One cannot understand the numbers in our number system without understanding place value. Simply put, where a digit is situated in a multi-digit number, or its place in the number, determines the value of the digit. Thus, although 32 and 23 both are made using the same two digits (a 2 and a 3) we know that 32 and 23 are not equal. The 3 in 32 is in “ten’s place” and actually carries a value of 30 (or 3 tens) whereas the 3 in 23 is in “one’s place” and has only a value of 3. Each of the 3’s have a “face value” of 3, but one has a “place value” of 30. The same can be said for the other digit 2 in each of these numbers: in 23 the 2 carries a value of 20 (or 2 tens) whereas the 2 in 32 has only a value of 2 ones or just 2. Since 3 tens has a greater value than 2 tens, 32 is greater than 23.
We have traditional algorithms for doing each of the four main operations (addition, subtraction, multiplication, and division). An algorithm, according to Wikipedia, is a “step-by-step procedure for calculating.” It goes on to say, “More precisely, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating.” Thus, for each operation there is a list of rules or instructions which, when followed, effectively produce the solution. That is a good thing, right?
Well, the answer to that depends on whether the goal is to produce answers quickly or to understand mathematics. Following rules requires little or no understanding of the mathematical concepts underlying the procedure. Worse, all of those who do research regarding the brain and learning report the same thing: the brain remembers things that have meaning; isolated bits of knowledge that are not meaningfully connected to other knowledge that we have are discarded. Sadly, the rule-based approach to mathematics is rather devoid of meaning. The user often has no idea why the rule works. It is no wonder, then, that teachers spend weeks in the beginning of a new school year “reteaching” things that have been forgotten over the summer. Without meaning behind the rules, the rules are easily forgotten.
Let’s look at the thinking processes involved in some computational strategies using the problem 26 + 39. Most of us were taught in such a manner as to think the problem through like this (see example a): “6 plus 9 is 15, so put the 5 under the 9 and carry the 1. Now 1 + 2 + 3 equals 6, so put the 6 under the 3. Reading the bottom numbers together, the answer is 65.” Certainly that set of rules works, but do you notice that the “1” being carried does not, in reality, have a value of 1. It has a value of 10 — we just neglect to refer to it that way. In like manner we ignore the fact that the 2 and the 3 are worth 20 and 30 respectively, but we again never refer to this fact. You might argue that this is not really a problem — we are adding two 2-digit numbers and it makes sense to get a 2-digit number as an answer. But I will say, from many years experience as a classroom teacher and as a mathematics specialist, that the more complex the adding (or whatever operation) gets, the further removed from logic students get. It is, in fact, quite easy to make a mistake leaving out a whole place value spot and get an unreasonable answer, but students do not notice such things. They are not looking at the place value of any of the digits in the problem, and thus have no reference for really thinking about the answer.
I like to have students work with non-traditional methods of computing that keep the place value of digits. I offer three other such methods for your consideration (all of which, by the way, have been used to good effect by young students whom I have taught, and as recently this year!). There are many other ways of non-traditional computing — this is not meant to be an exhaustive list, but rather to give some examples of other possibilities.
Looking at example b, the thought process might sound something like this: “20 plus 30 is 50, so I will record that. 6 plus 9 is 15, so I will record that. But I see that 15 includes a ten which I can combine with the other tens, giving me an answer of 65.” For example c, the thinking might go like this: I see that 39 is very close to the friendly number 40. If I add one more I can use that friendly number. Now I can mentally add 26 plus 40 (adding 4 tens and 2 tens) and get 66. Since I added one extra to make the friendly number, so I must remove it**. My answer is 65. Finally, for example d, the thinking might be this: 39 is very close to the friendly number 40 — just one away. I can take one away from 26, leaving 25, and move that one to add it to 39 to get 40**. Now I just have to add 25 plus 40 (adding 4 tens onto 2 tens) to get the answer of 65.
Examples b, c, and d all keep place value of each number, a critical component, especially when first learning about the operations or different levels of the operations. Later, when understanding is firmly in place, students can move to the “traditional algorithms” and not lose sight of what is happening conceptually. But conceptual understanding needs to be developed for the operations — which takes time. It is not something accomplished in flash.
I could give similar examples of meaning-based methods of calculating for each of the other three main operations, and maybe in a later post I will do so. The point, however, is that we want to students to be thinking about the meaning of the numbers that they work with, to be thinking about the place value of each digit in the number and not just the face value of each digit. It is an important component of understanding mathematics.
What will you do tomorrow to ensure that your students working meaningfully with numbers?
** Both of these examples involve using the “zero principle” — that for any number when you and and then subtract the same amount, you do not change the number. Thus 39 + 1 – 1 = 39, because + 1 – 1 combine to be zero. This zero principle is actually a very powerful principle in addition and subtraction and can be used effectively to simplify many problems.