Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
At the beginning of the month I had the gratifying experience of attending ICME-9. The International Congress on Mathematics Education -- known as ICME -- convenes every four years, each time in a different country. This year the Congress was held in Makuhari, Japan. An interesting fact: people who attended all the congresses starting with the first one are known as old hands. Three old hands were in attendance at ICME-9: J. Becker (USA), C. Gaulin (Canada), and E. Wittman (Germany). For me it was the first time in Japan and at ICME. My thanks go to NSF and NCTM for the travel grant that made this trip possible.
Makuhari is located some 35 km from Tokyo's main subway hub. How far is this? (I just remembered the musings of Sei Shonagon -- a court lady in 10th century Japan -- about things distant and near.) It depends, of course. By train it's about a 30 minute ride, but driving may take more than 2 hours. Traffic is very heavy there.
Yet another reflection on the distance. The weather there was very hot and humid. The sun was scorching even with dark and heavy clouds making their way across the sky. Twice during the congress, the CNN forecasts promised rain in Tokyo and environs. And indeed we were close enough to hear the rumble of thunder and catch glimpses of lightning, but rains never reached Makuhari. (Not until the last day of the congress when the weather finally relented did a few drops reach the ground.)
With all the reservations and registrations completed over the Internet months in advance, getting a registration bundle was a breeze, the weather notwithstanding. On browsing an information sheet, a few thoughts crossed my mind. The Congress drew 1840 mathematics educators from 75 countries. The former Soviet Union was represented by four now independent republics: Russia (11 participants), Ukraine (5), Latvia (2), and Estonia (1). 7 participants from former Czechoslovakia were evenly distributed between the Czech Republic (4) and Slovakia (3). Three participants came from Poland - formerly of the Warsaw Pact, now a NATO member. Another 3 came from Slovenia -- a republic in Yugoslavia until a few years ago. Here are some additional statistics:
Many saw significance in the fact that the Congress was held in Asia for the first time. While mathematics education evolves in every country on a national background, research is truly international. Emphasis on problem solving skills seems to be a common denominator for various schools of mathematics education. In Japan, The Three Open Methods will be included into the national high school curriculum starting with 2002. The three are open process, open-end product, open problem formulation. The first captures the intention to pursue multiple ways of solving the same problem. The second refers to the problems with more than one right answer. The third encourages students to pose and solve their own problems.
Aside from four plenary and 45 partially overlapping regular lectures, the work at the congress was organized into 13 Working Groups for Action and 23 Topic Study Groups. TSGs ran in a more or less standard format: lectures followed by question-answer sessions. WGAs split their allotted time between presentations and small group discussions. I took part in WGA1 (Mathematics Education in Pre- and Primary School) and TSG7 (The Use of Multimedia in Mathematics Education.)
According to Mogens Niss (Denmark) who gave the first plenary lecture on Key Issues and Trends in Research on Mathematics Education, the field of mathematics education research has reached its first stage of maturity. This is especially manifest in the breadth and scope of current research. In the 1960s and 1970s, the first researchers were preoccupied with the mathematical curriculum and ways of teaching it. Next, the goals and objectives of mathematics education became objects of discussion and investigation. Various teaching aids and information technologies fell into the research domain as soon as they were available. Investigation of the curriculum development process grew out of consideration for societal and institutional matters. Issues related to the education and profession of mathematics teachers were included into research agendas since the 1970s (education) and 1980s (profession). Research into students' actual learning processes began gaining momentum in the 1980s. In the 1990s, students' notions and beliefs with respect to mathematics were added to research programs. Today, research on the learning of mathematics is probably the predominant type of research in the field of mathematics education. Current research also covers classroom communication, social, cultural, and linguistic influences on the teaching and learning of mathematics, and of course its assessment. According to Ness,
In elementary education there is an overwhelming concern with psychological aspects of child development. The prevalent research trend embraces Jean Piaget's constructivism and Lev Vygotsky's social constructivism. The former refers to Piaget's doctrine that knowledge cannot be transmitted to an individual but must be constructed by the individual through personal exploration. The latter emphasizes the crucial role of conversation and, more generally, social interaction between individual learners.
