Note: The following discussion included within it a summary of a position paper called "Habits of Mind: An Organizing Principle for Mathematics Curriculum," to be published by the Journal of Mathematical Behavior. It was followed by a discussion of the history of the high school math curriculum -- its beginnings, rationale, and where it might be going.

Previous Topics || geometry.pre-college

From: Michael Keyton
Subject: Re.: 1 Year of Geometry
Date: 11 Apr 1995 01:37:39 -0400

On Thu, 6 Apr 1995, Linda Dodge wrote:

> Do we really need a full year of geometry, anyway?

I ask somewhat facetiously the following: suppose we had three years of geometry and one year of algebra in the curriculum, would not the question "Do we really need a full year of algebra, anyway?" be appropriate?

Yes, we need a full year of geometry, but we need a full year of geometry and not some year wasted without mathematics. We need more years of investigations using thought and fewer years of learning meaningless algorithmic processes that are more easily forgotten than learned. We need years of having students learn to think through a problem, to understand, and to develop rather than to mimic.

Do we need a full year of geometry? Yes, and more. Let's not bail out the students, let's not make their lives easy, but rather let's get inside their heads and rummage around, expunging the inert while getting them to begin generating fruitful thoughts.

If I had a choice of having students study 3 years of geometry and only 1 of algebra as opposed to the present, I would have guessed heaven had arrived on the wings of a TI-92.

From: Anthony D Thrall
Subject: Re: Re.: 1 Year of Geometry
Date: 11 Apr 1995 06:14:11 -0400

I understand you to say that geometry is a good training ground for thinking, and this, independent of any additional benefits, justifies at least one year of high school geometry. Do I understand you correctly so far?

Now I have two questions for you. First, do you claim that geometry, or mathematics more generally, is better suited for exercising and training student thinking than, say, history? If so, in what way? How would you respond to the assertion that it is better to train student thinking in the contexts in which students are and will be engaged, e.g., issues of public policy?

Second, what do you say about the importance of the content of geometry as opposed to other mathematical topics? And within geometry, what criteria would you apply to decide which topics to present and explore within a limited amount of time?

I ask you these questions not as an adversary, but because I have come to respect your comments in this forum, so I want to hear what you have to say on these issues of curriculum.

From: Bill Marthinsen
Subject: Re: Re.: 1 Year of Geometry
Date: Wed, 12 Apr 1995 09:15:07 -0800

I would venture to say that Michael Keyton is not alone in wishing that geometry was emphasized more in the mathematics curriculum. Currently the secondary mathematics curricula in the U.S. are strongly dominated by algebra. Aside from a year of geometry, there is little geometry content in a student's mathematics education. I fear that the push toward an integrated curriculum will further erode the amount of geometry students see in high school in favor of heavier algebra content. (Linda Dodge's original question tends to validate this anxiety.)

How was the current balance of algebra to geometry determined? What was the rationale behind it? Was it always so? What other balances have there been? I would like to see answers to these questions before jumping to conclusions about whether to zap either geometry or algebra from the curriculum.

These two areas of study are actually complementary: algebra focuses on developing in students convergent thinking patterns (simplifying and solving) and the ability to manipulate abstractions (variables), while geometry encourages the development of divergent thinking patterns (specific example to generalization) and the visualization of abstractions (geometric modeling). Each seems to have a different function, and both are needed.

I worry when teachers focus narrowly on a single learning or teaching style and become intolerant of other ways to learn or teach. Individuals have different styles; for learning, thinking, teaching. One way of doing things will not work for everyone, nor should we expect it to. We need to develop an understanding of our own styles and a tolerance and understanding of the styles of others. Disagreement is healthy, but needs to be tempered with understanding. Our history is littered with tragedies resulting from intolerance.

Previously on the Forum, a long discussion on two-column proofs covered a lot of ground. There were strong feelings about the topic, and opposing views were aired. On top of that, consensus was reached on some aspects, and some history of the procedure was unearthed.

I would like to hear what others have to say in the algebra v. geometry debate.

From: Judy Roitman
Subject: Re: Re.: 1 Year of Geometry
Date: 12 Apr 1995 15:39:32 -0400

There's a terrific paper by Al Cuoco, Paul Goldenberg, and June Marks called Habits of Mind: An Organizing Principle for the Mathematics Curriculum, which deals with the differences between algebra and geometry. Of course there are other parts of mathematics with distinctive patterns of mind -- probability and statistics, combinatorics, etc. -- and not everyone will agree with everything they say, but it's a good place to start.

