Math Education-Is There More To It Than Just Numbers?

Rasha asks: "I am taking a class this semester which discusses different models of the human mind. One topic that has caused much debate is the nature of math education and its goals. I am writing a paper for this class which will attempt to discover what exactly we try to accomplish by teaching math to young students. Are we trying to give them skills or is there more? Is math our attempt to access the more abstract parts of students' brains and develop them? I have sent a survey to a bunch of teachers (mainly elementary and high school) but I am curious what Slashdotters think. I think that most Slashdot readers are probably more mathematically inclined than the average and might have a greater insight into the issues that I'm addressing. Also if anyone knows any previous research on the topic let me know (this is not the focus of the paper though, I'm just curious)." (There's more...)

"Here are some of the questions I sent to the teachers so you can get a feel for what I'm looking for:

• Of all the subjects which is the most important for the development of the student? That is, which subject gives the most skills to the student beyond the actual information taught? Why? What is the goal of teaching Math to children? Is it to give them skills to manipulate numbers or does it accomplish something else (or maybe both)? What are those skills?
• People often say that math teaches abstract reasoning. Is this so, how and why? Could there be a better way to accomplish this?
• With the development of small computers and calculators do you see the role of math education declining? Why or why not?
• Why are children often forced to memorize multiplication tables and do long division?
• Why is it that students who have some deficiency in math are stigmatized as "not so bright" more often than children who fail to do well in other subjects? Conversely, why are children who excel at math considered gifted (more so than other subjects)?"

Comments

A few answers on math education

[Matthew Weigel]

• I think that good math and science courses, coupled with plenty of social interaction, are the most important individual elements (although I would actually consider a more balanced curriculum better than one in which math/science is tops, and everything else is non-existant). The reason is that it teaches a way to think that is both abstract and directly verifiable, so it's more critical of the thoughts. Then again, I think all of it is important.
• Math doesn't just teach abstract reasoning, but it teaches abstract reasoning with a starting point in reality. A calculus, after all, is simply a stone (and it's stil used that way in the medical community, I believe) -- and to calculate is to play with stones, counting and measuring. Addition and subtraction are pretty clear concepts, but you can abstract them into multiplication, and then you can abstract the "undoing" of multiplication as division, and so on, until you're dealing with things that are apparent nowhere in the real world (not really true, but true prima facie).
• I don't see calculators as being at all useful for teaching until after the skill has been taught. Addition is easy -- especially if you do it sufficiently concretely or abstractly that you use the way different people think, rather than work against the way they think -- and other arithmetic follows from it. As in all things, computers are useful so long as they don't try to give skills, but merely use them faster. What's the point of hiring a human to do something if all he does is push buttons unknowingly?
• Memorization of multiplication tables is, IMO, a good example of 'testing' gone wrong. I mean, clearly anyone who can do multiplication with relative ease can reproduce multiplication tables as requested, so it appears (to my mind, which isn't very familiar with the history of math education) to have been a goad to get kids to learn to multiply. But the skill itself is not tested, and hence not learned. As for long division -- is there something wrong with learning it? How is "long" division different from "regular" multiplication when lots of digits are involved?
• And finally, first get rid of the idea that the kids who get good grades in math are "gifted" or "inclined" to math; the two sets intersect, but are not the same. The reason they are not, of course, is because different people learn different ways, and in a lot of schools, everything is taught one way. The reason children who do well in math are considered gifted is because a lot of people have trouble with math, precisely because it's taught the wrong way to different kids! So, by teaching it only one way, they are creating an artificial elite out of the children who are basically competent and happen to learn things the "right" way. Bah.

Math=logic (well, not really)

[Ian Bicking]

There's two types of math with much different goals. Arithmetic and some simple problem solving (like learning to balance a checkbook) is one. These are the things people need to be functional in the world. Calculators don't make this unnecessary, though they make certain parts of it unnecessary. I never use long division and neither does anybody else. And a lot of programs aren't teaching it anymore.

