From: Johnny Hamilton
Date: 23 Aug 1995 20:47:46 GMT
Organization: Construction Trades Press
Subject: Re: Love vs. money

Dear Andrei,

Here are a couple of practical problems with similar dimensions.

You have been hired to paint a design on the wall of a 150' corridor. The basic design will be a band of color 12" wide with the center of the band running three feet above the floor.

There is a set of steps in the exact center of the corridor that changes the level of the floor by 16.5". The designer wants the band to run parallel with the angle of the steps and the angles for the band to be changed with arcs, with the center radius of the arcs to be 1.5 times the wide of the band.

If the steps have a rise of 5.5" and a tread of 12", what is the angle of drop? (Round to nearest hundredth of a degree.)

How much before the first step should the arc start? (Give answer to nearest sixteenth.)

How long is the straight part of the band between the two arcs?(Give answer to nearest sixteenth.)

If the corridor is 150' long with the center of the steps being in the center of the corridor, how long are the straight lengths of the bands, excluding the offset? (Give answer to nearest sixteenth.)

Part two:

After you have finished, the handrail crew comes in. Their rail is 1.25" steel pipe and will also be 3 feet from the floor. Their elbow is also 1.5 times the nominal size of the pipe. Please provide the answers for the same questions above.

Johnny

About Johnny's problem:

If a team of schoolchildren actually HAS been hired to paint a design on the wall of a 150' corridor (&c), then this problem could be of real interest to them. Certainly the ability to solve such problems is a valuable one.

But am I alone in thinking that otherwise it's just a tiny bit boring, and might not be the best way to turn students on to mathematics?

John Conway

About this particular problem:

Imagine that this problem is included into curricilum of high schools and/or colleges. Of course, it is impossible to find enough place to have every student solve it physically, actually painting some hall. So students will be expected to solve it ON PAPER. Then it will be a nightmare for teachers. Imagine all those confused essays they will have to read, UNDERSTAND and grade.

Then it will be a nightmare for students: they will not know what to write to get a good grade. Then teachers will find a way out: demand strict following a certain script (probably prepared as a handout) and penalize every deviation. Then students will find a way out: to follow this script and never think.

Now let me open my cards: what I wanted to prove is that all the present fad about 'real-world problems' in the classroom is silly and dangerous. Real world problems can be solved only in vivo, not in vitro.

Andrei Toom

I did the problem, or at least part of it, and I'll give my answers right or wrong, but then you have got to tell me, what would a student get from this that he wouldn't get from any other "story problem"? Some good trig, and lots of number crunching in it, but (Johnny) why is it somehow preferable to mixed nuts or coin problems? How many of my high school kids do you think would feel this is "REAL" for their life?

If I asked our school engineers to do this, they would do just what I did to check my work, they would walk over to a CAD machine and draw it and NEVER figure out any math... Sorry, doing the numbers on this just doesn't seem like "real world math" to me.

Now here are my answers and if they are really wrong, you can use it to reinforce either side of this argument.

I pictured the hall running from left to right with the right hand side being the 16.5 inches higher. I disobeyed the instructions (this is real world, Johnny) and computed everything in hundredths of an inch and got:

the straight section on the left (lower hall) is appx 864.21";
The straight part of the incline is appx 31.74";
The straight part of the right (upper hall) is appx 891.93"

I don't do carpentry and have no idea what the pipe sizes mean, and was offended that anyone would spoil my beautiful hall with plumbing fixtures so I molded a 3" glass tube with internal fiber optics to act as both a hand rail and emergency lighting.

Pat Ballew "lovin' what I do"
Edgren HS, Misawa, Japan
ballewp@emh.misawa.af.mil

I'm on the side of love, of course, although I have a lot of sympathy with Johnny's concerns.

The trouble is that REAL real-world problems are likely to be boring except to those who are intimately involved, and often even to them. The "real-world" problems one meets in school may be interesting, but if so, this is usually because when the "real-world" cover has been stripped off, the underlying problem that remains really IS interesting.

From my own schooldays, I remember the problems that were dressed up as "real-world" to be the most tedious ones, and I have a sneaking suspicion that kids still find them so. In truth I believe that the pressure to include them comes from above - from parents, legislators, and so on, who are not intimately involved in teaching.

I think there is a confusion between "real-world" and "interesting" in these people's minds. They recall lots of boring old stuff from their schooldays - just as I do from mine - and think that by asking for problems with real-world connections they might help to reduce this boredom.

I'd like the practising teachers to help out here by trying to recall the mathematical problem or project that has most turned on their students over the last year (say). My guess is that if this was a problem, it WON'T have been a "real-world" one. If it was a "real-world" project, it probably was a REAL one, where the kids were actually DOING something practical (and probably it was this activity, rather than any underlying mathematics, that they enjoyed).

