The myth of "the basics"

What's WRONG with math education?

Before I talk about why I want to define what I mean by the phrase "doing mathematics". Stephen Wolfram defines doing mathematics as a 4 step process ad I agree with him. Doing mathematics is...

1) Finding a problem in the real world.
2) translating the real world elements into the language of math using formulas and symbols.
3) Computing the answer.
4) Reinterpreting the answer's value back into the real world to help solve the problem.

I think we have lost sight of what mathematics is all about. We now have machines to free us from the drudgery of step 3. We no longer have to fixate on it as we did in the past. In the days of pencil and paper it was extremely important to focus on step 3. There were no machines to do step 3 and it was very difficult to compute answers. Procedures were made and proven to allow mankind to extend his ability to compute ever more difficult mathematics. Some Mathematicians have built their careers almost exclusively on creating and proving abstract computational procedures for step 3.

Mathematics has developed and grown as a field because it solves problems in the real world, and computing is just a part of that process. From here I will be talking about what I observe in the classroom and how it compares with "doing mathematics". I will also analyze some of the arguments used to perpetuate and justify the status-quo of mathematics: procedural step 3 thinking and technological obsolescence of content.

I want to point out that what I am advocating is that we switch how we handle and do step 3. Instead of using paper and pencil, we use computers to do the computing work. Every other step in doing mathematics will remain the same. That may mean instead of making the assignments like "solve for x" we ask questions with higher cognitive demand like "write a real world story problem that uses the equation below and a person would need to find x". The difference here is that to answer the second question, you really have to know what the symbols of math mean. Students often limp their way though the procedures of math by memorizing processes without attaching meaning to the procedure. A student may memorize the procedure of finding an average as "add all the numbers together then divide by how many numbers there are." That's great, but what good is it if they don't know that the average is the exact center or balancing point of a set of numbers? Faking math by memorizing a bunch of steps superficially is common crutch for students. Asking questions like the second one above ensure that the purpose and concept behind a math idea is learned because in order to answer the question you need to understand math, not just be good at memorization.

"What if the computer is wrong? How would anyone know?" is another excuse that is often given for the "reason" we need to do things by hand. This is a silly argument when you think about it. Why stop at computers? How can we trust your own hand calculations? How can we trust mathematicians? How can you trust the internet? How can you trust your eyes, you could be delusional couldn't you? You can be forever agnostic about everything if you want. The fact is, its easy to point to a time when computers failed while ignoring that they are thousands of times more accurate and faster than humans at the same computational tasks. If some person can find an error with Mathematica, does that mean all computers everywhere are dangerous and will destroy math? It doesn't seem to be true of hand algorithms. When we get those wrong we acknowledge it was a mistake and correct it. We don't abandon hand algorithms and move back to stones and abacuses for counting, so why would computers be any different?

"What if everyone just trusted computers all the time? That's dangerous to trust computers to such a high degree all the time. If we don't teach our students how to do things by hand how will they verify the accuracy of computers? People won't be able to think!" You might think I am crazy, but I have heard this argument more than once so I will address it. Forgive my sarcasm but in society we have books, a primitive form of data storage. Should all of the world fall into the chaos that will "surely" follow by trusting a computer to solve for X and it failing to do so, people could read these books. Some people even learn things like math by reading books alone. In fact, I think some people will want to learn the hand algorithms because they find them interesting.(Wouldn't teaching that class would be a math teacher's dream!?!) Unless all math books are destroyed these hand algorithms will not be lost.

I think its questions and lines of reasoning like the last one that really tell it like it is. People do not know the difference between knowledge that everyone should know and knowledge that only a few people need to know. The current math procedures could be taught in a math history class where the inventive and ingenious people who developed them would get the honor in history they have earned. Now most of their names are hardly mentioned because we are to busy drilling everyone on ALL the algorithms.

"The basics" in math is this undefined hazy ideal that seems to impede progress in math education. No one would EVER put up with in a class that was not so abstract. What would you say if you wanted to take a class to basic computer class and on the first day the teacher gave you some computer chips a soldering iron and began teaching circuit dynamics? In the afternoon you were taught binary and machine code. You wanted to learn how to use Microsoft Windows, but when you tell the teacher this he teacher assures you that in order to REALLY understand Windows you must first master machine code and later MS DOS, then you might be ready to learn Windows NT then Windows 2000, then Windows Vista, then Windows 7, but only if you get your doctorate. Would you pursue such a program or would you look at all the people you know, realize that none of them know machine code but still know enough about computers to use them and then drop out? This is currently what happens in mathematics education. Is it any wonder why our students don't want to learn math? THIS IS UNACCEPTABLE! No one would take a computer class like the one above unless it was a special course in computing history. So long as that was the only program available, those that did would probably make a lot of money, but their success wouldn't be because machine code was the bedrock of the industry or gave them new insight into programming. It would be because few people would have the patience to deal with that kind of program. Fewer people in the program means higher demand for qualified people and thus higher wages, but students learning of machine code and soldering could have been better spend on other more important concepts like object oriented programming, system design, program architecture, etc.

If this program were like math education today students would complete their mostly irrelevant coursework, the professors would see that the few students who completed the program were successful and confirm their beliefs about just how important it was for students to know "the basics" of computer design. Even a few of the students who liked their machine code classes would extol how useful it was that they learned the basics. Since their were no other programs out there that didn't obsess over the basics, there would be no way to tell just how useful the basics really were in creating competent computer operators. If no another program was skipping the basics to teach more useful and practical information would we assume that there was no other way to teach it as we currently do in mathematics education today? Is it really necessary to get good with a soldering iron and circuit boards for you to use a computer? Is this knowledge everyone needs to know or some people need to know?

