#math teaching

In the previous post I set up the rules of arithmetic for addition, subtraction, multiplication, and addition. If you paid really close attention you would have seen one thing pop out that wasn't ever described, which was an exponent. It just shows up all of a sudden in my explanation of the multiplicative inverse. In the post before that, I gave a very basic explanation of what an exponent is, but then said I'd be covering it in more depth. This is that, at long last.

I: Couple Bits of Terminology

We label mathematical statements with some terminology in order to more accurately communicate what we're talking about.

In an addition problem, like 3 + 5 = 8, 3 and 5 are the addends or the summends (some people differentiate the numbers as: 3 is the addend and 5 is the summend, because 5 is "being added to" 3 like it was an action). Generally people will just refer to both numbers as the addends. The result, 8, is the sum.

In a subtraction problem, like 8 - 5 = 3, 8 is the minuend, 5 is the subtrahend, and the result, 3, is the difference.

In a multiplication problem, like 8 ⋅ 5 = 40, 8 and 5 are the multiplicands or the factors (again, most people are gonna use just one of those words, and the word most commonly chosen is factor because, let's face it, multiplicand is a mouthful; so, from now on IMPORTANTLY!!! we will only use the word factor to mean "a number we're multiplying with another number"), and the result, 40, is the product.

In a division problem, like 40 ÷ 5 = 8, 40 is the dividend, 5 is the divisor, and 8 is the quotient.

And finally, in an exponential problem, like 2³ = 8, 2 is the base, 3 is the exponent, and 8 is the product -- just as it is with multiplication, because exponentiation is multiplication:

BASEᵉˣᵖᵒⁿᵉⁿᵗ = PRODUCT

II: Exponents/Powers

I'm aware that this is going to be repeating myself a bit, but I think it'll be good to read through anyway!

In the same way that multiplication is repeated addition, exponentiation (raising a number to an exponent or power) is, at first glance, repeated multiplication. We'll say it that way because there's some complication later and even then it's basically true, but it's absolutely good enough for your first exposure.

We write exponents as a superscript number, meaning we are writing it smaller and above where we normally would write, kind of at the top-right corner of the thing before it: in 2³, the 3 is the exponent. The number it follows, 2, is the base that the exponent is attached to.

The exponent tells the base that there are that many factors of itself in this term; in this case, 2³, there are three factors of 2, like so: 2 ⋅ 2 ⋅ 2

There can be more than one base in a term, and not every base will have a visible exponent. If there is a base with no exponent attached, assume its exponent is 1, meaning there is just that one factor of that base in the term. Some examples:

5⋅2³ has one factor of 5 and three factors of 2, making it 5⋅2⋅2⋅2

7³⋅5 has three factors of 7, and one factor of 5, making it 7⋅5⋅5⋅5

2⋅3⁴⋅5² has one factor of 2, four factors of 3, and two factors of 5, making it 2⋅3⋅3⋅3⋅3⋅5⋅5

IIa: How to Read Exponents

So how do you say 7³ -- you know, like, out loud, so other people can understand what you're telling them?

First, we can say "7 raised to the exponent of 3." But that's a bit stuffy. We can replace that word "exponent" with "power," as we saw briefly above.

So now we can say "7 raised to the power of 3." Still stuffy. Change the ordering a little and we can then say that it is "7 raised to the 3rd power."

But if we're only ever "raising" a number to a "power," do we have to say that it's being "raised" every time? No. So, we can then say it is "seven to the third power."

But when else do we ever say something is a power? Never! So, finally, we can reduce this to "seven to the third."

7³ - Seven to the third

2⁵ - Two to the fifth

etc.

Also, we have a couple of shorthands for the specific 2nd and 3rd powers, which is useful since these are the exponents you will end up using the most often!

When something is raised to the second power, like 9², we can just say that 9 is being squared -- "nine squared." This is because if you find the area of a square whose sides are a length of 9, you would do 9⋅9, which is 9².

Likewise, if something is raised to the third power, like 5³, we can just say that 5 is being cubed -- "five cubed." This is because if you find the volume of a cube whose sides are all a length of 5, you would do 5⋅5⋅5, which is 5³.

IIb: Raising to the Power of Zero

This is actually one of my favorite lessons to teach!

So, when we have 5³, that means we have 5⋅5⋅5, which is 125. In the following, think of the left side as the exponential form of the term, the middle side as a demonstration of the factors in the term, and the right side as the value of the term.

5³ = 5⋅5⋅5 = 125

If we take one factor of 5 away, we now have two factors of 5, which makes it 5².

