With the publication of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989), a national reform in mathematics education began which though it has found resistance in New York City's high schools, is slowly bringing about much needed change. The Standards encouraged a greater emphasis on concept development, reasoning and conjecture, problem solving with interesting real-world applications, and in seeing the connections of mathematics to other subjects in the curriculum. They de-emphasized the role of the teacher as the sole `expert' in the classroom, placed greater importance on developing verbal and written skills in the process of problem solving, and on changing the role of the student from passive recipient of `chalk and talk' to that of active participant. In short these were all things alien to the three-year math sequence we were then teaching. But the Standards were also encouraging educators to look ahead to the twenty-first century and to consider what mathematics and the mathematics classrooms of the next decades should be like.
Among the 54 standards were clear recommendations for the inclusion of discrete mathematics (NCTM 1989, p. 176): In grades 9-12, the mathematics curriculum should include topics from discrete mathematics so that students can
Behind these recommendations was an acknowledgment of the effects of computers on the study of mathematics and the real-life applications of mathematics demanded by telecommunications, networking, robotics, scheduling, social decision-making, cryptology - even computer-chip design and production.
All of which indicate the huge change in the manner in which mathematics is used in the modern world. The standard for discrete mathematics included these sentences (NCTM 1989, p.176):
Whereas the physical or material world is most often modeled by continuous mathematics, that is, the calculus and prerequisite ideas from algebra, geometry, and trigonometry, the nonmaterial world of information processing requires the use of discrete (discontinuous) mathematics. Computer technology, too wields an ever-increasing influence on how mathematics is created and used...In light of these facts it is crucial that all students have experiences with the concepts and methods of discrete mathematics.In 1989, the same year the Standards appeared, Rutgers University in Piscataway, New Jersey received a grant from the National Science Foundation (NSF) for a residential summer Leadership Program in Discrete Mathematics. Each year from 1989 to 1994, under the direction of Professor Joseph G. Rosenstein, the high school institute of the Leadership Program brought new groups of teachers from all over the country to Rutgers for intensive classes and workshops in discrete mathematics. In addition, teachers from previous summers returned for follow-up institutes. The purpose of all of these institutes was the development of a group of teachers knowledgeable in discrete topics who could implement discrete mathematics in their schools and teach other teachers in their districts. I was a participant in the Leadership Program in the summer of 1990 and began teaching topics in discrete mathematics when I returned to the lower Manhattan high school at which I was then teaching.
By 1993 a number of Manhattan high school teachers had attended the Rutgers program. We felt that our introduction to discrete mathematics had opened a wider view of mathematics to us and a greater understanding of the use of mathematics in the real world. When we introduced discrete topics to our students we rarely heard the question most hated by math teachers, `When are we ever going to use this?' In teaching discrete topics in our own classrooms and seeing our students' excitement at doing real mathematics (Picker, in publication), we came to the conviction that we should develop a discrete mathematics curriculum.
The director of instruction at the superintendent's office for Manhattan high schools was now my immediate supervisor, and she and our superintendent had become increasingly enthusiastic about the possibility of creating this new course. They felt that discrete mathematics presented advanced mathematics relevant to our students' lives with widespread uses in a variety of fields. In the meantime, New York State had changed its mathematics requirement from two years to three. Many students who had been successful through the first two years of their math classes were, for a variety of reasons, having a difficult time with the third year. Some ended up repeating the third year more than once but with no more success. We wanted them to be given the option of studying mathematics they had not seen before instead of taking the same course over and over. There were also students who did not want to take calculus in high school or who felt that they weren't yet ready to take it but who knew that they should continue to study mathematics. We felt that discrete mathematics would be an excellent course for these diverse students because with two years of mathematics as a prerequisite, all of them could begin on an equal footing.
During the 1993-4 school year we created and began to pilot a discrete mathematics curriculum which was designed as a two-semester course with five core topics each term. The topics for term one included graph theory and coloring, apportionment and voting theory, fair division, tessellations and combinatorics, with algorithms as a theme through the year. In the second term, students study coding theory, matrices, fractals and chaos, and sequences and recursion. The teacher has the option to add topics of particular interest to them, such as knot theory, bin-packing, or game theory, and they are given the freedom to decide how long they wish to spend on a topic. As yet there is no textbook for this course which further keeps it from becoming lock-step and codified. Teachers rely instead on modules from a number of sources and create some materials themselves.
