There is a community of high-school math nerds, united in our enjoyment of the game of SET, which revolves finding groups of cards that share certain properties. The game itself does not require any mathematical knowledge, simply the patience to understand and improve in it. (In my experience, this means it is best taught to children.) I have a copy of the deck, and, soon after purchasing it, I taught the game to my elementary-school cousins. They loved playing and soon bought the game to build their skills and strategy at home. When my cousins returned they surpassed my ability. But I still had one more trick up my sleeve...
The game is designed in such a way that I can predict the last card using modular arithmetic. My cousin, awe-struck by this trick, begged for an explanation, and I showed her the theory behind this prediction. Having learned remainders in school but not modular arithmetic, my cousin struggled to understand at first, but persevered despite her dislike of mathematics. The joy on her face when she was able to figure out the final card for herself in the next game was priceless.
Mathematical Standards Achieved
working with remainders
doing modular arithmetic
looking for patterns
persevering in problem-solving
(And she built up all these skills with a smile.)
Why can’t our schools do this? I ask myself this question each day, when I attend my accelerated Algebra 2, Trigonometry, and Pre-Calculus class, where the textbook lesson gives a formula for the day that we memorize by completing the homework set consisting of identical “problems.” Our task is simply to plug numbers into these formulas in order to prepare us to repeat this ritual in standardized tests.
My school district (apparently in complying with a new Common Core standard) has replaced the old honors program with an accelerated program, skimming concepts faster instead of going over any topic in depth. This is a conflation of the image of a difficult math problem and what a true good challenge should be. A problem that seems hard to us students because we don’t know the necessary formula is not necessarily a hard problem at all. We can see this in problems outside math too; I don’t know the answer to “In which year did Napoleon retreat from Russia?” - not because it is a difficult question, but simply because haven’t memorized much about Napoleon or Russia. A genuinely difficult problem may require lots of knowledge, or it could be knowledge-sparse, but what makes it difficult should be the reasoning required to get the problem into a state where you can produce the answer.
Power rounds in math competitions, a problem set based on writing basic proofs about a new topic, are a great example of how to achieve this desirable difficulty. They generally provide all the necessary definitions for the given problems, which focus on reasoning from those definitions. In my opinion, this is the difference between an honors class and an accelerated class. While an honors class allows the teacher to go deeper into each concept and ask challenging questions based on those ideas, the accelerated course fits so many concepts and formulas into a short time that the teacher is forced to give only a brief overview. This idea that more is better forces the teacher to skip teaching for understanding or deeper elements of problem-solving.
The teacher is forced to stick to what the textbooks say and what the standards cover; in short, what the students are going to be tested on. It is difficult to test logic or problem-solving skills. But it is easy to test formulas.
This is not to say I haven’t had teachers who tried to teach beyond the script. I am extremely grateful for the dedication of the wonderful teachers I have had. However, they are forced to work around the curriculum instead of with it. Why should a good math education only arrive when a teacher is willing to be exceptional, and is willing to work around the curriculum that was supposed to support them, far beyond the time they are paid for?
Knowing what it is we are memorizing is unnecessary, so skipping understanding brings students to meet the standards necessary for their course and the SAT fractionally faster.
The textbook is supposed to be a tool to help the students, not a cut-and-dry list of formulas. Yet it is sometimes unclear what image the textbook authors have of math. Math is seen only as a tool for formulas in different subjects. We are told to memorize Newton’s Law of Cooling, or the Decibel Scale, or the Richter Scale, without much understanding of what the law means or how it works; simply:
Here is another formula to memorize, with five different variables. In the test, you will be given four of these and are expected to find the fifth.
I understand that mathematics appeals to many through its application, but where is the application in the above? How is this different from memorizing another formula? Knowing what it is we are memorizing is unnecessary, so skipping understanding brings students to meet the standards necessary for their course and the SAT fractionally faster. It often looks as though the textbook even encourages this. After giving a formula for the transformations of sine and cosine (the third set of function-specific transformation formulas they gave), the textbook from CENGAGE Learning gives the formula for harmonic motion as a real-word example. This is the exact same formula as before, simply with a few changes in the names of the variables. Yet, following the textbook, my classmates memorize it as a completely different topic, with the only common thread between the two being the chapter in which they are located. Of course, the homework problems rarely lead the student to discover this for themselves. Besides a lonely “critical thinking” problem that may flitter by (without further discussion connecting it to the material, to confuse students more), the exercises at the end simply ask the student to repeatedly plug different numbers into the formula. There are no problems to solve here. There is no thinking to be done.
Surely, though, there would be some rebellion against such a cut-and-dry curriculum? This was a question I only thought to ask about six months ago, when I stumbled upon Lockhart’s Lament. What shocked me was not how well Lockhart described some of the complaints I personally had about the curriculum, but the specific details he brought up against the program. Many of the examples Lockhart had about high school education were situations I could vividly remember, having been through them sometime in the past few years. This pushed me to explore the world of math education more, which is how I found the wonderful work being done by the people of Q.E.D.
I also bought a book called Building a Better Teacher that discusses the current movements in math education and teaching as a whole. The author, Elizabeth Green, speaks of teachers who were naturally “good” at their job and figured out how to dissect what we often consider a natural talent in order to teach it to other teachers. These people, led by Magdalene Lampert , pushed for a reform of the education standards. There were many obstacles in their way, and it did not seem possible to push effective reform. However, they eventually did gain victory in the form of the Common Core standards. I had to reread this to make sure my eyes weren’t fooling me. How could the amazing ideas I was reading about match with the accelerated classes I was forced to take instead of honors, or the textbooks taught formula by formula, instead of idea by idea, or the entire system of teaching only the formulas necessary for standardized tests? I have not yet found the answer to this question, but I do see its damaging impact on the students around me.
According to the conversations I have had with my fellow classmates, math is a set of dry formulas that come from nowhere and mean nothing, useful to know for passing the class, doing well on the SAT, getting into a good college, and maybe some complicated physics thing somewhere.
I wish I could say that the greatest impact of this disconnect is that students do not learn math. However, students are not even given the opportunity to experience what math really is. According to the conversations I have had with my fellow classmates, math is a set of dry formulas that come from nowhere and mean nothing, useful to know for passing the class, doing well on the SAT, getting into a good college, and maybe some complicated physics thing somewhere. It is no wonder that they are disgusted with the subject; if I had only learned math from the official curriculum, and never found true problems, with beauty and logic and insights, I would have disliked the facts and figures of math just like I disliked the names and dates of history. And the problem with the math curriculum is more than just fostering dislike.
Our brains are not built for memorization without context. This makes the math curriculum increasingly difficult, as students are expected to memorize more and more formulas while retaining their knowledge from of previous units’ definitions. As they attempt to surmount this overwhelming task, people like me, who learned mathematics outside the traditional curriculum, are seen as magicians — wizards who possess the superpower of “intelligence” that allows us to complete the task at hand with ease. Meanwhile, my classmates see themselves as simply “not smart enough” or “not math people.” Both these roles are completely false, as I know my classmates to be extremely intelligent. However, while they are struggling to memorize formula after formula, I have the much simpler task of solving problems given the understanding of ideas I have built up over time, making the task of keeping up with the curriculum increasingly difficult for my classmates. It is imperative that children learn the reasoning and problem-solving that actually makes up mathematics.
After all, a good math education can begin with a quick game of SET.
