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#### The Physical, the Image, and the Abstract

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From: Pat Ballew
Newsgroups: geometry.pre-college
Date: 27 Mar 1995 02:05:01 -0500

I think that the relationship between the physical, the image and the abstract are important in learning for conceptual reasons but the conjecture that physical, the image and the abstract are forever tied together in the student's mind is more difficult to assume if we progress beyond the more simple operations. It may be that a student visualizes 8 stacks of 5 to multiply 8x5 but I think this quickly evaporates into a mechanical process which is independent of origin. I think there are few people who have a clear geometric picture of 5 ^ (7/3) or (2+3i)^(3-2i) in mind when they do the calculations.

I think the same process is true in the basic operations for most daily calculations. We may be able to explain why the methods we use "without thinking" are based on some more primitive geometric structure, but in fact our skill is based on the fact that we can do them "without thinking" about the underlying structure. We have internalized the procedures and accept them.

I really doubt that most people who ever solve 730/10 visualize 730 of anything ( or 10 of anything either) . We see the problem, cross out the zeros in our mind, and announce the answer. If we didn't believe in the truth of the abstractions, none of us could balance their own check book... "Okay, I remove 3729 counters for the check you wrote at Sears and 400 for the one you wrote the paper boy, and Add 15006 for the tax refund and...... Honestly, no matter how many times we tell students to think, we still do most of our math by rule of thumb, and eventually so should they.

Pat Ballew "lovin' what I do"
Edgren HS, Misawa, Japan

From: Judy Roitman
Newsgroups: geometry.pre-college
Date: 27 Mar 1995 11:33:10 -0500

I have to disagree with Pat Ballew that "abstract" means "rule of thumb" or "not thinking." "Reify" means "to make real." One goal in mathematics teaching is to have students make the abstractions of mathematics real. Certainly for most people "2" is real, as is "1/2." When the child says "half of what?" we know they have not reified "1/2." Geometric pictures are not the only reality, although they are certainly very helpful. "5^(7/3)" is quite meaningful -- I even have a sort of geometric picture of it -- and real to me, as is "(2+3i)^(3-2i)" -- where I have no instinctive geometric picture, although folks who think comfortably in 4D can come up with one. They make sense within an abstract system that I have a fairly good understanding of.

Also real to me are the first uncountable cardinal number, and the cardinality of the set of real numbers, because I have done a lot of work with these infinite numbers. Less real to me are some of the very large cardinal numbers, because I have not thought about them very much, although I can listen intelligently when someone tells me something about them.

The analogy with language might be helpful here. Much as we might like to deny it, language is a semi-free-floating system that we understand on its own terms. There are many connections with reality, but also many ways in which language misleads us if we take it too seriously. (Think of the notion of "race" which is finally being deconstructed in the mass media.) So too with mathematics, a semi-free-floating system that has many (in fact, astonishingly many) connections with reality, but which can also mislead us if we take it too seriously (think of the controversy about race and IQ).

The fleeting images that help us understand mathematics are just that, fleeting images -- as Pat points out, we don't manipulate markers to figure out our checking account balance. Tremendously useful fleeting images, which are fantastically helpful when used well in the classroom, but fleeting images nonetheless.

Judy Roitman, Mathematics Department
Univ. of Kansas, Lawrence, KS 66049

From: Joshua Silver
Newsgroups: geometry.pre-college
Date: 28 Mar 1995 16:06:29 GMT

I want to enter this discussion, being a student who, having reached a certain level of mathematics, was no longer able to follow because I was taught by and with people who easily understood the concepts on an abstract level, whereas I did not.

I am a big proponent of using "real" concrete knowledge to help students understand concepts. At least at a certain level for all students, and at all levels for some (myself as an example), it is extremely difficult, if not impossible, to understand mathematical operations without some clue as to what they actually mean. In Ginsburg's book Children's Arithmetic he gives an example of a child unable to add two and three when asked "How many is two and three?" However he gives the right answer when asked how many are two lollipops and one more. Then the original question is repeated, yet he still gives the wrong answer.

The numbers and symbols used in mathematical operations don't exist in the real world, except when we right them down. You can't go to the store and by a 3, or a +, or any operation. For this reason, people need something in their understanding which they can tie the processes, thereby helping them to see what the operation does.

However, I agree with Pat Ballew that as a teacher you do want to eventually bring the student to the abstract understanding. This brings us back to what Hannah Freedberg was talking about with the three levels to help comprehension. This concept has been suggested by a few authors who have helped in the field of education. Bruner is the first to come to mind. The point being that students can understand the abstract better after they have mastered the concept at the concrete and representational levels.

I want to add another twist. What I would have found helpful as a student of math would have been something of a loop. After the abstract is taught, I would find it helpful if the concept is brought back around to its real world use, thereby cementing it even further in the students understanding.

Joshua Silver
Swarthmore College

From: John Conway
Newsgroups: geometry.pre-college
Date: 28 Mar 1995 18:55:53 -0500

I remember my own nephew, who was having difficulties with fractions. He could say that 3/4 of 12 pence was 9 pence (he lives in England), 3/4 of 12 pigs was 9 pigs, but (even straight after these) 3/4 of 12 seventeenths completely floored him.

I got him past this as follows: "Stephen, you obviously know what a penny is, and what a pig is, but aren't quite sure what a seventeenth is. That doesn't matter. Do you know what a snark is?"

Stephen: "No."

"Well, that doesn't matter either. I do, so let's just agree that a snark is something I know about, but you don't. Then what's 3/4 of 12 snarks?"

