maths-screaming

[maths-screaming]

I Got Distracted by Internet Security and So Should You

Taking a scream break to fix my messy digital footprint, per this extremely useful and usable guide:

HIGHLY recommend.

[maths-screaming]

Known and Unknown

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.

I did not experience math education that way. And that is a big reason math lost my attention very early in school. That would have cost me even without dyscalculia gumming up the works.

[maths-screaming]

ACCURATE

If I were writing a math textbook I’d have a section in the beginning called “A Note on Notation” where I introduce all the notation Im using. And everyone would want to fuck me sooooo bad

[maths-screaming]

Soroban

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

"Japan's Ancient Secret to Better Cognitive Memory," BBC Reel - video on Japanese soroban classes.

[maths-screaming]

she screm

Starting to feel like my elementary math classes both over- and under-estimated our abilities, fam.

Still in Maths - No Problem Textbook 3A, about halfway through. If I'm reading the book intros correctly, this is what they have you doing about halfway through what we in the US call "third grade" (age 8-9ish):

This is pre-algebra.

To me, this is sixth grade (age 11-12ish). This is "you grew up so much we moved you to a whole new building" math.

I screamed at this one because my brain sees this as algebra, so it tries to do it as algebra. With variables and so on.

But this textbook hasn't introduced the concept of variables! This textbook thinks eight year olds can do this with basic arithmetic!

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. ?????????

Anyway, I tried to do these as algebra problems, failed, and made myself go back several pages to re-practice basic multiplication and division. And then I did them with basic arithmetic, even though now my brain is jumbly because a bunch of half-remembered crap is shoved into the same slot as new arithmetic skills now.

I am confuse.

[maths-screaming]

Things That Are and Are Not Changing as a Result of Re-Teaching Myself Basic Math

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.

Still, if we can fix "poor math education" and "math anxiety," I'll be much further ahead than when I started - and more willing to live with the dyscalculia.

[maths-screaming]

hi mathsscreaming! ive been casually looking at your posts (mostly in the mathblr tag) essentially since its creation and i just wanted to reach out and say im enjoying the project immensely! im currently studying maths at uni (but, contrary to popular belief, also often struggle with actual calculations) and your blog is a fascinating insight into the process of learning! I especially love when you find some pattern (like the 7s thing) and notice something more fundamental about the structure of numbers; because thats exactly the kind of thing that draws me to math! (and also might be useful to me doing mental math) also its just great how open you are about your struggles and discoveries, without being negative about math as a whole, its great.

essentially: 10/10 blog, keep doing the good work, and thank you

~🦐

Thanks! :)

Part of this whole quest was that I'm tired of being negative about maths as a whole. I'm tired of being anxious and hating basic calculations and pretending to be proud that I got away with not having a math class after about age 16. When in fact (a) all that is a big hindrance to my ability to understand other topics that fascinate me and (b) perpetuates the whole "if you don't understand maths you're dumb" myth.

Also, I now work with teens daily, and nothing bruises my ego more than when they ask a question I cannot begin to answer, lol. I need to understand the basics so I know which resources to recommend if nothing else!

[maths-screaming]

Have you seen this short book on the problem with maths education? I think it explains quite well why a lot of people enjoy maths and why a lot of people hate it.

I haven't, but it's on my reading list now. Thanks!

[maths-screaming]

...what.

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.

ANYWAY, I still make dumb number mistakes, but at least now I know why long division looks the way it does.

[maths-screaming]

In Which I Lawyer a Bit

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.

---

*I can't remember the math-world definition of "proof," so I'm avoiding that word. But based on my vague memory of high school geometry, that might be what I mean: a step by step approach from first principles (or basic agreed-upon principles) that reveals how the result is the result.

[maths-screaming]

oh my godddddddd

"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.

---

*by "you" I mean my former math teachers

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

***I still want that cheesesteak

[maths-screaming]

[swears]

(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.

---

*I'm betting every math person in the world is going "actually it's very concise and useful notation, let me explain the many ways in which it is neither weird nor dense." I appreciate your passion, but I won't believe you until I can navigate it myself. Please feel free to recommend some good introductory materials in the comments.