Initially, when we brought up the idea of a discrete mathematics curriculum at a meeting of Manhattan's mathematics supervisors, we were unnerved when it was met with dead silence. But we soon realized that we were dealing with two things: the pre-Standards mind-set which insists that only mathematics which leads to the calculus can be legitimate; and fear of the unknown. Since it is new, few mathematics teachers and supervisors had studied discrete mathematics, and it was difficult for many of those people who saw themselves as the subject experts to admit that they had no idea what discrete mathematics was.
So we began little by little to change the minds of many of them. It helped a great deal that the NCTM's conferences and their Mathematics Teacher began to have an increasing number of workshops and articles about discrete mathematics. We began piloting the course in 1994 with teachers who had studied discrete mathematics and were committed to its implementation. In September of 1995, Lehman College of The City University of New York and its Institute for Literacy Studies' Mathematics Project began sponsoring a graduate level course in the implementation of discrete mathematics in the classroom in order to prepare more teachers to teach it. And at the time of this writing in the spring of 1996, the superintendencies of the boroughs of Queens and The Bronx are interested in implementing discrete mathematics in the fall.
An important thing that discrete mathematics does for students is to enable them to see mathematics more accurately, as having many problems which are still unsolved. With only a small amount of technical background students can understand enough about a class of problems such as graph coloring to understand that no algorithm exists which will always guarantee the most efficient coloring. It means that students can stand at the cutting edge of mathematical knowledge with the real possibility of making a contribution; something they could not have previously believed.
There are a number of other student beliefs about mathematics that discrete mathematics seriously challenges. Such widely held beliefs (Mtetwa & Garofalo, 1989) as mathematics problems can only have one right answer, that doing math means memorizing formulas and giving them back in the right order at the right time, that most math problems can be solved within minutes, and that mathematics can only be created by geniuses, are disproved in a discrete mathematics class where there may be many answers to a problem, there may be no formula to follow, students may work on a problem over several days, and where students may see something that the teacher has never seen that way before. It was a telling moment when at the end of a class in which the teacher had been reviewing graph coloring problems from the previous term, a student new to discrete mathematics wondered out loud, `But where are the numbers?'
Recently I visited a discrete mathematics class in one of our high schools where the students were studying coding theory. That day they were creating Greek scytales, models of the earliest known coding device dating back some 2500 years in which a long strip of parchment had been wound around a cylinder with a specific radius. The students were using empty cardboard tubes from paper towels or toilet paper which they wound with masking tape. Their messages were written across the tube on the tape and the rest of the tape was then covered with meaningless, or null, letters. When the tape was removed it appeared to be a strip of random letters and it was passed to another student to decode. The difference in the radii of the two types of tube was barely discernible, but the students saw very quickly that it made all the difference in the world when it came to deciphering their classmates' codes. If a tape was not put back on a tube of the same radius with which it had been encoded, the message could not be read.
It occurred to me how different this class seemed from the classes in which the traditional curriculum was being taught. The room was noisy though students were on task, the teacher was not standing in the front of the room but rather assisting individual students who had questions by asking further questions, students were learning history as well as mathematics and they were helping each other. The atmosphere was bright and good-natured as students succeeded in decoding each other's messages.
Students really enjoy discrete mathematics and many have said that they hope to study it further in college. `In the other math classes', a student named Marissa told me, `it's just formulas they throw at you. But here you get the chance to explore new areas of math'. Another student related that she was always showing what she had learned in the discrete math class to her father because he found it so interesting. She had never done this with her previous math classes. As time goes on, and the topics of discrete mathematics become better known to our teachers, we hope that discrete mathematics can become integrated into the three-year mathematics sequence so that all students can experience this level of interest and excitement in their mathematics classes.
References
Mtetwa, D. & Garofalo, J. (1989(. Beliefs about mathematics: an overlooked aspect of student difficulties. Academic Therapy, 24(5), 611-618.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
Picker, S. (in publication). Discrete mathematics: giving remedial students a second chance. In D.S. Franzblau, F.S. Roberts & J.G. Rosenstein (Eds.), Discrete mathematics in the schools: making an impact. American Mathematical Society, AMS/DIMACS series.
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