I don't believe it's appeared anywhere, but if you're interested you could write to them at EDC.

From: bo3b Overkamp
Subject: Re: Re: Re.: 1 Year of Geometry
Date: 12 Apr 1995 23:14:15 -0400

Geometry in particular and mathematics in general are better suited to exercising and training student thinking than history because the fundamental idea of mathematics is "you could have figured this out". If this principle is absent from a course, I submit that course is not mathematics. History, on the other hand, seems to admit both interpretations of "history is fiction with the truth left out".

If one cannot reason in the simple universe of geometry, then one is very unlikely to be able to reason in issues of public policy.

Our school has predictable class reunions every five years, and I attend many of these. It is amazing and gratifying to me to have dozens of grads tell me every time that they are reminded of episodes from our geometry classes at least weekly. And their testimony has no correlation to how well they thought they were liking geometry or their geometry grades or their current occupations.

From: Bob Hayden
Subject: RE: A year of geometry
Date: 12 Apr 1995 04:01:05 -0400

I begin by assuming that most of the people who subscribe to this list probably think geometry is pretty important and should be taught . I know there are a few who only are on this list to catch our errors in grammar or politically correct speech, but except for them we are all pretty much Pro-geometry. So let's try to rephrase the question to get a different slant on it, to wit,

"should geometry be taught in a separate course (predominantly focusing on geometry) or should it be integrated into the "Algebra" sequence, where it could probably bear a new name like MATH, course 1, etc.?

And Is it really true that Colleges Don't teach Geometry courses now? I took a really interesting one back in the dark ages and thought it was a great course. Can we get survey data from the top? How many geometry courses and what kind are taught in the different colleges?

From: Bob Hayden
Subject: geometry in college
Date: 12 Apr 1995 12:55:03 -0400

I did not say that no geometry was taught in college. I was only discussing what high school mathematics was taught in colleges. However, not much geometry of any sort is taught in colleges. We have just one course that is mainly for future teachers so that they can teach geometry. Of course, there may be some geometric content in calculus, linear algebra, group theory, etc.

From: William T. Webber
Subject: geometry in college
Date: 12 Apr 1995 15:39:34 -0400

In regards to the question of what geometry is taught in college I have a few observations from recent experience.

From the time I was first introduced to Geometry in the 10th grade I have not been able to get enough of it. In fact I have had a hard time getting any of it. U Mass offered one course in geometry for prospective high school teachers. It covered some advanced topics in Euclidean geometry, then some spherical and hyperbolic geometry. None of the material was on the level of what I learned in high school (Although I know of many high school teachers now that do some non-Euclidean stuff in their classrooms). When looking for a graduate school I had the following conversation many times.

"What do you want to study?"
"Geometry."
"Differential Geometry?"
"No, Geometry."
"Algebraic Geometry?"
"No, Euclidean Geometry, polyhedral theory, stuff like that."
"Oh, Combinatorics!"
"No."

Needless to say it was rather discouraging. It seems to me that "geometry" at the college level means either "differential" or "algebraic." These certainly are major areas of research and require the knowledge of high school geometry. I've heard Branko Grunbaum refer to what we do as "Geometry without an adjective." Many (probably most) colleges offer courses in differential or algebraic geometry. But only a few offer anything more than the course for prospective teachers in the lines of Adjectiveless Geometry. Many colleges also have an "advanced geometry" course. Usually the topics included in such a course are chosen by the professor teaching it. So it is quite likely to again be a differential or algebraic course. There are some that offer more but they are few and far between.

My last measure of the scarcity of Adjectiveless Geometry is the job market. To say that the academic job market is tight is an understatement. But try finding a college that is looking for someone with a Ph.D in Geometry. There are a few - but not many.

From: John Conway
Subject: Re: Why teach geometry
Date: 13 Apr 1995 01:10:21 -0400

College professors are people, too (at least some of them are), and can have opinions which might not always coincide with the policies of the institutions they belong to!

This one loves geometry, and also thinks it's one of the best ways to introduce mathematics to those who might not yet know that they love it too; high school students in particular, whatever else they are studying, and whether or not they will later go on to a college education. It also happens to be the least frightening mathematical topic to many people.