Eventually, maybe this stuff won't be very necessary. When there is ubiquitous and omnipresent computing ability (chips in our heads?) and if we feel we can always trust these, then complicated arithmetic will be unnecessary. But kids spend a lot of time learning 5+3, and when you become an adult you need to be able to figure out simple math like that without an interruption in thought. These things that must be learned are mostly "math facts" - the basic bits of math that we must learn to solve without reducing them, i.e., you can't be very functional if you figure 6+3 by using your hands, but you must simply Know that 6+3=9.

Many (most?) kids don't really get much else than that. They are introduced to other things, but the introduction is poor and their attention is difficult to maintain on something that requires hard thought.

For instance, I say 6+3=9, but really people think "6 and 3 make 9". The deeper notion that 6+3=9 implies that 9-3=6. If you really understand this, answering 4x+2=10 is pretty easy (though is still requires a certain gestalt to achieve the specific value of x from the infinitite possibilities). But equality, timeless and hopelessly nonimperitive, is a very difficult concept. C's x=x+1 is far easier, though from a mathematical perspective is looks terribly confused.

Really, all these hard bits of math are philosophy, not skills, and certainly not science. There's not a lot of philosophy until college (and even most of that is dumb, but I won't digress). That's probably not the way it should be... and that the most inaccesible bit of philosophy -- math -- is the most emphasized is perhaps a bit peculiar. A question like "can killing be justified" is something you can relate to life. But when you really start thinking about probability, say, and the fact that there's a 50% chance that a coin will flip heads before you flip it, but that chance becomes 100% one way or the other after you've finished... it's awfully abstract.

So people say math is about teaching abstraction, and maybe it is. Being able to resolve the infinitude of possibilities into one solution.

So I think good math education is about exploring the intuitive (gestalt/right-brained) solution of problems, abstract (numerical, symbolic) and concrete ("real world"). Ironically, the most successful math students are rigorous and left-brained, because teachers like this and give problems that can be solved this way -- even though real problems that people actualy want to solve are seldom so easily solved. (this bias is by no means isolated to math, though)
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My answers

[Otter]

I'm not sure exactly what you're asking us for, so I'll respond to the specific questions you mentioned:

Of all the subjects which is the most important for the development of the student? That is, which subject gives the most skills to the student beyond the actual information taught? Why?

Reading. Developing ability in reading and the habit of reading makes learning a lifelong activity instead of something you're forced to do in school, that ends the moment you graduate.

What is the goal of teaching Math to children? Is it to give them skills to manipulate numbers or does it accomplish something else (or maybe both)? What are those skills?

For the large majority of students, the goal is to get them to learn math skills and how to apply those skills in real life situations. For a minority of students it's more, and it's crucial to make sure that all the students who might benefit from more get the opportunity to show it.

With the development of small computers and calculators do you see the role of math education declining? Why or why not?

A few years ago, I made a $16 purchase in a supermarket. I gave a $20 to the assistant manager (!) working the register. He rang it up as $200, and started counting out the $184 the machine told him to give me. That's why you need to learn to do math yourself.

Why are children often forced to memorize multiplication tables and do long division?

In theory, you could learn math by principles rather than by memorization. In reality, that's a disaster because 1) many, if not most, students are not capable of learning math that way and 2) many, if not most, elementary school math teachers don't understand the principles well enough to teach that way.

I find it odd that people are so convinced current methods of math teaching are wrong. This isn't something new, like "diversity education". Formal math education has been systematically refined since Euclid. Why is it so hard to believe it works well?

Why is it that students who have some deficiency in math are stigmatized as "not so bright" more often than children who fail to do well in other subjects? Conversely, why are children who excel at math considered gifted (more so than other subjects)?

I'm not sure that's true. On the contrary, at least for educated American adults, saying "I'm terrible at math" is not considered embarassing. If anything, people take a sort of pride in it. But you'll never hear an engineer or physicist brag, "I have poor reading comprehension." or "I'm a terrible speller." Well, except for Rob Malda...

Real life use for math (algebra & geometry)

[cr0sh]

One use for mathematics I have always found handy has been for those stupid "how many jelly beans are in the jar" contests...

