Philosophical Natterings: Abstraction

Background

One of the thoughts I find myself returning to frequently is this: There is the belief shared among high school mathematics teachers that the struggle students have with algebra is that it’s the first time they’re exposed to abstraction.

This isn’t true. The first abstraction in mathematics comes at such an early age, and so quietly, that it tends to pass without notice. This is the abstraction away from units.

I’ve written about this topic before, but I’m revisiting it now in order to check in with how my thoughts have developed.

The First Abstraction

In the earliest years of mathematics education, we teach numbers in terms of concrete objects: Three ladybugs. Four apples. Two bears. If three ladybugs visit two dragonflies, there are now five bugs playing together.

At some point, we start emphasizing that it doesn’t matter what items we’re counting, the numbers work the same way. If three books are on the table and two more books are placed on the table, there are now five books. If John has three apples and Peter has two apples, the two (!) of them have five apples.

We can say that 3 + 2 = 5, regardless of what it is we’re counting.

While objects are still used throughout much of elementary school to represent the natural numbers, it is made clear to students that these are merely conveniences. It doesn’t matter whether it’s dinosaurs, pieces of fruit, cars, or whatever else the worksheet maker feels like illustrating: The mathematics works the same way.

In some cases (as with John and Peter above), we don’t even consciously note what it is we’re counting. It is implied that there are two kids, but we probably don’t make that overt. There are simply “two of them”.

And now… multiplication!

This abstraction allows us to gloss over a complication with multiplication that contributes to future confusion, in high school courses (both math and science).

How is multiplication taught in elementary school? Well, if Jimmy has three apples and Sally has three apples, then Jimmy and Sally have six apples. Children are shown that \(3 + 3 = 6\), and that there are two “3”s there, so \(3 \times 2 = 6\) is a nice shorthand for that.

If five bears each have three honeypots, then how many total honeypots are there? Let’s count! 1. 2. 3. … All the way to 15. So \(3 + 3 + 3 + 3 + 3 = 5 + 5 + 5 = 5 \times 3 = 15\). Indeed, my six year old son insists on saying “5 3 times” instead of “5 times 3” for multiplication.

But… here’s the problem. With addition, we’re combining like items to create a bigger group of the same items. With multiplication, we’re combining groups of like items. This is visible in the pictures, but not in the actual operation, because we’re not using units.

Let’s say that history had created a symbol for the abstract unit, \(u\). Addition might work thus: \(3^u + 2^u = 5^u\).

But multiplication does not involve multiplying the same sort of unit. In the examples, we’re not multiplying honeypots with honeypots, we’re multiplying bears with honeypots per bear. If we were going to write this with units, we would write: \[5 \text{ br} \times 3 \frac{\text{hp}}{\text{br}} = 5 \times 3 \frac{\text{br} \times \text{ hp}}{\text{br}} = 5 \times 3 \text{ hp} = 15 \text{ hp}\]

Abstractly, we’re multiplying abstract units by abstract groups of units. Perhaps history had given us another notation, \( ^g\), to indicate abstract groups. Multiplication might then work thus: \(5^g \times 3_g^u = 5 \times 3_g^{ug} = 15^u\).

There are solid reasons for not writing these abstract units. I’m not recommending that we try to start teaching them as a general policy. But at the same time, as I’ve said, ignoring them entirely allows us to gloss over what multiplication involves.

To review: Addition involves combining quantities of the same unit; the base unit does not change, and addition can only be done when the base unit is the same. Multiplication involves combining different units; the base unit always changes, even if that base unit is implied. Multiplication can only be done when the combination of the base units is meaningful.

Fractions

The first time this glossing over becomes a serious problem is generally fractions. Consider: \[\frac{3}{7} + \frac{2}{5} \\ \frac{3}{7} \times \frac{2}{5}\]

Common Core, to its credit, wants to teach fractions as numerators and unit pairs. The units in fractions are the inverses of the denominators. Addition can thus only take place if the units are the same.

This would be an excellent transition… if we were already stressing the notion that “addition involves combining quantities of the same unit”. But by the time fractions come along, students have either entirely forgotten about apples, books, and honeypots, or have at least glossed over the real-world relevance. So 4 is 4, not \(4^u\).

Regardless, with the addition of fractions, the base units have to be the same. However we teach that, it must be taught. Traditionally, this is taught as “fractions can only be added when their denominators are the same”, so we have to adjust the denominators. This is turn is most often taught in terms of the mighty and mystical Least Common Denominator, so students who simply convert \(\frac{a}{b} + \frac{c}{d}\) to \(\frac{ad}{bd} + \frac{bc}{bd}\) and then to \(\frac{ad + bc}{bd}\)—a far simpler algorithm—are told they’re doing it wrong.

Meanwhile, frustrated secondary teachers not only resort to that algorithm in order to try to fix what previous teachers have apparently broken, but come up with excessively cute mnemonics to reinforce it.

Multiplication, meanwhile, is much simpler if we see a fraction as a numerator times a unit: Multiply the numerators, multiply the units. This perspective in turn is complicated by division of fractions, which then requires us to perceive a fraction not just as a value and a unit but also as a unit which is the inverse of the value and as a value that is the inverse of the unit. For instance, \(\frac{3}{4}\) = 3 “one-fourths” while the inverse is 4 “one-thirds”.

Algebra

If students manage to make it past fractions, they hit another bump in algebra. Consider: \[3x + 2x \\ 3x \times 2x\]

In the first case, we can simply treat \(x\) as a unit. So, huzzah, just add the values and keep the unit.

In the second case, we can do the same, but because students have generally not deeply learned that multiplication involves changing the units as well as the value, they might not think to multiply the units together. In this case, the unit of \(x\) becomes the unit of \(x^2\).

There’s a deeper matter at foot here. The issue isn’t that there are “values” and “units” but rather that, for the various operations, we select a part of the measurement to treat as a unit, while the rest is treated as a value. The “unit” can itself contain a numeric aspect.

But this is high-level philosophical thinking, which demands the question: How do we best communicate this to students, particularly younger students? The explanation in the previous paragraph would likely go far above the minds of most school age students.

Geometry

If the issue hasn’t already come to a head with fractions or algebra, geometry will be the make-or-break point. Either students “get it” (consciously or unconsciously) in geometry, or they’re likely to spend the rest of their mathematics experience is a fog.

What is the perimeter of a square with a side length of 1? 4.

What is the area of a square with a side length of 1? 1.

Why oh why, keens the tragic voice of the confused geometry student, is the area less than the perimeter? Look at all that area! Look at how little perimeter there is!

The answer, of course, is that the 4 and the 1 have different units. Sometimes we’ll make those units overt (four inches, one square inch), but oftentimes we won’t.

Geometry is the first place that we need to consistently rely on different units for the same problem set: Length, degree, area, and volume are the basics, and of those, all but degree are often expressed implicitly.

Using my established pattern above, I’ll use \(^l\) for an abstract unit of length and \(^a = ^l \times ^l \) for an abstract unit of area (\(^a = ^l \times ^l \)). So the perimeter of our square is \(1^l + 1^l + 1^l + 1^l = 4^l\) while the area of our square is \(1^l \times 1^l = 1^a\). Said that way, it’s clear why an object might have a perimeter of \(4^l\) but an area of \(1^a\)… they’re not the same unit.

Where to go from here?

I’m still thinking about the ramifications of all this. I have so far worked mostly with high schoolers, who have been riding their mathematical bicycles for quite some time. I’m not sure how to best address this, but knowing the problem is, allegedly, half the battle.