Logic for Mathematics

I have collaborated professionally with the IMACS principals over many years, almost half a century in some cases. I have participated with them in brainstorming, drafting, writing, and polishing large quantities of stimulating mathematics curriculum materials, some of which now form part of the Elements of Mathematics Program.

It is my considered opinion that the IMACS curriculums in general and the Elements of Mathematics Program in particular do a wonderful job of captivating young minds and provoking within them a lifelong thirst for learning, particularly in the areas of mathematics and computer science. The Elements of Mathematics Program and the other IMACS courses to which it gives access produce graduates with an unusually mature insight into mathematics, both as a corpus of knowledge and as an intellectual activity. These are students that university mathematics departments recruit like star quarterbacks.

Students who succeed best in the Elements of Mathematics Program are those with a flair for mathematics, who do not hold back from devoting time and effort to their studies. Willing and enthusiastic participation pays huge dividends. The few who do less well are typically those to whom success in mathematics has come easily, with little or no effort, and who are unwilling or unable to discipline themselves to thinking long and hard about their mathematics studies.

In view of the impressive effects that the Elements of Mathematics Program has on the mathematical maturity and confidence of the students for whom it is designed, I heartily endorse the Elements of Mathematics program to students, parents, teachers.

Sincerely,

Vincent Haag
Professor Emeritus of Mathematics
Franklin & Marshall College
Lancaster, PA

November 15th, 2000

The State University of new York at Buffalo Gifted Math Program has provided mathematics instruction for a total of over 1200 bright secondary school students in Western New York since it was founded twenty years ago and one of the strongest features of our curriculum has been the logic content that is part of the IMACS Elements of Mathematics curriculum.

Our students study this content in grades six through eight but we have also found it suitable for still younger children who are highly motivated. At the other extreme, I have used this same content with university graduate students. I cannot speak too highly of these materials. The content and the pedagogy are both of highest quality and I recommend Elements of Mathematics without reservation. In the new internet format, I believe that this content can be studied either with teacher or parent supervision or in the case of well motivated students on their own.

Sincerely,

Gerald R. Rising
Distinguished Teaching Professor
State University of New York at Buffalo
Buffalo, NY

December 20th, 2000

This is an extract from an open letter recommending the EM program to the chairperson of the School Board of Broward County, Florida.

The EM program is the outgrowth of more than thirty years of effort to improve the teaching of mathematics in the public schools. It is far and away the most ambitious of the many curricular reform programs that have been instituted in the public schools of this country, and it has been successful. We have here at Harvard a young man who came out of this program. He was able to begin his mathematical studies at a very advanced level by taking a course usually taken by juniors and seniors, and he has been a straight A student. Our program in mathematics is the strongest in the nation and for him to be able to jump in with our best students and perform at the highest level is ample testimony of the strength of the program. I know of no other program conducted by a public school system that has done as well. There are a few comparable programs run by large universities as add-ons to the regular high school curriculum, but no public school, not even such famous schools as the Bronx High School of Science, Stuyvesant High School in New York, or the North Carolina High School of Science, achieve the level of the EM program.

There are, of course, other "accelerated" programs. But many of these have not worked well, often because they have not demanded a high standard of performance from either their students or their teachers. We have many students here at Harvard who have taken advanced placement calculus and passed the College Board's AP examination, but who do not do well in our regular second-year course. EM is special in that it starts early, takes a more serious view of mathematics than traditional high school programs, and demands a high standard of performance from the beginning. Most accelerated programs merely try to squeeze the usual high school curriculum plus one year of calculus into a three-year program, often making serious compromises with quality.

As you know, it is a national goal, articulated by the President and endorsed by the National Council of State Governers, to improve the standard of instruction in science and mathematics and to make our students "first in the world". The EM program is a beacon showing what can be accomplished with our schools.

Sincerely yours

Andrew M. Gleason
Hollis Professor of Mathematics and Natural Philosophy
Harvard University
February 25, 1992

This article was written by Natasha Chen, who took the Elements of Mathematics course through her middle and high school years at a Project MEGSSS (Mathematics Education for Gifted Secondary School Students) in Florida. It reflects in a thoughtful and lucid manner on the advantages of the unique mathematics education offered by this curriculum.

The Benefits of EM

A year and a half ago, I ventured out into the sinister world on my own for the first time. I traveled to the peaceful city of the angels to begin my studies at college as a math major. I had just completed the most enriching seven years of my academic life as a disciple of the Elements of Mathematics curriculum under the guidance of Burt Kaufman, Edward Martin, and the other instructors.

The first thing I realized was that I had led a sheltered life in the EM program. My classmates at college obviously weren't so lucky. I sometimes wonder why these people are at college if they are so negative about learning. I find this attitude to be more acute in the area of mathematics.