One can usually get fairly close via figuring out how many jelly beans will fit into the area of the jar's cross section (sometimes they make the jar round to make it more difficult, but hey, A=PI(r^2) for a circle - no sweat), then multiplying by the number of "thicknesses" of the jellybeans that the jar is in height (if you understood that, then you know what I am saying), to get a number that is close. Add a little fudge factor ('cause those damn jelly beans never manage line up properly in even layers, like they would in an ideal system), and you're set.

Sometimes a jar/vase is used with varying area cross sections - these can be figured out individually, then totalled at the end.

Heaven forbid they use a sphere (oh no - spherical volumes - don't make me work).

Of course, I may be just too much of a geek...

Then again - anybody got ideas/info on obtaining better values for this kind of close fit problem?

Re:The Role of Calculators in Math Education

[radja]

After a nice discussion with my (now former) math-teacher, we both came to the conclusion that the use of calculators in school has actually made math-exams a little harder. Before that widespread use of calculators in schools, math-exam-ansers tended to give 'pretty' answers, i.e. answers with round numbers. Answers like: 42, 2*pi, 3*sqrt(2), 2/3. This made it easy to spot mistakes, since the form of the answer would stand out. If you got an answer 2.31172, it was probably wrong. The use of calculators however made such numbers just as easy to handle as the pretty ones, which closed off the quick'n'dirty approach to spot mistakes. It's not a big thing, let alone one to really complain about. Was a nice discussion though :)

//rdj

Get real!

[Greg Merchan]

Because public education in this society was designed around the time of the Industrial Revolution, when factories needed just a few simple things from their workers:
1) Do what you are told without question
2) Do it again and again and again (ie repetitive tasks) without getting fidgety

Bullshit! Don't try passing off your political views as knowledge about education. Arithmetic is learned by rote because is so simple and basic that it should be done quickly with little thought. Or do you really want to do 5*2 as (((((((((1++)++)++)++)++)++)++)++)++) ?
The basics have to be memorized so you can move on to more interesting things. What's ridiculous is how much time is spent memorizing - nay, trying to teach kids "how it really works". Think "hash table".

As a CS/Math major, I feel that arithmetic is basically useless, in the sense that nobody needs to know long division (or what 11 * 15 is, etc). That's what calculators are for. I feel the same about Calculus (yes, somewhat more deep and much more complex than addition and subtraction but basically just computation; no actual thought required if you know the right techniques). The interesting problems are the ones that computers can't solve.

Then you're a fool and a stooge. If you can't do arithmetic then you're a slave to however can do it and uses that knowledge to make the calculators and computers. The reason you think "no actual thought is required" is that you have learned it by rote. You've done exactly what you think shouldn't be done. Yeah, most anyone can compute a derivative algorithmically, but that's not the point and you miss completely any appreciation of either the mathematics or the real world. Go read Newton, philistine!

As for "higher level math" which you imply starts after arithmetic, everyone uses it every day. If we didn't then every dog would be completely unique; you could never form a concept of dog - i.e., you could never define a variable to be an element of the set "dog" and then recognize memebers of that set. The math professor you heard was dead wrong and probably pushing his own political agenda to get more funding. High schools should be starting with, at least, calculus; that they don't is indicative of what a failure the education in the USA is.

So now get religion.

[Greg Merchan]

OK, so you learned it explicitly by rote. (I was refering to the rote learning that takes place through continued use, not the actual course content.)

If you're a math major, you should get religion; start seeing how all the parts tie together.

religion - res ligare - things tied
-or-
religion - re ligare - regarding ties

(I don't remember which; and my latin grammar is poor from lack of use.)

For example: Study the history of calculus. I think it starts with the method of exhaustion of Eudoxus. Learn what problems they were trying to solve and why and what they tried. See how the methods have evolved. Look into their connections with other sciences - not just today, but throughout history. Even delve into the lives of the people. See what they learned that led them to try what they did. Find what non-mathematicians did for the science and what mathematician did for the other sciences.

All of this is your education. (voice of yoda) "Not this base computation".

ex - out from
ducere - to lead

Education is a leading out: of your notions into ideas, and of yourself from slavery to the minds of others.