Stephen (hesitantly): "9 snarks?"

"GREAT!!!! What's 3/4 of 9 boojums?"

Stephen: "What's a boojum?"

"I'll tell you later."

Stephen: "9 boojums."

After a few more such questions, we got back to seventeenths, and this time, Stephen was fairly happy that the answer was 9 seventeenths.

(He was really annoyed when I told him that snarks and boojums were just mythical objects in a poem by Lewis Carroll, but later on he really loved that poem!)

I think this illustrates some of Joshua Silver's points, and also one that I made some time ago about using words rather than letters when introducing algebra. What was holding Stephen up was that a "seventeenth" was just a meaningless blank to him, and he thought he had to understand its meaning. I think the letters in algebra have the same effect on most kids. They just aren't used to things that have no particular meaning, and so can't operate at all with them.

So "(the number in the) BOX" is a better name for a variable than "X". So is "PIG", so is even a nonsense word like "BOOJUM".

The last one might seem to contradict what I've just said - because nobody understands the meaning of "BOOJUM", since it hasn't actually got a meaning!

But the point is that we know it's not supposed to have a meaning. What worries the kids, I think, is that they think that "x" has a meaning they're supposed to know, when the truth is that is hasn't.

(What IS "x"? An unknown number? Give me an example of an unknown number? Is 3 one? No. There isn't any unknown number!!! That's roughly what I mean by saying that x has no meaning. ) "The number in this box" is much better.

"I know a number I'll call boojum, but I won't tell you what it is" is also better, and might be more fun.

John Conway

From: Pat Ballew
Newsgroups: geometry.pre-college
Date: 28 Mar 1995 22:27:32 -0500

John Conway's remarks about using "Box" instead of a variable or blank box remind me of a suggestion I once made to elementary teachers about adding fractions. I suggested that before they ever gave students problems like 3/8 + 2/8 they should give them practice with problems like 3 "eighths" + 2 "eighths". I felt that students in fourth to sixth grade would do these almost automatically from their previous experience with units. If they had difficulty, I suggested writing out problem sets with mixes of 3 feet + 2 feet and 6 dogs + 4 dogs and others where the units were in words rather than some shorthand.

They seemed to think I was totally insane and reminded me that I had no experience with elementary teaching. They were right so I went on my way, but still I wonder.....

Pat Ballew "lovin' what I do"
Edgren HS, Misawa, Japan

From: Judy Roitman
Newsgroups: geometry.pre-college
Date: 29 Mar 1995 12:27:48 -0500

When I taught in a program affiliated with Project SEED 25 years ago in regular Berkeley elementary classrooms, we used boxes instead of variables with absolutely no trouble, as early as second grade.

Cheers.

Judy Roitman, Mathematics Department
Univ. of Kansas, Lawrence, KS 66049

From: Harry Sedinger
Newsgroups: geometry.pre-college
Date: 29 Mar 1995 14:15:23 -0500

With all you people drawing rectangles and calling them boxes, no wonder students have difficulty in three dimensions.

Harry Sedinger
St. Bonaventure University

From: F. Alexander Norman
Newsgroups: geometry.pre-college
Date: 29 Mar 1995 18:30:04 -0500

By the way, my New World Dictionary (2nd College Edition) makes a similar reference in its first definition for box:

1. any of various kinds of containers, usually rectangular and lidded ...
Clearly they meant "rectangular parallelepipedal" or "rectangular prism" or some such thing.

I assume your comment is more jest than criticism, but I see no problem with referring to a rectangular image as a box. I've seen cylindrical boxes, heartshaped boxes, octagonal prism-shaped boxes, I even have a spiroid toolbox. Rectangular prism is only one of many kinds of boxes. Perhaps, instead of worrying about its colloquial usage, we should enjoin the use of the word "box" as too imprecise for descriptions of 3-dimensional objects.

Then again, maybe not.

Sandy Norman
University of Texas at San Antonio

From: John Conway
Newsgroups: geometry.pre-college
Date: 29 Mar 1995 23:41:30 -0500

"Box" is a damn' sight more convenient than

"rectangular parallelepiped"
which must be one of the longest phrases for such a simple shape that has no official shorter equivalent.

I don't know whether I've already put on the forum my comments on the origin of the second word here? It was a bit longer until about 1850:

parallelepipedon
which breaks up into 4 or 5 bits:
para + allele + epi + pedon
meaning beside . other . upon . ground

The "al" of the second bit is the same as in "alternative". (The entire word has been adopted in genetics for the alternative genes that may appear at a given locus.)

The Greeks very early combined the first two bits to get "parallel" for two lines that kept beside each other, i.e., were parallel.

The meaning of the entire word seems to be that however you put this shape down, the top face is parallel to the ground.

I remember the meaning of "pedon" as being the thing you put your foot (ped) on.

The Greek and English words here are really the same indo-European word

f oo t
p e d
(indeed, that e is often an o, as in "tripod").

So for "upon" and "epi"

So also for "allele" and "other", which is easier to see if you interpose the Latin:

allele (gk)
aliter (lat)
alter (lat and eng)
other (eng)
and so, I think, even
"para" and "by".
So we could translate 5 pieces into essentially the same words, but now in English!
par allel epi ped on
by other upon foot-on
Sorry for taking your time, but I just love the way that words work out like this, especially the mathematical ones, and this is a particularly good one. You should hear me on "parabola" and "parlour" some time! (They're the same word!)

John Conway