I have worked with math majors who, simply put, do not know how to think and were never taught how. If logical reasoning means anything to them, it does not show in their questions, thoughts, or proofs. I do not doubt that my peers are intelligent. They just seem to have trouble articulating their mathematical knowledge in an orderly fashion. I find my ability to think and think clearly an indispensable benefit, one I would not have had without the EM Propositional Calculus [logic] course. Not only was the availability of such a course important, but also its placement at the beginning of the program which seems, well, only logical. This background prepared me for courses to follow not only in math but in all other subject areas.

For almost all the math majors I know, college is the first time they have been exposed to math theory. While teaching true mathematics is more the exception than the rule in the standard classroom, the EM program makes this the rule. This prepares students for what no advanced or even A.P. course can do: dealing with the abstraction of college math. I know students who have changed majors because of this shocking realization. Because of my background, I am ready for the type of study that is required of me in college. Abstraction, rigor, and formalism do not surprise me at all because I have been taught to expect them and how to deal with them.

If the deception of standard high school math causes trouble for college students majoring in the subject, one can only imagine how much worse it is for non-math majors. Most that I know either dread taking a math course because they believe the material and the instructor will be dry and confusing (as they were in high school), or they cannot imagine what there is to know about math beyond high school calculus. This is the kind of ignorance that worries me the most. It pains me to see someone at that extreme when I know how rewarding it has been for me to be in a program like EM. Every time someone asks me what there is to study "beyond 2 + 2 = 4", I appreciate more the continual efforts of Burt Kaufman and his curriculum development team to make my education and all their students' knowledge as complete as possible. While each EM student may not leave the program with as much enthusiasm for math as I have, none will go away disliking math without really knowing what it truly is.

After a year and a half, I realize that what my experience in EM taught me about college is that . . . well . . . it stinks. That's the first thing. The second thing is that I have the power to change that because I have the experience of a better way of learning, and not just relative to math. The best teachers come from the best students, not judged by grades or class ranking, but by their respect, enthusiasm, and desire for learning. These are qualities that the MEGSSS program cultivates in a student. If all subjects in school were taught the way math is taught in EM, imagine the success of our students. Imagine what their passion would be for knowledge and how many more may be so generous as to want to share it with future scholars.

Natasha Chen
Sophomore, Mathematics
University of Southern California
January 1993

In April 2001, Ms. Chen sent the following:

A Decidedly More Adult Retrospective!

The invitation to my 10-year high school reunion arrived recently. After initial thoughts of "Oh, no! The BIG 3-0 is almost here," I started reminiscing about the good old days of big hair, blue eyeliner and the time I spent in Project MEGSSS.

Now that I have been a so-called working professional for six years, I can see just how important the critical thinking skills I developed under the EM curriculum are. Capital markets are never dull with investment bankers always concocting some new financial product that analysts such as myself are paid to understand. Being able to make sense of complicated details is a core part of my job. Whenever I interviewed potential new hires, I always looked for people who could think logically through challenging problems. In many ways, that ability is a better indicator of successful job performance than related experience.

I feel very fortunate to have had my EM experience and know that, in a very fundamental manner, the way I learned to think then helps me do my job well now.

Natasha Chen
Assistant Vice President and Analyst
Moody's Investors Service
April 2001

While a junior at Rice, David Broman wrote:

Being in college now, looking around at other students in my classes, it's amazing what an advantage I have over them in any theoretical course, in computer science, mathematics, or logic. The things they consider the most difficult - constructing proofs or understanding abstract concepts - are the things I find the easiest! I find myself able to help people taking abstract algebra, which I learned in tenth grade, and complex analysis, which I never even took, simply because I have learned how to organize my thinking about mathematical problems in a logical way. I was easily able to skip an introductory CS theory course required for my CS major, and was asked to take a graduate level CS theory course my freshman year (in which I got an A) solely because of the material and the techniques I learned in EM.

David Broman
Houston, TX, USA
January 1993

After completing his degree in computer science, Mr. Broman took up employement with Microsoft in Seattle, where he continues to work.

This letter was received in March 1987. The author studied the Elements of Mathematics (EM) at a Project MEGSSS (Mathematics Education for Gifted Secondary School Students) site in St. Louis.

Dear Students,

I would like to share my perception of the EM program's impact on my life and life choices. Of course, my achievements in EM augmented my applications to several universities, including Brown which I choose to attend.

