In order to learn calculation procedures, give students ample opportunity to explore and analyze key patterns and relationships
Here’s a nifty trick. Take a look at the following calculations:
1/3 + 1/5 = 8/15
1/2 + 1/13 =15/26
For most people, addition calculations involving numbers with unlike denominators (bottom numbers) are somewhat difficult to do. But calculations like the one above are actually quite easy to do in one’s head. All you have to do is add the denominators to get the numerator of the sum and multiply the denominator together to get the denominator of the sum. In other words:
1/3 + 1/5 = 3 + 5 / 3 * 5
1/2 + 1/13 = 2 + 13 / 2 *13
With just a little practice, one can get good at doing these calculations mentally. Try calculating these sums:
1/10 + 1/9
1/100 + 1/3
(I’ve put the correct sums at the end of this blog post.)
Learning this technique is quite useful, and, moreover, it definitely possesses some sort of whizz-bang appeal because it greatly simplifies an otherwise complex calculation. For most people, however, I suspect that following the description of this technique was a bit difficult. And that, in fact, is why I presented it, because I actually want to discuss its serious limitations in spite of its obvious attractions.
In particular, I want to point out that nowhere in the short-cut procedure I presented is there any need to understand the underlying number relationships that make this technique possible. In this regard, my presentation of this technique was similar to lessons that still predominate in math classes throughout the world, whether the subject is adding multidigit numbers with carrying, or subtracting multidigit with borrowing, or long division, or dividing fractions, or solving rate problems or using the Pythagorean Theorem. Teachers often present new calculation procedures by saying, “Follow these steps,” without allowing the students sufficient opportunity to explore in some depth the number patterns that make efficient calculation possible.
When students learn calculation procedures without having sufficient time to explore fundamental concepts in depth, several problems arise. One of the most significant is that students become needlessly confused and make numerous errors. I’ve already mentioned one example of this problem in my previous post when I spoke about the the problems of the first grade teacher introducing subtraction to her students. There is a considerable literature on common and persistent errors of students, and skipping steps or not remembering sequences properly is extraordinarily common. Anecdotally I can share that many of the students I see for math tutoring find the presentation of a procedure in school difficult to follow.
Another common problem with this method of instruction is that students have limited opportunity to develop and use problem-solving and analytical skills during math class. The lion’s share of their time is spent on memorizing and practicing calculation procedures. As a result, students remain highly dependent upon their teachers and have difficulty making even simple judgments about whether a particular calculation they have made is correct. They thus have less time to develop their number sense, and without adequately developed number sense, they have difficulty applying what bits and pieces of math they do know. For many students, even when they learn to apply calculation procedures correctly, math remains a subject that frequently doesn’t make much sense to them, a set of rules and procedures whose internal logic remains opaque. Indeed, many studentts learn to de-activate their sense-making skills in formal math classes because what they do day in and day out rarely makes much sense to them.
Fortunately, there is another way to help students learn efficient calculation procedures, namely, to give them ample time to explore number patterns using calculation tools that they already have. For example, to help students learn how to subtract multidigit numbers efficiently, it helps to have them use snap cubes to represent 2-digit quantities, with a “stick” of 10 cubes representing a ten, and individual cubes represent one. Students can practice representing numbers with the cubes and then taking away quantities and recording these. It's worthwhile doing such calculations using other models as well, such as number lines. After students have had the opportunity to perform and record many such calculations, the teacher should then help the students analyze any patterns in their calculations and give them the challenge of figuring out how to do a subtraction calculation without using the snap cubes.
When a teacher helps students acquire new calculation skills in this way, many good things happen. First of all, the work gives students important experiences that deepen their understanding of important number patterns and relationships. Adults often forget or are oblivious to the tremendous amount of experience with numbers it takes for a person to see important patterns. Having good models helps, certainly, but there is no substitute for time exploring numbers with tools which one is already familiar and comfortable. Additionally, this approach requires that students engage with the task actively looking for patterns and trying to solve problems based on the patterns they have observed. Thus, they build up their problem-solving skills.
Their view of math as a subject changes in the process as well, because their work is something they have to puzzle over and make sense of, rather than merely memorize. Students who learn calculation techniques this way will also be less confused and better able to apply their new knowledge in different contexts. This is not to say that students will not make errors. But when they do make errors, the teacher will be able to help them identify the error with models and calculations that make sense to them and the error correction will be integrated into the overall sense-making quality of the investigations.
So if you want your students to learn calculation techniques efficiently, use curriculum materials that give them lots of opportunities to explore and analyze the relevant number relationships, and keep your lessons more about making sense than about memorizing steps.
In my next blog post I will talk about the role of error in lessons.
Here are the answers to the beginning problems.
1/10 + 1/9 =19/90
1/100 + 1/3 = 103/300