Anyway, I think the answers will be more interesting, and more useful to teachers, than any amount of discussion of educational theory!

John Conway

Well, the one "project" I did that absolutely electrified the class was certainly a real-world problem. Unfortunately for this forum, it wasn't very geometric.

I asked if the kids knew anything about the California Lottery (they were 7th graders), and we pieced together basically how it works -- you choose 6 numbers from 51 and if the lottery folks choose the same 6, you're a millionaire.

So I said that I would run a lottery in the class, but 51 numbers seemed a little high, and I'd just use 12. The kids each picked 6, and anybody who got the same 6 that were later picked out of a bag would win $20. (I guess I said that multiple winners would split the $20, just like in California.)

Of course the kids were all certain that they were going to be rich (after all, all they had to do is pick 6 of 12 -- what's that? a 50% chance of winning, maybe?).

With a class of 30, you, as teacher, are running a slight risk -- you do get a winner 1 time out of 924, so you'll lose in one class out of 30 or so, so you're basically paying 60 cents per time to play the game.

Of course when everybody loses, I ask, "OK, the first game was free, but how many of you would pay me 25 cents to play again?" Almost all the hands go up, and you're off to a great class.

I told a friend about the excitement in the class when I did this, and told him that I felt a little left out since the kids were all so excited, and for me the threat of losing $20 didn't quite churn up the same amount of adrenalin. My friend suggested that next time I make it $100 -- then the drawing would be exciting for me as well.

Teacher's hint: collect the kid's choices BEFORE you pick the winning numbers -- the kids aren't idiots, you know.

Dear Mr. Conway,

Boy, I'm glad you came back with this second post. It is the ability to strip away the context of the problem or any similar problem, thus exposing the underlying nature of the problem, that is important. That is education, while solving the problem for the problem's sake is training. The solution for this problem and every other problem that uses curved lines and straight lines together is mathematics and geometry. Most people really do lack that understanding, which is my real concern.

Again, I think it is the ability to link all problems, such as this, to the core math involved that creates interest and shows that math is the solution for real world problems in any medium. The reason for using the painted band and the handrail was that they both had the same center line and the same right triangle of offset, but the tangent lines of the curves and the curves themselves had to be based on the thickness of the media involved and the radii required. The exact same formulas are used for both, with just the dimensions being different. It should be a simple problem. It surely doesn't rank as one of our harder problems in the trades, yet very few people have the math tools to work it. If they had the tools to work with this simple problem, they would have the basic mathematical tools to deal with most of the problems we face, but they simply don't.

Parts of the solution are being taught to some students, but seldom are the parts linked together in a way that allows for solving simple problems like this. Most math teachers know a lot about the different parts and if they sat down and worked at it for a little bit, they would find the answers, but most do not teach how curved and straight lines are used together, nor do they see their use in everyday products around them.

Neither I, nor anyone else, can write a practical problem that cannot be torn apart. I can and did write a real problem that requires more mathematics than most people are ever taught.

Johnny

P.S. I really enjoyed your explanation on soccer balls.

Dear Andrei,

Unfortunately what you have proven with your obvious trap is that academic mathematics must be sterilized. It can only be taught in a vacuum.

Your postings also seem to indicate that it is not proper for mathematics to have any connections outside of itself. When you take away the paint and the pipe, the problem is not theory or philosophy, but purely a simple geometry problem. No more, no less.

Whether it is solved on paper, the wall, or a computer or even in a different, perhaps a more acceptable, context has little meaning. What is important is that people outside of the math community recognize what geometry skills are needed in such situations. If geometry operates in a vacuum, how will that ever happen? It simply won't happen any more than it ever has for most people.

The real problem is that you failed to recognize a genuine problem, even slightly disguised with a little paint. Thousands upon thousands of people did similar offset problems today, though most of them would never think of it as geometry - it is just work. Geometry is a different subject.

Johnny

The distinction of project versus problem is an interesting commentary as regards 'real-world'. However, I have known many real world projects that have led to the formulation of interesting (IMHO) problems.

When Andrei began his search for good real world problems, I thought about sending him some that I felt my students had enjoyed. One reason for waiting was the discussion that seems to be developing, i.e. is a particular problem good. So I asked Andrei if he might post a good problem for a particular age group (real or not) and perhaps the discussion could start from there. However, it appears that he is too busy at this time.

I often, as a teacher, find myself taking problems that I think are interesting and dressing them up for my students. Often they can be embedded in a 'project' or at the least made somewhat 'real' world. So I would be most happy even to see GOOD problems (I wonder if a consensus is possible :)) much less good 'real world' problems.

Ed Wall

On Thu, 24 Aug 1995, Ed Wall wrote:

> When Andrei began his search for good real world problems,

It was not really a search. Johnny Hamilton is right: it was a trap. Because I knew in advance that there would be no good `real-world' problem suitable for classroom.

