The Danger of the One-Way Approach

As I mentioned before in this blog, when I was a student, I was taught math in school with the one-way approach. My teachers were focused on our procedural understanding, and taught us specific ways to find answers using a particular method or formula.

In other words, I was taught that that 4 x 5 = 20. 

I saw math as something that was structured in a very specific way, that had to be memorized, and that was linear. You followed the instructions step by step, and if you couldn't find the answer, you must not be mathematically minded. There wasn't room for creativity or exploration in math. I developed a fixed mindset in math, and began to seriously dislike math.

What if I had been asked 20 = ? 

Suddenly, math becomes creative. The answers are endless!

20 = 4 x 5
20 = 10 + 10
20 = (8 x 3) - 4
20 = -20 + 40
Etc.

This may not seem like a big change, nor may it seem to be teaching any curriculum expectations, but this shift in thinking is important. This shift from robotic plugging in numbers to exploration and curiosity adds intrigue into the math classroom, and frames math as a wide, interconnected web rather than as a single pathway. If math is supposed to be all about problem solving, how can we expect students to think in different ways if they are only ever shown math linearly?

But seeing math as linear wasn't the only way that the one-way approach impacted me. It also affected my ability to do mental math.

For example, when I see a question like 18 x 5, I immediately want to grab a calculator. My immediate explanation is because when I was in elementary school, we didn't learn multiplication facts past x 12, so I don't have 18 x 5 memorized. Looking back, I realize just how much emphasize my teachers put on memorization, how little time we spent on exploring the relationships between numbers, and how we were not taught to do any mental math beyond the math facts that we had memorized. Also, we were always taught to use the fastest, most efficient and direct way possible to get to the correct answer at the center of the problem, usually through formulas, instead of exploring around the perimeter of the problem and discovering our own path to the answer.

So when I was asked to solve 18 x 5 mentally this past week, after I resisted my initial urge to grab a calculator, it took me a moment to realize that I didn't have to directly answer 18 x 5, and that I could travel around the problem to find an easier way to answer it mentally. I decided to round up 18 to 20, since I knew that 20 x 5 = 100. Then I removed 5 twice to find 19 x 5 = 95, and then 18 x 5 = 90.

What fascinated me afterwards was the fact that there were other ways that people took to answer the question, such as breaking the question up into 10 x 5 = 50 and 8 x 5 = 40, then finding that 50 + 40 = 90. I was amazed at how many ways there were to solve the question, and at how they could each be visually represented.

In school, I was presented mostly with closed tasks, where we had to find a single answer to a problem, but as a teacher I will be striving to present my students with rich tasks, where my students will be encouraged to figure out multiple answers and explore math while doing so.