So why all the confusion? Why is it that math never seems to move forward?
What's wrong with mathematics education is simple: everyone is totally obsessed and hung up on step 3. Since step 3 is taught almost exclusively in the classroom from day 1, often the people who get into math fall in love the the procedural elegance and exact nature of the subject. Even though we now have computers to to the drudgery of calculation, they still extol computation by hand as some kind of virtue, while ignoring the actual application of math to the real world by glossing over steps 1, 2, and 4. Simply put we have not moved math into the age of computers; We must teach all 4 steps of mathematics in balance. Bringing math into the computer age will also make math accessible to the general population, democratizing math and opening exciting possibilities. No longer would mathematical analysis be restricted to the few who learned hand procedures. Anyone who had an understanding of math terminology and relationships could analyze data or use data to make decisions.

The age of computers has made hand calculation obsolete. If you don't believe me go to wolframalpha.com and punch in an equation try "3x+2=8" Magic. Check out the introduction video here.

http://www.wolframalpha.com/screencast/introducingwolframalpha.html

It is amazing what will happen in the future! We could be awestruck by its possibilities or fearful what will math teachers do when free websites like Wolframalpha are available to them 24/7 on smartphones. In 10 years, most US citizens will have a smartphone, are we going to force more algorithms down the throats of people who don't want to learn them or will we strengthen the rigor of our math curricula and make mathematics accessible to people who don't have the patience to learn the computational end of math but thrive on steps 1, 2, and 4. Which is better for the economy: fear and no change or change and possibility?

Math is one of those subjects that people think does not change over time. I would agree that the theoretical foundations of math have not and may never change, but how we teach math, how we learn math, where math is applied, and technology are factors that do change the game of math education. Fortunately, technology has long changed the game for mathematics for scientists and engineers, but it has not for students. Technological advances are being made at an exponential rate and we need more mathematically competent people to operate and understand these complex machines.

Are we producing people fluent in the concepts of math? The answer is a resounding NO! From my experience, most people have no concept of math beyond it is working with numbers or doing problems like "solve for x", "simplify the expression" or "rationalize the denominator". You would think these procedures were all made of gold and absolutely necessary to understand math the way some math teachers and professors talk about them. "They need to know the basics" is appealing rhetoric, but are students able to do the basics?

A simple look at the NAEP, TIMMS or PISA (international math and reasoning tests) we see US students' scores doubled by their rivals in other countries. It's pretty obvious students are not getting the basics (or much else) out of math classrooms today. I don't think foreign born students are twice as smart as us or that some sort of genetic advantage evolved in the last 100 years. There is nothing wrong with the American student!!!! They have every right to be bored to tears with hand algorithms they are forced to learn in exchange for a letter on a sheet of paper. (A,B,C,D,F) Especially after they have been shown that a computer can do it in a second. Nothing highlights our dismal failure to advance math curricula forward like the poor scores we see in the US today.

The students we do produce are can barely compute all the rigorous procedures and algorithms we teach them at all much less explain why and where we would apply those procedures. Basically, we cram computation down their throats, claim is important to know, and punish them when they choke on it. We busily teach them that doing math like solving equations by hand is very important, but then after we spend hours and days in the classroom teaching them these pencil and paper procedures, we then show them how it can be done on a calculator in 20 seconds. After all that, its more than unfair to be expected to be taken seriously by students who rely on technology everyday. Shouldn't technology make these manual calculation units easier and shorter? Shouldn't the questions we pose to students be becoming more conceptual and real-world based as technology frees them from the drudgery of manual calculation? According to most mathematics educators today, the answer is no.

I will end with some good news. We may be behind as a country, but no one is where they need to be in this math debacle. Other countries are busy teaching their students the same "by hand" procedures as we are. Some like Japan do not allow calculators, and I have even seen Japanese students wasting hours working abacuses with lightning speed and precision. (apparently while calculators are cheating abacuses are not)

Why do other countries seem to destroy us in math? Its a good question that demands an answer. This could be a thesis, but I will briefly discuss some of the reasons. Most countries scores benefit from the educational systems they have in place that test students after elementary school and then place children in different educational "tracks". Often only the students who are on an academic track take the tests for the country. Students who are on a good track and like mathematics do not have to contend with the poor behavior of students who don't want to be there. This is a huge advantage. Another factor I feel is not often discussed is why Asian countries do so well on tests is language. Many Asian countries have languages that are made up of thousands of symbols. The order of symbols and their placement in relation to each other acts a lot like an algorithm to change word meaning. Math symbolism may come easier to them because many of the learning techniques they use in language (memorization and word order) happen to be useful in math symbolic manipulation.

Another reason for the decline of education in the US is the abundance of supplies Americans have. Most poor families in the US still own a TV, have enough to eat and drink, and have access to shelter. To them physical survival is not really the issue.
In other countries the poor have much less. It is obvious that education creates opportunity to move up in social standing. People who are rich are educated, and vis-versa. To our uneducated and media brainwashed youth, learning seems a lot less appealing then waiting for your big chance and making it big. Their ignorance of probability aside, it is a matter of values. Whenever our youth have the good life already they take it for granted and don't look at the factors that may have bought their family to their current status. In China the children of the middle class are already refusing to work in school. In Japan there is the increasing problem of Japanese youth not learning the full Japanese language. In Japan this language is what all academic books are written in!

Teachers keep hope, math education in the world is not much better than in the US. We now have a choice: will we move forward by using technology in place of hand algorithms, or will we forbid it until students can do everything by hand? Will we move forward or backward?