From: bmarthinsen@keypress.com (Bill Marthinsen)
Subject: Re: 1 Year of Geometry
Date: Mon, 17 Apr 1995 09:23:16 -0800

I am really interested in the rationale behind the existing division of curriculum in high schools. Was it done with something in mind? I guess it is a question for historical research, and I throw it out here because the historical background about two-column proof came out of that discussion earlier. Maybe someone has a tidbit about this also.

From: king@math.washington.edu (James King)
Subject: Re: 1 Year of Geometry
Date: 17 Apr 1995 14:39:39 -0400

(I am making up this answer from memory and from anecdotal info, so if someone who actually knows something about the history of the math curriculum wished to apply a corrective, I will be happy to defer.)

My recollection is that there was a period in the earlier part of the century when the high school curriculum was 2 years of algebra, 1 year of geometry (11th grade), and 1/2 year each of trigonometry and solid geometry. This curriculum was compartmentalized, but the balance between the geometric side of math and the symbolic side was quite different from now.

When the New Math came in after Sputnik in the late 50s and the 60s, one of the reform ideas was to base a lot more on the idea of function and to aim more at the mathematics that would lead to calculus and linear algebra. The room for this innovation was made by reformulating all parts of the curriculum, but the practical effect seemed to be to remove solid geometry and to convert the trig course into part of a course about functions which would include the trig functions (this became what is now called "math analysis" in high schools, I think).

The next phase of change was the gradually widening penetration of advanced placment calculus into high school. This had the effect of moving algebra into the 8th grade for more students but also in squeezing out geometry or making it an island in a sea of functions and symbols.

Now we are in a time of new reform with the NCTM Standards which should result in less compartmentalization. Also, there is a movement in many areas towards an explicitly integrated math curriculum.

If done right, both of these have a lot of promise for connecting parts of math which are currently taught in isolation. However, there is a danger that if the changes are made without sufficient thought, that the knee-jerk approach to the introduction of new topics such as data analysis will be to throw out some important topics from the traditional curriculum (especially in geometry) rather than trying to connect these topics with other mathematics and to teach them better.

From: hayden@oz.plymouth.edu (Bob Hayden)
Subject: Re: 1 Year of Geometry
Date: 17 Apr 1995 18:36:03 -0400

> (I am making up this answer from memory and from anecdotal info, so if
> someonewho actually knows something about the history of the math
> curriculumwished to apply a corrective, I will be happy to defer.)

My dissertation (15 years ago) was a history of the new math. I also looked at prior history or math. ed. as background.

> My recollection is that there was a period in the earlier part of the
> century when the high school curriculum was 2 years of algebra, 1 year of
> geometry (11th grade), and 1/2 year each of trigonometry and solid
> geometry. This curriculum was compartmentalized, but the balance between
> the geometric side of math and the symbolic side was quite different from now.

This was common, though I don't recall geometry being in 11th grade. I graduated from high school in 1962 and we did the above except for no solid geometry and plane geometry was in 10th grade.

> When the New Math came in after Sputnik in the late 50s and the 60s, one of
> the reform ideas was to base a lot more on the idea of function and to aim
> more at the mathematics that would lead to calculus and linear algebra.
> The room for this innovation was made by reformulating all parts of the
> curriculum, but the practical effect seemed to be to remove solid geometry
> and to convert the trig course into part of a course about functions which
> would include the trig functions (this became what is now called "math
> analysis" in high schools, I think).

I think solid geometry was already gone by the time the "new math" became widespread. Trigonometry slipped into one of the chapters many classes never got to in the Algebra II book. Around 1978 I did a study of the trig. backgrounds of calculus students at Iowa State. I found that most had heard of trigonometry, but for many it had been only a passing acquaintance, far short of the half-year course I had taken.

On the other hand, the "new math" put more analytic geometry into the high schools. (It had been a college course. Many of the calculus books of the 60's were titled "Calculus with Analytic Geometry".)

> The next phase of change was the gradually widening penetration of advanced
> placment calculus into high school. This had the effect of moving algebra
> into the 8th grade for more students but also in squeezing out geometry or
> making it an island in a sea of functions and symbols.

I get the impression that some students went from Algebra II (or whatever replaced it) into calculus. This might work if you never got beyond polynomials in calculus. At Iowa State, these kids would whiz through the first little bit of calculus and then cradhed badly when we got to stuff involving the trig functions. They had taken calculus INSTEAD OF trigonometry in high school.

> ... there is a danger
> that if the changes are made without sufficient thought, that the knee-jerk
> approach to the introduction of new topics such as data analysis will be to
> throw out some important topics from the traditional curriculum (especially
> in geometry) rather than trying to connect these topics with other
> mathematics and to teach them better.

I think data analysis can be integrated into the curriculum in ways that help the mathematics and also shows the students some useful quantitative skills that 90% of them are more likely to use than, say, factoring quadratics. However, I said can be. What I actually see is disintegrated "add-ons" that do replace older math. topics, much as happened with sets in "new math".

