I’m sure your approach to teaching math is sound, but “you can’t do it” is just another way of saying that zero is special, that it can’t be used in all the same ways that other real numbers can (“real” in the mathematical sense). Numbers existed long before its invention. Indeed, the concept of “zero” was invented at different times in different cultures; according to Sci-Am, zero was “devised in India in the mid-fifth century, spread to Cambodia near the end of the seventh century, and into China and the Islamic countries at the end of the eighth. Zero reached western Europe in the 12th century.” Pretending that it’s just an ordinary integer like any other can be confusing.
Yes. Except that it gets a little confusing because doubling the radius means quadrupling the area, and octupling the volume if all dimensions change proportionally.
Zero is special, not because you do things differently with zero, but because when you do the same thing with zero, it gives interesting results. “You can’t do it” isn’t “just another way of saying that zero is special.”
Division by zero is one of the very few “you can’t do it” cases in mathematics. Extrapolating from that edge case that zero is treated differently by other operations is unambiguously incorrect.
Or to put it another way: Zero is certainly weird when it comes to division. But when it comes to addition or subtraction, it’s absolutely the most ordinary number there is.
True, but zero also behaves oddly in multiplication. Cite: the OP, and this resultant thread. When a number can never be used as a divisor, and that number as a multiplier always produces a result equal to itself regardless of the multiplicand, then you’re dealing with a number that behaves unlike any other in two of the four basic arithmetic operations. That’s not an “edge case”, that’s special.
On first fast reading my brain is kind of going: “ding, ding, ding, ding ding!”
But I’ll follow up with a closer examination…
For now, between my post of last night and this one: what is necessary for my mind-of-meat to achieve genuine comprehension, vs. an ability to robotically feed back what I have been told: (Someone mentioned their daughter “learning” math via “brute force” tactics. This recalls the method I had to resort to, with the invaluable tutelage of a still-dear friend, to escape high school with a “Proficiency” certificate in 1976. Six months later, I am certain I couldn’t have done 80% of what I had to do for that test.) it is crucial to eliminate as much mathematical language as possible, because it is no more meaningful to me than trying to explain it to me in Mandarin.
What you did here is what works: make it as a real world-touch/taste/see/hear/smell as possible, using language that I comprehend completely at the start.
I also wanted to address some speculation about my early education.
I was an absolutely stellar student from the moment I entered school. I could read at probably a third or fourth grade level when I began first grade, well into college level/adult reading by the time I entered middle school. (It was like Christmas morning every time the Scholastic Books order came in: nearly all of it, usually a couple of feet high, was mine!)
Arithmetic was no sweat when it was just adding and subtracting. I recall with a shudder of shame and frustration how mortified I was when multiplication was introduced and my brain failed me. And division! Oy!
My self-worth was almost entirely based on being a star student, teacher’s pet, and my father’s greatest source of pride due to my brilliance (also my beauty, but that was starting to be obscured by my weight around the same time I was introduced to multiplication. Gee, do you suppose they might be related?) This “math” bullshit represented an existential threat and I very effectively erected impregnable barriers to facing it at all.
As an adult, and, for all intents and purposes, an autodidact driven by innate thirst for knowledge as well as a total fascination with computers, I came to understand what others have said, which is that there was nothing fundamentally wrong with my brain’s ability to comprehend basic mathematical concepts, and even more advanced concepts. My absolute favorite thing to do on computers, (aside from this message board stuff, which absolutely dominated and even directed the path of my life for probably 20 years.) is dick around with spreadsheets, databases, and automation. Love it! But thanks to that impregnable barrier erected so early in life, I have run into a lot of walls that prevented me from fully exploiting my fascination to the degree that I would like.
And the topper to that autobiographical caca nobody really gives a shit about: I found out at the age of 50 that I have extremely severe ADHD. I’m practically the poster child and once I devoured all the information I could about ADHD and reviewed my life through that lens, decades of terribly confusing struggles became perfectly clear. First and foremost, my report cards through middle school and high school, which mostly reflected A’s in English while failing everything else because I just never showed up.
But thanks to the proficiency exam previously mentioned, I was able to attend Los Angeles City College a few years after high school where I maintained, effortlessly, a 4.0 in subjects like psychology, philosophy, journalism and my main reason for attending, which was broadcasting.
And I wanna thank you all for your attempts to help me understand so many things! It’s been almost 25 years of hanging out here, off and on, even through some brutal times, and I still love you guys. When I have exhausted the Google, I know where to go. May it never change. 
You’re not alone.
I wasn’t bad at math (I got good grades in school) but it was one of my worse subjects relatively speaking. And when I took the SAT my verbal score was significantly higher than my math score.
My brain doesn’t do well with abstractions. It does better with practical stuff. That’s why chemistry was painful; the weird crap with molecules and valences and whatever, you’re describing stuff I can’t see.
Math was awful until my senior year of high school when I took physics. Then all of a sudden math was awesome. I was learning stuff I could see. I was learning formulas that could predict things in real life. I was fascinated by it all and I used to challenge my teacher all the time with questions and extrapolations that sometimes stumped him, and he literally used to be a rocket scientist for the government.
Also, I’m an IT guy, that’s what I do as a career, and 99% of what I know was learned on the job by doing stuff. Not from school where you just learn theories about things. (I was a computer science major at first in college then swapped to business administration because I realized I could never be happy writing code for a living; ironic that through total chance I ended up in IT anyway.)
You’re not alone. A lot of people are like you and need to see something tangible to really understand it.
I’m so glad! It’s definitely the way I have to think about math in order to understand it myself.
