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Last week we had an interesting little debate about math education. It does happen every once in a while that we talk about maths, and I always find it interesting. I have quite a few mathematicians in my groups and in general like to know a little about my group participants’ fields of expertise. I also used to like maths in school before I had the wrong teachers. And yes, there is no doubt about it: your success in maths has little to do with your brain (or gender) and all with your maths teacher/s.
The question that came up concerned arithmetic i.e. basic calculations. The debate was about whether you should, or should not, teach elementary school kids the order of operations as early as possible, or, in other words, when and HOW should you do so (what is meant by order of operations will become clear in a minute)
To give an example:
2 + 2 x 3 can be 12 or 8 depending on what you do first.
So if we count 2 + 2 is 4 and then multiply by 3 we have 12 and if we multiply 2 x 3 first and then add 2 we get 8.
In the elementary school of one group participant’s kid, the teacher had allowed both results in the beginning of the childrens arithmetic education. The father, though, was opposed to this practise, and would have preferred the teacher to teach it ‘correctly’ from the beginning, assuming that allowing both solutions would be wrong and confusing when they had to learn that you always do multiplication/division before addition/subtraction. I wasn’t sure I agreed. In any case, the question was about ‘the right way to learn something’ and that always gets me going (I am a teacher afterall, and I know about suffering from ‘the wrong way’.)
Common convention in our Western societies is to read and interpret something from the left to the right, so at first sight, the second calculation would seem ‘unnatural’, unless we rewrite it to 2 x 3 + 4. So if we follow this convention, we could claim there wouldn’t really be a problem at all. Just calculate from left to right. This is actually how I would prefer it to be presented to school kids (or to me as a school kid if I could go back in time).
Once learners have mastered the basic four operations and the teacher starts mixing them all, it could be demonstrated how a different order of calculating would lead to different results. And how this could lead to problems once the calculations become longer or more complex. (I never liked the didactic practise of just being confronted with a rule without the reasoning behind it. I strongly believe that every rule has a reason.)
So in the context of the above scenario, I was more in favor of the teacher’s position.
In the following discussion, the father of the school kid wanted to demonstrate, how doing multiplication before addition also made more sense from a mathematical point of view; how it was the ‘more logical thing to do’ and not just an agreed upon convention.
I must admit that I couldn’t follow his argument, but wasn’t sure if this was because the mathematical part of my brain was so dormant it couldn’t grasp his explanation, or because the explanation was maybe too constructed and not convincing; or maybe correct, but only in a certain arithmetical context that had nothing to do with the underlying question if the order of operations was based on convention or logic.
In the following I consulted my math books (I do have a few as I was never happy with my disrupted and interrupted math education. However, like learning a new language, there is never as much time for learning complex systems than when we were young/er).
And I went on an internet search. Here I found another wonderful website: The Khan Academy, an extensive treasure trove of teaching. There are loads of the academy’s educational videos on youtube. The one I watched was on the ‘Order of Operations’.
Most of us will be familiar with this ‘rule’ – called ‘Order of Operations’ – that you first multiply or divide and then add or subtract. However, I wanted to know if this was just a matter of convention or if there was a mathematical or logical reason for doing so.
I have found in the past that in many respects mathematics is like language: there are certain rules and conventions that need to be follow if mathematicians want to be able to arrive at the same results, or – to keep the analogy to language: want to communicate sucessfully.
One of these conventions that was agreed on (it doesn’t seem to be totally clear when and by whom) was, indeed, to fix the order in which basic calculations were performed. The complete order being: PEMDAS – Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
Mark Zegarelli, a mathematician and author of several “For Dummies” books also plays with the analogy to language when he says (and I quote from chapter 4, Basic Math and Pre-Algebra Workbook for Dummies,2008: Wiley Publishing Inc.):
It’s just an Expression
An arithmetic expression is any string of numbers and operators that can be calculated. In some cases, the calculation is easy. For example, you can calculate 2+2 in your head to come up with the answer 4. As expressions become longer, however, the calculation becomes more difficult. You may have to spend more time with the expression 2 x 6 + 23 – 10 + 13 to find the correct answer of 38.
The word evaluate comes from the word value. When you evaluate an expression, you turn it from a string of mathematical symbols into a single value – that is, you turn it into one number. But as expressions get more complicated, the potential for confusion arises. For example, think about the expression 3 + 2 x 4. If you add the first two numbers and then multiply, your answer is 20. But if you multiply the last two numbers and then add, your answer is 11.
To solve this problem, mathematicians have agreed on an order of operations (sometimes called order of precedence): a set of rules for deciding how to evaluate an arithmetic expression no matter how complex it gets. In this chapter (…) (page 49)
In the following week, I showed the group the web site of the Khan Academy and we watched the Introduction to Order of Operations together.
(to be continued)