kitchen table math, the sequel: Real world vs mathematics for math's sake

Saturday, September 8, 2007

Real world vs mathematics for math's sake

Arguments about how math should be taught frequently include the issue of "real world" math problems. I.e., students need relevance, otherwise they'll tune out. This attitude excludes a whole host of problems that one might find in a geometry book, let's say. So a problem that is real world and makes use of the Pythagorean Theorem is OK, but a proof of the Pythagorean Theorem is not. Well, no, people might object to my extension. So let's use another example. "For any point within an equilateral triangle, the sum of the perpendicular distances of the point to each respective side of the triangle is constant, and equal to the altitude of the equilateral triangle."

This wonderful theorem probably wouldn't qualify as "real world" and even though understanding the theorem and how it is proven opens up higher order thinking skills and mathematical reasoning, a math reformist would not teach it in its pure form. They would have to find some application, however contrived, to provide relevance.

Math for math's sake is out. All math must be relevant. In fact, when the Fairfax County Public School Board (in Virginia) was adopting math textbooks back in 2001, they argued about the criterion for "real world" applications. During the ensuing debates, two school board members found the following criterion too narrow: "Materials and concepts are related to real world situations". They argued for, and lost, the following (excerpted from the minutes of a School Board meeting found here):

"Mrs. Brickner moved, and Mrs. Thompson seconded, to amend the main motion to add the words, “mathematics and” to the eighth bullet under criteria #2, so that it would read, “Materials and concepts are related to mathematics and real-world situations.”

"Mrs. Brickner said that a mathematics textbook could not relate everything to a real-world situation; that there should be a balance between the presentation of math concepts and their relationship to the real world to help students understand the need for those concepts; that applications of mathematics should come from both within mathematics and from problems arising from daily life; that a strict application of the criteria, as originally written, would cause an evaluator to find a textbook less effective than they otherwise might on the basis that the text did not wholly focus on the real world; and that the objective was clearly to teach math concepts and skills.

"The motion to amend the main motion to add the words, “mathematics and” to the eighth bullet under Criterion #2, so that it would read, “Materials and concepts are related to mathematics and real-world situations” failed 4-7, with Mr. Braunlich, Mrs. Brickner, Mr. Reese, and Mrs. Thompson voting “aye”; with Mrs. Belter, Mrs. Castro, Mr. Frye, Mr. Gibson, Mrs. Heastie, Mrs. Kory, and Mrs. Strauss voting “nay”; and with Mrs. Wilson absent. "

So, the upshot is that math textbooks must base all examples and applications on real world situations; mathematics for mathematics sake does not count. That would make calculus textbooks rather challenging to write!


Unknown said...


Is it just me? I can't put those minutes down. I'd love to get a hold of more of them from different schools/boards.

Anonymous said...

We are in deep trouble when the notion that the teaching of math should be related to mathematics cannot get a majority vote.

VickyS said...

That would make calculus textbooks rather challenging to write!

Well, I guess it's a good thing, then, that few reform-math educated students are ready to study calculus in high school...

Anyway, Barry, you are so right about this. I have bristled against the "real world relevance" mantra for a long time. It's one of the persistent memes of reform math: you cannot develop interest, motivation, or competence in students without real world relevance.

And look: the result (because reform math HAS been implemented...this is another message we must get across...we are now, today, reaping what the reformists have sown) is nothing but further declines in motivation, interest and competence in math and science.

This morning I showed my 11 year old son how those three perpendiculars added up to the altitude. His eyes lit up and he said, "Cool! That's useful for knowing things about triangles!"

"Cool!" apparently isn't good enough anymore. But "cool" is what motivates a lot of good students to continue studying.

I for one hated the occasional math "applications" we delved into. They were tedious, because they were inefficient and boring. Most interesting real world applications are called jobs, and they use a whole compendium of (cool) skills and concepts that need to be developed in school.


Anonymous said...

"That would make calculus textbooks rather challenging to write!"

Unless the teacher was also certified to teach physics you are correct. Calculus was co-invented by Sir Isaac Newton so that he could handle motion problems without running into the paradox of zeno. Physics and Calculus grew up together.

Trouble is there are very few tachers certified in both math and physics. At the school I used to teach at before going back to engineering I was always having to set students straight on how to use calculus in physics since the math teacher could never get the physics correct. I also had students that were in my physics classes tell him straight out that he was getting the physics wrong.

We were fortunate, before he came, the woman he replaced was certified in both and we worked together great as a team by reinforcing each other.

I will say that I learned more about calculus when I actually had to use it in my physics and engineering classes, but I also had a solid grounding in it before I took those classes. They just helped by giving me a lot of "aha" moments.

