Content is the Engine
The problem begins in elementary school. What I have found in 40-plus years of teaching upper elementary and middle school math is that students’ difficulties in high school and college math stem from their failure to understand the most basic mathematical concepts.
Students’ problems with addition and division of algebraic fractions stem from their lack of understanding of the same operations on fractions formed from integers. They have weak understanding of geometry; their understanding of fractions/decimals/percents, and their application to the real world, is quite poor. On top of that, they have not internalized basic algebra to the point of automaticity.
What is more important is that they don’t seem to grasp what it means to understand mathematics. They mostly think it is all recipes — formulae to be plugged into their graphing calculators, without understanding the derivations of the formulas themselves, and geometric theorems to be used, without understanding their proofs.
I have been teaching elementary and middle school math since 1969, and have a pretty good idea of what students are missing when they are having trouble. A teacher can make an honest attempt to correct their misconceptions so that they will genuinely understand, or the same teacher can streamline the exams and homework so that they can memorize templates without understanding what they are doing.
I have heard the problem stated in the following manner: If you need to smooth out a rough piece of wood, you can take a piece of sandpaper and press, or you can take glue and cover up the rough spots.
I don’t have much optimism about the ability of mathematicians and adult practitioners of science and engineering to alleviate the situation. They have very little experience dealing with children, at least as it comes to teaching math. They have little practical experience about knowing the limits of their concentration. As parents, they may have been involved in teaching their own children (which is a valuable experience), but it is not the same as explaining something like division of fractions to a room full of 11-year-olds with widely different abilities and knowledge.
One problem that confronts elementary schools is that the wide differences in abilities in math present far more difficulties than the wide differences in reading. With reading, a teacher may often allow students to choose their own books. It is tougher to do this with math. While there are plenty of students who could easily understand algebraic concepts in fifth grade, while others may still not understand “borrowing and carrying,” it is hard to devise a one-size-fits-all curriculum.
Perhaps what is needed is better cooperation between college mathematics departments and schools of education. The creation of undergraduate and graduate degrees in math for middle-school teachers is one such solution, implemented at a small number of colleges.
The attempt to set up of such programs, however, often results in their becoming victims of the age-old issues about how much math would be in the curriculum, beyond the usual arguments about methods courses verses content courses.
There are also disputes about courses the school of education wanted on “multiculturalism” and on “political correctness” within a school building. Goodness knows how many innovative program ideas have succumbed to the “collegiality” of American higher education.
Teachers, often with the financial encouragement of boards of education, spend a lot of money and time in the acquisition of master’s degrees for the purpose of salary improvement and/or advancement in the educational hierarchy. So, they invest little or no resources on coursework that will actually result in improved education for our students.
Elementary teachers, more often than not, are the purest of practitioners of their craft. They don’t need to re-learn how to teach; they do that well enough already. What they need is to learn what to teach, and why mathematical relationships work the way they do.
Ask a typical fifth-grade classroom teacher to explain why some fractions may be expressed as terminating decimals, while others yield repeating decimals. Most know the what, but not the why. Only by knowing WHY things work in mathematics will teachers become gung-ho, passionate deliverers of mathematical knowledge, understanding, and wisdom.
Yes, being passionate about mathematics – strange concept eh? Almost an oxymoron. Unless a teacher demonstrates such passion, our students will continue to regard doing mathematics as a chore, a painful exercise, rather than the beautiful exercise of the mind that it can be.
Teachers exclaiming, “Wow!” must precede students’ similar exclamations. Teachers need content in the what’s, how’s, and why’s of mathematics. Content, not method, is what should drive mathematical education and, eventually, high achievement in the subject.