Early Operations by M. Small

This article explains the four basic operations in numeracy: addition, subtraction, multiplication, and division. These are such basic concepts in mathematics that they are difficult to comment on. Yet, what seems simple to an adult, may be quite complex for young children learning it for the first time. Even adults may have something to learn about these operations.

Small explains addition and subtraction in detail before going on to the more complex operations. He helpfully explains each rule, gives examples of common errors and how to correct them, and gives good examples of manipulatives that can be used to illustrate these concepts.

The only thing that frustrates me about this article is the way Small describes the rules as things that were made up in a committee as opposed to things that naturally exist. He even says of the associative property that it exists because of the need for rules for adding three or more numbers to be "created". Nobody "created" the associative property, they just put a name on what naturally exists. If you have three sets of objects and combine them into one, the result will be the same regardless of the order you combine them because that's how many there are in the three groups combined. Similarly, I've always been peeved by encouraging students to look for "keywords" in word problems. It always felt to me like an oversimplification that stands in the way of proper linguistic and numeric comprehension to teach students to associate key words with operations in lieu of comprehending what the word problem is saying and why "less than" means to subtract. Rules exist for a reason, they aren't just decided on in a committee.

The second half of the article deals with the rules of multiplication and division. The part I find interesting is the explanations for why it is impossible to divide by zero. Because mathematical principles usually have multiple explanations and different ones work for different people, Small gives two explanations. First he uses repeated subtraction, explaining that no matter how many times you subtract zero from a number, it will never reach zero. The other explanation is that if a number, let's say 5, divided by zero equals n, then 0n equals 5, which is obviously impossible. This has helped me understand why it is impossible to divide by zero. However, I have one question about it: is there a reason imaginary numbers cannot be used here? In any case, students don't learn about imaginary numbers until high school so I guess my question is irrelevant to primary and elementary.

One of the books Small recommends is about a group of 25 bugs that divide themselves different ways and usually have one remainder. It got me thinking of all the ways 24 divides neatly. This gave me an idea for an activity that might be suitable for older grades or perhaps gifted students in younger grades: have students try to come up with a number that divides evenly into 2,3,4,5,6,8. and 9. Hopefully, they will come up with 360, the number of degrees in a circle, and this can introduce them to circle geometry and angle measurement.