Catherine Johnson said...

You read my mind.

I was just looking up the reviews for Glencoe Geometry.

They are a hoot.

All one-star except for the lone review written by the real-world person.

Here's a happy parent:

In my home state of Tennessee, this is the text that unsuspecting school kids must decipher. It should have been easy - it's structure (and I use the term loosely) broadly resembles MTV with its numerous subjects per page, chats about unrelated subjects, some of the worst use of the English language ever and a colorful, multi-font, in-your-face appearance. There's pictures and graphs and arrows and charts and big text and small text and cartoons...Yet it cannot be understood much less absorbed - at least by anyone looking for something rational.

Like others, I noticed that the "team" of writers (it's almost as if they took turns writing paragraphs) continually introduced material BEFORE it was studied. Then there were the "examples" - just pitiful. Proofs were confusing, redundancy is taken for granted and the number of sub-subjects - review, standardized tests, chapter study, real life example, etc all ran together in a mushy mixture of words and concepts. IF YOUR CHILD MUST USE THIS BOOK GET EITHER A TUTOR OR PURCHASE "Geometry the Easy Way" by Lawrence S. Leff. I picked it up for $2.99 from Amazon.

A one star is overly generous.

Barry Garelick said...

Physics and Calculus grew up together.

Excellent point. One could make the same point, though, about geometry, which was developed to help with measurement and in design. Like many branches of mathematics that had their initial motivation in solving practical real-world problems, the mathematical theories that grew out of it were often math for math's sake rather than for practical applications.

Me said...

Barry this is excellent, thanks so much for writing it.

I think that even "math for math's sake" has practical applications in that it teaches you to think and reason in certain ways.

Also, now that there is research suggesting that exercising your brain can stave off Alzheimers, it might even be that studying "math for math's sake" is good for your health!

le radical galoisien said...

Well, they simply have indirect practical applications, and are not just for merely exercising the brain either.

I mean, does knowing that the number of prime numbers is infinite have any direct practical application? Not really. But in realising this, you think about a dozen other interesting properties of universe, life and math that for example, may help you make a good password (breaker).

I don't think it's just about developing generic higher order thinking skills. I am sure when studying art nobody thinks about the length and position of the chord in relation to the area it slices off. In fact I don't think it really has a direct application to art. But in exploring this fact, you might find it easier to consider how much to overlap those circles for aesthetic effect, consider the artistic beauty of fractals.

Applications and utility and practical applications are quite two different things. Like, being a skilled orator is probably not a practical skill per se. But that doesn't mean it is of great utility.

Otherwise we should stop teaching Austen, Socrates or even the Holocaust in schools. I mean, does learning about the massacres of people in a faraway place have any practical application? Why learn about the origin of civilisation? What practical application does it possibly have? Is there a practical application in those children's books detailing how peanut butter is made?

Had humans focused only practical applications we'd probably be still living in caves.

Karen A said...


Your level of knowledge is amazing! I am curious as to how much of your knowledge comes from school courses and how much comes from your own intellectual pursuits.

le radical galoisien said...

Let us stop teaching any practical application in the real world. There is no practical application in knowing the age of the universe, that all the world's continents were once stuck together, that your eyes have receptors for three different colours (and that different wavelengths trigger them in varying quantities), or that Saturn has rings.

And what practical application can art and music possibly have? We should wipe those subjects from the syllabus. Who really cares about the golden ratio and harmony anyway? Does Van Gogh and Monet put food on the table?

We should be very practical people. Let us be only people that eat and sleep, and not make merry. The irony! That in trying to make a syllabus contain "practical applications" only, you have made math exactly what you were trying not to make it into: dull and boring.

le radical galoisien said...

*correction to my above post: let us stop teaching anything that doesn't have a practical application in the real world

oops. Sort of ruined the effect.

le radical galoisien said...

karena: I'm flattered! :-)

A bit of both. To me, I've attributed my style of learning to both my migrant background (I have moved my residence between Singapore and the US several times). I have observed that immigrants (pardon for any political incorrectness) that survive the original barrier of (partial) assimilation often have a different outlook to learning.

So it's a bit of mixture of both plus the element of culture shock [it makes you think harder about things].

le radical galoisien said...

Also, there should be a more extensive covering of proofs, not less.

During the (frenzied!) AP Calc course, sometimes I didn't really get the proofs and my teacher would sometimes oblige. Some proofs are "this must be true" and you see that it must be true, but you don't exactly see exactly why it is true. Sometimes she said, "we are on a tight schedule for the AP exam, so look up the proofs yourselves".

