6 = 4" as a "check" of "10 - 4 = 6" - those feel more like saying the same thing twice than like counterbalancing facts. "6 + 4 = 10" makes more sense to me as a check there because it does the opposite operation but expresses the same relationship among the three numbers. Counting "9, 8, 7, 6, 5, 4" and noting there are six digits makes more sense to me to, as it relies on the basic relationship among digits to get from 10 to 4.

I grabbed "material" out of the vocab bin because it's a term lawyers throw around a lot, and I used to be one. Lawyer Brain does not find "I said the same sentence but the other way around" to be material evidence that a statement is true. "She and I made a contract on Wednesday" does not prove that "on Wednesday, she and I made a contract" is true.

"On Wednesday, she and I agreed that she would pay me $5,000 and I would deliver to her 5,000 widgets" gets us closer. This statement reveals that at least a few basic contract elements (offer, acceptance, consideration) occurred. This is a more meaningful or material proof: it proceeds toward the conclusion "between us, she and I met all the elements of a contract on Wednesday" and thus "she and I made a contract on Wednesday."

I have had Lawyer Brain long enough to be extremely skeptical of anything that can't show its own work.

My standards for "show your work" are higher as well. My math teachers wanted to see the numbers we carried when doing addition or the borrowing we did in subtraction, for instance. Writing 10s or 9s above the zeroes was what they meant by "show your work."

I don't trust the absolute walnut who crosses out zeroes and puts 10s or 9s in my arithmetic problems. I know that walnut. That walnut is terrible at arithmetic.

I do trust methods that demonstrate to me that the answer I reached inexorably has to be reached because that is how numbers number. Like counting on. Or rearranging piles of beans. 8x4 definitely is the same as 4x8 because I need the same total number of beans to make both four piles of eight beans each and eight piles of four beans each.

One thing that has frustrated me for decades is that we weren't taught material ways to check our own work. We really were just expected to look at an answer and go "wait, that can't have three digits in it."

Which is not something a walnut with the number sense of a peanut can do.

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"Ms. F, are you just taking photos of every math thing that makes you scream now?"

yes. yes, I am.

(Still Maths - No Problem, Textbook 3A)

So I bought the tens unit bricks because I know I am notoriously bad at adding/multiplying/etc. that involves tens. zeroes. orders of magnitude. whatever you call it. If it has zeroes at the end, I'm going to muddle it up.

And sure enough, I got anxious again looking at these sample problems. Until I looked at the helpful cartoon children (my personal math saviors) and their talk bubbles.

ARE YOU SERIOUSLY telling me that ALL THE YEARS I HAVE HATED AND FEARED QUESTIONS LIKE THESE, I could have just said to myself "three times four tens is twelve tens"? Or "four times eight tens is 32 tens"? And then written down "12" or "32" and just stuck a zero behind it??

I didn't have to keep track of that zero? I could have just turned it into a word??

Because HERE'S THE THING, y'all: I did just fine in chemistry when the math had units attached.

Not even kidding. Stoichiometry was my favorite part of chemistry classes, bar none. It looks like math but it doesn't require me to manipulate numbers; it requires me to line up units (moles, milliliters, whatever). If the unit tags all lined up correctly, the calculator did the rest.

ARE YOU SERIOUSLY telling me that if I treat the zeroes like a unit tag (linguistically speaking), THE REST COMES OUT? JUST LIKE THAT??

ALSO, ARE YOU SERIOUSLY telling me that THIS WORKS WITH ALL THE ZEROES??? My number sense still sucks but I strongly suspect this works with all the zeroes.

ARE YOU SERIOUSLY TELLING ME RIGHT NOW, RIGHT THE ACTUAL CRISPY CHEESESTEAK EFF NOW, that I can just say "three times four hundreds" in my head and write down 1200 and call it a day??????

I DID FORTY OF OUR GOOD LORD JESUS HECKING CHRIST'S OWN YEARS FREAKING OUT SIDEWAYS EVERY TIME I HAD TO MULTIPLY BY SOME TENS OR HUNDREDS WHEN I COULD HAVE JUST DONE WORDS??!!?!?!?!?!

I hate you.* I hate you all.**

You owe me cheesesteak.

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*by "you" I mean my former math teachers

**who are nearly all dead now and thus immune to my displeasure

(Maths - No Problem, Textbook 3A)

So far, I've been annoyed that my primary math education:

deprived us all of concrete and pictorial representations extremely early,

withheld alternative means of doing problems,

neglected to teach us simple methods of doing proofs that would have both built our number sense and taught us ways to check our work meaningfully,

expressly prohibited us from using methods that relied on number patterns and relationships in favor of brute force rote memorization of disconnected nonsense, and

blamed the student's "laziness" or "poor parenting" or "being a girl" or whatever other excuse was convenient when students lacked neurotypical-grade number sense.

Today, I am annoyed because I would probably have really enjoyed algebra if it had been presented like the above.

I sucked at numbers, but I really enjoyed logic puzzles as a kid. I checked out every book of logic games our library had. I cleaned up in every game of Mastermind. When I took the LSAT, years later, I scored highest in the logic games section.

The logic games section of the LSAT is notoriously designed to make test-takers reconsider their decision to go to law school. CPAs and dentists only exist because the logic games section forces aspiring attorneys to embrace their fallback plans. Nobody scores highest on the LSAT in the logic games section.

I was today years old when I realized algebra is just logic games written in unnecessarily weird and dense (to me) notation.*

The downside? I'm screamy again.

The upside? If it's just logic games written in a weird language, I can figure it out. I'm good at logic games and languages.

Herein marks the first time in 42 years I have said the words "I can figure out algebra."

Thanks once again, cartoon children.

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