5² = 5⋅5 = 25

Now we'll take another factor of 5 away, so we now have one factor of five, which makes it 5¹.

5¹ = 5 = 5 --- WAIT!!!!

Something is funny here. If you didn't notice, our middle section, our demonstration of the factors in the term, has a subtle issue. And that's the word "factors." What are factors? Remember from earlier in I: Couple Bits of Terminology, I said, "IMPORTANTLY!!, that factors are numbers that multiply with other numbers." So, uhhhhhhh.... what number is 5 being multiplied with?

Well, remember way way back in the previous post when I was talking about the Properties of Arithmetic? There was one there that was kind of important: the Identity Property. This states that any number multiplied by the multiplicative identity remains unchanged -- anything times 1 is itself. And any equality in mathematics that works in one direction, must work in the other. This means that if you have a number, it is secretly being multiplied by one.

Every number. All the time. Even the ones you're multiplying by secretly, are secretly multiplied by one.

We just don't write them because, wow, what a hassle it would be to write literally infinite multiplications of 1 every time we want to do math. We'd (literally) never get anything done.

So that 5? Is actually 1⋅5. And, to be honest, the 5¹ is also 1⋅5¹

1⋅5¹ = 1⋅5 = 5.

And this holds true for every row above as well!

1⋅5³ = 1⋅5⋅5⋅5 = 125

1⋅5² = 1⋅5⋅5 = 25

1⋅5¹ = 1⋅5 = 5

This important bit of setup will help us understand what's coming.

Because what if we take another factor of 5 away? Well, we now have zero factors of five: 5⁰

And if we take that factor of 5 away from our demonstration of the factors in the term, we just lose the five from our 1⋅5 leaving us with... 1.

1⋅5⁰ = 1 = 1

Every number (except for zero -- math nerds fight me! we'll see more about this later) raised to the power of zero is 1. This is the zero property of exponents:

x⁰ = 1 while x≠0

IIc: Raising to a Negative Exponent (Pt. 1)

Continuing our trend:

1⋅5³ = 1⋅5⋅5⋅5 = 125

1⋅5² = 1⋅5⋅5 = 25

1⋅5¹ = 1⋅5 = 5

1⋅5⁰ = 1 = 1

What if we take another factor of 5 out? Well, there aren't any other factors of 5 in the term, so how would we do that?

If we look at the results of our experiment so far, from one row to the next (top to bottom), we might see that removing a factor of 5 is effectively the same as dividing by 5:

125 ÷ 5 = 25, 25 ÷ 5, = 5, and 5 ÷ 5 = 1.

So the process of removing a factor is the same as dividing by that factor. What's more, we know that every time we remove a factor, the exponent reduces by 1. So, from 5⁰ to the next row where we remove a factor, we'll get 5⁻¹, and we'll have to divide by 5 to get what that is!

1⋅5⁻¹ = 1 ÷ 5 = 1/5

So we get the fraction one-fifth as the result. We'll talk more about why this becomes a fraction in a moment, but first! Notice that this means that 5⁻¹ = 1/5, whereas 5¹ = 5. 5 is actually a fraction whose denominator is 1: 5¹ = 5/1. Comparing the results from 5¹ and 5⁻¹, which are 5/1 and 1/5, we see that these are fractions that have the same numbers but are sort of upside down.

When you take a fraction and flip it over like that, you have found its reciprocal. So:

A negative exponent is the reciprocal of the result you get from a positive exponent!

Okay, so, why would any of that be? Well...

III. Some Facts About Division

Fractions are division problems. Whenever you see a fraction, you are actually looking at a division problem. Reading 6/2 as "six halves" or "six divided by two" makes no mathematical difference; these two things have the same value, which is 3.

IIIa. Division Sign

You ever wonder why the division sign looks like that? I mean. Probably not, right? Why does the addition sign look like that? We made it up! But it turns out, there's a specific reason for the division sign's appearance!

It turns out the division sign, ÷, is just a way of formatting a fraction on a single line! We can do that too with the slash, like in 1/5, but 1÷5 is in fact the same thing! It turns out the dot on the top of the horizontal line in the division sign is the numerator of a fraction, and the dot on bottom is the denominator! So it's telling you to take the number to the left and put it on top, and the number on right and put it on the bottom!

So, yeah, you can think of the fraction line (the "vinculum") as the actual division sign, and the commonly-used division sign as a formatting shorthand to make it easier to write on, for example, a printing press or typewriter.

IIIb. The Word "Negative"

What does the word "negative" mean?

The first instinct of most people is, "less than zero." This is not correct. That is the definition of a "negative number." But that's not the definition of "negative."

