Part 1i: Place Value/Arithmetic Models/Arithmetic Algorithms
Suppose I asked you to solve the following two problems.
- Compute .
- Suppose you are cooking dinner for six people. The recipe you are using calls for of a cup of water, however the recipe will serve only four people. How many cups of water do you need to use?
I would imagine that a vast majority of students, teachers, and random people on the street would say that the first problem is far easier than the second. And it certainly is. Problem 1 involves a straightforward calculation; to solve the problem correctly one simply needs to know the correct procedure for fraction multiplication (as well as the answers to and ). However, to solve the second problem correctly, one must understand the operation of multiplication at a level deep enough to know that is actually the correct computation needed to solve the problem. Since the second problem is far more likely to occur in day-to-day life, it is clear that such a deep conceptual understanding of arithmetic is just as important as being able to calculate accurately.
In this section we completely develop an understanding of whole number arithmetic (the models and interpretations we introduce here will also be used in the section on fraction arithmetic). Since place value notions are essential to conceptual understanding of the arithmetic algorithms, we start there. But as discussed above, before we learn the computational algorithms we must develop a deep understanding of each of the arithmetic operations. We do so through use of careful and precise definitions, and we also examine the operations through use of models and interpretations. Once all of this is done, explanations are given for each of the algorithms; they are not and should not be taught simply as an arbitrary list of steps.
- Clips 1 vi and 1 vii involve arithmetic in base 5. This is not necessarily an elementary school topic (although it could be for advanced students), however it can be useful for elementary teachers to examine how arithmetic works in a different number base, as such exercises can shed some light on difficulties their students might have performing computations in base 10.)
- Clip 1 viii is a twenty minute introduction to multiplication; including the definition, an explanation of models of multiplication, and a discussion of how properties of multiplication can be used to learn multiplication facts. One of the important properties discussed here is the distributive property, so if you are just interested in a discussion of that topic, clip 1 ix is a much shorter clip that focuses solely on this property.
- Clip 1 xi is a fifteen minute development of the multiplication algorithm that begins by using a chip model to demonstrate how to multiply by a one-digit number. The clip goes on to extend the algorithm to the general case; for those solely interested in the chip model, see clip 1 x.