I've taught calculus at the college level and, now that my daughter is taking algebra in middle school, I see why students have so much trouble with college math: they aren't taught math in middle school. Let me clarify this statement by first illustrating what I mean when I say that too many students have trouble with calculus and why you should care about that. Calculus is generally considered to be the first college math course. It's nice if students have taken calculus already, and it's even possible that their high school calculus class was good enough to allow them to skip the first quarter of college calculus, but it's not absolutely necessary. Students generally take calculus from the beginning in college, they get credit for it, and degree curricula are designed with the assumption that they'll take calculus. On the other hand, few pre-calculus or earlier classes count for college credit -- they are offered on a supportive basis to make up for background that students should have already had.
Moreover, calculus has become a gatekeeper course for many degrees in the sciences, engineering, and even business. For many degrees, calculus is an absolute prerequisite for later core classes. In other degree programs, calculus may only be a prerequisite for certain electives, but the faculty have made the decision that they don't want students moving through the program who are unable to take these electives. So, a student unable to succeed in calculus (notice I didn't write "pass"; students need to become facile enough with the subject matter to apply it naturally and without effort in classes that build upon calculus) will not become a science, engineering, or business major. I cannot emphasize this point enough: if a student has not been prepared to succeed in calculus by high school graduation time, it is much less likely that he or she will be able to have some of the most rewarding careers.
So, why are students having trouble in calculus? While there are many reasons, the biggest is a lack of facility with algebra, and by this I mean the ability to solve algebraic equations by symbolic manipulation. A simple example of this might be to solve the equation 3x + 12 = 45. A symbolic approach would be:
3x = 33 (subtract 12 from both sides)
x = 11 (divide both sides by 3)
I'll explain why this is so important for calculus in a moment; let me first describe what children are taught instead.
Basically, they're taught something more like math appreciation than math. Let's use as an example my daughter's textbook, which is Contemporary Mathematics in Context: A Unified Approach, part of the "core-plus mathematics project" from McGraw-Hill. First off, much of this book is statistics, rather than algebra, even though this is supposed to be an algebra I (eighth grade honors, or ninth grade regular) class. The kids spend enormous amounts of time in groups, measuring each other's fingers, counting which thumb is on top, etc., so that they can gather some of their own data. They spend even more time writing paragraphs describing histograms. Then they move on to linear equations, motivated by linear fits to data -- linear regression -- rather than just algebra. Part of the reason for this is that they have yet to see an equation in the book!
Finally, they get to linear equations, though first through a little detour in which the words "NOW" and "NEXT" are used, instead of letters, for variables. Why they do this is mystifying, since all of these kids have seen problems in elementary school in which shapes were used instead of variables. Along the way, they learn about slopes, even of slopes of nonlinear functions, which is a calculus problem. Of course, they don't need to actually do any math in that last case (thought they're given the quadratic equation involved), just write about what the slope of a curved graph might mean, how rate of change relates to slope, etc. By this point, they've note seen more than three equations on any page of the book, and most pages have none at all. No techniques have been presented for solving linear problems; students are asked to solve such problems by inspection of graphs or tables of (x, y) pairs. Finally (precisely 2/3 of the way through the book), a symbolic approach is mentioned (in a chapter entitled "Quick Solutions"). Here's what the book says:
...it is often possible to solve problems that involve linear equations without the use of tables and graphs.
This is absolutely untrue! It is
always possible to solve such problems this way (i.e., symbolically). In fact, the fraction of problems that can be solved with tables or graphs -- the only method used so far in the book -- is
vanishingly small.
Let me disabuse you of the idea that the book will now introduce methods such as adding/multiplying both sides of the equation by the same value. Instead, what the book does is say that different people might reason about solving such equations different ways. A couple examples are given, and student groups are asked to try to figure out what the reasoning process is in each case. Then, kids are invited to solve equations in whatever way makes sense to them. That's it!
So, to sum up, symbolic manipulation is erroneously introduced as a specialized shortcut that can work in some cases: a sort of mental trick with no fixed methods that you just have to figure out for yourself. This is in a five-page section, after more than 200 pages that are taught with graphs, tables, and calculators in the context of statistics.
So, why is this a problem? Because calculus requires facility with the symbol manipulation approach. Unlike simple algebraic problems, which can (again, in simple cases with integer solutions or the like) be approached graphically, calculus is the mathematics of change. Graphical aids in calculus involve computer programs in which you interactively move things around and watch how solutions change. While solutions are numbers in algebra, they are equations in calculus. If the symbol manipulations of algebra aren't as easy as breathing for a student, he or she will have a tough time in calculus. I cannot see how a student who has used the Core-Plus curriculum can pass calculus, unless his or her math education has been supplemented outside of school. Thus, in Washington state we are preventing most of our children from getting the degrees that lead to the highest-paying, fastest growing career paths that exist: science, technology, and business.
And who is to blame for this mess? Why, the Superintendent of Public Instruction, Terry Bergeson. She's the one who has pressed this kind of instruction for a decade now. She's the one who has instituted a special test for Washington state that prevents our children's performance from being compared to children in other states and countries -- a test that costs much more than commonly-used alternatives.
That's why I'm advocating in this post for Randy Dorn for Superintendent of Public Instruction. If you're a Washington state voter, click on the title of this post to go to his web site. Read about him. Compare what he plans to do to the current abysmal state of education. It's time for a change in this Washington, too.