I think it would be disastrous if it were to be dropped or further downgraded in the high schools. (By the way, I was shocked to learn by reading a message this morning that geometry is only taught inside one year in American high schools. (I should have known this already, I suppose; but didn't. It astounds me, but I suppose does help to understand a little bit more just why mathematics is in such a poor state in this country.) How can this possibly have come about? Is there any other subject in the high-school curriculum that's taken up for such a short time and then just dropped? (I hope not.) It's a ridiculous way to arrange things.

The people who arrange remedial courses, or fix the rules for accepting students, are usually not the same folk at all. They're more likely to ask things like "how many of our introductory courses need such-and-such an amount of algebra/geometry/... ?"

These aren't very inspiriting questions. They are more concerned with the efficient and practical working of such-and-such a college, rather than with (say) the way we can try to ensure that a larger proportion of the population enjoys and appreciates and understands some mathematics.

There's no inconsistency if I think differently to the way my employer acts.

From: Bob Hayden
Subject: Re: why teach geometry (fwd)
Date: 14 Apr 1995 00:40:01 -0400

I can't speak for you or your institution, but I've rarely seen things quite as divided as you describe. Here my dept. wrote the placement test for years, and now they use one made by the MAA, so the content is determined by mathematicians, not the administration.

From: Michelle Manes
Subject: Re: Re.: 1 Year of Geometry
Organization: Education Development Center, Inc.
Date: Fri, 14 Apr 1995 19:54:31 GMT

Thanks for the plug, Judy. :) How about I take over?

I work with the above-mentioned triad on a curriculum development (funded by NSF) called Connected Geometry. We are developing a series of (at least 5) geometry books for high school. (Our hope is that the introductory book, of which Judy is a reviewer, could eventually be used at the middle school level.) Three of the books could be used together to form a one-year course in geometry, or they could be used in more modular form to integrate more geometric content into other classes (whether they have titles like, "Algebra II" or "Integrated Mathematics I"). Al, June, and Paul have written a position paper called, "Habits of Mind: An Organizing Principle for Mathematics Curriculum." The paper will be published by the Journal of Mathematical Behavior. In the meantime, I'd be happy to snailmail a draft copy to anyone who sends me their address. (You'll probably get a standard information packet about the project, including the Habits of Mind paper.) I'll post another article with a brief (?) summary of the paper.

From: Vince Devlin
Subject: Why algebra **VS** geometry?
Date: Fri, 14 Apr 1995 22:51:06 -0500

I have been reading the recent discussions concerning "algebra vs geometry" with great interest. However, contained within these discussions, I have sensed some other conflicts:

• high school math education vs college math education
• isolated math courses vs an integrated math curriculum
• subject-based curriculum vs an integrated school curriculum
Each of these controversies is an issue in itself, but using them as an overview of the concerns of "isolation vs integration" in education, we can attempt to focus on the primary issue of algebra vs geometry.

By looking at algebra and geometry as separate fields of study, do we create the same situation as considering math and science as distinct disciplines? The division that we imply within these two intertwined areas of mathematics tends to reinforce the artificial barrier that has existed between algebra and geometry for many years. A typical high school mathematics student studies Algebra I, Geometry, and then Algebra II. They realize that some topics are "algebra-focused" while others are "geometry-focused". This "shifting" encourages them to identify with one subject vs the other and to formulate opinions such as "I like geometry better than algebra" or "I like algebra better than geometry". What does this type of isolation do for the study of mathematics?

Rather than wrestling with questions such as:

• How many years of geometry should we teach?
• Should geometry be a full year course?
• etc.

Should we be looking at issues such as:

• Is this isolation beneficial or harmful, both mathematically and attitudinally?
• Should high school students "put aside" their algebra skills in order to study geometry?
• Is this isolation being caused by the names of the courses themselves (geometry, algebra)?
• Should algebra and geometry be "blended" into more of a continuum of study?

As a high school teacher, I listen to students compare algebra to geometry. Many like one better than the other. Although we may not see this distinction in the subjects, if they do, then it's real!

Do you feel that there is a need for:

• an integration of algebra and geometry into a single continuum?
• maintaining the study of algebra and geometry as separate courses?

From: John Conway
Subject: Re: Habits of Mind: Paper Summary
Date: 16 Apr 1995 13:33:12 -0400

On Fri, 14 Apr 1995, Michelle Manes wrote:

{Lots of very interesting things, which I've edited out}

{A paragraph which I summarize as "It's important to be able to work with fuzzily posed problems"} . . . and then lots of absolutely lovely stuff about

> Mathematical Approaches to Things:

which which I wholeheartedly agree, finishing up with

> Why Habits of Mind?
>
> If we really want to empower our students for life after
> school, we need to prepare them to be able to use, understand,
> control, and modify a class of technology that doesn't yet exist.
> That means we have to help them develop genuinely mathematical
> ways of thinking. Our curriculum development efforts will attempt
> to provide students with the kinds of experiences that will help
> develop these habits and put them into practice.

