# Struggling to Learn the Breaststroke

In the introduction to our book Traditional Math, I state that “I try to stem struggling with a problem and aim to have students be successful. My guiding principle is that struggling to learn the breaststroke is not the same as struggling to keep from drowning. The latter doesn’t teach you how to swim.”

So how do we address struggle in traditional math? First, by acknowledging that learning anything new involves some struggle in the beginning. The initial stages of learning a new procedure or problem solving technique usually involves imitation. And as anyone knows who has learned a skill through initial imitation of specific techniques, such as learning a dance step, bowling, golfing, playing an instrument—watching someone else do it and then doing it yourself are two different things. So too with math. What looks easy often is more complicated than it appears. Students will struggle to get the procedure down correctly at first, and then as they get better at it, I introduce some complexity to the problems. This requires students to make small but significant leaps of reasoning. Let me give you an example, again from algebra.

I will provide basic and explicit instruction on a type of distance/rate problem, using diagrams and other techniques. At first I might ask "Two cars go in opposite directions from the same spot; one going 60 mph and the other 70 mph. How far apart will they be in 3 hours?” Since they have been given the distance = rate x time equation, this is a relatively straightforward problem to do.

After a few like this I will give a similarly structured problem, but this time a different part of the problem is missing. Specifically: “Two cars go in opposite directions from the same spot; one going 60 mph and the other 80 mph. How long will it take for them to be 420 miles apart?” Students will definitely struggle with this but with some prompts from the teacher, they can put it together based on the prior knowledge they have. I might ask “What are we trying to find? How did we solve the previous problems? Can you set up a similar equation? What are you going to let *x* represent?” and so forth. Amanda Vanderheyden in her interview with Anna Stokke on her podcast, calls this “acquisition instruction” which is that stage at which students are learning and understanding new things. It is fluency building in something they have just learned, so will tend to be difficult. At this point in learning it is important to provide appropriate support. Leaving them entirely on their own might work for some, but too much struggle will result in labored responses on their part and is “not teaching them to swim”. Teacher guidance is especially essential here. Ultimately, students set up the equation as , and will get hours.

Once students have reached mastery in a particular aspect of math, and are in what Vanderheyden refers to as the generalization and adaptation stage of learning, it is appropriate to have them work on problems that require more work. For example the following problem (taken from an AMC-8 competition) is a multi-step problem that requires no algebra, but is challenging, for both seventh and eighth grade students:

“Karl's car uses a gallon of gas every miles, and his gas tank holds gallons when it is full. One day, Karl started with a full tank of gas, drove miles, bought gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day?

This may cause some struggling, but they have the tactics and prior knowledge to solve the problem. They may very well need some guidance but it will likely be reminders and hints of what they’ve learned and mastered at the acquisition stage.