A branch of knowledge is more than just lumber; it is part of a living thing and a home to other things.

Don't forget your looking glass.

Social and political implications of innumeracy

[Tau Zero]

I once heard a quote from a mathematician (IIRC he was a professor somewhere). He basically said that most people will not use much more than basic arithmetic (say 6th grade level) in their lives.

Very interesting, indeed. I must be one hell of an outlier, because I found myself using differential calculus on a summer job between my junior and senior years, to calculate the optimum diameter of a circular weight for balancing a roller. It had to fit inside a cylindrical envelope (the diameter of the roller), of course. I found that the circular weight of minimum thickness was 2/3 the diameter of the roller (it would have gone past the shaft, clearly impractical) but that told me something very useful. I was actually subtracting mass by drilling instead of adding mass, but that's another story.

There are so many important things in everyone's lives which really require algebra or better that it's sad most people can barely handle arithmetic. For instance, the optimum first-year depreciation deduction for a business vehicle may not be either the flat deduction or the straight-line or double-declining balance figure, but some proportion of one and the balance of the other to reach the depreciation limit. To determine what the proportions should be, you need algebra and differential calculus. It's a simple formula, but you need to understand what you are doing. Another example is home mortgages and retirement planning. If you don't know what your loan balance will be 5 years from now, you have no way to plan. If you can't calculate your IRA portfolio's value based on projected rates of return, and the income you can expect to take from it, you have no idea what you have available to live on and what kind of lifestyle you can expect... nor what to do to get to where you want to be. This requires knowledge of compound interest, which is a fairly simple derivative of the formula for the sum of an infinite series.

This last is very important in politics. A great debate is going on in the US presidential race, and it is almost entirely uninformed by numbers. Only a tiny fraction of the populace would understand, so the news media does not publish them, and ignorance is perpetuated. This certainly does not serve either the body politic or posterity.

Algebra and calculus would be useful to a huge number of people, far more than have a good command of them. Unfortunately, those who have the need for it often have no idea what their problem is or that some knowledge of these matters would improve their lives.
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This post made from 100% post-consumer recycled magnetic

Interesting Statement I heard once

[randombit]

I once heard a quote from a mathematician (IIRC he was a professor somewhere). He basically said that most people will not use much more than basic arithmetic (say 6th grade level) in their lives. However, high schools _should_ teach higher level math, because otherwise people with an interest and ability in mathematics might not discover their talent otherwise.

Why are children often forced to memorize multiplication tables and do long division?

Because public education in this society was designed around the time of the Industrial Revolution, when factories needed just a few simple things from their workers:

1) Do what you are told without question

2) Do it again and again and again (ie repetitive tasks) without getting fidgety

Doing stuff like mutiplication tables is good practice for #2. And at the elementary school my younger brothers went to, there was a rule: "You will obey an adult without question". My Mom got mad at that one: an adult? Any adult? One who just walks of the fscking street?!?!? "Come with me, little girl?" "OK, I wouldn't want to break the rules". So you see #1 is well handled in most lower education (and in some cases taken to a dangerous extreme).

As a CS/Math major, I feel that arithmetic is basically useless, in the sense that nobody needs to know long division (or what 11 * 15 is, etc). That's what calculators are for. I feel the same about Calculus (yes, somewhat more deep and much more complex than addition and subtraction but basically just computation; no actual thought required if you know the right techniques). The interesting problems are the ones that computers can't solve.

importance of H.S. geometry

[b_pretender]

A rigorous mathematical curriculum enhances many areas of child development. IMHO the best high school course is a geometry course.

In a standard high school geometry course, theorems and proofs are introduced well for a HS level. Childeren are taught to look at complicated problems/theorems and then break them up into small steps in order to prove it. One of the main concepts is to teach the students to solve a problem by breaking it into a linear series of steps all leading towards one goal. This is a major part of the abstract reasoning that you mention in the second bullet.

I have four years of tutoring experience at a college level and in every college class from basic algebra to differential equations, all problems can be solved/understood by breaking them into a linear series of smaller problems. (This is also true of the classes I took that were past Diff Eq.) Something that should have been developed in HS geometry.

