In an email exchange involving several former students (now at college) and a colleague, I said,
This discussion reminds me that one of the great challenges as a math teacher is to combine the necessary (at least in our current curriculum) skill-set with a wide-ranging mathematical curiosity and a systematic approach to helping people discover what it means to think “mathematically.”
A recent article from eSchool News caused me to think about that comment: what do we teach in math classes. And why?
I wonder how many mathematics teachers realize that one of the purposes of algebra is to enable people to take an arithmetic observation and answer more general questions about it. Or, to use the eSchool News writer’s example:
We all learn early in life that two plus one equals three and that two times three equals six–that’s simple arithmetic. A natural follow-up question would be whether there are any numbers other than two that, if you multiply them by the number that is one greater, gives a result of six.
It’s algebra that lets you answer the latter question. But it’s very rarely taught that way…
One of the mixed blessings of the reform winds blowing through curricula (Common Core, standardized tests) is a greater emphasis on using math to solve problems, usually posed in English. Eg.:
“The recommended daily calcium intake for a 20-year-old is 1,000 milligrams (mg). One cup of milk contains 299 mg of calcium and one cup of juice contains 261 mg of calcium. Which of the following inequalities represents the possible number of cups of milk m and cups of juice j a 20-year-old could drink in a day to meet or exceed the recommended daily calcium intake from these drinks alone?” — SAT sample question.
So on the plus side, these kinds of questions require students to translate English into math. The presumption is that the translation skill will allow students to use math to solve problems. Yay!
On the minus side, reform is being packaged as a greater emphasis on familiarity with families of functions (“the better to model you with, my dear”) and a de-emphasis on symbolic manipulation (“since computers can do that for us”). OK, that’s a trade-off we can make. The cost is that Algebra — groups, rings, fields, modules, vector spaces — is not about families of functions. It’s about the logical structure of mathematical thought. In the world of algebra, the symbolic manipulation IS the utilization of a highly refined mathematical language to generate and prove results about prime numbers and monster groups and whatnot.
Families of functions used to model phenomena really belong to the Analysis branch of mathematics — which I love and have loved since Dr. Sharp’s classes, but is ultimately different from Algebra.
So the downside of math reform is that analysis has swallowed up algebra proper. We’ve become better at modeling, but worse at cryptography and rigorous proof. Is that an OK tradeoff?