So we'd end up memorising a bunch of formulas without proofs. We knew they worked. Some of the formulas we had strong ideas (or at least moderately strong grasp of). Sometimes we had some vague idea of why some formula was true (but the proof would be only explaiend once so we tended to forget the idea), and sometimes we had little or no idea at all.

But on we plowed, memorising our formulas, sometimes using mnemonic aids or the nifty fact that the derivatives and integrals of cofunctions differed in sign only (the fact, not the reason).

Arguably, the class was a success. I got a 5; my classmates got 4's to 5's too.

But four months on, when I need the formulas for my external Calc II class (taking it while in high school), I struggle to remember what the formulas are.

I got a 5 on my exam, but I regressed so much by having built on such a hasty foundation that now I have to teach myself all over again (but luckily familiarity prevents me from being murdered by my homework).

Young children should encounter the idea of proofs early. They should also try to prove things "hands-on". Elementary school is the age of scissors and construction paper. What age is better?

Take a fixed length ruler as a hypotenuse, and get children to raise and lower it's incline (increasing one side at the expense of the other). Add up each side's squares and compare them each time.

This approach is not rigourous, as it cannot exhaustively prove every case, but I find that on top of having a proof showing that it must be true, children would also benefit from having the idea behind the proof explained to them. (Thus it's okay for the second part to be less than rigourous -- it's only to show how the pattern works.)

For younger children, they should be encouraged to prove why multiplication is repeated addition, or why division is an inverse process.

When they grow up, they will rarely have to memorise formulas at all. As long as the proof's concepts are explained to clearly (and reminded to me regularly), they'll never have to memorise formulas. They'll know the elements of the formula that make it what it is.

And the practical application? Come their science classes, when everyone else is mugging their formulas, the rest will have the formulas well intact inside their heads.

To do this, more than textbook proofs must be rehearsed. Too many teachers do, "Well there's the proof. [Understand it at your own risk!] Let's begin our homework."

Barry Garelick said...

Steve Wilson, a mathematician at Johns Hopkins, stated the following about the 'tude of math having to be "relevant" in order for kids to stay tuned in. (By permission):

"In many circles it is hard to defend math for math's sake: "they just want everyone to be a mathematician, and that's unreasonable".

All the math done in K-12 really is for "real world situations",
it is just that they aren't yet ready for those problems.
I think there is a famous quote for this situation that
can be said the students: "You can't handle the truth!"

Thus, the math that some might say appears to be just for math's sake, is really to prepare kids for Maxwell's equations (talk about real world), or even lower level physics (and, of course, all sort of other real world
things students aren't ready for because they don't have the math).

Perhaps the appropriate response is:

"You want I should have them work problems with Maxwell's
equations in Algebra I? That's why we are learning this

Or, if they continue, tell them you know an undergrad pre-med who would love to take out their appendix for real world practice; the student claims his pre-med
studies just aren't "real world" enough.

Doug Sundseth said...

"Most interesting real world applications are called jobs...."

Absolutely right.

I'll add that most real-world applications aren't very interesting. Somehow, I can't remember ever figuring out the price of truckload of grain, the appropriate tip for a mediocre waitress, or the number of board feet of lumber in a bookcase because the answer is just fascinating. Similarly, I can't remember writing a detailed procedure for installing an enterprise software application because I couldn't live without that thrilling experience.

All of those things are important, all use tools that should be developed in primary and secondary school, and all are important. None is interesting or fun.

SteveH said...

How many times have you gone to your child's school and have been flabergasted at their assumptions and attitudes? My favorite example (that I haven't had topped yet) was my son's first grade teacher who said: "Yes, he has a lot of superficial knowledge" when I said that he loved geography. Later that year, my son had to show the student teacher where Kuwait was on the map during a thematic unit on sands from around the world.

How can we even talk about best practices or good education if we're on different planets. I am constantly struck dumb at schools because there is nowhere to begin. In the first packet of information that came home this year, we received a questionnaire asking us to asses our child's "multiple intelligences". They gave us a list.

If I went to them to argue against a strict diet of real world math problems, I don't think they would have the ability to understand. They would just fall back on their simplistic understandings. They would say times are changing and that I just want what I had when I was growing up.

I've argued before about their love of top-down education because it's more fun and interesting. They claim that they work down to the needed low-level skills, but it never happens. Their "real world" approach is really all about lower expectations and the unlinking of mastery (hard work) with understanding.

Karen A said...

Well, calculating the tip for a waitress, particularly if the service was mediocre, tends to be quite entertaining in our family. Assume a bill of approximately $27.00:

Paul: How much should we leave for a tip?
Karen: I think $3.00 would be fine (having already calculated in my head the tip for 15%)
Paul: Well, 15% is standard, so that would be $4.00.
Karen: Well, the waitress didn't really do much, since it was a buffet.
Paul: Yes, but she's probably a poor college student; we should leave 15%.
Karen: Yes, but $3.00 just feels better.
Paul: I give up!