To answer your other point about Multiplication tables and long division...
I believe students should memorize them merely to teach them that some things they have to sit down and rigorously go over until they know it. What other area in education forces students to rigorously memorize anything? In schools around Okemos (MI), they have discontinued multiplication tables from the curriculum, and I fear that the students are never going to learn that sometimes they need to sit down and go over something until they know it. There's no easy way to learn multiplication tables and that is it's strength.

One last interesting point is that I have tutored a wide range of math classes at college, and the biggest difference between the math-challenged people in College Algebra 101 and the engineering students taking differential equations is self confidence. My observations show that Mathematical self confidence almost always correlates to personal self confidence. In the Diff Eq room, you help a student by guiding them, and they would trudge through the details, making mistakes, but ultimately solving the problem. In the Algebra 101 help room, many of the students need you to solve the problem while they watch, they seem afraid to perform basic steps by themselves.

One of my most rewarding experiences in the math help room was with a girl that would come in 2-3 times a week. In the beginning of the semester, she was constantly frustated because she didn't understand things, and I wouldn't just sit down and solve it for her. I would tell her how to do it but make her do the thinking. One day in the middle of the semester, she stopped me in the middle of a problem that I was helping her with. She said that she wanted to figure the rest of it out herself. I was happy for about 3 days because of that, and she probably never even realized it.

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Math and English

[CFN]

I strongly believe that the two most important courses are Math and English (not really English, but the language of the country you are in.) They both develop two very different but equally important skills.

Math is very important because it develops abstract thinking. It is where you learn to break a problem down and think logically. Yes, the average person probably does not use much more than arithmetic during their day, but the skills of critical thinking are used all the time.

Recently there has been reforms in the math curriculum, at all grade levels, including the misnamed 'Harvard Calculus'. These reforms aim to encourage people to devote more time to 'understanding' and 'exploring' the math, without being rigorous: relying on calculators and estimation.

Although, on the surface, these reforms sound like a good idea, they do not work as expected. Students become reliant on these techniques, can pass all their exams by graphing a curve and looking for the maxima, and never gain any real understanding of what is going on.

It is only through working through laborious problems (be they long division or finding integrals) that a student will finally, really understand what is going on. Students do a few problems, step by step, until it just 'clicks' and they then understand the purpose of each of those steps, and what the problem really 'means'. Just typing something into a calculator, and looking for the minima, does not really teach anything: it does not explain why there is a minima, and why it is at that point.

English is the other very important skill. It teaches communication, which is the most important skill one can have. Being able to read with understanding, and communicate ones thoughts clearly, is a skill surprisingly very few people have.

It is only through spending a lot of time reading until you gain comprehension, or repeatedly re-writing something until it is clear and concise, that you can learn to communicate clearly.

So often, I am surprised at office memos (and even newspaper articles) that re logic-less and make absolutely no sense.

Clearly, these are the two skills most important for success, and the teaching of them should not be softened.

Sentience == Abstract thought === Math

[seldolivaw]

The difference between the thoughts of sentient life and the thoughts of other kind of life are, IMHO of course, abstract thoughts: thoughts about the thinker, thoughts about thoughts, thoughts about things which do not exist in the real world. Math is one of the subjects that is closest to pure abstract thought, and this is why I feel it is also one of the most valuable subjects.

Granted, practical subjects, which teach you actual physical skills in how to manipulate objects, are useful, but it is essentially true that you could also teach a monkey to do those things. The difficult part is the thought behind those actions; knowing whether to build a table, rather than how to build a table.

Students who are good at math (and other abstract subjects, like music for example) are treated as more "intelligent" because they are in fact more sentient, assuming it is possible to be "more sentient".

I would also say that students would be better prepared, and better educated, if more attention were paid to developing their capacity for abstract thought with
(a) greater attention paid to math
(b) some or more attention paid to topics like music and, yes, computing where abstract, logical thought is often key (unless you're using a Microsoft product :-)

Elementary school teaches Arithmetic, not Math

[Louis_Wu]

We teach math to enable students to live in our society. Beyond that, it's just frosting on the cake. Some people (like my mom, who teaches math to junior high kids [11-14 years old]) maintain that kids do not have abstract thinking ability before ~13, for the most part. So teaching it would be hard in elementary school.

