Math is not a form of ritual magic

My title is what my editor calls click bait but its also a real problem that Occupy Math encounters as a math professor. Today’s post arises from a brisk, helpful argument between myself and my editor. The key point of this argument was her incisive question “So you’re mad at your students for not following directions, but you also don’t want them to follow someone else’s directions?” At the time she asked the question my answer was “yes” which is a little sad for me. In this post, I want to try to explain what the problem is. My explanation involves teaching to standardized tests, substituting Pavlovian conditioning for problem solving, and the low-grade horror called “Transition to University”.

Ritual (noun): a religious or solemn ceremony consisting of a series of actions performed according to a prescribed order.

A serious problem not only in math, but in science education in general is that, at the lower levels, math and science are treated not as a way of thinking and a set of tools for understanding the world and solving problems. They are treated exactly like extra chapters in the holy book of your choice. Bill Waterson absolutely nailed the problem in this wonderful cartoon:

Suppose that your school district’s funding depends on getting good scores on standardized tests that use the same kind of problems every year. Then, at least in mathematics, the strategy of teaching students math runs a distant second to another strategy. Instead of teaching math, you teach reliable sets of instructions for recognizing and solving the problems that appear on the standardized tests. In other words…

You teach the students a series of magical rituals for nailing the problems that appear on the standardized tests.

At this point I am seriously at risk of offending high-school math teachers, my natural allies in improving math instruction, and I need to couch my statements. In my part of the world this counter-productive, abusive teaching strategy clearly originates with the province of Ontario and some teachers are absolutely members of the rebel alliance. Here’s what arises from this teaching method:

  • Students that are taught this way can only solve the problems that appear on the standardized tests. These are problems that seldom, if ever, arise in practice, lending support to the idea that math is pointless and useless. These problems are chosen for their simplicity and brevity. If your life is brief and simple, you’re a mayfly, not a person.
  • Many students arrive at University thinking that the correct way to get good grades in their math class is to build a catalog of types of problems and their corresponding ritual. This means I need to burn my short supply of class-time running a de-programming camp (not a fun activity).
  • The glory and power that potentially reside in the minds of each student is stunted and discouraged. Occupy Math has already shown you fractals and flowers, the origin of geometry in art and the exquisite statistical work of Florence Nightingale that saved million of lives. If I write a blog a week for the rest of my life I will expose only fraction of the peripheral glow of the storm of beauty I am privileged to learn about and discover.
  • The standardized tests are rendered completely worthless. It is an assumption of such tests that those being tested were not prepared for the specific content – it is like Asimov’s fictional psychohistory. It does not work on people that are aware of it. This is bad primarily because we should assess our math education techniques and secondarily because wasting time and money on tests that tell us nothing is, itself, a bad thing.

Transition to University

This simple phrase represents a comprehensive orientation activity that we either make available to or force upon entering students at the University of Guelph. At its core, it is an attempt to convince the students that a number of things, turning in assignments late, having six chances to do make-up work, extra credit that substitutes for skipping core parts of a course, and flexible, optional deadlines are not as available in University as they were in high school.

The province has permitted all of these things in high-school at different times during my tenure at Guelph, without consulting or even warning the poor people downstream. This means the student who says things like “I’m not going to have time to finish this homework until next week,” come as a horrible surprise to the instructor. This student is understandably upset when we tell them there is no credit for turning in the assignment next week – which in turn impairs the professor’s ability to do his job.

Ritual versus following directions.

On my first homework this year I gave the problem “Solve: y=(2x-1)/(1-x) for x”. A fairly large number of students, rather than re-arranging the expression to get “x=(y+1)/(y+2)”, first set y to zero and got “x=1/2”. Instead of reading and understanding the three word directions “Solve for x”, they chose a technique for solving a different type of problem and applied it. It was not an appropriate technique for the problem posed. If this was an isolated incident, involving a few students, no problem – I spend five minutes correcting that problem in class. It is not isolated. Dozens of students did it and something like this happens every year. It is clearly, inarguably, a result of their previous education.

In the view of Occupy Math, directions should specify a problem and also which of several interesting aspects of that problem are to be addressed, illuminated, or discovered by the student’s efforts.

In contrast, rituals are directions, but they are micro-managing, hyper-specific, and depend on conditioning rather than understanding to achieve apparent mathematical competence. If I do my job correctly, then the students I train will imagining a collection of possible solution techniques for a problem, make quick mental checks for their practicality, and arrive at a short list that need to be checked by trial solution on paper (or their iPad).

Many of the students I get fresh out of high school do not even know this is an approach to solving math problems. If they cannot recognize a problem as being of a known type with a solution ritual stapled to it’s back, they are paralyzed. Again: I have hundreds of samples and this is a highly repeatable phenomenon. My three children all went through the Ontario math curriculum and all got dinged over and over for getting the right answer “the wrong way”.

It is the opinion of Occupy Math that the Province of Ontario should dismiss the individuals in charge of figuring out how to teach math and adopt a system like that of the State of Iowa where the curriculum is developed cooperatively and locally by teachers and parents. This opinion is based on the fact that the results obtained by Iowa, a small, rural state, are far better. Remembering that standardized tests have been rendered meaningless, I speak as a professor who has taught several different sorts of first year math in a major university in both places for more than a decade each.

Do you have an opinion on this issue? Are Ontario math teachers oppressed by provicial guidelines? Could we do better? How about the impact of deleting grade 13 in Ontario? Please comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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3 thoughts on “Math is not a form of ritual magic

  1. I’m in the States, so I don’t have any idea of Ontario specifically. (I wandered in here by way of Charlotte.)

    What killed me repeatedly in grade school was executing calculations, and the “rituals” (or is it algorithms?) I was taught to use (‘carry the 1’, ‘borrow from the tens column,’ etc.).

    I was an education student in college/uni– I didn’t become a teacher, but that’s a long story. The issue of standardized testing alone is a can of worms unto itself for ALL coursework, more especially when corporations like Pearson are creating and grading the tests.

    Like

    1. The idea of measuring progress with a standardized test is a good one – but you are absolutely correct that having a single for-profit authority create and grade tests is a bad idea. I will go farther – it is corrupt. From the most recent to sex-education lessons in Ontario to the math Common Core in the states, newness always creates a strong negative reaction. Pearson, wanting happy customers, is very likely to make only cosmetic changes creating the perfect environment for teaching to the test.

      Liked by 1 person

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