Please don’t be intimidated by the calculations below! Occupy Math needs them to make a point — part of which is that lines two through six are completely not needed. This post was written directly after grading a midterm in a first-year university calculus class. What that grading taught Occupy Math is that the students have been deceived about the very nature of math!
One of the questions on the midterm was “Find a sixth degree polynomial with exactly two roots”. Occupy Math thought that this would be a freebie question. There are many correct answers, one of which is the first line of the calculations above. About 20 of the 100+ students at the exam took this correct answer and drove it right off a cliff by trying to multiply it out. Occupy Math has multiplied it out to show how much work that is! Here’s the punch line: not one of the students got the arithmetic correct. As far as Occupy Math can tell, they have been indoctrinated to believe they have not answered a question until they have done a good deal of calculation? They feel that math is not math without the busywork!
There is an inaccurate stereotype of millennials as lazy, entitled, self-absorbed lumps who live with their parents. When Occupy Math’s editor graduated from Bard College, it took her a couple of internships and several years to find a good position. Her grandmother could not understand why finding a job took so long and was concerned that lack of zeal might be the problem. What’s actually going on is that good jobs are far rarer than they used to be and affordable housing is also really hard to find, hence living with their parents. This inaccurate stereotype is an example of a smooshing error. This is an error where you combine a whole bunch of diverse factors (job availability, housing prices, the impact of automation and the internet) into one thing in your head and then map it onto your own experience, drawing a wildly incorrect conclusion. In the late 1940s (when the above-mentioned grandmother was hunting for a job), inability to find a position probably would have required laziness. It was a different world then.
This week’s Occupy Math is on a familiar topic, the effective teaching of math, but it was sparked by a number of recent media reports on why students in Occupy Math’s part of Canada are doing worse into the teeth of increased spending and emphasis on math. The problem has two parts. The first is the elementary teachers are not required to learn math themselves and are often afraid of it. The second is they are being handed a teaching strategy that cannot work and then getting no help trying to make it work. This is a big multi-part problem and Occupy Math hopes we can all dive in and help.
For the past three years Occupy Math has been working on a calculus text for the integrated first-year math and physics course he co-designed. The regular books did not have the right topics in the correct order. This post announces that the book is ready. It is the second edition (now with far fewer errors!) We used the book last year and found a need to revise and extend. The book is not a standard calculus book and the rest of the post is about that. The big points are: we got the cost down to less than half that of the book it replaces. It covers all the topics of a standard first year calculus course for science majors. The presentation is different based on decades of experience with what does and does not work.
One problem that Occupy Math has in teaching his first year courses is that many of the students have been trained, by their high school experience, to believe that a math class is a game that is scored in points with the goal of a grade. If it weren’t actually true in some of their high school classes, it would be nonsensical, and the whole notion is counter-productive to the goal of getting an education. Every year, Occupy Math has some students who are trying to do just enough work to pass the course, no more. Many of them flunk because a math course builds technique upon technique. What appears to be the correct level of effort for a D near the beginning of the term is actually preparation for an F or an F–. These grade-management tools from high school are also used as fear-management tools; instead of engaging with math, the student tries to scam a passing grade and so avoids the math. This has all sorts of bad downstream effects. In this week’s Occupy Math, we want to look at the issues of effort, fear, and effective teaching and learning.
You’ve probably been lied to. It might be a direct and intentional lie, as with high school students Occupy Math met at a girls math conference (not Occupy Math’s name for it). Teachers told these women that they couldn’t do math — often before grading any of their work. The lie might be implied, as when the person teaching you math in fourth grade was clearly scared of it herself. We also have a distressing cultural context which paints mathematical ability — or even just being smart — as being anti-feminine. Men are not immune to this sort of idiocy either. Occupy Math has encountered far too many students who, when faced with a difficult concept in math said “I can’t do math, I’m not smart!” With many of these students it turns out they could do the math and they were pretty smart. For some reason, however, running away was a much more comfortable proposition than, oh, saying “can you explain that again a different way?”
Occupy Math finds the term “math person” pretty frustrating. If you think you’re not a math person then you’re taking a whole hamper full of issues and putting a two-word sticky note on it. Do you mean you will never win a prize for mathematical research or do you mean you can never learn to add? “Not a math person” lumps together a huge spectrum of ability levels and training in one small place. Here’s the scoop: unless you meant the “not winning a prize for original research in math” extreme, you’re probably wrong about not being a math person. I don’t mean you’re good at math right now or that you’re not afraid of it. I mean that it is incredibly unlikely that you cannot do math at some basic (but useful) level and probably you can do more. In this week’s post we will look at both fear and the case for trying to overcome that fear.
At this point some of Occupy Math’s readers are thinking “But I don’t want to be a math person!”
“Do not worry about your difficulties in mathematics. I can assure you mine are still greater.”
A perennial topic in Occupy Math is the difference between how hard math is and how hard students think it is. The biggest part of this difference lies in the fear we call math anxiety. One of the most painful things that happens to Occupy Math is to have a student come to office hours for help wound up in such a state of fear that the question “what is 4+7?” is answered with a pained cry of “I don’t know!” The student could add four and seven (she did a few minutes later) — but she was too afraid to try at that moment. This is something that has not only happened, but happens more semesters than not. This interchange comes when Occupy Math tries to help a student by going through a problem with them one step at a time. If Occupy Math provides each step, it impairs learning. If the student is asked to provide each step, they sometimes panic. Based on these experiences, Occupy Math has concluded the following.
Understanding the source and psychology of math anxiety is a critical job skill for any mathematics instructor.
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.
This week’s post is about the role that creativity plays in mathematics and the role mathematics plays in creativity. The post started as an angry rant, but Occupy Math wants to limit the amount of time spent shaking his fist at the sky, so the rant part of this post will be brief.
A political scientist named Andrew Hacker has published a book entitled The Math Myth and Other STEM Delusions. The thesis of this book is that too many students drop out of high school because we are teaching them math beyond arithmetic, which they don’t need. I thank Abigail Ann Young for pointing out this controversy. Others have already taken a good shot at refuting Dr. Hacker’s pack of arrant nonsense in places like the Atlantic and Slate. I think they did a pretty good job explaining why he’s horribly, tragically wrong and a quick search will net you hundreds more items on this debacle, er, debate. Here is a cartoon about possible dark motives for teaching only arithmetic, with thanks to Elizabeth Knowles for pointing it out.
Occupy Math has complained about the quality of math education on a number of occasions. Last week Occupy Math went to the 40th meeting of the Canadian Mathematics Education Study Group at Queens University in Kingston, Ontario. He met lots of math education researchers and several classroom teachers. The conference was a rewarding experience and Occupy Math will be going to the CMESG conference in future years. Next year the meeting is in Montreal, but Occupy Math will bring himself to go anyway.
One of the things Occupy Math obtained at the conference, from a classroom teacher, was a Ministry of Education booklet explaining how to teach fractions for K-12 teachers. Occupy Math has linked the PDF so you can take a look if so inclined. The teacher wanted Occupy Math’s opinion on one of the points the booklet made, treating fractions as operators, which was technically a mis-use of the term “operator”. That was far from the biggest problem in the booklet.