Occupy Math is writing this week as a break from grading the initial homework turned in by his first-year calculus students. These students are not stupid – this is a group that self-selected to take the much harder course that Occupy Math runs. In spite of this, Occupy Math had to grade nine papers to find a paper that gets a passing grade. This happens every year like clockwork. The problem is not the students; they are just the ones that are paying the price. What is going on?
- The students have not had much graded homework before, because their teachers are given too many classes to teach and not enough time for prep or grading.
- Most of the students have never had a math class. Instead they have been taught to look for a template and apply it, effectively math as ritual magic. No thinking about the problem, no asking if their answer makes sense, no practice with problem solving.
- The math curriculum the teachers are asked to present is badly designed and poorly presented because it is created by people that are both out of touch with mathematics and steeped, nay marinated in useless jargon. Here’s an example.
- The goal of the curriculum these students were subjected to was not the teaching of mathematics. It was Pavlovian conditioning to get better scores on (useless, more on that below) standardized tests.
Current math education in many high schools produces failing university students.
To start with an example of the problem, let’s look at something several students did with a standard high-school problem. The problem was this:
I kept the typo because that’s the problem I assigned: d’oh! With a problem like this, it is good – but not necessary – to sketch the track of the ball. Many of the students own calculators that can make the picture for them. Here is Occupy Math’s version of the picture. I added the answer to the picture after I solved the problem.
If you use a technique called completing the square, in about 40 seconds you get the answer that the ball’s highest point occurs at four-tenths of a second and is 24.8 meters above the ground. You don’t need the picture to do this.
What did Occupy Math’s students do, instead?
- Plugged into the quadratic formula to find the point where the ball hit the ground.
- Make an arithmetic mistake to get that, when the ball hits the ground, it is actually eight meters in the air. Antigravity ball?
- Make multiple arithmetic mistakes (with their calculators!).
- Use calculus, in spite of the directions. Occupy Math was using this problem as algebra skills practice – one of the points of the next lecture is that calculus makes this type of problem easier.
The point here is that a majority of students – with an enrollment over 100, so this is a good sample – could not solve a routine high-school math problem less than a year after finishing high school. With a calculator. And a week to do the problem. And a help room staffed with math graduate students. And office hours with Occupy Math. And class time for asking questions about the homework. And dozens of examples on the internet.
The students had been taught, over and over, for years, that finding a formula that shares a few symbols with the problem and plugging into that formula was the correct way to solve math problems. They cannot even choose the correct formula to solve a problem (when more than one is available) because the meaning and intuition contained in those formulas are outside of their experience.
Many of the students got answers for the greatest height of the ball that were under ten meters. When time is zero, the ball is 24 meters high. None of these students asked the critical question “does my answer make any sense at all?” None of them thought about the problem for a minute.
Math is the right of all free people! These students have been betrayed.
Occupy Math is not the only person to notice a problem. The Toronto Star tells us about a drop in math scores on Ontario’s standardized test. This is with increased time in math class! MacLean’s Magazine (The “Time Magazine” of Canada) looked into this and concluded that this is because of a flawed mathematics curriculum. The Toronto Star, on the day of this post, tells us math scores have been in a fifteen year slide in Ontario. Occupy Math has complained on several occasions about the way math is taught in Ontario (he lives there) but, in fact:
Deeply flawed math teaching is becoming a planetary problem.
The flaws in current-day math education are numerous and assorted, from a minority of math teachers being terrified of math and unable to teach it to completely insane workloads and lack of support for all math teachers. The flaws include arcane curricula and a pervasive culture of fear of math. Occupy Math wants to focus on one pervasive problem: teaching to standardized tests.
A standardized test is a collection of questions and problems that is given to a broad group of students in order to get an apples-to-apples comparison of the student’s skills at different institutions. With some care in how the problems are selected from year to year the tests can also be used to compare students from different years. This seems a fairly simple, sensible way to track student’s mastery of the material they are learning and to track the impact of educational policy changes. So what’s the problem?
- If the curriculum teaches to the test, it invalidates the test. A standardized test cannot possibly asses the entire skill set in its area – only parts of it that fit on a not terribly large piece of paper. Coaching students primarily on the subjects that are tested makes the test’s estimate of overall skill somewhere between wrong and meaningless.
- The test focuses attention and effort on trivial problems. Serious problems, those that require understanding and originality, subject mastery and reasoning, do not fit on a standardized test! Worrying about the standardized test pulls the curriculum away from any math skills that would be valuable in life, in employment, and in the sciences.
- Teaching to the test bores the garments off the high-talent students.
Endless repetition, intended to get a majority of the students to where they will score well on the standardized tests, drives the natural future teachers and mathematicians, scientists and thinkers away from mathematics. They come to believe that math is trivial, boring, and repetitive. This policy sacrifices the future to achieve meaningless goals in the present while it poisons the field of mathematics, killing its hopeful children, and depriving the sciences of recruits as well.
Occupy Math does what he can. When there is a chance to speak in a school, run an activity, or write a blog about the problem, he steps up. That means we have 0.0000043% of the problem solved. In other words help is needed. Volunteer in the classroom, coach your kids at home, if you have the talent run a math club at your local school, sponsor and run “Math Nights” for students and parents in your community. Most of all – don’t let the drivel merchants stay in charge of your children’s education. Tell your government officials that the current situation is not okay. I’ve linked the Ontario Ministry of Eduction – only bug them if you’re from Ontario. A quick web search may tell you where the relevant office is for your part of the world.
Occupy Math would like to propose a modest group project. Do you have a math-education horror story? Please comment, tweet, or (I will keep these identity-private) e-mail Occupy Math at firstname.lastname@example.org. Be sure to mention why you are e-mailing, that address gets over a hundred messages a day on a broad variety of topics.
I hope to see you here again,
University of Guelph,
Department of Mathematics and Statistics