As I walked into an eighth grade math class with the students last week, the teacher was projecting the day’s Challenge Problem on the board. As I took a seat, the students plopped their books on their desks and grabbed personal slates and markers. They sat down and started working on the problem: “If N is the square of an integer, what is the square of the next integer?”
The students began finishing their work within the traditional five-minute time period allotted to answer that day’s iteration of the daily math challenge. I was frozen.
As they paraded to the front of the room to show the teacher their solutions, I was still searching my memory for the right algorithm to solve the problem. I was taught that if I didn’t know the algorithm I couldn’t solve the problem. I was taught to memorize. They were taught how to THINK ABOUT the problem.
It turns out that what I needed to do, and what the students were doing, was to read the problem over and over and consider the language/ the concepts/the symbols. Not the numbers. (See below for the solution)
Algebra fundamentally is not about numbers, but about symbols representing mathematical relationships. Strikingly those abstract symbol relationships do represent numbers and physical patterns in the real world. The sine curve can describe the times of the high tide over a yearlong time period, for example.
Starting each math class with a five-minute math challenge is an old tradition at SF Day, an iconic representation of the School’s teaching methodology. We are not trying to force information into young minds, but to help those young brains welcome challenges they have not been explicitly taught and to grow into confident critical thinkers. It is a philosophy we have embraced at every grade level and – after I overcame my personal sense of “math panic” – I was thrilled to see this example of our pedagogy expressed in the eighth grade math class.
Students today are not asked to memorize the algorithm and apply it to a dozen identical problems.* [That was my high school experience.] Students today are asked to understand the concepts and apply them to a variety of novel and changing specific situations. For example, “create a sine curve equation that predicts the times of the high tides at the Golden Gate Bridge for the next six months.” We wouldn’t ask that of our eighth graders, but they might be asked that in high school.
The difference is profound. Students today have to understand and think about the concepts. The little problem about the square of the integer following the integer really required turning the ideas expressed in language over in your head, representing the words in algebraic notation, and then manipulating the ideas. It wasn’t about numbers. It wasn’t about a memorized procedure for a pre-determined and known situation. It was about taking what you know and what you can think about and applying it in a new and unknown situation. It means learning to live and learn in the unknown. And that requires flexible thinking and some pretty serious emotional strength and resilience.
THE ANSWER: For those of you who froze like I did here is how the teacher helped me drop my search for an algorithm and think about the problem.
She simply asked: “What is the integer, if the square of the integer is N?”
I could have been thinking, “So, if N is the square of the integer, and I need to think about the next integer, how could I express the integer and not its square?”
“Oh,” I said to myself, “That would be the square root of N: √N.”
“Oh the next part is easy,” I said to myself. “The next integer is √N+1 and the square of the “square root of N+1” is (√N+1)( √N+1).”
“Why don’t I just multiply that out to express it as a polynomial, (N+2√N+1).”
Without the teacher’s incisive question, I would have stayed stuck in my old mindset, trying to remember the approved way to solve the problem with the algorithm. With the teacher’s simple but incisive question, I started to think and have the words, symbols and concepts actually make sense to me. What a thrill! It is never too late to understand algebra! Fortunately SF Day students won’t have to wait until my advanced age to “get it.”
*A Note about Memorization – Of course, there are many situations within the SF Day School curriculum and instruction where memorization is called for. Students do need, for example, to memorize their math facts. You cannot do Upper School math topics of ratio, proportion, percent, and common denominators without having quick recall of multiplication and division math facts. Students also, among other things, benefit from memorizing poetry, the lines in Shakespeare, and the major river systems of the world.
