Whole Math Nonsense

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Whole Math Nonsense

by Ari Armstrong


I became most concerned with the state of mathematics education in this country about a year ago when I attempted to help my sister with a homework assignment from her political-school math class. She was in 8th grade then. The assignment consisted of about two pages of problems. Some of the problems were "traditional" math problems which I could solve and which I could teach my sister how to solve. (The math book made very little attempt to actually show students how to solve the problems, however.) The rest of the problems - around a quarter of the total - were literally impossible to solve, however.

I completed the assignment with about 10 pages of text and figures (typed, single spaced). Of course I explained why some of the problems could not be solved, because of insufficient information, and I wrote in a number of qualifications to other problems under which I could offer some solution, given (my own) added parameters. I have never met an 8th grader capable of the skills and knowledge needed to qualify and approach such problems.

What struck me about the assignment was the wide number of "practical examples" presented which weren't actually relevant to anything practical. One of the problems which was un-solvable, for instance, was an economics problem; I knew it to be un-solvable because I had earned my bachelor's degree in economics. The book was written by perhaps 20 people; apparently it suffered from the "too many cooks in the kitchen" syndrome. It was gibberish, at best.

I unfortunately do not have the citation of the book available. A few days ago, though, I obtained a copy of some of the materials used in a class currently taught at Boulder High School in Colorado. The materials are similar to those my sister was trying to use. I'll here quote from this material and explain some of the implications of this type of "education."

The teacher of the class, in her "Course Outline," explains the purpose of the materials:

"The Interactive Math Program replaces the traditional algebra 1 - geometry - algebra 2 - trigonometry - precalculus sequence by integrating each strand of mathematics into the curriculum. The program challenges students to actively explore open-ended situations in a way that closely resembles the inquiry method of mathematicians and scientists. The Interactive Math Program is supported, in part, by the National Science Foundation."

This sounds pretty superficially, but what exactly does it mean? "Integrating each strand of mathematics" means "presenting mathematics in a non-hierarchical way." The result of trying to teach math without its natural hierarchy is that students have absolutely no idea what they're doing at any time. Students are presented with, say, trigonometric functions but are not given the foundations in geometry necessary to an understanding of what the trigonometry means. This results in students performing math by rote-memorization, for they never really learn what the symbols and formulas mean. The students can apply the symbols and formulas in only a narrowly circumscribed arena - the math book exercises - and never gain a rich understanding of the math. Hence, students good at "faking it" excel; students who dig for meaning are left discouraged.

"Open-ended situations" largely means the type of un-solvable problems of which I wrote. While it is true that we frequently encounter real-life problems to which no solution exists, it is equally true that the only way we're able to recognize these problems for what they are is by calling forth our knowledge of mathematical principles. I.e., we have to *learn math* in the first place to be able to solve math problems and recognize problems that cannot be solved. Simply presenting children with un-solvable problems without helping them build a foundation of mathematical knowledge leaves the children without any means whatsoever to think about math.

And that bit about this "Interactive Math Program" being similar to the "inquiry method of mathematicians and scientists" is truly disgusting. Scientists and mathematicians build on real mathematical knowledge by applying that knowledge to new theoretical and experimental problems. They do NOT shun a basic education just so that they can confuse themselves with theorems which have existed for thousands of years. If we're really interested in encouraging our children to pursue mathematics and science, we need to help them learn how to think critically and logically and how to master the body of knowledge in math and science which has been building for scores of generations. If we wish to smother the spirit of scientific inquiry in our students, we could do nothing more consistent with this end than present them with dis-integrated formulas, unexplained symbols, and nonsensical "practical examples" which have no solutions and no bearing on real-life.

In a unit on quadratic equations, students are informed, without any background information, that in order to figure out how far a fire-work rocket will shoot, they will need the formula "h(t) = 160 + 92t - 16t(^2)." This is a "practical example" of a quadratics equation, you see. But where does this formula come from? What does it mean? "You will learn more about this [next year] when you study *High Dive*," students are told.

In addition, go the materials, "[T]he team can find the horizontal distance the rocket travels with the following formula: d(t) - 92 x cos35(degrees) x t." Again, the students are told that they'll learn something about what the formula means in a year.

The exercise consists of the following:

See whether you can help the soccer team [trying to figure out how far the rocket will fly] find the answers to their questions.

1. Draw a sketch of the situation.

2. Write a clear statement of the questions the soccer team wants answered.

3. Describe how you might use Hilda's formulas to help answer the questions you stated in Question 2.

4. Using whatever methods you choose, try to get some answers (or partial answers) to the questions you stated in Question 2.

Whatever it might be, this is not math education. Students can learn nothing from this. At best, they can memorize a couple formulas for a short time, without ever knowing what the formulas mean, and perhaps run a couple numbers through the formulas to attain "partial answers" to a math problem (without understanding the significance of these partial answers). And we wonder why American children have trouble in math and science?

In case a student still has any desire left to actually learn math, any spark of excitement associated with mastering difficult concepts, the _Interactive Math Program_ presents other exercises for students to refrain from solving.

I'm going to quote most of an entire exercise, also related to quadratics, and then explain why the exercise is completely ridiculous, for those to whom this fact is not immediately obvious (perhaps those who actually relied on political-schooling for their education).

Do you remember the housing developer and the city planner from *Solve It*?

As you may recall, the housing developer had submitted plans to the city planner for some houses she wanted to build. But the city planner thought the plans were boring because the lots were all square and all the same size.

After some discussion, the planner and the developer decided that the lots should include other types of rectangles. So the developer proceeded to change the lengths of some of the sides of the lots.

The developer wrote plans for the new lot sizes, but put them in code, using the variable X to represent the length of a side of the original square lots.

For example, if the original square lot was extended 4 meters in one direction and 3 meters in the other, the developer represented the new lot by the factored form expression (X+4)(X+3).

1. a. Draw a diagram illustrating the lot represented by the expression (X+4)(X+3).

b. Find an algebraic expression without parentheses for the area of this lot. That is, write (X+4)(X+3) as a quadratic expression in standard form.

In *Solve it!*, you helped the city planner decode some of the developer's expressions. Now it's time to do some more.

2. Here is what the city planner found written in the developer's plans one day:

"Build a lot whose area is X(^2) + 4X + 2X + 8."

Please help the city planner, by finding out what the developer planned to do with the original square lot.

(Note: Question 3 (a) and (b) and Question 4 (a) through (d) are merely variations of Question 2 but with different quadratics)

THIS is what we are to take for a real-life example?? A "city planner" and a "housing developer" arguing over whether square houses are "boring"?? This is the "open-ended situation" intended to "replace" the "traditional curriculum"?? May god help us, for our students will surely be unable to do so.

Besides "integrating" a ridiculous story into the problem, the exercises are utterly meaningless.

Presumably, the point of Question 2 is for the students to factor the quadratic X(^2) + 4X + 2X + 8 into (X+2)(X+4). But there's no reason to think, as the problem seems to suggest, that the quadratic is to be interpreted to mean, "One side of the building is to be X+2 units long while the other side is to be X+4 units long." Such an assumption would take more qualifications than are provided in the exercise. Taken at face-value, the quadratic merely specifies an AREA, which might take on an infinite number of dimensions.

So I suppose the students must proceed in a touchy-feely "maybe this is right" manner to come up with the answers in the teacher's answer key. It's meaningless, it's arbitrary, it is, taken as written, un-solvable and yet it implies that it is solvable.

This sort of education is horrible at preparing students to understand mathematics and the sciences. But I suppose it does a fair job at preparing students for the mushy sort of "thinking" so important to remaining a complacent ward of the State.

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