December 28, 2015

Horse 2050 - All Things Being Equal But Not Necessarily Equivalent

What is three times five?

This sounds like a pretty easy question to answer and you'd be absolutely correct if you said 'fifteen' but if you were to write this out as a number sentence, there is only one correct answer.
3 X 5 = 15
Again, this might sound like teaching your grandmother to suck eggs (though in all honesty I don't know why she already has that skill - such a thing serves no useful purpose as far as I can tell) but three times five is not 5 X 3=15. That is five times three equals fifteen.

Three times five equals fifteen tells us something fundamental about the grammar of that number sentence. If we take the assumption that multiplication is fast addition then three times five in a slightly more expanded notation reads 5+5+5=15. Whilst it is perfectly true that 3+3+3+3+3=15, these things are different identities.
In mathematics there is a difference between equality and equivalence. Two statements might be equal but not necessarily equivalent. The operations of multiplication and addition confuse the issue because they are both commutative in nature and whilst you can argue that it doesn't matter what order you do a binary number sentence in, anything more than three terms clouds the issue.

Grammar in any language, is the set of rules that you use to read said language. Granted that all languages are by definition human inventions and therefore arbitrary, a set of rules have to be agreed to in order to convey information or else chaos rules supreme.

The clause "I had the car washed" conveys a different meaning to the clause "I had washed the car". The first implies that you had someone else wash the car but the latter says that at some point in the past the car was clean and now it might not be. I can imagine the second statement being said after your sparkling shiny car has just be sprayed by a wall of muddy water because a truck has just driven through a particularly deep puddle.
Likewise, 3 X 5=15 conveys different information to 5 X 3=15.
2 X 12=24 might tell you how many eggs you bought in two cartons but 12 X 2=24 seems to imply that you're buying your eggs in pairs.

Before we even get to any mathematical operation the basic rule is that we read from left to right. 531 is five hundred and thirty one. Even that has that basic rule embedded inside it.
531 is 5 X 100 + 3 X 10 + 1 X 1. With a binary operation like 9 - 6=3 the number sentence also reads from left to right. 6-9 tells you that you are in deficit rather than having three things left over.
All of this seems pretty straightforward and indeed trivial but when you start dealing with things like multiplying matrices together, where the orientation starts to become important, matrices might be equal but not equivalent or even multiplyable.
Matrices are labeled using a row by column notation, m x n. To multiply matrices together, you multiply the rows of the first matrix by the columns of the second. The number of columns in the first matrix must equal the number of rows in the second, or else you cannot multiply them together.

The objection which is raised is 'what does it matter if I get the right answer?'. The issue might not be whether or not the answer is fifteen but whether information has properly been conveyed. An 8X10 bolt will have a different width and thread pattern to a 10X8 bolt. One will not fit into a bolt hole which has been designed for the other.
Of course as with any language, the the grammar of a number sentence is important, if you don't understand the grammar then you can really screw things up or in the case of nuts and bolts, not screw up anything at all.

Grammar in a number sentence as indeed any sentence asks questions like:
- What term belongs with what?
- What order should you work in?
- What relationship do things have with each other?

For a simple number sentence like "What is three times five?" that grammar reads:
multiplier x multiplicand = product
multiplier x thing to be multiplied = product
3 is the multiplier, 5 is the thing to be multiplied and 15 is the product. There is no negotiation and nor will any correspondence be entered into. These are the rules. That's it. 

If it is just pure integers we are playing with then a sum or a product of numbers might very well be enough but the world is far more complex and mathematics itself is far more than just simple arithmetic. The thing is that unless students are taught the structure and grammar of simple number sentences, they will never understand the structure and grammar of complex number ones.

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