Karen A said...

Of course, then the students arrive in junior high, or perhaps it's high school or college and have instructors who expect that they have mastered the fundamentals.

Karen A said...

Speaking of Geometry, my ninth grader was working on her Geometry homework earlier today and I froze when she said she had a question for me (hubby is the "go to guy" for Geometry in our household).

They are currently working on the measurement of segments and angles, and this was the problem she asked about:

"Draw a number line and label points F, G, H, and J with the coordinates -4 2/3, 2, 5, and 3.5 respectively. One of these points is the midpoint (the halfway point) between two others. Which is it?"

Her question? "Does respectively mean that F goes with -4 2/3 and G with 2, and so on?"

Ah, yes, a question I could answer! "Why yes, sweetie, it does. I'm glad I could be of help."

Catherine Johnson said...

Math for math's sake is out.

This is the core progressive move; this is why the liberal arts are in a state of constant peril and have been for 100 years.

This is why Latin and Greek are not taught; this is why formal grammar is not taught.


This is why students study social studies, not history.

We have Diane Ravitch & Chester Finn to thank, amongst others (probably including Ed to some degree), for the fact that our kids study more history today than they did 15 years ago.

Catherine Johnson said...

We are in deep trouble when the notion that the teaching of math should be related to mathematics cannot get a majority vote.

Ditto that.

As far as I'm concerned, we are in deep trouble when we have school boards voting on subject matter in this manner at all.

It is chronically astonishing to me how little "input" the public has.

Speaking of which, we got yet another letter from the middle school principal thanking us for our ongoing cooperation and support.

Catherine Johnson said...

Trouble is there are very few tachers certified in both math and physics.


A parent I know, who was teaching at a magnet school, begged our math department to interview a superb veteran teacher at the school who was certified in both math & physics.

Our school refused even to meet the guy, I believe on grounds that it would be "too hard" to coordinate math & physics.

Catherine Johnson said...

My friend told me she put so much energy into persuading the school to interview this guy that she finally had to stop.

"They were going to think I was sleeping with him."

le radical galoisien said...

I think making Latin and Greek optional was a good thing. (Otherwise I'd be totally slaughtered as an immigrant student.)

Singapore doesn't have those subjects in school (save special tutors who teach them), although we do have a compulsory bilingual policy.

Well, or perhaps replaceable with some sort of classical language. Sanskrit is literally the Indic cousin of Latin and Greek (on the other side of the Centum-Satem isogloss).

What I dislike is the reduction of Latin to reading/writing only. Classical Latin when spoken is aesthetically pleasing. A lot of teachers don't seem to focus on pronunciation and vowel length any more. Little emphasis on the spoken aspect makes the significance of good orators hard to understand.

Anonymous said...

First thing I want to clarify my stand on this. I am not sure if it came through in my first post I was in a hurry when I wrote it. Yes I did learn a lot about calculus in my physics and engineering classes but these came after I had learned the theories involved in calculus. With that background I was able to make the connection later.

I was certified in both physics and math and so was the one calculus teacher I mentioned. The reason, we were both former engineers. The school I taught at had enough physics students so that all I did was teach physics. She taught several sections of calc and pre-calc. We were able to coordinate and reinforce each other, it really was not that difficult. We did not get together and write specific lesson plans we just looked over the chapters I was covering and what concepts the students would need.

Not all mathematics has to be learned with practical examples and if the examples are trivial or wrong then the situation can even be worsened. Having students learn in the abstract allows them a better chance to see how to apply it in various situations, not just the few narrow practical examples they get.

On the subject of practical versus knowledge for the sake of knowledge, I remember interviewing with a school, a good suburban school near where I live, and I showed them a lesson plan that involved coordinating with another school either several hundred miles north or south of us. The idea was to collect data to determine the radius of the Earth. I was astonished when they said in effect. "That's nice but it is of no practical use so why bother?" I wound up not accepting their offer. any school that devalued knowledge for the sake of knowledge was a school I did not want to be at.

At the school I did wind up at I got into Relativity, both special and general (though not too deep into general due to the advanced math required) and quantum mechanics. Neither is practical to everyday living but the students loved learning about it. They thought it was so cool because it was so counter intuitve and strange. Some of them even went down to the local university when Brian Greene of "The Elegant Universe" fame came in to speak. They loved it, they even went out and bought a copy of the book, had him sign it and gave it to me the next day.