Wednesday, January 14, 2009

Wednesday Math, Vol. 56: First thoughts on math education

Over the next few weeks, I'm going to talk about math education. The question I have been thinking about is what we should expect in terms of mathematical knowledge from high school graduates. Many states have exit tests from high school now, and most of these tests go up through Algebra II or some other set standard. Personally, I think most of the standards I've seen set are too high. Students who are going on to college need to know that much and probably more, but what is a reasonable standard for all high school graduates, including those who don't plan to go to college? What kinds of things should be known by citizens of a democracy and workers and consumers in a modern economy?

I'd like to see a basic educational framework that focuses on numeracy, a word coined in the late 1950s as the mathematical version of literacy. One place I'd like to start is to take the fear out of word problems and to make them more practical.

Here's an example.

1. Solve x^2 - 10x + 21 = 0.

2. Find two numbers whose sum is 10 and whose product is 21.

3. You have 10 feet of fencing to make a pen for chickens that needs to enclose 21 square feet of floor space, and you can use the corner of a rectangular barn as the back wall and left wall of the enclosure. How long and wide should the pen be?

What I would like to see from high school graduates is not just the ability to solve these problems, but the understanding that these are really three statements of the exact same problem. The first time it is stated algebraically, the second time colloquially and the third time in the form of a word problem.

What do you think should be the standard for high school graduates in the 21st Century? Clearly, what I have stated here isn't the only thing I'd like them to know, but I think it is a good example of level of competence I'd like to see.

jolie said...

do you have any thoughts on why, for learners like me, question 2 is so much easier to understand and thus solve than either 1 or 3? is it a left-brain, right-brain thing? or is it simple familiarity with a concept?

I can do theoretical for complicated cause-effect economic or political issues but find it hard to parse through abstract math problems.

John V said...

That's actually a pretty complicated word problem -- you have to translate your geometric knowledge of area and perimeter into a system of two equations in two unknowns, which reduces to a quadratic equation. If you'll pardon my saying so, it's also not very practical -- a farmer would most likely want to make the largest enclosure possible with a given amount of fencing, which requires you to find the value(s) of x for which the curve y = 10x - x^2 has its maximum y value. I don't know how to do that except by plotting the curve and inspecting it (or, equivalently, evaluating the equation for each value of x).

Anyway, I'm not convinced that an average citizen/consumer needs to know more than basic arithmetic, geometry, and maybe some trig. Certainly in my everyday life, or even at my job as a computer programmer, I seldom have occasion even to use algebra, let alone calculus (which is good, because I've forgotten most of that).

What I do think would be immensely valuable to citizens and consumers is a solid grounding in probability and statistics, not as it's usually taught as a collection of abstract concepts like standard deviation, chi-square test, Poisson distribution, etc., but its real-world applications -- gambling, evaluating risk, polls, safety and effectiveness of drug trials, and so on. Most of us don't have to perform the mathematics, but we do need to evaluate the "lies, damn lies, and statistics" that we come across every day.

Tara Mobley said...

What does it mean that I was able to solve them, using different methods, in my head?

Oh, although problem 2 had the easiest solving method, I find quadratic equations more fun.

Matty Boy said...

Jolie: The second statement is the easy way to put it and can be solved by guessing and checking for anyone who can multiply one digit numbers or can factor 21 in his or her head, which I hope is a solid majority of the population. It is the "easy" statement, but all three have the same answer.

John V: it does take some knowledge of how algebra and geometry work together, but I think that can be considered stuff we can teach nearly anybody, now that the basics are more than 300 years old.

Sometimes a space won't allow for the optimal solution.

I agree that the basics of statistics should be part of the high school curriculum, which I will be writing about in later posts.

Tara: Any math problem worth its salt has multiple ways to solve it.

Mathman6293 said...

A form of your problem is one of the GA state tasks for our new math1 course. It is presented as a fence problem where the students must explore perimeter - linear and then area - quadratic. In order for our students to be successful they must know Area, Perimeter, how plot points and linear equations - all middle school standards in GA. The problem introduces the idea of quadratics and continuous vs discreet values.

I think that our standard expectations, in GA at least, that all kids must take 4 years of math is too high because not all kids even if they go to college will need all that math. Also, it will eventually keep many kids from graduating. I think that we need to ensure that kids understand basics like fractions and how data can be represented.

Now, that our 9th graders know, for the most part, the first half of Algebra I they are more receptive to more difficult concepts but they still are weak in basic concepts of fractions, multiplication and graphing tables and functions.

Nan said...

As a nonmathematician I'd lean towards numeracy in the form of being able to understand and solve word problems similar to the one you give as an example. That seems to be the area where out in the real world people fall down the most. I have known many people who could have solved the problem easily the first two ways it's expressed, but would just stand there looking dazed if it's phrased as a word problem.

Matty Boy said...

Mathman: Thanks for the info. Like you, and even though it means taking money out of both of our pockets, I think four years math for students not planning to continue on to college is too much. I'd like to see less classes, but more useful ones.

Nan: You are absolutely right. Translating from a real world experience to a math problem is a skill a lot of educated people do not possess. Sadly, a lot of textbooks make word problems very tricky and the setting even less "real world" than the one I concocted.