In Alberta, the government certifies elementary teachers, but not with regard to what they are qualified to teach. Elementary teachers are generalists, and all of them are expected to be able to teach mathematics.
In the past, the Faculty of Education at the University of Alberta encouraged Elementary Education students to take one or more half-term courses offered by the math/stats department. Some opted for a rudimentary Calculus or Statistics course (neither of which is still offered by our department). Others took Math 160, a course which I taught many times. (It went by the spiritless and dismaying name Higher Arithmetic, and it was restricted to Elementary Education students.)
Back then, math was a bad thing for aspiring elementary teachers. Previous encounters with it had intimidated them. Math avoidance had become part of their lifestyle. The exasperating fact was that they were actually very smart, and abundantly capable of understanding the math that they needed.
Of course, not all of my students had spent their lives sidestepping math. Some were very strong mathematically, and they actually liked the subject.
So, there was a large variance in the students’ readiness to take on more mathematics, and it raised some concerns: Should I cater to the math avoiders and reteach them “the basics”, or should I dip right into the “higher level” topics for the stronger students? Or is there some middle-of-the-road compromise?
Here are three options which appear to offer some resolution and which I occasionally found quite tempting.
Option 1. Treat Math 160 as a sieve.
In a sieve course, the professor usually has in mind standards that should be achieved, and he or she sets up the course accordingly. The standards are generally quite high and inflexible, and only those students who approach a predetermined level of attainment get a passing grade.
Rationale: Somehow, the weaker students gained admission to our university, and that’s not our fault. Teach the course at the appropriate level and let the students sink or swim. Weed out the weak ones before they become teachers.
Counterargument: It’s not always clear what the the words “appropriate level” mean. Besides, this won’t actually weed out the so-called weaker students. Remember, many of them are expert at math avoidance. If pushed, they’ll find a way to circumvent the math requirement. And when they become teachers they’ll still believe that math is a bad thing and will transmit their aversion to another generation.
Even worse, a sieve math course will weed out some brilliant arts and humanities teachers.
Caveat: I do not advocate a complete absence of standards.
Option 2. Treat Math 160 as a math appreciation course.
By a math appreciation course, I mean one that discusses the accomplishments and personalities of mathematicians and explores how math has played a significant role in civilization.
Rationale: Such a course shows students the human side of math and de-emphasizes the hard mathematics that frightens so many. It will show students that math is something that is worthwhile learning and will encourage them to learn more.
Counterargument: Adulation of great mathematicians spreads the myth that you have to be a “math brain” to learn basic mathematics. An appreciation course for students with a weaker math background will be high on memorization, low on developing mathematical thinking & problem solving skills. The course may end up doing exactly what the weaker students are doing, namely avoiding math.
Caveat: I do think that university math departments should offer courses that deal with the sociological aspects of mathematics, but, I also think such courses should have a fairly heavy dose of mathematics and therefore should be restricted to students with strong math backgrounds.
Option 3. Treat Math 160 as a remedial course.
Ahh! The teach-them-the-basics approach. By “basics” I mean stuff that usually involves the so-called standard algorithms along with lots of shortcuts and tricks and strange rules like cross-multiplication or FOIL or BEDMAS1. The course should include lots of drill and practice so that the students become fluent with the material that they should have already mastered.
Rationale: Too many Math 160 students lack mastery of the basic skills—for example, some cannot even carry out the long division algorithm2. It’s our job to remediate the situation—the students should know the basics and should be able to perform the standard algorithms automatically.
Counterargument: For a large number of people, the basic arithmetic facts and routines are very abstract and highly cryptic. Drill, practice, and contrived exercises will not only fail to alleviate their view that math is impenetrable, but will actually reinforce that belief. Math avoiders will become even more reluctant to learn mathematics. And the course would be a great disservice to the stronger students. To them, Math 160 would become a Mickey Mouse course. They would see it as an easy “A”, and would derive no benefit from it.
Caveat: In spite of my displeasure with the pedagogical advice by the back-to-basics people, I do believe that students should understand the fundamentals of arithmetic and be comfortable working with them.
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Teaching problems arise in any course where there is great disparity between the levels of the students3. I think the counterarguments above show that none of the three options resolve that situation for Math 160. In reality, all three skirt the issue by tilting the content towards one end or the other of the spectrum of student abilities.
When I began teaching Math 160, I had very specific ideas about how I should present the course. To me, the key to successful instruction was to concentrate on the clarity and explicitness of my lectures. It turns out that I was singing from the wrong songbook.
In a subsequent post I will describe an approach for Math 160 that actually worked (at least for me).
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[1] Incidentally, when I went to school, BEDMAS was called BOMDAS. You may called it PEDMAS or PEMDAS. The four algorithms don’t always produce the same result. James Tanton has an illuminating essay about the order of precedence for arithmetic operations.
[2] This shocking state of affairs occurred in the era before there was any commotion about “discovery math”, so don’t go placing the blame there.
[3] Judging by what I see in teachers’ tweets and blog posts, the disparity in K-12 courses is even greater than in university courses.
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