I knew it because I am aware of one of the most important and basic aesthetical laws, which may be called `the law of economy'. It says: Culture CANNOT describe reality one-to-one, it always has to condense. So all the movement to exclude from classrooms `problems by type' and introduce `real-world problems' instead is based on aesthetical illiteracy.

> I asked Andrei
> if he might post a good problem for a particular age group (real
> or not) and perhaps the discussion could start from there. However,
> it appears that he is too busy at this time.

This is a misunderstanding. I did not recognize that I was asked. If you want a good problem, here is one (non-real-world):

Given two triangles. Every side of the first triangle is shorter than the corresponding side of the second one. May we conclude that the area of the first triangle is less than the area of the second one ?

Andrei Toom

> No. Criteria of understanding are quite different.
> However, we can, of course. By using good old and
> useful 'problems by type'.
> Andrei Toom

I disagree. The criteria for understanding ideas are independent of the ideas themselves. One who understands an idea demonstrates that understanding by explicating and manipulating that idea in context with other ideas. The confused essays that mathematics students write about mathematical ideas do not differ substantially from the confused essays they write about other ideas. (I've read both kinds of essay.) Those who write well about other kinds of ideas typically write well about mathematical ideas. Those who can't handle mathematical ideas typcially can't deal well with other ideas, either.

It seems to me that we are addressing a two-pronged problem here (although it is not a dilemma, because the horns of this problem aren't alternatives -- they're parallel problems). The culture of mathematics education as established in the USA today suggests that we should read and grade students' answers. Hard-working instructors may actually scan the students' processes to see if they are appropriate and correctly performed. But the culture denies that we should ask students to present their reasoning, and it denies this because we know that it is hard to evaluate reasoning so we don't want to do so.

The other prong, or horn, of the problem arises from the fact that we have built up a structure of mathematics education that allows us to get away without evaluating reasoning. This has made us into a cash cow for our institutions -- which are glad to pile 30, 40, 50, 60, or more students onto a single, unaided instructor. (Not to mention the hundreds dumped into mass lectures in our state universities.) After all, math teachers have it easy -- all they have to do is look at the answer to see if it's right.

Reform *must* address the issue of reasonable class sizes for instructors who will evaluate ("assess" is the current buzz-word, I suppose) their students' reasoning in a meaningful way. And it must address the problem of dealing with instructors who will not do so.

--Lou Talman

When I write to this list, I implicitly assume that the present level of civilization will remain approximately the same in the foreseeable future. This includes:

- The competence of teachers will remain the same
- The motivation of students will remain the same
- The value society attributes to education will be the same etc.

What you describe is some kind of utopia. However, this utopia has a record of implementation in real life - on a small scale. I mean math olympiads. What is going on at olympiads is just what you dream about: Students write `reasonings' (I use your word) and organizers read, understand and evaluate them. Is there a math olympiad for high school students in your state ? If yes, you can participate as an organizer. If not, try to organize one.

Andrei Toom

Andrei Toom wrote:

> In is in the context of these assumptions you should understand
> my insistence on the value of solving `problems by type'.

But, in a recent post that I've misplaced, Andrei also decried the very phenomenon that "solving 'problems by type" leads to, the phenomenon of students devising "scripts" that they then use blindly to get grades.

I think word problems are valuable when students are taught to approach each problem as a problem in its own right. I think they have very little value when students are encouraged to classify them and, upon encountering a word problem, solve it by ascertaining its class and responding with the script for that class. It is the latter strategy that the Standards suggest we should move away from, and I heartily agree.

> What you describe is some kind of utopia.

I must disagree again. English Departments can offer freshman writing courses in sections that are limited to 25 students. Why can't Mathematics Departments insist on equal treatment? It is a matter of standing up for ourselves and for what is right. (It is also a matter of teaching our administrators that there is more involved in reading a students' attempts at mathematics than just looking to see if the last number in each problem is correct.)

--Lou Talman

On Fri, 25 Aug 1995, Lou Talman wrote:

> But, in a recent post that I've misplaced, Andrei also decried
> the very phenomenon that "solving 'problems by type'" leads to,
> the phenomenon of students devising "scripts" that they then
> use blindly to get grades.

I never did this. On the contrary, I am sure that the phenomenon of devising `scripts' is inherent in human nature and essential for effective behavior. All civilization is based on such scripts.

> I think word problems are valuable when students are taught
> to approach each problem as a problem in its own right.

We absolutely have no time and other resources to approach each problem in our life as a problem in its own right. We do develop scripts or stereotypes whatever we do: drive, clean our teeth, solve equations, greet people, make love, take cash in ATMs, use computesr, clean our asses etc etc. For example I have a stereotype or script to use e-mail. Sometimes my son introduces some innovations (e.g. changes the autoexec.bat) and the old script does not work. I become very angry.