[maths-screaming]

All The Reasons I Can Think of That Basic Multiplication is Easier for Me Than Basic Addition or Subtraction Ever Were

In no particular order.

Mm, donuts.

Elementary school spent a lot of time, over multiple grades, making us memorize every multiplication question from 0x0 to 9x9.*

For a kid who couldn't manipulate numbers to save their life, I was preternaturally good at memorizing long strings of them for no reason - and multiplication tables are "long strings of numbers for no reason" to someone without the number sense to see the patterns in them.

Literally decades of music and dance classes, followed by 11 years and counting of being a part-time semi-paid marching arts choreographer.

My parents had me in music lessons by the time I was three and in dance by age six. So much of music and dance are measured in groups of 2, 3, 4, 6, or 8. Occasionally 12 or 16, if your choreographer is particularly picky about beat stress.**

In hindsight, I'd guess 90% of my being fluent in basic multiplication is from music and dance classes, 1% is math facts drills, and 9% is donuts.

---

*Weirdly, they also explicitly forbade us from using multiplication tables. We were also not allowed to "count by" to reach an answer - so if you forgot, say, 3x4, counting "3, 6, 9, 12" was an automatic "fail" of that question. It was "rote drill these out of order and for no reason" when every single one is a pattern.

**I always swore I'd never make my performers count in 12/8...and then I choreographed sabre for the Harry Potter theme. Sorry, kids, but y'all move wrong when you hear this as 1,2,3,4. I need 123,456,789,101112.

[maths-screaming]

"Check Your Work"

All my elementary teachers wrote the same thing on my report cards:

"Makes careless mistakes because she does not check her work."

I could write a book about how having undiagnosed, untreated ADHD was the primary problem here. But I'll skip that for now.

Instead, I want to talk about a secondary problem with "does not check her work."

To do it, let's look at this example from a few posts ago:

Here, I made the mistake of adding the tens instead of subtracting them, which temporarily gave me "84" instead of the correct answer, "4." I caught it when I started a number line to check that 43 is 4 away from 39 and realized that there was no way I'd need to write out 80+ numbers to figure that out.

At the time, I wrote: I guarantee you that 7 year old me would not have caught that mistake, because 7 year old me would have been expected not to do something as concrete as a number line to check their work.

Here's what I want to talk about:

I'm not sure what 7 year old me was expected to do to "check their work."

I didn't know then, and I don't know now. I think we were supposed to look at our work and find obvious mistakes? Like, I was supposed to look at 43 - 39 and say to myself "wow, that cannot possibly be 80something"? I guess?

But were were never given any strategies to do that.

I'm new to the "counting on" number-line approach; we were never shown how to do those. We weren't even told "add the answer to the bottom number to get the top one" until middle school. I know, because I vividly remember that being a revelation to me. You can just add the numbers back together to get the number? Why did no one tell me this??

So when told to "check my work," I didn't, because I didn't know how. I couldn't tell from looking at it when it was wrong; I needed concrete alternative "proofs," which I was never taught to do. I lacked the number sense to figure them out myself. And, at age 7, I lacked the self-awareness to say "what do you mean by check your work, because I have no idea how to do that."

(I also grew up in an era, and with parents, that would have seen "what do you mean? I don't get it" as impermissible "backtalk," worthy of a "go to your room without dinner." But that is yet another entire book.)

Again, I'm pretty sure non-dyscalculic kids can look at 43 - 39 and say "that can't be 80something if you subtracted." Non-dyscalculic kids can probably also say "I checked by counting and 43 is 4 away from 39." An expert would likely read this post and say "yeah, you have weak number sense."

The problem is that at age 7, there was no way for me to know my number sense wasn't on par with my peers'. And no one with the knowledge or perspective to deduce that fact appears to have done so. So I kept on "making careless mistakes because she does not check her work."

[maths-screaming]

Confession

I'm not going to do the math I'm doing.

Example: I spent a lot of yesterday on all the rules of "borrowing" when one subtracts from a number with a lot of zeroes in it. Like, say, 4000 minus 1724.

But this isn't how I'll be doing these problems in real life.

For practical purposes, it's a lot easier to do 3999 - 1723 than it is to do 4000 - 1724. It's easier because there is no borrowing. And they yield, literally, the same difference.