1)

Of all the subjects which is the most important for the development of the student? That is, which subject gives the most skills to the student beyond the actual information taught? Why?

Reading. Period. After that, she can teach herself. But "A, B, C" isn't enough, which is why English classes are so key, they give practice in reading. English class doesn't teach anything about reading; for that see "How to Read A Book" by Mortimer J. Adler.

2)

What is the goal of teaching Math to children?

So that "the future of America" will be able to live in the "America of the Future(TM)".

3)

Is it to give them skills to manipulate numbers or does it accomplish something else (or maybe both)? What are those skills?

Again, until high school, it's just coping skills. Then, higher thinking is slowly introduced. Slowly.

4)

People often say that math teaches abstract reasoning. Is this so, how and why? Could there be a better way to accomplish this?

Math teaches abstract reasoning, arithmetic does not.

From "Mathematics Dictionary" 5th Ed.

Arithmetic n The study of the positive integers (1,2,3,4,5, ...) under the operations of addition, subtraction ,multiplication, and division, and the use of the results of these studies in everyday life.

Mathematics n The logical study of shape, arrangemant, quantity, and many related concepts. Mathematics often is divided into three fields: _algebra, _analysis, and _geometry. However, no clear divisions can be made, since these branches have become thoroughly intermingled. Roughly, algebra involves numbers and their abstractions, analysis involves continuity and limits, and geometry is concerned with space and related concepts.

Thought that I would clear the definitions up a bit. Basically, you don't hit mathematics until high-school. So elementary 'math' doesn't teach abstract reasoning, though it may teach reasoning on some level.

5)

With the development of small computers and calculators do you see the role of math education declining? Why or why not?

You always need a gut-level check of whatever you are doing. If you don't know that 1882*1000 should be bigger than 1.9, you won't realize that you divided instead of multiplying.

In engineering we occasionaly finish a complex analysis which has many possibilities for making mistakes by doing a "sanity check" where we use a less precise but simpler method to check our answer. Stress analysis of a spring using elasticity methods is a good example. I had a 3/4" stack of paper for my analysis, with the pages covered in calculus and static analysis. When I was all done, I checked my spring constant equations against a handbook equation, and I was close. So I assume that I was 'right'. Without the sanity check, I wouldn't really know.

6)

Why are children often forced to memorize multiplication tables and do long division?

Because it is actually useful. Not just for engineering students like me, but for checking the high-school dropout who is ringing up your groceries: if he puts the decimal in the wrong place, your loaf of bread is $10, not $1. That is much easier to check if you know that $10 is 10 times $1, and that multiplication by 10 can be done by moving the decimal point. An ability to do basic arithmetic cannot be thought unnecessary when our society is ruled more and more by numbers. (Politicians use polls, we all use prices, homeowners use mortgages, nearly everyone uses credit cards. To understand all of this, we must understand arithmetic so well that we don't have to check to see if we did it right; arithmetic must be nearly second nature.)

7)

Why is it that students who have some deficiency in math are stigmatized as "not so bright" more often than children who fail to do well in other subjects? Conversely, why are children who excel at math considered gifted (more so than other subjects)?"

Because it seems that our society thinks that math is hard (to quote Barbie), so if you can do math, you must be smart. That one is mostly societal.

BTW,
Are the people in your class primarily from the sciences or the humanities? I ask because I have noticed a trend at my university that the students who use math in class regularly (physics, engineering, chemistry, etc.) think that math is an essential life skill for everyone to know, and the students in the humanities (psychology, english, history, etc.) see math as useful in balancing a checkbook, but beyond that, it seems to have little point. "Why did I have to take algebra? I've never used it?" When this comes up, the science types insist that math is an essential skill, but are hard pressed to find "real life" examples of how algebra or geometry could be useful. And we aren't even up to basic Calculus in the discussion! I would like to find a way to convince people that math is useful, not just arithmetic.

Louis Wu

Thinking is one of hardest types of work.