However, there is something about Michelle's paper that worries me tremendously. I'm sure it's mostly accidental, but it doesn't seem to mention what I regard as the most important thing of all about the mathematical experience, from both the practical and theoretical standpoints.

This is the habit of precise thinking about precisely worded problems. This is the most important thing to teach, and should precede any kind of thinking about fuzzily worded problems, in my view (and I'm speaking here as a teacher, rather than as a professional mathematician).

I think this is desperately important, so much so that although I think I agree with everything in her paper, I also think that the paper has negative value as it stands, because this most important thing is left out. [Michelle - I know that you did have some words about precise thinking, but they occurred too late, and too little.]

I know this is a very strong criticism, but it's meant to be a helpful one, and I really do think that this is important. So, although this will repeat some things I've said in this forum recently, let me explain.

The art of precise thinking is very difficult to teach. It cannot be acquired by starting to think about fuzzily worded questions. It can only be acquired by starting to think about very precisely worded questions, so that you know exactly what each word means, and exactly which types of arguments are "legal".

It is also very important (though not quite as important) to be able to give clear answers to the fuzzily worded questions we are likely to meet in real life. The correct way to do this is to learn how to analyse these fuzzy problems so as to locate the precise ones that might be inside them, and then to think precisely about these precise questions. [This is why it is important that precise thinking about precise questions should come first!] There is no way to think precisely about a fuzzy problem other than to convert it to one of several possible precise ones first.

Thinking precisely about fuzzy problems is therefore harder than thinking precisely about precise ones, and one shouldn't teach people hard things before one's taught them easier ones. [And even precise thinking about precise problems isn't all that easy!] Thinking fuzzily about fuzzy problems is nothing to do with mathematics, and is not a worthwhile skill to teach. Thinking fuzzily about precise problems is useful, if only because it's often all we can do.

Let me summarize:

fuzzy(fuzzy) - WORTHLESS!

fuzzy(precise) - DOESN'T NEED TEACHING!

precise(fuzzy) - VALUABLE & HARD!

precise (precise) - MORE VALUABLE & EASIER & FUNDAMENTAL & MUST COME FIRST & IS NEEDED BEFORE YOU CAN DO precise(fuzzy) !!!!!!

So please, everyone, emphasize the value of precise thinking about precise questions. Yes, we do also want to teach the very valuable skill of precise thinking about fuzzy questions, but put precise thinking about precise questions first in your teaching, first in your writing about teaching, first in your hearts. If your students learn that, they will have absorbed the most important thing in the mathematical experience, and will be able to apply it to all the problems they will meet in their lives (they might also go on to become interested in mathematics, and maybe even become mathematicians, but that is totally unimportant).

If our students don't learn how to think precisely about precise questions, then they will learn no mathematics (very sad, though perhaps not the worst thing that could happen to them), and will never be able to think precisely about anything (which I think is a disaster), and we have failed as teachers.

John Conway

From: John Conway
Subject: Re: Why algebra **VS** geometry?
Date: 16 Apr 1995 13:48:27 -0400

On Fri, 14 Apr 1995, Vince Devlin wrote:

> I have been reading the recent discussions concerning "algebra vs
> geometry" with great interest. However, contained within these
> discussions, I have sensed some other conflicts:
> - high school math education vs college math education
> - isolated math courses vs an integrated math curriculum
> - subject-based curriculum vs an integrated school curriculum

--- I'm editing a lot out ---

> the primary issue of algebra vs geometry.

--- when was there a war between algebra and geometry?
was it before or after the Franco-Prussian one?
I don't recall! ----

> . . . The division that we imply within these two intertwined
> areas of mathematics tends to reinforce the artificial barrier that has
> existed between algebra and geometry for many years.

--- there's no barrier between algebra and geometry -
I presume you mean between the teaching of these two
subjects in some benighted places! ---

> Do you feel that there is a need for:
> - an integration of algebra and geometry into a single continuum?
> - maintaining the study of algebra and geometry as separate courses?

--- Well, I've made my opinions plain! (I recognize that of course we are agreeing with each other, and have phrased my opinions as I did just for fun!)

John Conway

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