The root of "negative" is "negate." To negate something is to reverse the effect of that thing; to cancel, undo, reverse, or revert. A negative stands in opposition to a positive. Simply put:

"Negative" means "opposite"

A negative number is the opposite of that number. What do we mean by that? We mean that it stands on the opposite side of the number line, with zero being at the center. 6 and -6 are both 6 away from zero, but in opposite directions from zero.

Subtraction is negative addition -- the opposite of addition. To subtract is to add a negative number. 9 - 5 is the same as 9 + -5. Here we can see the concept of "negative" can affect operations, and the negative operation is always the inverse operation. The inverse of addition is subtraction.

IIIc. Negative Exponents

Likewise, the inverse of multiplication is... division! But in what way is it negative? It's not that the numbers we're dividing become negative. Remember we have described negative exponents as "removing factors." This is the thing that is becoming negative: the exponent on the divisor!

6 ÷ 2 = 6 ⋅ 2⁻¹ = 6 ⋅ (1/2)

So this is where we get to finish our "table" from earlier:

1⋅5³ = 1⋅5⋅5⋅5 = 125

1⋅5² = 1⋅5⋅5 = 25

1⋅5¹ = 1⋅5 = 5

1⋅5⁰ = 1 = 1

1⋅5⁻¹ = 1÷5 = 1/5

1⋅5⁻² = 1÷(5⋅5) = 1/25

And again and again, they try to explain that math is very much within everyone's reach and can be fun and, at least in western countries, education was to blame, messing up this very doable and fun thing by teaching it wrong

But I still gotta wonder - why math? If it is really just education messing this up, why does it mess up so much with math, specifically? I'm sorry but I still cannot shake the sense that even if it's just bad teaching, math is especially vulnerable to bad teaching.

#1: I think they are fundamentally not a helpful part of the learning process and that the benefit of them having more free time so they can be happier and more willing to work during class is more important.

(Bonus points if you can name the anime from whose blooper reel I got the title of this post. No really - it became a part of my personal meme lexicon in the early 2000s from an anime dub blooper reel, and for the life of me I cannot remember which anime it was. Now that we've gotten the depths of my sad nerdery out of the way, let us talk about how much I suck at math.)

Like everyone else in a US high school the last week before break, I did not do much actual work the last week before break. I handed out a lot of candy canes and shelved a few books. I made a to-do list of projects that are Next Year Me's Problem. And I Googled "dyscalculia."

This isn't the first time I've looked up "dyscalculia," but it was the first time I did so on a mission: to do anything at all other than throw my hands up and go "welp, brain broken, too bad no numbers for me."

So I started looking at books and articles on dyscalculia. I read about how to build adaptation skills in kids that have it and how adults can adapt to it too. I learned that not everyone with math anxiety has dyscalculia, but everyone with dyscalculia has math anxiety.

And I learned that, with or without a numbers learning disorder, the way I was taught arithmetic was probably the worst possible approach offered in the entire previous century.

Let me back up.

The current "gold standard" for teaching math to kids is the CPA approach. It stands for Concrete, Pictorial, Abstract - the sequence by which new ideas are introduced. (It's also called the CRA approach, for Concrete - Representational - Abstract.)

Teaching a kid how to count to 10? Hand them 10 of something. Have them move the pieces around, line them up, make groups. Eventually, add pictures of groups of things (apples, kites, washed-up baseball players). Later, add the number symbols we all know and love (in the US, 1, 2, 3, and the rest of the squad).

This is the cornerstone of the Singapore method, which consistently lands Singapore's students in the top math scorers worldwide - by a lot.

Above all, this method stresses, do not withdraw concrete and pictorial representations too soon. We're hard-wired as a species to like concrete counting methods. We have ten fingers, and we use them for more than picking our noses!

Flashback to my first grade classroom.

My first grade classroom had four or five abacuses - the kind with the rainbow beads on horizontal strings. Being a child whose favorite color was "rainbow," I loved these things. I would choose them over toys during playtime.

I was not allowed to use them for math lessons.

Why? Because the abacuses were reserved "for the slower kids" (my teacher's exact words). I was "very bright"; I "just needed to apply myself."

(As a multiply-neurodivergent adult who wasn't diagnosed with any of it until adulthood and as someone who works with high schoolers every day, let me just say: the phrase "you're smart, you just need to apply yourself" should be a one-way ticket to a full LD screening. #justeducatoropinions)

I had math anxiety by the end of first grade. Because "applying myself," whatever that meant, didn't bridge that gap between the concrete and the abstract. Numbers were weird alien symbols I could decode only occasionally. Sometimes they just wouldn't talk to me. I couldn't explain why. (I was six!)