> I think they have very little value when students are encouraged
> to classify them and, upon encountering a word problem,
> solve it by ascertaining its class and responding with the script
> for that class. It is the latter strategy that the Standards
> suggest we should move away from, and I heartily agree.

I would like to call it `revolutionary romanticism'. Nothing good ever came out of it. Believe me, an immigrant from a Communist country.

In this semester I teach `college algebra' to a group which contains less than 25 students. (I did the same during the last two years.) So what ? I am happy if at the end of semester my students develop a few `scripts'. Because this is some start.

You seem to see `scripts' as dead ends or as alternatives to productive, creative learning. I see scripts as milestones (or stepping stones) on the way of productive learning.

Andrei Toom

Andrei Toom wrote:

> I am sure that the phenomenon
> of devising `scripts' is inherent in human nature and essential
> for effective behavior. All civilization is based on such scripts.

There is some truth to this.

> I am happy if at the end of
> semester my students develop a few `scripts'.
> Because this is some start.

If the *students* develop scripts for themselves, that's one thing. If you provide them with a template and teach them to solve word problems by responding with your template after they classify the problem at hand, that's quite another. In the first case, their scripts are indeed "milestones (or stepping stones) on the way of productive learning". In the second, I think they are what you accused me of thinking them: "dead ends or...alternatives to productive, creative learning". It seems to me that teaching in this way is the antithesis of teaching to solve problems, and I read the Standards' injunction to move away from teaching "word problems by type" as agreement with this point of view.

--Lou Talman

Exactly. So you admit that the difference is made by the manner of teaching. Same old `problems by type' can be used productively or unproductively depending on the mentality and manners of the teacher (and students).

> It seems to me that teaching in this way is the antithesis
> of teaching to solve problems, and I read the Standards' injunction
> to move away from teaching "word problems by type" as
> agreement with this point of view.

No. Standards misplace and misinterpret the problem. They say nothing about the manner of teaching and pretend that all the evil came from `problems by type'. Or, more exactly, they explain nothing. The explanation comes from you and others.

Anyway, 'problems by type' are used as a scapegoat and sent away to the desert. Instead some mythical `real-world problems' are declared as a panacea. Nothing good will come out of it. What will pass in practice as `real-world problems' will be again same problems by type, only overburdened by irrelevant details. And they will only aggravate the situation.

Andrei Toom

In the long run, the teaching methods may not be as important as the learning methods of the student. It seems that what has been ignored is the different ways students will see and grasp a problem, real or mathematical. No one method will work for all students. If teachers have a good grasp of how mathematics or geometry is used in manmade products around them, they can point out or demonstrate the uses or mechanics of math or geometry in so many ways that they provide students with a plethora of examples.

One of the important things about the example I gave is the number of invisible right triangles involved. To correctly answer the problem, there are three right triangles that must be calculated. You also must know where all nine right triangles in the two offsets are located and how they affect your calculations. The essence of a problem like this is that any time you use an arc together with a straight line, there are right triangles and tangent lines involved that are not indicated by the wording of the problem. You are teaching students to visualize and calculate invisible geometric shapes essential to solutions of the obvious shapes.

When the students grasp the idea of how the arcs, tangent lines, and right triangles fit together, I would extend the problem. I would say that the designer liked the way the color accented the wall and decided to continue the banding up the wall with different size bands. The calculations would get boring through repetition, but another aspect comes into play. How do you keep the different size bands and their different radii parallel and maintain the correct spacing throughout the offsets? Again, the solution involves calculating a different set of invisible right triangles. To be correct, the center points of the two repeating sets of arcs must be parallel lines, which in turn creates a symmetry of geometric shapes. If not, you will notice the difference visually.

If you want to relate this same geometric problem to more real life problems, then go into the mechanical room of your building and look at the home runs of electrical conduits going back to the electrical panels. Look at an industrial commercial site for piping or ducting run in series. Look at the roads you drive home on or on freeways and notice the lines of the offsets in the pavement or painted lines. Look at circuit boards and the series of circuits. These patterns are absolutely everywhere. You can see visually whether the people who installed them knew how to do the geometric calculations necessary to create the correct symmetrical patterns. It is very obvious to the eye when the symmetry is off.

From there, I would suggest looking at and calculating compound 90 degree turns made by two 45 degree individual turns. The patterns of the center points of the repeating arcs for the compound turns are not parallel, but radiate from a single point. A geometer would see that this configuration is one quarter of an octagon with one whole sector and two half sectors. Anyone who adds to the series of lines must follow the symmetry of the concentric regular polygon or it will show up visually as wrong.

Much of the problem we are having out here is that people are not relating their work to the geometry of it because they have never had geometry or they have never been shown how the geometry they did take relates to anything beyond the geometry course itself. It does relate and the relationships should be shown, no matter how you want to word the problem.

Johnny