Both with regard to instruction and assessment, the logistics of a constructivist educational system is anything but simple. Judging from the WGA1 discussions, teachers are more comfortable with a solid curriculum that offers well-defined goals. The reasons are many. In the context of instruction, letting students construct their own knowledge at their own pace makes taxing demands on teachers' mastery of the subject matter. As was noted by Margaret Brown (King's College, UK),
However, making connections on individual levels of understanding calls for flexibility of content knowledge that comes with a reasonable proficiency in the subject matter. The latter is usually not assumed of elementary school teachers who instruct in several subjects.
In the context of assessment, individual treatment of students in one-on-one informal, unstructured sessions is on one hand prohibitively time consuming, while on the other necessitates competence in aspects of child psychology traditionally not required of elementary school teachers.
Lastly, education is a multistage process. Every stage assumes a certain degree of knowledge -- presumably acquired in previous stages. A well defined curriculum plays the role of a goal setter for admission to and successful performance in the next stage. Unless the whole system undergoes a fundamental overhaul, we may expect that teachers will be more comfortable with curriculum as a framework for specific content and target skills than as a set of open-ended activities.
Mogens Niss noted:
In another plenary lecture, Erich Wittman (Germany) introduced the idea of substantial learning environments (SLE) as a tool for bridging the gap between theory and practice in mathematics education. The lecture was preceded by a video segment showing two girls probably 6-7 years of age playing the game of Scoring.
The game of Scoring is very simple: the board is a strip of several squares, on which there are placed several chips. Taking turns with your computer you drag chips (one at a time) leftward until all are collected in the leftmost square. The last fellow to make a move wins. (The loser is the first unable to make a move. Are the two conditions really the same?) The "heap size" parameter controls the number of squares, and the "max step" parameter determines the largest step you are allowed to make in a single move.
Wittman defines SLE as a teaching/learning unit with the following properties:
Scoring can be used to introduce and practice counting (forward and backwards), counting by grouping, division with remainder, modular arithmetic, binary system, logical and bitwise operations, impartial games, and more. The right strategy and winning and losing positions may be empirically discovered by students in early grades.
Wittman warned however of a danger inherent in focusing mathematics education on SLEs:
Seymour Papert [The Children's Machine, BasicBooks, 1993, p 1] offers a thought experiment. Imagine two groups (surgeons and teachers) of time travellers from 100 years ago brought into our time. Surgeons are shown to an operating room, teachers to a classroom. Who would be more surprised by the observable changes in profession? His answer: the surgeons would probably not know what their colleagues were trying to accomplish with those unfamiliar devices. The teachers might be puzzled by the changes. They might disagree whether the changes were for the better or for the worse, but other than that they would be right at home in a modern classroom.
There are many reasons why that would be so. Instead of discussing them, I only want to remark that, as rumors have it, there always have been and there still are good teachers out there whose students grow to love and appreciate mathematics even without becoming mathematicians by trade. These good teachers achieve their results in a more or less traditional classroom framework. I believe that the field of mathematics education could greatly benefit from documenting and classifying experiences and practices that reliably achieve very desirable results. If the field of mathematics education is to become a science in its next stage of maturity, the future science will be classified as natural rather than speculative. I strongly believe that, as a side effect, concentration on gathering and classifying past and current experiences may well help close the gap between research and practice in mathematics education.
In his book (pp 151-153), Seymur Papert gives a short description of Piaget's three developmental stages (sensorimotor, concrete operations, formal thinking). He notes that Piaget failed to recognize that concrete thinking and knowing is not confined to children in a certain age group:
At the closing of WGA1, participants were asked to suggest topics for discussion at ICME-10. As I am sure that teachers also do and benefit from, concrete thinking, I suggested a discussion on good teaching practices. Hopefully, somebody will respond to this idea.
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at email@example.com
Copyright © 1996-2000 Alexander Bogomolny