Can you do Division? Divide a loaf by a knife – what’s the answer to that? ~Lewis Carroll, Through the Looking Glass
Once students are fluent in place value they are ready to start using those concepts in computation. The Utah Core Standards are pretty explicit in how computation instruction is sequenced. If you read the standards carefully you will note that they speak of having students use strategies to learn how to compute. Then, when they understand what they are doing, they can be taught the standard algorithms. What is the difference between a strategy and an algorithm? Here are the definitions given in the Numbers and Operations in Base-Ten progression document:
Computation strategy. Purposeful manipulations that may be
chosen for specific problems, may not have a fixed order, and
may be aimed at converting one problem into another.
Computation algorithm. A set of predefined steps applicable
to a class of problems that gives the correct result in every case
when the steps are carried out correctly.
When most of us were in school algorithms were the instructional recipe of the day. We were taught the steps for solving addition, subtraction, multiplication, and division problems. Most likely, we were taught them without understanding, meaning that we were just taught the steps and not why they were used or how they worked. That is a main cause of much of the confusion from and dislike of mathematics in general. Unless you have a very logical and organized brain it is difficult to remember all those steps. Now, that is not to say the algorithms are bad. They are not. They are efficient means of solving problems and, as the definition says, you get the correct result every time you follow all the steps correctly. We want students to be fluent in those algorithms.
But that fluency comes at the end of an instructional sequence where students use strategies to solve problems before they move to the algorithm. Why do we do that? It leads to better understanding. Let’s explore why.
Van de Walle (2013) lists three significant differences between strategies and standard algorithms:
- “Strategies are number oriented rather than digit oriented.” When students add 43 + 25 using the standard algorithm they tend to think of 4 + 2 instead of 40 + 20. In some ways, you could say the standard algorithm unteaches place value. This is especially true of the division algorithm. Students (and adults) in a problem such as 345 ÷ 7 will think “How many time does 7 go into 3? It doesn’t, so how many times does 7 go into 34 ?” In reality, we are speaking of how many groups of 7 are there in 345. That is what 345 ÷ 7 means. To use a place value strategy students might do something like this (thinking through the process): Well, I know that 300 ÷ 7 is about 40 because 4 times 7 is 28 so 40 times 7 would be 280. So there are 40 groups of 7 in 345. So I subtract 345 – 280 which means I have 65 left. Now I can use my math facts. The closest multiple of 7 to 65 is 63, which is 9 x 7. So there are 9 more groups of 7 in 345. 65 – 63 is 2. So my remainder is 2. Now I add 40 + 9 and that equals 49. So the answer is 49 R2.” In long division that would look like this:
That looks a lot like the standard algorithm, but the computation was done with place value strategies. Another place value strategy for division is called partial quotient division. Take a look at the video on this page to see what it looks like.
2. “Strategies are left handed rather than right handed”. Often, when students use a strategy to compute, they will begin with the largest values rather than the smallest. In my last post I gave this example:
A student when explaining this strategy might say, “First, I added the hundreds. 400 + 200 = 600, so I wrote that down. Then I added the tens. 50 + 60 = 110, so I wrote that down. I made sure to put the hundreds digit in the hundreds place. Then I added the ones. 7 + 3 = 10, so I wrote that down. I put the tens digit in the tens place. Then I knew I had to add those all together to get my answer. 600 + 100 = 700, 0 + 10 + 10 = 20, and 0 + 0 + 0 = 0. So my answer is 720.”
3. “Strategies are a range of flexible options rather than ‘one right way’.” Students tend to change their strategies based on the numbers they are using. For example, try adding 465+230 and 526 + 98. Did you use the same strategy? I would have computed these two problems like this:
465 + 230 is pretty straightforward. 400 + 200 = 600, 60 + 30 = 90, 5 + 0 = 5. So the answer is 695.
526 + 98 = well, it’s easier to add 526 + 100. That is 626. Then I have to take the 2 I added to 98 back off. So the answer is 624.
Of course, when students are first exposed to a new concept the strategies they use should include using concrete objects, such as base ten blocks, counters, linking cubes, and so on. Using those manipulatives gives them the opportunity to explore concepts with things they can actually touch and manipulate. Students may then move to the representational phase, which means they will use pictures, diagrams, visual models, and so on to take the place of the concrete objects. Then they will move to using abstract numbers, like those we have been using above.
This is true even when students are learning algebra. I was watching a seventh grade class not long ago working on linear equations. The teacher allowed the students to use whatever strategies they felt comfortable with. None of the students chose to use concrete objects. But some used tables, some used graphs, and others used equations. When the students explained what they had done they were asked to tell how and why they got their answers. Those using equations explained what each part of the equation meant and why they had written it as they did. Then the teacher tied all the strategies together to show how each illustrated the same thing.
The point is that students should never use a strategy nor an algorithm unless they can explain what it means, what they did, why they did it, and why they feel their answer is appropriate.
I am going to try to write posts in this blog every two weeks rather than once a month. Just so I don’t leave you hanging, next time we will talk about computational strategies for addition and subtraction and how students can learn the standard algorithm as the core requires.
Van de Walle, J. A., Karp, K.S., & Bay-Williams, J. M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally, Pearson Education
Numbers and Operations in Base-Ten Progression, http://ime.math.arizona.edu/progressions/, accessed on 2/25/2014