I was made aware of another major failure of the U.S. education system a few weeks ago, when someone I know made the statement that she knew how to do the math, but she was not good at story problems. It turns out that this is extremely common. I feel I need to make something clear, if you cannot do "story problems," knowing the math is worthless.
Math is about solving problems. I don't mean that it is about solving problems that look like "3 × 4" or "32x + 3y". These are the kind of math problems you get in school, on homework and tests. They might be good for passing an utterly awful math class that does not actually teach anyone how to use math. The fact, however, is that there is absolutely no value in learning this kind of math, if you cannot apply it to real problems. In short, we could eliminate math entirely from the U.S. education system, and the negative effects would be minimal, because the way we currently teach math is not actually worth anything! (This is a mild exaggeration, as occasional students figure out application on their own, as they learn the abstract math.)
Consider this math problem: You have a standard room in a house (that means the ceiling height is 8 feet, and the corners are all right angles). The floor dimensions are 10 feet by 12 feet. You are buying paint that claims to cover 350 square feet per gallon. How many gallons of paint do you need to paint the walls and ceiling? If you cannot figure out how to solve this problem, you do not actually know math. What I mean is, this is a real math problem. All real math problems are like this. Not only that though, but this problem is also a trivial problem. If you cannot solve problems like this, you don't actually know math. Being able to solve abstract problems like 3 × 4 is nice, but it is purely academic. It is worthless by itself. In fact, it might be argued that it is utterly worthless either way, because computers can do that problem faster and more accurately than any human. Real math is all word problems, and if you cannot do word problems, there is no value in knowing how to calculate the answers.
Now that you feel that I have insulted your intelligence, let me explain something else: None of this is your fault! Yes, I am absolving you of any fault, because it is not your fault. Now, I have heard math teachers say that the problem is the laziness of the students, and maybe this is a small part of it, but it sounds like an excuse for poor teaching performance to me. The fact is that the students would be a lot less lazy, if they understood how to use this math in real world applications (this has been demonstrated on many occasions). In other words, the laziness of the students is probably largely attributable to the teachers who don't know how to teach real math.
Of course, even the teachers are not entirely at fault. This problem can be traced back to the industrialization of education. There was once a time when math was taught purely in terms of application. Merchants learned addition, subtraction, multiplication, and division in terms of money transactions. Farmers learned basic math in terms of crop yields and taxes. Alchemists and other scientists learned more advanced math in terms of their specific disciplines. No one learned "2 + 2 = 4". Instead they learned "Two dollars plus two dollars is four dollars", "two bushels of wheat plus two bushels of wheat is four bushels of wheat", or "two ounces of mercury plus two ounces of mercury is four ounces of mercury". The industrialization of education (which came with the creation of the public education system) changed this, attempting to generalize math by reducing to the most abstract form. The problem is that removing the context from the math makes it worthless for everyone. Knowing this abstract math does not matter if you don't know how to use it.
How do we fix this problem? To start with, teachers need to learn to teach math differently. Math should be taught using a variety of applications, from money, to calculating how much paint is needed, to figuring out the best unit price for buying a product. Our current system teaches abstract math and then expects students to figure out the application on their own, and it clearly fails at this. Teaching to the application instead of adding it as an afterthought is the key to teaching real math. Unfortunately, there are things in place that hinder this goal. The good news is that they are easily fixed. The typical math text has a list of 30 homework problems at the end of each chapter. There are usually 2 to 4 kinds of abstract problems, and 2 to 4 word problems at the end. Math tests are similar, with 15 to 30 problems, where only 2 to 4 of them are word problems. Given this balance, it is clear why teachers don't bother making sure students actually know how to use the math. When around 80% of the problems are abstract, and our current education system is funded largely based on test scores, even competent teachers who know how to teach real math feel backed into a corner. So, I propose that students cannot pass a math class without knowing how to apply the math. Instead of 20 to 30 math problems for homework, only give students 5 to 10 problems, but make them all story problems. For tests, no more than 30% of the problems should be abstract problems, and no more than one abstract problem of each type should be allowed. As with homework, reduce the number of problems accordingly. A 10 problem test with 7 or 8 story problems will tell far more about a student's ability to use math than a 30 problem test with only 2 to 4 story problems. In addition, this can leverage the propensity to teach to the test to encourage teachers to teach students how to use math, instead of just teaching them to compute problems that any computer could do for them.
Now, I don't want to imply that abstract math is entirely worthless. Memorizing addition and multiplication of all one digit numbers will make learning pretty much all higher math much easier. Learning abstract algebra concepts is essential to certain kinds of higher math problems. The fact, however, is that even all of these abstract concepts have real world applications, and teaching to the application is not that hard. Within a single math class, teaching significant amounts of abstract math may be essential, but anything the student is graded on should require a demonstration of application of the abstract concepts. The abstract math should be stepping stones, not the end goal.
Abstract math has no value without application. Research has shown that teaching application motivates students to learn math better, and students who learn math through application come away with a better understanding of the math and much better ability to use it. The first reform we need in math education is to change the homework and tests so that teachers teach application instead of abstract math.
If you are still confused with the above story problem, I will explain how to solve it here. A standard room has 6 surfaces (4 walls, a ceiling, and a floor). We only want to paint the four walls and the ceiling. The dimensions of the room are 8 × 10 ×12, but it is important to know which dimensions go with which surface, because we don't need to buy paint for the dimensions of the floor. The ceiling and floor are 10 × 12, two of the walls are 8 × 10, and the other two are 8 × 12. We need the surface area of the ceiling and all of the walls, so (10 × 12) + 2 × (8 × 10) + 2 × (8 × 12) = Total Area. This gives us 120 square feet for the ceiling, two walls with 80 square feet each, and two walls with 96 square feet each, for a total of 120 + 2 × 80 + 2 × 96 = 472 square feet. Now, a single gallon can of paint covers 350 square feet, which is not enough. Two cans cover 700 square feet, which is too much. Since we cannot buy a fraction of a can, we will have to buy too much (which is perfectly fine, since now we will have some extra in case we need to make repairs in the future). So, we will need two gallon cans of paint for this project.
This problem may seem complicated if you have not learned to translate real world problems into abstract math, but honestly it is incredibly easy. If you really want to know what is complicated, calculate the exit velocity of a projectile from a rail gun given the materials and dimensions of the rails and the projectile, the initial input voltage and capacitance of the power supply, the friction between the rails and the projectile, and the resistance of the wires from the power supply to the rails. I'll give you a hint: This problem require calculus, but the calculus is not the hard part. The hard part is translating all of that data into the abstract math problem!
