Back to BlogTaking the Shortest Path: On Multiple Methods

Why is it so important for students to learn multiple methods for solving a math problem?

I have a quick and painless math question for you to start.

It's not a trick question. We're using plain old Euclidean geometry here, where all the lines are straight and everything is on flat planes. You will, of course, travel the least distance between A and C if you walk along AC.This "problem" probably didn't feel like a problem to you at all. That's because determining the shortest distance between two points is an ability we (and even other animals) are probably born with, or learn at a very young age.

But this ability—to immediately seek out shortcuts and shortest paths—can also probably explain, in part, why many of us didn't have a positive first reaction to studying multiple methods in math class. If I know a really quick path, like AC, and it gets me from A to C every time, and I'm pretty sure I'll never have to use any other path, why should I be forced to know another way to get to the same place?

People have asked this question often enough that we educators now have a stockpile of answers ready to go, most of which we have heard before. So, let me try to answer it a little differently.

First, multiple methods are not an absolute good in math class. That is, if you told me that some topic was taught one way in a first class, but in a second class they taught it two ways, I can't assume that the second class has it right just because they use multiple methods. What are those methods? What is the topic? What prior knowledge are the students, in general, bringing to the table? Questions like that (and many more) have to be answered and the contexts worked out before we can make a judgment about multiple methods.

Second, there are methods in math that are more efficient than others. **Not all ways of looking at a problem are created equal. **This is not to say that there is necessarily some rigid hierarchy of methods, from best to worst. But it is also not the case that every method is equally deserving of our and our students' valuable time and attention.

Finally, having said all that, we have to be careful. We adults grew up with math mostly being about using methods to solve problems. So, when we hear "multiple methods," we might naturally assume that what is meant is that our students will be using multiple methods to solve problems. But math has more than just this one purpose—of applying techniques to solve problems. **Mathematics is also useful for gaining new insights into natural and other mathematical phenomena. And often the way we arrive at these insights is to analyze these phenomena using multiple methods—or, we consider them from multiple perspectives.**

As an example of this, consider these three segments. Will they form a triangle? What about any three segments? How can we tell if they will form a triangle?

Well, let's try to think about this question from a different perspective—one we have already seen above. Think about drawing a line segment on a piece of paper, from a starting point to an ending point. Now place your pencil on the starting point and think about all the ways you could draw two straight segment paths to get you from the starting point to the ending point without going along your original path. You will have to draw triangles like below. And, since you started by drawing the shortest path between two points, the two other side paths you draw, together, must be longer than the first path.

So, to form a triangle, each of its sides must be shorter than the other two added together. That's a nice conjecture! That means that if we could measure those line segments above, and each pair was longer than the third segment (say, 5 cm, 6 cm, and 9 cm), then they could form a triangle. If this were not true (with segments of, say, 5 cm, 6 cm, and 12 cm), then the segments would not form a triangle, because the path 5 cm + 6 cm between two points would be shorter than the straight-line distance of 12 cm between the points, which is impossible.The switch in perspective here is to think about the sides of a triangle not as sides but as paths that we can walk along. We know that any segment on a triangle is the shortest path between its two endpoints, so the sum of the lengths of the other two sides—any other two sides that form a triangle—must be greater than the side we want to walk along.

What we have just seen—that, for any side of a triangle, the sum of the lengths of the two other sides must be greater—is one way of stating what's called the Triangle Inequality Theorem. Notice how we can get there using multiple perspectives: by thinking about the triangle's sides both as sides and as paths, and then using what we intuitively know about the shortest distance between two points. That's pretty cool!

**Getting used to looking at things from multiple perspectives is a powerful habit of mind that can serve you well when learning mathematics.** Doing so usually takes you off the shortest path, but knowing different paths is often worth the extra effort.

*Check out our Middle School Course 2 textbook lesson on Constructing Triangles Given Sides.*

- Josh Fisher
- Instructional Designer
- Carnegie Learning, Inc.

Josh is an author and instructional designer with Carnegie Learning's Instructional Design team. He's been an instructional designer for more than 20 years and has worked with Carnegie Learning for 7 years, most recently helping to author the Middle School Math Solution series and upcoming High School Math Solution series. Josh and his wife, Vicki, both write for Carnegie Learning and have 4 children, which keep them pretty busy outside of work.

Explore more related to this author- April 13, 2018

We adults grew up with math mostly being about using methods to solve problems, but math has more than just this one purpose. Mathematics is also useful for gaining new insights into natural and other mathematical phenomena. And often the way we arrive at these insights is to analyze these phenomena using multiple methods—or, we consider them from multiple perspectives.

Josh Fisher, Instructional Designer, Carnegie Learning

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