Learning at age 42 that I was systematically deprived of the tools required to succeed in math from the age of 6 is doing a number on me. On that first day, I had to take a fast three laps around the high school track to calm down before my lunch break ended.

When I got back to my desk, I of course went looking for better textbook options.

I find the artwork the Maths - No Problem! series, part of the UK National Curriculum, weirdly soothing. It's cute and friendly. And the fact that every single lesson is modeled with little cubes reassures me that no one is going to make me rely solely on those devious little "digits" until I'm good and ready.

The series is also available for free on the Internet Archive. Since I'm not looking for the best books so much as something better than I had as a kid, I've decided to start here. With Book 1A.

Let's see what happens.

Further reading:

What is the Concrete Representationa Abstract (CRA) Approach and How to Use It In Your Elementary Math Classroom

What is Singapore Math?

Singapore Math (Wikipedia)

Euler's Identity and the 3D unit helix

Sine and cosine are the 2D projections of the unit helix.

When I first saw the concept of the unit helix some years ago, trigonometric functions made so much sense.

Unfortunately, I never heard of any of it in school. Trigonometry was taught as a very unintuitive thing, as something one could not have a connection towards, as something not really 'integratable' into one's own mind. There were no elaborations about 'why' equations are the way they are, nor how the methods are used in relation to the "why", and what these methods are in concrete - in a manner which can be plotted in one's own imagination.

The abstraction appeared alienated from the own thinking process. No inner comprehension was really possible.

Yet, seeing only this concept with no further words added, it was an "Eureka-moment" for me.

I'm so good at getting kids excited about math to the extent that they're annoyed that we're finishing!!

I have a number line.

So, I'd like you to imagine that you've sat down across from me on your first day in class, and I put some kind of math worksheet with this number line on it that gives you a very simple-sounding task: "Circle four."

If you are like nearly every one of the older students I've had, you'll circle the number 4 on the line. And I'd say something like, "That is a perfectly reasonable response to this request. There is nothing wrong with what you've done. But aren't you suspicious about how simple this question sounds? Did you see that there was a twist coming?"

On a number line, the place where we've written 4 is not actually the value 4. It is more like the position 4; we would call this the ordinal number (a number used to track the positional order of things; another example of an ordinal number is the word "fourth," which tells you that there are three other things before that one).

So, if you just circled the number 4, you definitely circled the position, but the thing you circled isn't worth 4. If you were going to circle the thing that has the value (or cardinality, since we're counting individual objects) of 4, it would look something more like this:

This is because in order to have the value of 4, you need everything that came before it! Imagine counting out candy while your kid sister steals and eats the candies you counted already. When you get to your final number, that number is not how many candies you have, but rather how many you counted cuz your sister's got 'em all! For the value of your candies to stay the same, you need all of them to be there!

"Four" could also look, less straightforwardly, like this:

The distance from the start of the circle to the end is four; and since we're circling the line, the length of the line that is circled is four. If we imagine the circle starts (from the left) at the position of 2.3, it would have to end at the position (from the right) 6.3. If we find the difference of the numbers at these positions, we would be doing the equation 6.3 - 2.3 = 4.

A quick summary:

The position of a number is shown with an ordinal number.

The actual physical quantity of the individual items is shown with a cardinal number, which we more typically call the value.

We can find a value inside the ordinals by finding the distance between any two ordinals, via subtraction.

Okay... But notice we've used two words here that seem to imply the same thing: difference and distance. Let's clear this up real quick:

Difference is the result of subtraction. Generally, if we want to find the difference between two things, we subtract the first thing from the second thing:

second - first = difference

But that's not a rule, so much as a norm. In reality, if you wanted to find the difference of two numbers, the person asking had best be very clear which way they want you to do it, because if you reverse this you will get the negative of the result above!

For example, if you were asked to find the difference between 6 and 15, you would do 15 - 6 = 9. But if it's not clear and you instead did 6 - 15, you get -9.

If, however, someone wanted to know how far apart these two numbers are, they would not be asking the difference but rather the distance between these numbers. The result would therefore be the absolute value of either one of these subtractions:

|15 - 6| = |9| = 9, |6 - 15| = |-9| = 9 Same result!

This is in actuality the real use of absolute values. Absolute value is typically shown as "the distance from zero," which is kind of true? But it's really for finding distances of any kind -- in fact, it's later used to mean exactly that when you get into complex algebra (that is, algebra with imaginary numbers) and linear algebra (matrices and vectors).