While at Brown I took several pre-med science courses including biology, chemistry and physics. Because of my EM math background, none of these courses were overly grueling. In fact, I maintain that nothing compares to a EM take-home test. Nothing. The EM program enabled me to cultivate excellent study habits: prolonged concentration, the ability to "work through" unknown problems, and a high level of self-efficacy. This math background enabled me, with only a 10th grade chemistry course, to be number one in a class of 400 chemistry students at Brown. I did almost as well in physics with no high school preparation. The math department at Brown allowed me to transfer my EM credits, equivalent to an entire year of college study. I only needed to take four upper level math courses in order to obtain a B.A. in mathematics. I could have graduated in three years, but I chose to stay for three and a half years to get a second degree (SC.B.) in psychology.

Presently, I'm applying to clinical Ph.D. programs in psychology. Psychology faculty are aware of the importance of a strong math background. EM prepared me well for future studies in a variety of fields. I worked as a research technician in two capacities (pharmacological and physiological) as an undergraduate. I obtained these jobs because of my math background and subsequent ability to analyze data and follow logical thought. My present interest is research-oriented within the field of psychology.

In essence, EM enabled me to develop the strong cognitive abilities necessary to undertake any (possibly overwhelming) project. I believe that I can learn from and contribute to any field of study that I choose. This vision of self-efficacy is directly related to my math training; it challenged me and allowed me to recognize my abilities. For this I am forever indebted to the program that Burt Kaufman and his curriculum development team has created and to the most inspirational teaching I have ever had.

Sarah Erickson
Brown University
March 1987

Dr. Erickson completed her post graduate studies and now has her own practice as a Psychologist.

Derek Richard studied EM through middle and high school. Just before his senior year the family moved from Florida to Virginia. This letter was received from his father in November 2000.

Dear IMACS,

An incident happened a couple of weeks ago that reminded me of IMACS. So I thought I'd use this opportunity to bring you up to date on my son's educational adventures!

Derek decided to attend Gonzaga University in Spokane, WA. Derek chose Gonzaga because it ranked high (19th last year) in the US News ranking of domestic Engineering and Computer Science colleges and universities without a graduate program. I'll leave out the part about how far away from home it is!

During his last year of high school in Virginia, there were essentially no math courses offered that were any challenge. In Virginia, as in Florida, education seems to be about barriers, not assistance, for the administrations and teachers. So Derek has not had a math class in over a year since he left IMACS.

Derek's advisor cautioned him on taking Calculus III as freshmen often have difficulty especially if they have not had math in a while. After five weeks, Derek's Calculus III class finally reached a point where he observed “I'm now where I stopped in math at IMACS”. That was nearly three years ago!

I feel you give yourselves too little credit for the magnificent job IMACS does in teaching children in a pleasant yet demanding manner. It is too easy to say “we only take the talented” to offset the achievements. Even the talented do not understand until later what a tremendous advantage IMACS is for their development and their capability to achieve success as they go forward in their education and life. When they are really young, no one really knows who is good at math and who can become good at math with the IMACS experience.

Anyway Derek is doing very, very well. He hopes to double major in computer science and computer engineering with an emphasis in math. He started last summer the day after he got his driver's license. I'm not sure which he was more excited about.

Jenny and I hope IMACS is continuing to prosper and succeed.

Sincerely,

Robert M. Mason Jr.

Robert M. Mason Jr.
776 Snead Road
Manakin Sabot, VA 23103

Craig Citro, who majored in computer science at Indiana University, took the EM curriculum while in middle school, and writes:

I’d like to thank you all for the very existence of EM. I only had the chance to study the curriculum for two years, but I can honestly say that I wouldn’t be half of what I am today if it wasn’t for this program. I didn’t realize how valuable it was at the time ... but the lessons I learned there were more incredible than anything else I learned in all of my schooling.

Having been in EM, any other math or computer science class I chose to take was a breeze. When I first started to take programming classes, I discovered that I knew how to do many of the things that we were doing as a class, but I didn’t know why. I watched the rest of my class fumble over recursion while being able to do it in my sleep, and realized it was because of the work I had done with IMACS. Math classes were much the same; I found myself coasting through upper level math classes, based on knowledge I had earned in the sixth grade! I can assure you that EM provides a foundation that is second to none.

I know I've come to realize it very late, but I wanted to relay just how much EM meant to me, and how profound of an impact you’ve had on my life. To all of you who gave me this chance, I thank you from the bottom of my heart.

Craig Citro
September 2000

The following is an excerpt from a letter from Dr. Matt Kauffman to Burt Kaufman, then director of Project MEGSSS in South Florida:

In my opinion, the EM curriculum in mathematics and computer science is truly impressive, way beyond anything I had ever imagined for high school, and even considerably beyond what many US mathematics majors would study for a bachelors degree. If only a few students per year go through your program, that should be considered a monumental success achievement. It is a program of which you can be proud.

Dr Matt Kauffman
Senior Computing Research Scientist
Computational Logic Incorporated
Austin, TX, USA
February 1992