“What’s the formula?”
Easily one of my biggest pet peeves to hear a student ask. Whether when it was while I was teaching middle or high school or now in college, the pervasiveness among students that every math problem can be solved via an algorithmic approach is a sad reality perpetuated in large part by teachers and textbooks. Many of you are aware of Dan Meyer’s groundbreaking math talk at TED several years ago (if not, you can see it here) where he so eloquently points out gaping flaws in the paradigm of mathematics education today.
Students are often given formulas and shortcuts and the “brass tacks” of a problem after a teacher’s initial exposition, or sometimes without any exposition at all. Students are then left to simply “plug and chug” into the formulas to arrive at the answer. Textbooks do not help this issue at all. In almost every standardized textbook out there, the problem sets are simply cookie cutter templates that follow the formulaic examples earlier in the lesson. The end result is that students are left with only a surface-level understanding of the underlying math, and the beauty, elegance, and more importantly the application, are left out.
The sad truth is that students and teachers do not engage in productive struggling enough. It is the first Standard for Mathematical Practice in the Common Core State Standards that students are to “persevere” in solving a problem. Often enough, though, students and teachers do not even know what this persevering even means or looks like.
This is a wonderful example from the Mathematics Assessment Project that really digs into statistical sampling techniques and estimation. This task nicely captures student attention and draws them into a struggle for a correct answer. I have used this task multiple times, and students develop a wide range of techniques to solve the problem, and about 5 minutes into it, they are all clearly struggling. But it is so wonderful! By the end of it, they are proud and confident in their answer and have a sense of achievement. This is the case on many problems I pose to my students where they must “dig into” and investigate the material, play with it, coerce the data, and devise their own methods that aren’t given forth in a formula or plan by myself or a textbook. And students have come to me months or years later and reminded me about such a problem and how happy they were they solved it. I am convinced that this is much more effective than a formulaic approach.
Another favorite problem that I pose to my students follows, and to further fuel the drive for an answer, I inform them that an elementary school child could solve this if they thought about it! It involves no math beyond elementary school, only logic. The problem is:
- I am an even 4-digit number X less than 9000.
- The sum of my digits is a second number Y.
- Y squared is a 4-digit number less than 1100.
- The last two digits of Y squared are the middle two digits of me.
What number am I?
The math here is deceptively simple, but the logic is a tad tricky, and it forces real perseverance to solve.
It is these types of brain teasers and open-ended low-entry-level math questions that I love to induce a productive struggle. Students learn much more using their machete of wit to chop down the jungle of confusion before them than giving them a map and which roads to use.
It is just as much about the experience as it is about the math.
And until we teachers all can stop and realize that, we will be dealing with middle school, high school, and college students that think struggling is wrong, a sign of weakness, and that they’re bad at math. But any math worth learning is learned through productive struggle, not a formula sheet.
Until next time.