MathLand, Connected Mathematics, and the Japanese Mathematics Program
Some Comments on MathLand, Connected Mathematics, and the Japanese Mathematics Program
R. James Milgram
Department of Mathematics
MathLand and Connected Mathematics, (CMP), are the elementary school and middle school mathematics programs used in the Mountain View School District.
The supporters of these programs defend their selection using three arguments.The first is that these programs work better for minorities and girls than other programs. To justify this assertion they quote statistics supplied by MathLand for six unidentified elementary schools in southern California that used MathLand. They assert a dramatic rise in scores on nationally normed exams in these schools after the introduction of this program.
Unfortunately, repeated attempts to obtain the names of these schools from the publisher have been unsuccessful. However, by accident, one of them was identified. What was found was that these gains came in the third year of the program, following a dramatic drop in scores after the first year and the introduction of supplementary material. Moreover, the gains did not bring the scores for this school up to the level they had been at BEFORE MathLand had been introduced. But even more is true - Mathematics Professor Wayne Bishop, who discovered the identity of this school, reports in more detail on the students at the school who were not counted in the MathLand statistics:
"This school was heavily Hispanic, some 80% of its students were taught and tested in Spanish using the Spanish language edition of MathLand. These students continued to decline in their third year to the 15th percentile nationally, so low that the overall performance was not recovery but continued decline. This information was not included in the MathLand information on student performance nor in the information that they supplied to the U.S.Department of Education as evidence for their rating of "Promising" that was made public in October of 1999."
The second argument is that these programs are modeled after one of the best and most successful programs in the world. The Japanese mathematics program is held up as a model for programs of the MathLand, Connected Mathematics type. It is asserted that these programs work in the same way and at the same level as the Japanese program. It is also pointed out that the Japanese mathematics program propelled them to their remarkable success in all international comparisons of mathematics achievement - second only to Singapore - for the last 20 years.
While everything that is asserted about the Japanese program is true, the assertion that programs of the MathLand, CMP type are modeled after the Japanese programs is so far from true as to be a sort of sour joke to those of us who actually know something about these issues. In the following three paragraphs we explain how these programs actually measure up against the Japanese curriculum.
The current California Mathematics Standards are aligned with those of Japan, Singapore, and Hungary, three of the highest achieving countries in mathematics in the world. Indeed, in the writing process, it was these three standards that were used as references for both the topics in the California Mathematics Standards and when they would appear in the curriculum.
In 1999, Connected Mathematics was submitted by its publisher to be evaluated for approval as a PARTIAL PROGRAM - a program which, while perhaps not quite at the level of the California Standards, met a significant number of the grade level standards and could be used as a supplemental program. The mathematicians and teachers who reviewed it rejected it, asserting that it was at least two years below grade level and that it contained numerous and significant mathematical errors.
After the 2000-2001 academic year, state monies can no longer be used to buy Connected Mathematics for local school districts, as CMP was not even submitted for evaluation this year for the full program evaluation. Nor will state funds be available to buy MathLand.The publishers of MathLand did not submit their program for evaluation either as a partial program or as a full program. Perhaps they understood that MathLand was so far below the level of the California Mathematics Standards that the fact that the reviews would be publicly available could seriously cut into their sales in other states.
The third argument used is that programs of the MathLand, CMP type teach children to be "problem solvers" and make mathematics relevant.
While an attempt is made to introduce mathematical reasoning as part of the curriculum, it fails badly in programs like MathLand and Connected Mathematics. The main reason for this is that the authors really do not understand the process themselves.
The ideas about problem solving that were used in developing these programs rest primarily on a serious misunderstanding of the work on mathematical problem solving by the mathematician, G. Polya. Polya was at Stanford during the 1960's when mathematics educators were starting to quote his work and use it to shape their attempts to introduce mathematical reasoning as a component of the K - 12 mathematics curriculum. He repeatedly tried to get them to stop, explaining that his work had been done with juniors and seniors at Stanford majoring in mathematics, and was not appropriate for use until students had a deep grounding in the subject.
Interestingly, the author and creator of the Japanese mathematics curriculum - the great mathematician Kunihiko Kodiara - was also at Stanford during the 1960's. Problem solving is very strongly a part of the Japanese text books, but it is introduced in a very measured and controlled way. The Japanese method is completely at variance with the process used in programs of the MathLand, Connected Mathematics type, where the underlying assumption is that students will learn everything they need to know by working in groups and devising their own solutions to partially posed problems.
A typical problem at third grade level in texts like MathLand is the following: "Marta and Akoshi are in different third grade classes in the same school. They want to know which classroom is larger. How do they decide?"
As stated, this problem is not well posed, hence is not a problem in mathematics. It is necessary to assign a precise meaning to the word larger in order to make a mathematics problem out of it. However, since the meaning of this term is not specified, the only correct answer would consist of the entire collection of all answers associated to all possible meanings of larger - an infinite set. Such an understanding is unlikely to be achievable at the third grade level. Indeed, it is unlikely to be achievable by most elementary school teachers. Thus one has to proceed in a much more careful and structured way in order to inculcate the desired skills and understandings in our students. These are among the reasons for the differences between the approach to problem solving in the Mountain View school system and that in Japan.
It is, perhaps, interesting to get some idea of the true success of the Japanese program. Here is a quote from the introduction to the UCSMP translations of the Japanese books for grades 7 - 11. These books were published in Japan in 1984, and the translations were published in 1992.
"Let us take a brief look at the schooling behind much of Japan's economic success. The Japanese school system consists of a six-year primary school, a three-year lower secondary school, and a three year upper secondary school. The first nine grades are compulsory, and enrollment is now 99.9%. According to 1990 statistics, 95.1% of age-group children are enrolled in upper secondary school, and the dropout rate is 2.2%. In terms of achievement, a typical Japanese student graduates from secondary school with roughly four more years of education than an average American high school graduate. The level of mathematics training achieved by Japanese students can be inferred from the following data:
"Japanese Grade 7 Mathematics (New Mathematics 1) explores integers, positive and negative numbers, letters and expressions, equations, functions and proportions, plane figures and figures in space. Chapter headings in Japanese Grade 8 Mathematics include calculating expressions, inequalities, systems of equations, linear functions, parallel lines, and congruent figures, parallelograms, similar figures and organizing data. Japanese Grade 9 Mathematics covers square roots, polynomials, quadratic equations, functions, circles, figures and measurement, and probability and statistics. The material in all three grades (lower secondary school) is compulsory for all students."
It is also worth noting that when Japanese students continue on to upper secondary school, 40% of them elect a lower track, and 60% elect an upper track which culminates with a serious course in Calculus in 12th grade. The number of Japanese high school graduates who have had calculus is more than 50%. In the United States, the best estimate is less than 6%, and this number appears to have been dropping nationally with the introduction and growing adoption of programs of the MathLand, CMP types.
(The author is one of the four Stanford mathematicians who rewrote and revised the California Mathematics Standards for the California Board of Education. He and H.-H. Wu were the mathematicians who largely wrote and put together the new California Mathematics Framework for the California Board of Education. He has consulted on state standards for numerous other states, and is currently part of the Achieve Panel creating national standards for this country. He was also one of the members of the Content Review Panel for the California 1999 Partial Adoption and is currently a member of the Content Review Panel for California's Full Adoption. He is one of the authors of the letter to Secretary of Education Riley that pointed out the problems with programs of the MathLand, CMP type, and he has testified before congress on these issues.)