This page is not about answers, it is about defining the questions.
The educational system is not as bad as people make it out to be. A lot of good goes on in our schools. Somehow, with all its flaws, our schools turn children into (mostly) productive adults who learn how to survive and thrive in the real world.
Except in one area: Mathematics.
OK, I admit I am blowing things out of porportions a little bit. We hear college professors bad mouthing freshman all the time: "They can't find Germany on the map", "They can't spell 'business' without a spell checker", "They think the Ottoman Empire is a grunge rock band". But, when it comes to our math skills, the complaints are not just from academia but the business world as well: "They can't make simple change for a twenty", "They can't figure out a 15% tip", "They can't split the bill three ways".
I present for your perusal, a debate. A debate between the ideals of math education (the way it should be), and the reality of math education (the way it is). This paradox haunts every secondary math teacher in America. As a former teacher myself, I have heard the complains and all the arguments and all the worthless rhetoric. Most of it is too detailed to put into a short article, so I will just mow through the highlights.
| The Idealist: | Question: Why are there so many lawyers in America? Answer: Because it is the only well paid career that does not require math. Lets face it. Math is important to our lives. Want to be a pilot? you need math. Want to be a doctor? you need math. The careers that require Algebra or even Calculus are countless, and growing. If America wants to compete we need to emphasize math education in our schools. What is the cost of not training American students in math? More and more computer and engineering firms are going to Asia to recruit for high paying skilled workers, because not enough can be found to meet the need on this continent. People complain about declining real wages in this country, yet most people are unwilling to do what it takes to get the skills they need for the good paying jobs that are out there. It is a simple equation: Math skills = high wages. But, it is just not a wage issue. The lack of understanding of math concepts, leads to the lack of understanding of basic science concepts, which in turn leads to gullibility, superstition, and just plain ignorance. When most Americans do not understand basic proven scientific truths, this is ultimately disastrous for society as a whole. |
| The Realist: | Schools, school districts, and state boards of education are very well aware of the need for math in today's economy.
What have they been doing about it? raising standards, that is what. Is this improving math education? No, in fact
what it is leading to is lower graduation rates and higher failure rates. For the past 10 years, The State of Arizona has been arguing over the math standards for the state. For the most part, the new standards follow the recommendations of the National Association of Teachers of Mathematics. Click here for the latest version of the standards. A test was developed, part of the Arizona Test of Basic Skills, which, starting with the class of 2002, will be required to graduate high school in Arizona. A trial version was given to the class of 2000 in order to "means test" the exam. The results were embarrassing. Less than a third of High School juniors came anywhere close to passing. Update: Final figures were released on Nov. 15 (delayed two months because they were so embarrassing). Only 11% of those that took the test (sophomores and juniors) passed the math portion. Even worse results came from a similar test soon to be required for Teachers. Only 25% of "fresh out of college" future teacher prospects managed to pass the math test containing only beginning Algebra. Ask yourself these questions: Are we really going to stick with the standards if it means less than half of our students will not graduate high school? How are we going to teach math skills to our students that most of our teachers do not even have? |
| The Idealist: | Of course you stick with standards. The problem is that it is just not enough to test for a diploma. we need to
hold standards at every grade level. A surprising number of students come into the 9th grade not knowing their
times tables or how to add large columns of numbers without a calculator. How do you start a student on Algebra
without even basic arithmetic skills? In order to improve math education standards, we have to start in grade 1. If the goal is to have every student through Algebra 2 or Geometry by graduation, they had better be at pre-algebra levels before even getting to high school. Improving math education requires standards be set and met at every grade level. Even if this means holding students back a grade or two early on. Sure it hurts self-esteem, but what is better: self-esteem at age 9, or career potential at age 19. |
| The Realist: | The self-esteem movement is not the first trendy problem facing math education. One could make the case that bad
school improvement schemes are responsible for the whole mess we find ourselves in. Go back to the late 60's and
look at the failure of the "new math". Then came "humanistic education" where students were
encouraged to discover math on their own. This was closely followed by open schools and open classrooms, cooperative
education (or organized cheating), main streaming low level students with high level students, and the demonization
of rote learning. I am not saying any of this was bad for education in general, but it spits in the face of what
is needed to learn math. Learning math is not like learning facts. It is like learning a new language where meaning is moot. Rote and repetition is the only means of understanding in the beginning. Students need to memorizes the multiplication tables, otherwise they will never take multiplication or division seriously. Students need to be divided by their skill level, otherwise the slow will get lost and the smart will get bored. Math is not like any other subject, different rules for successful teaching apply. But, there will be those that say such methods are old fashioned. Math has been taught in places of learning for at least 2700 years now. The successful teaching methods of Thales, Pythagoras, and Euclid are the same ones used by my 1890's math primer I keep in my library, which are in turn used by the Saxon method (based on test results the most successful method used in schools today), and virtually every Calculus professor at the college level. Successful math education does not fit with what is trendy it fits with what is proven. And, anyone who says otherwise needs a refresher in logic. |