So why am I bothering to make my brain learn how to borrow all those tens when I have an easier shortcut I plan to use all the time in real life?

Partly it's stubbornness. Skipping the part where I finally re-learn, or figure out, "the way I was taught" feels like admitting I'm too dumb to learn subtraction the way my peers learned it or my teachers taught it.

Partly it's because, thanks to the tens unit blocks, I can see where the "borrows" come and go, and I like seeing that. It gives me the same satisfaction as checking my bank statements does. Yep, I spent that. Yep, I have that much left.

Mostly, though, it's the stubbornness.

[maths-screaming]

A Hypothesis as to Why My 80s/90s K-12 Math Ed Sucked

K so today is "going down the line until someone has a spare set of tens they can loan you" subtraction. AKA problems like "300 - 214" or "4000 - 738."

As usual, I'm doing these with the unit blocks because the primary school textbook told me to:

I'm also doing these with the blocks because I am historically terrible at problems that involve multiple "borrow" steps. Specifically, I can never remember when to borrow and when not.

My fifth-grade teacher once told me I'd make "a terrible banker" because "you borrow too much money!" I think he meant this as a joke. As a kid, I felt weirdly insulted; as an adult and a teacher, I wonder why he didn't do a single thing to help me stop "borrowing too much."

That specific teacher aside (he was legit bad at most aspects of teaching), using unit cubes for multiple-borrow borrowing also has me wondering: Was 80s/90s math education bad because we thought we were imitating good math education?

When the Common Core standards first dropped, the Internet was awash in examples of number lines, "counting on," and other CPA-esque methods of teaching primary math. A lot of these examples got pushback from their posters, who didn't grasp that there are many ways of thinking about numbers.

A friend of mine, a marine biologist, replied: "What most people don't get is that these Common Core examples are how people who are good at math think about math."

(Said friend was very good at math and regularly surrounded by people who are good at math, so I believe her.)

When I was taught basic math, we were deprived of concrete manipulables and even pictorial representations almost immediately. If we demonstrated that we understood what a number symbol stood for (7 = "that's a seven" *holds up seven fingers*), we were deemed "too advanced" for concrete or pictorial representations.

By the time we got to the borrow-ad-infinitum-minus-one type subtraction, most of us had had concrete tools withheld for at least two years (first and second grade) and sometimes 4-5 years.

People who are good at math can look at a problem like 4000 - 1228 and get an accurate answer fairly quickly in their heads. There are no manipulables involved and there may be no jotting answers down, either.

I'm no historian of math education, but I wonder if the prevailing hypothesis was "if people good at math can do this invisibly, then to make kids good at math we should make them do it invisibly too, and as soon as possible."

Anyway I still suck at borrowing repeatedly, so back to the clicky blocks it is for me.

[maths-screaming]

Arithmetic as a Spatial Problem

Doing some subtraction with my new friends the dopamine-fueled clicky unit cubes zeroed me in on one thing that makes math hard for dyscalculia brain:

Even basic subtraction demands spatial reasoning.

To-wit: We read/remember numbers left to right (324 = "three hundred twenty-four"), but when we add or subtract them, we do so right to left (286 + 38 = 4 ones, 2 tens, and 3 hundreds, aka "324").

The unit blocks get me around the "hold this abstract symbol for specific units in your working memory" problem by showing me in concrete terms what the numbers look like.

The unit blocks can be ordered either right to left or left to right, but only one at a time. So if I order them left to right (300, 20, 4), I cannot simultaneously order them right to left (4, 20, 300). Those lineups cannot exist side by side. Those are two different configurations of matter at two distinct moments in time.

Yet holding these numbers in my brain to add or subtract requires me both to remember "286" and "38" and to manipulate them as "6 80 200" and "8 30." And then to reorder the results, which I calculated as "4 20 300", back into "324."

People with dyscalculia frequently struggle with spatial reasoning as well - things like learning to tie their shoes, remembering left and right, and following travel directions. I can't peer-reviewed-study prove that "not knowing my right from my left" and "getting lost when I have to keep a three-digit number in my head in two directions at once" stem from the same source, but the cotton candy feeling in my brain sure seems similar.