Friday, June 21, 2013

Request: Can we please stop teaching cross-multiplication?

I don't usually write blogs out of annoyance and frustration (not since I grew up anyway), but if there's one thing that annoys me more than most things in mathematics education, it's the retention of such a worthless "tool" as cross-multiplication.

Cross-multiplication is the quintessence of why people are crap at math in America.

At its heart, cross-multiplication tempts us with ease of use and reliability.

...But you're not buying a bloody car, you're trying to learn and educate! Why on Earth do we spend so much time trying to get kids to emulate a calculator? They're never going to be as good as a calculator is at calculating!

But you know what they can be good at? Mathematics. Real mathematics, not the ABC crap we learn and teach in primary and secondary schools. Arithmetic and algebra are to math what the ABCs and grammar are to English.

Do you think people would enjoy English if they spent 12 years learning to read? and then never having read any narratives, only a paragraph or a page at a time?

When does reading for fun start? Grade 1? 2? BEFORE THEY LEARN TO READ?!

When does math for fun start? Calculus? Something most people never get to? And then only if they have a halfway decent teacher?

Real mathematics is asking "Why?", not stating the value of x. Real math is problem-solving, not solving word problems and worksheets. Real math is the universe, not bloody cross-multiplication!

If you doubt this and have gone through a few levels of math yourself, try and ask yourself if you know why we add in long multiplication.

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Well? Did anyone question the teacher when they learned long multiplication? Or was it, like cross multiplication, handed down from on high and left with the promise that you will be able to multiply big numbers and is just a technique that we use because it works, you don't have to worry about how it works.

You should always ask why. (I explain it in Section 1 of ZAM, "Arithmetic and Number Sense: The 'Basics' of Math We Are Never Taught", but the short answer is the distributive property)

Why does cross multiplication work? And what on Earth IS cross multiplication?  At its most base, cross multiplication is a statement that is always true, a/b = c/d if and only if ad = bc, b, d ≠ 0.

It is usually presented as a way of solving proportions in the sense that if we know 3 out of the four in the fraction statement, it is easier to solve the multiplication statement.

But why does this work? and why do I find it so infuriating?

Well, let's scale back and ask how to evaluate 3/4 + 2/6?

Now, I hope, that most of us know that we convert this to twelfths and then add. 9/12 + 4/12 = 13/12 (if you don't know why THIS works, maybe you should start reading my math books? or feel free to ask).

But what about if instead of using 12ths, we used 24ths instead. Now, you might be saying, but 12ths worked out find and the answer is in lowest terms~! To that I'd counter, but 24ths is easier to find than 12ths. You just multiply by the other denominator. We get 18/24 + 8/24 = 26/24

Easier's better, right?

How is this related to cross multiplication? Well, let's start with how cross multiplication works.

In algebra, one of the techniques you learn to help you with equations involving fractions is that if you multiply by the least common denominator (or LCD) then all of the fractions magickally disappear (or rather, their denominators are cancelled out). This is the same idea in a cross multiplication problem. Let's do an example.

x / 4 = 9/12

Here, the least common denominator is 12ths, so we multiply both sides by 12 and get:

3x = 9

Thus, x = 9/3 = 3.

To put it back in the original statement, we now know that 3/4 is the same as 9/12. 

No trickery needed, no magickal techniques. We just cancelled out the fractions, but even that is less than what we could have done if we hadn't turned off our brains when we saw fractions. This says x divided by 4. To solve you just need to multiply both sides by 4!

x / 4 = 9 /12
 · 4         · 4
      x = 3

In ONE step! Now, not all of these can be done in one step, but the fact that we can gives us at least some pause before we turn off our brains.

Now you might be asking yourself, what about cross multiplication? Cross multiplication to this problem is the same as if we had multiplied not by 4, and not even by 12, but if we had multiplied by $*!!!  (note: $* is a capital 48 on the keyboard). By forty-bloody-eight! Not even the LCD, but just a lowly common denominator!

Let's see how it works.

x / 4 = 9 / 12
 · 48      · 48
  12x  = 36

Look familiar? It's the result of the cross multiplication step. Cross multiplication is just a way to hide the truth of multiplying by the common denominator instead of the least common denominator in clearing fractions and completely puts aside the idea of not even needing to clear fractions in this problem in order to solve it.

Algebraically, we can prove the cross multiplication rule using this fact now:

 a / b = c / d
· bd      · bd
abd / d = bcd / d
      ad = bc

Again, the only place this would fall apart is if b or d were zero.

So why do we use something so clunky as the cross multiplication rule? Because students can get the correct answer by following the rule blindly and turning off their brains.

Why is that desirable? Students won't ever be able to be as good of a calculator as a calculator, so why do we focus on that? I'm not saying we should go back to "new math", but this math is just crap.

Why do we want crap?

There has to be a paradigm shift in how we teach and what we expect in mathematics. Getting the answer is like finishing a book, it's next to worthless without the story behind it. If we keep teaching kids to skip all these steps they're going to end up at the bottom of the stairwell. The fastest way to get to the bottom of a building is to jump off the roof but I am pretty sure most people aren't ready for that.

So why are we teaching this bastardized technique that is slower and less efficient than what it comes from and does nothing for understanding?

Mathematics education in America is ridiculous.

Students will fight back against any sort of paradigm shift also. They don't want math to be interesting, they just want to get through it. To endure. Mathematics is everything in the world. Mathematics IS the bloody world. The turn of the Earth is described in math. Math is the language of the universe.

Deal with it.