| The Idealist: | But, there is such a thing as too old fashion. In the past, we used slide rules and abacuses, today we
have calculators and computers. Is it not more beneficial to cater to this technology? Math is so complicated the
higher you get, that in order to get to the higher levels fast, maybe we should abandon those concepts that could
be done on a calculator. Lets worry about those concepts needed in higher math, and not sweat the small stuff. We have math programs in place today where algebra students are learning to visualize math problems by starting with a graphing calculator. Geometry is taught in the context of computer graphics. Spreadsheets become statistical analysis laboratories. These are successful programs. Granted they are mostly geared towards the high level student, but why can't similar programs be used in lower levels. Calculators are faster and more accurate, why not emphasize their use? |
| The Realist: | Quick, you are a nurse in a trauma ward, a 250 pound man is having a heart attack. If you do not treat it with
Adrenaline in 30 seconds, he is dead. You know 5 ml of adrenaline is needed for a 150 pound man, how much do you
need to give him? There is no calculator handy! Times up! The answer is 8 1/3. If you gave him less than 8 it is not enough, he is dead. If you have him more than 8.5, he overdosed and he is dead. Your lack of math without a calculator just killed a patient. Low level math requires rote to learn, high level math requires creativity and actual understanding. What does it mean that 4 times 3 is 12? What number comes after ½? Why is it that when you multiply two negatives, the answer is positive? Why are there more infinite real numbers than infinite rational numbers? Understanding only comes with knowing, not with processing. |
| The Idealist: | Lets say I agree with your premise that a mixed skill level classroom is a bad environment for learning math. There
is something else to be considered as well. One of the problems in education is different learning styles. It is
obvious to anyone who teaches math that those who learn math the easiest are those that think abstractly. Good
math students are more likely to be into music composition, than painting or auto mechanics; more likely to be
on the Chess team than the Football team. Is it possible that the lack of good math students has to do with the
lack of natural abstract thinkers? If so, are there better methods of teaching math to those students who think more concretely? Actually, there are. One program that has shown promise is the Interactive Mathematics Program (IMP). It presents mathematical concepts from a problem solving and game playing perspective. Best of all, it is not dumbed down at all. It is a rigorous class, just presented in a different way. Students who normally do not get math are showing success in programs like IMP. No "cure all" exists. But, if this program is given as an alternative to those that will benefit, why not offer it? The point is that there is no one ideal solution to teaching math. Improving math skills requires offering choices and catering to the learning styles of the students. What must be kept in mind is that Mathematics, at its core, is really easy. It is made up of a few basic rules that have to be followed. The problem we humans have with math is not the complexity, it is the lack of meaning. When concepts are meaningful and have a purpose, we can understand. When concepts lack meaning, as mathematics does, it is literally alien to our nature. Give a high school student a complex relationship question like: Julie was first attracted to Brad because he was fun and spontaneous, but now that they are committed to one another, Julie is starting to find Brad is unreliable at keeping scheduled meetings. Could it be that the thing that attracted her to him is also the thing that is now causing trouble? You will find students writing essays on the subject. Give the same student a simple math problem: Julie has x number of dimes, Brad has y number of quarters. Together they have 15 coins totaling 3 dollars, How many coins does each have? You will probably find the student cowering in the fetal position. |
| The Realist: | As wonderful as many think programs like IMP are, there are many
good arguments why such programs can do more harm than good. Non-traditional math programs have one goal in
mind: Make math fun. If students enjoy math, they will learn math. Such ideas often get too blown
out of proportion. IMP works for some students, it fails for others, and it is closely tied to learning styles.
Many districts are blinded by its apparent success. Conclusion: IMP for Everybody. Result: Students who succeeded
in traditional classes find IMP to be lacking. If our goal at the secondary level is to prepare students for college mathematics, scientific fields, or at the very minimum prepared for the SAT's, we are doing students a disservice with "fun math". No college I ever heard of use IMP style programs in their lecture halls. No SAT questions involve pigs or pendulums. Making math fun is not the same as making math understandable in the real world. High school mathematics should remain traditional in their approach, because colleges remain traditional. This need to make our classrooms fun and exciting is a very modern development. Students had far better attention spans and fewer distractions in the days before television. Can schools compete with the entertainment industry? No. Should we even try? Probably not. |
| The Idealist: | If these reforms are so doomed to failure, why are they so popular?
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| The Realist: | I blame bonehead professional educators who are upset that 50% of our students are below average. :-]
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If you think the controversies above are disheartening, I merely limited myself to those that deal with teaching math. For a look at educational controversies in general see my highly biased philosophical essays on education.
It comes down to this: We have the knowledge needed to improve mathematics education in our schools, what we lack is the political and social will to pull it off. The mathematician in me realizes that this principle is a general one: We have the knowledge needed to [insert favorite social problem here], what we lack is the political and social will to pull it off. So the response from the general public on the need for better math education is perfectly understandable: get in line!
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This page dedicated to Scheherazade Haque (1967-1999), friend and fellow web hacker. |