Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

Why Not Geometry?

January 2001

This Fall Gov. Gray Davis has signed a law known as the Algebra Bill that makes Algebra 1 a graduation requirement in all California public schools. To graduate students will have to pass an exam which, for the class of 2004, will consist to a great extent of algebra questions. Students' future and state school financing will depend on the results. A high stakes exam indeed.

Last December, The Sacramento Bee newspaper came out with a 5 article series in support of the move. I take a critical look at the series on a separate page. Although I disagree with the paper on several accounts, the paper carried out the job admirably. The online version offers thoughtful algebra lessons and a discussion forum - a trademark of modern day publications. This is where I found an interesting post by al boothby:


Earned a "D" in High School Algebra. Learned some basics. Served in army in WWII. Illustrator and electronics instructor. Used algebra formulas. Designed buildings and airport runways for state. Learned more and used algebra. Earned BA degree Took Master's Record exam. Passed! Earned MA. Decided to retire and take Bee course to understand what I know about algebra! About time.

Sounds like learning algebra may be a life-long process. All it takes is maturity, motivation, and a little aptitude. However, stuffing algebra down the throats of unmotivated students will dry out any seeds of potential interest in the subject.

My main objection to the Algebra Bill is that it is bound to instill in students a wrong kind of attitude towards mathematics, thus having the opposite of the intended effect. Algebra 1 will be offered in the middle school. This is now a law in California. I have a very great doubt whether the majority of the students will be sufficiently prepared to face the challenge. However, the repercussion of failing an exam will weigh heavily on both students and teachers.

We read in an article Everybody Counts/Everybody else:


We have often heard the statement, "A student's last experience in mathematics is failure." This was driven home to one of us when a researcher added, "Even in research mathematics." Mathematics does not draw people in, it filters them out: from the cradle to the grave.

A student's last experience in mathematics is failure. This may be truer of a research mathematician than of a school student. Success and failure are often intermingled. Is it failure to get a "D" in a boring exam for a boring subject, but then at 70 to pick it up again? Is it success to pass a graduation exam with flying colors only to forget the whole thing in 2-3 years' time?

Not that the problem has not been discussed many times before. For example, in 1970, J. Piaget wrote the following about the end-of-school examinations:


... we are postulating that success in those examinations constitutes a proof of durability of the knowledge acquired, whereas the real problem, still in no way resolved, consists precisely in attempting to establish what remains after a lapse of several years of the knowledge whose existence has been proved once by success in those examinations, as well as in trying to determine the exact composition of whatever still subsists independently of the detailed knowledge forgotten.

With all the investment in mathematics education and in mathematics education research, it is absolutely unbelievable that Piaget's query remains unanswered up to this day. I suspect that the reason is that the answer is mostly known. Establishing the fact scientifically would probably shutter the whole hierarchy of mathematics education. A math education reform should start from the cradle and continue through K-16 in a concerted manner. As long as mathematics is taught for its pragmatic value, no such reform may be seriously contemplated.

The following passage from a 1777 letter by J. H. Pestalozzi - a famous Swiss educational reformer - to an affluent benefactor reads as a blatant anachronism:


A poor man is mostly poor because he is not accustomed to work to provide for his needs; this must be the main premise. The final goal of education of the poor must be, besides general upbringing, acclimatization to his circumstances. A poor man must be brought up for a poor life, and here is a stepping stone for a success of an institution for the poor.

The linchpin of our society and education is that every citizen must have an equal opportunity to lead a productive and happy life. In our society, for this sake, every child must be able to fully develop his or her natural abilities. In the very least, we strive to that goal. Still there is something to be learned from the above passage: practicality. Pestalozzi did what he could under the circumstances. I believe the best results in mathematics education will be achieved when both politicians and mathematics educators face squarely the present day circumstances: a relatively small percentage of the work force make use of any mathematics beyond common arithmetic. To boot, very little is learned in the absence of motivation.

David Lindsey Roberts describes an experiment reported in a 1919 publication from Teachers College, NY (G. M. Wilson, A Servey of the Social and Business Usage of Arithmetic):


In one notable study, some 4,000 sixth, seventh, and eighth grade students were asked to follow their parents around for two weeks, collecting mathematical "problems" solved by the adults in the course of business and household tasks. Most adults were found to use little except the most elementary arithmetic processes ...

What is then the goal of education? Is it to provide the poor children with the means of making a living as grownups? While that is an important goal, no self respecting educator would concede that nowadays this is what the education system should strive for. The means are available and the circumstances may be created to provide an environment conducive to full development of students' intellectual potential. Roughly speaking this may be accomplished in two stages. The first stage should help children realize their inclinations and help build up motivation to develop their natural abilities. Along the way children should pick up the little mathematics that is in common use. This stage should last through the middle school. I think of this stage as an early Liberal Arts education. On the second stage, the motivated students put more emphasis on the subject or subjects of their interest. Others receive reinforcement for the thus far acquired knowledge and further motivational assistance.

I would say that this is similar in spirit to Sol Garfunkel's proposal for an entry-level college math course but initiated and implemented starting at a much earlier stage. As a first approximation, we may talk of a library of what Erich Wittman has defined as substantial learning environments.

The important goal is to provide motivation, to convey the breadth and the beauty of a subject and interdependency between different subjects. I doubt this is possible with emphasis on algebra in an atmosphere of pending high-stakes examinations. Algebra, in the high school scope, is a collection of skills potentially useful to various degrees depending on the profession chosen. Other subjects should be considered as motivation builders. And the one that naturally comes to mind is geometry.

Geometry is more visual and more palpable than algebra. Geometry is often used to help introduce and explain algebraic concepts. Geometry is rich in structures, patterns and interrelations. And nowadays, technology makes geometric explorations more accessible and more intuitive.

The following page brings together the notions of discontinuous functions, altitudes and medians, similar triangles, iterative processes, and the Star of David.

  1. Why Not Geometry?
  2. A critical look at a newspaper article
  3. A nice piece of geometry


  1. T. R. Berger and H. B. Keynes, Everybody Counts/Everybody Else, in Changing the Culture: Mathematics Education in the Research Community, edited by N. D. Fisher et al, CBMS Issues in Mathematics Education v5, AMS/MAA, 1995.
  2. S. Garfunkel, The Curriculum Revisited: A Proposal, FOCUS, March 2000, MAA, p 4.
  3. J. H. Pestalozzi, Collected Works, v 1-3, USSR Academy of Sciences, Moscow, 1961 (in Russian)
  4. J. Piaget, Science of Education and the Psychology of the Child, Orion Press, 1970
  5. D. L. Roberts, E. H. Moore's Early Twentieth-Century Program for Reform in Mathematics Education, Am Math Monthly 108, n 8, October 2001, 689-696

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