My first “teaching math” professor gave me a guideline I think about almost every day that I teach math: start concrete, move to abstract, and if a student is confused, go back to concrete.
CONCRETE: That’s where you actually break out the beans in bowls.
SEMI-CONCRETE: Draw pictures of beans in bowls.
SEMI-ABSTRACT: Draw dots in circles instead of beans in bowls.
ABSTRACT: Write equations.
I teach algebra to fourth graders, using a phenomenal system called Hands-on Equations, which follows these principles. It’s incredible to watch these nine-year-olds, after a few months, solving 4(3x+2)-2x=6x+20. All because they started with a pretend balance scale and a bunch of pawns.
By the way, great book… I need to get a new copy and read it again. It’s been decades, but I remember I loved it.
Innumeracy: Mathematical Illiteracy and its Consequences
And why did this stupid board let me upload an image of the book and then tell me “you’re not allowed to embed media”? Why the hell does it let me do it if I’m not allowed to do it?
I was just beginning to kind of warm up to this board. This stupid stupid board. I miss vBulletin and its kin…
But again, how come other people have visuals showing up and all I seem to be able to do is just put a link (just like it was in vBulletin days 
) and why can’t I get a visual?
You can. You just need to know how. You have to paste a link to a picture, you can’t paste the picture itself.
There’s a discussion here on how to make it work.
I don’t remember being able to post pictures on the old board within posts, just links to them.
It’s sort of like trying to drive when you’re new to cars, and after driving an hour griping about how you miss your bike, even though your bike couldn’t have gotten you to where you drove.
That was amusing. Thanks.
I think I had the first response in the original post and mine was along the lines of how many times you count something: 1 frisbee x 5 is a count of 5 frisbees; 1 frisbee counted zero times is a total count of zero frisbees.
However, after reading this thread and seeing credits and debits mentioned, I have decided to multiply my debits by zero and divide my credits by zero. CHA-CHING!
They didn’t start testing kids who have trouble with math for dyscalculia till the late '70s. I suppose if you cared to you could get that done, though it’s probably expensive.
Unit cancellation may help with understanding. Along with multiplying the numbers, you can multiply the units. So:
5 days * 8 worms/day = 40 days*worms/day
Two things we know about numbers:
- The commutative property means that you can flip the two numbers in a multiplication and get the same result: 5 * 8 = 8 * 5 = 40.
- A number divided by itself = 1, so you can get rid of those two numbers and replace it with 1, e.g. 73/73 = 1.
The same things are true about units:
- you can flip two units around in a multiplication and get the same result: days * worms/day = worms * days/day.
- a unit divided by itself = 1. So days/day = 1.
So 5 days * 8 worms/day = 40 days*worms/day = 40 worms * days/day = 40 worms.
For calculating area, you can think of the problem in terms of rates like the worms-days problem above and use unit cancellation to arrive at the correct result. Suppose I have a rectangle 3 meters wide and 8 meters tall. What is its area in square meters?
I can express the height as a rate: for every meter of width, my rectangle contains 8 m2, so the height is 8 m2 per meter (note that the units for the height correctly cancel out to 8 meters).
So now my area is:
8 m2/m * 3 m = 24 m2/m * m = 24 m2 * m/m = 24 m2
This also works for triangles, but it implodes violently when I think about circles.
Huh. It might help for some folks, but it really really doesn’t help for me. While unit cancelation is a thing, it’s very difficult for me to attach it to what I could observe in the world, and the advantage of real-world stuff is that it acts as a bridge between what I know (e.g., eating worms) and what I want to learn (e.g., how multiplication works).
In hard-science disciplines (physics, engineering, etc.), unit cancellation is a critical part of attaching the math to what you can observe in the real world. If you’ve ever wondered why electrical power equals voltage times current, the answer lies in unit cancellation (along with the understanding that “volts” and “amps” are shorthand for more complicated units).
If you drive 60 miles per hour for 2 hours, why do you cover a distance of 120 miles?
I’m addressing this less from a “what’s mathematically true” perspective, and more from a “how do we get people to understand what’s mathematically true” perspective: the latter, not the former, is my specialty.
In this case, I think you can use this example to leverage someone’s understanding of multiplication as a way to teach about unit cancelation. I would be very surprised if you could leverage someone’s understanding of unit cancelation to teach them about multiplication. I can barely wrap my head around unit cancelation, but I immediately grasp how to solve that problem using multiplication; if I want to understand unit cancelation, that’d be a helpful sample problem for me to work with.
I have, alas, not.
Depending on how a person struggles with math, though, the cost may not seem so bad on paper.
(This is sort of a self-own because I mix up digits in numbers all the freaking time.)
The problem I see with ignoring unit cancelation is that it makes math into “just because I tell you” rather than a “how do we get people to understand what’s mathematically true” perspective.
@Machine_Elf discussed this but going back to your initial post on this:
So why does “day” as a unit drop out? What is special about “worms” that allows it to remain while “day(s)” does not? It can almost seem like magic.
I’ve enjoyed your explanations in the thread, and my son struggles with math. (As someone with an engineering degree, I’ve struggled to teach him at times because I just “got” math where he doesn’t.)
In real life math, multiplication is taking a number of something such as days, hours, people, rooms, etc. There is a number and a unit, such as 5 days, 2 hours or 25 students.
Second, there is a number associated with each unit such as “each day”, “every day”, “per mile”, “per person”, etc.
Looking at
25 students X 3 questions per student = 75 questions
this is
25 students X 3 questions / student = 75 questions
Multiplication only works if the “per (unit)” of the second number matches the unit in the first number.
(Should we tell @Stoid that you just introduced the Fundamental Theorem of Calculus?)