Introduction
Two weeks ago I shared that my colleagues and I have spent the past few years learning about the Cognitively Guided Instruction (CGI) approach to the teaching and learning of mathematics. A huge emphasis of this philosophy is the need for students to understand math concepts on a deep level and use strategies that make sense to them. Encouraging students to use a wide variety of strategies to solve problems is a practice that stands in stark contrast to the traditional way that most of us were taught. When I was a student, I learned a series of algorithms that I was expected to follow, step-by-step, whenever I needed to add, subtract, multiply, or divide large numbers.
The advantage of these algorithms, of course, lies in the fact that we can carry out calculations quickly. If we follow the steps precisely and avoid making mistakes with our number facts, we will consistently generate correct answers. The disadvantage of using algorithms is that genuine understanding of math concepts is less likely to occur. When using algorithms, we are merely applying a series of memorized procedures, whether we understand their rationale or not. I remember, for example, being taught as a kid to “carry the one†when multiplying a one-digit number times a two-digit number, but I never understood why. I just did it because I was told it was the way to get right answers.
A child using algorithms can quickly solve problem after problem, but one tiny computational or procedural mistake will lead to an incorrect answer, and the student applying that procedure may never notice this error.
Since my school adopted the CGI approach, I have been exploring with my students a wide variety of alternatives to these traditional algorithms. We are always on the lookout for strategies that are reliable, efficient, and rely on our number sense, not the application of memorized procedures that we may or may not “get.â€
The “Reverse Addition†Method of SubtractionIn this tip I highlight the “Reverse Addition†Method of Subtraction. No matter how much conceptual work my students have done with manipulatives in the primary grades, many still struggle with place value. As a result, they experience serious difficulty when attempting to subtract with re-grouping, especially when zeroes are involved. Consider the following example.
3,000
- 274 When using the re-grouping strategy, students who struggle with subtraction usually experience their difficulty when the top number is too small to subtract the bottom number. Instead of borrowing from the three, kids will often try to subtract the zero from the four in the ones place, the zero from the seven in the tens, the zero from the two in the hundreds, and then bring down the three to arrive at an answer of 3,274. On its face this solution can’t be possibly be correct because if one starts with 3,000 objects and takes some away, the difference can’t be larger than the original 3,000. Students who rely on algorithms typically overlook results such as these.
Other strategies exist to help kids solve subtraction problems more conceptually or, at the very least, break the problem down into smaller parts. In my classroom, the method that has worked most effectively for the most students is one we call the “Reverse Addition†method. This strategy capitalizes on the fact that people usually have an easier time putting numbers together than taking them apart. Similarly, we have an easier time counting forward than counting backwards. I am the first one to acknowledge that this method is also a type of algorithm, but I believe it offers a friendlier, more reliable alternative to the traditional method of subtraction that most of us were taught.
Returning to our sample problem, students using this method would re-write the problem so that it looks like this:
274
+_______
3,000
Step one is to look at the four in the ones place and see which digit needs to be added to the four to make ten. That number is six, and kids would then write a six under the four. Because four plus six equals ten, we would carry the one on top of the seven. Next, we would need to find the number to add to the seven and the one one we just carried in order to make ten. That digit is two, and we would write a two under the seven and carry the one into the hundreds place. Third, we would need to find the number to add to the two and the one we just carried in order to make ten. That number is seven, and we would write a seven under the two and carry the one into the hundreds place. Finally, since two plus one equals three, we would write a two in the thousands place to arrive at a solution of 2,726.
Our completed problem would look like this:
274
+ 2,726 3,000
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Mystery Writting 