From: jbens@casbah.acns.nwu.edu (John A Benson)
Subject: Re: 1 Year of Geometry
Date: 18 Apr 1995 03:07:26 -0400

O.K. We probably agree in principle on the idea. . So what are those things that we cannot throw out, and which should be thrown out, when we integrate the curriculum?

From: bo3b@oui.com (bo3b Overkamp)
Subject: Re: 1 Year of Geometry
Date: 21 Apr 1995 23:56:54 -0400

Nearly ten years ago I attended a talk which suggested (among many other things) that the U.S. high school mathematics curriculum is the fault of Harvard. Early in the nineteenth century, Harvard instituted an algebra proficiency test, and directly U.S. high schools taught algebra. Then Harvard instituted a geometry proficiency test also, and geometry became the typical post-algebra US high school mathematics course. Eventually people complained about forgetting the algebra they studied before geometry, and so algebra II was born.

I shall look for the notes I took on this and follow up when I find them. In the meantime, has anyone any corroboration or refutation to offer?

From: bmarthinsen@keypress.com (Bill Marthinsen)
Subject: Re: 1 Year of Geom
Date: 22 Apr 1995 01:01:21 -0400

Hmmm. If true, this could be a significant example of how testing influences curriculum. Some researchers on assessment have cited examples of how testing may generally act as a model for instruction. This suggests even more evidence for that view.

How much should we worry about the testing models that are being used today? What current models or methods will have the impact of the clout exercised by Harvard in the 19th century (if indeed this piece of information is accurate).

Is there any more detail or verification of this information? Did Harvard's proficiency tests direct the development of our curriculum?

From: mkeyton@tenet.edu (Michael Keyton)
Subject: Re: 1 Year of Geometry
Date: 22 Apr 1995 12:18:10 -0400

I'm not sure that this word "fault" should be used here, as it implies a negative aspect. The history of the relationship between curriculum at the college level and at the high school level in the U.S. shows that more and more is being moved down as time goes by. During most of the 19th century there was little "formal geometry" taught at the pre-college level. What was called "high school" geometry was mostly computation of area, perimeter, and volume. The significance of Wentworth was that his book in 1880 was the first "pre-college" book that addressed geometry from a proof concept.

For one, I lament that colleges no longer give admissions exams that address content, but understand the difficulties inherent there. Teaching in a school in which the average graduate has passed 4 AP exams, I also regret that too much of our time is spent teaching "college classes" at the high school level, and that the better universities tend to reject those who do not take a full battery of AP classes during their senior year.

One additional comment: if Princeton would reinstitute and administer an admissions test that addressed the identified skills that Prof. Conway laments not finding in current students, it could (would) change to a large degree what goes in high schools today. It possibly would also develop more of a collegial bond between teacher-student to prepare for college admissions, rather than the adversarial relationship caused from using grades as a basis for entrance. It also would standardize the students entering college, for they would be chosen on the basis of what they know and are capable of knowing rather than just the latter and how well they were at achieving grades.

From: bo3b@oui.com (bo3b Overkamp)
Subject: Re: 1 Year of Geometry
Date: 23 Apr 1995 18:56:40 -0400

Found the notes!

In July, 1986, Dr Stephen Willoughby (of Greenwich, CT) said to the participants in a Woodrow Wilson National Fellowship Foundation Summer Institute on Geometry:

The school sequence: arithmetic, algebra, geometry, algebra corresponds to the requirements by Harvard:

arithmetic
arithmetic and algebra
arithmetic and algebra and geometry

From: Marksaul@aol.com
Subject: Re: 1 Year of Geom
Date: Mon, 24 Apr 1995 16:39:31 -0400

> Some researchers on assessment have cited examples of how testing may
> generally act as a model for instruction.

This is obvious, if you teach. You don't need research to tell you that you should have the same goals for your job as whoever will judge if you are doing it.

I'm more interested in the reputed clout of Harvard, if this story is true. But then, what has happened in foreign countries? In Russia (and the former Soviet Union in general), algebra and geometry are also separated, although students often take them -- as separate courses -- during the same semester or year. A typical schedule might include "3 hours of geometry" and "4 hours of algebra" each week. Do others know about other countries?

>From where I sit, the division between geometry and algebra is historical.

Algebra evolved along with algorithms and arithmetic, in the near East and Renaissance Europe. It was always closely connected with extremely useful mathematics and with arithmetic. Geometry, on the other hand, was always viewed as something classical, and boasted -- or suffered from -- a single source, Euclid, which had undisputed primacy down to this very moment.

I tell my students that the geometry they study is the oldest curriculum in the school -- huge chunks of it have survived from Euclid's Elements. It's older, for example, than most of the history they study, and almost three times as old as the English language itself. Only the Odyssey and Oedipus Rex are older, and kids read these in translation (they don't usually read Plato in high school).

If we look at our methodology, our whole (traditional) mind-set on geometry, it is, consciously or otherwise, indelibly marked by Euclid's Elements. Kids whose intuitions let them slither right through algebra are bogged down in geometry because they are always thinking "proof, proof, proof," or "postulate, postulate, postulate." I would argue that even the paucity of geometry in lower grades, which are immune from most of the time-encrusted traditions of high school, is due to the influence of Euclid.

Great men cast great shadows.