JAPANESE TEACHERS OF MATHEMATICS
After World War II, the Japanese began education reforms, mostly to incorporate empirical observation and inductive reasoning; but reforms proved inadequate to the needs of the globalized world. As the 21st century approached, the Japanese decided to re-define education for the accelerating changes of the new millennium. In the midst of that re-definition, the teachers of mathematics underwent transformations. For the 20th century they had mostly taught math for calculations—to teach students to memorize formulae for solving specific math problems. All testing emphasized getting answers to different types of math problems. After the 1970s, the Japanese began observing teaching around the world, especially in the United States.
In the latter part of the 20th century, American educators experimented with teaching mathematics as a means of teaching different ways of thinking. [All subjects, especially math, were intended to be taught at both the practical and educative levels; but in the 20th century most math teachers concentrated on the practical.] Largely due to three books by: Dewey (Experience and Education, 1958), Polya (How To Solve It, 1945, rev. 1975) and the Manual of the NCTM (National Council of the Teachers of Math), several experiments attempted to change the ways math is taught in the U.S.—especially in Michigan, California and Chicago. Japanese math teachers observed those experiments which did not take root in the USA. Instead, those attempts to teach thinking through math transformed not only the teaching of math but of many subjects in Japan. This was partly due to the mutual, inter-active methods of Japanese teachers of math.
Japanese Math Teachers
East Asian teachers have long had practices of learning from each other’s methods—even when they taught mainly by memorization. The Japanese people are so interested in the education of their children that hundreds will attend public demonstrations of a math lesson taught by a master teacher. Though dedicated, Japanese teachers found their work tedious. When Japanese teachers of mathematics put what they learned from the three American books into practice (in their mutual learning, especially), they found joy in teaching once again! Bored with teaching mathematics by memorization methods for solving set problems, the Japanese teachers of math came alive with the excitement of teaching children to think through mathematics. This could not have been imagined a generation ago in Japan, or even today in most countries.
For Dewey, knowledge is based on a foundation of experience. After observing the world through experience, the processes of education lead the child, through induction, to arrive at generalizations about what they have observed. Those generalizations are then checked against further experiences in the education processes. A child’s thinking is thus educated by self-observing his or her experiences for making potential general principles (laws, rules), corrected by further experiences, in order to set those principles into the child’s mind as knowledge. That knowledge is subject to correction by further experiences that may prove it wrong. [Here are the empirical observation and inductive logic of Aristotle which empowered the Arabs’ thinking that brought about their Golden Age.] From Dewey the Japanese teachers of math learned the importance of relating learning math to thinking by giving children actual experiences of the math processes and relating them to the ways of learning: generalizations by observation and induction; abstractions through numbers; finding unknowns through algebra; spatiality through geometry, etc. and for solving problems in other aspects of life by thinking mathematically.
From Polya (Professor of Mathematics, Stanford) the Japanese teachers of mathematics found specifics of how math teaches thinking. In HOW TO SOLVE IT, Polya prints on two facing pages (turned ninety degrees as if one long instructions sheet) four steps:
First understanding the problem (What is the unknown? What are the data? What is the condition? Is it possible to satisfy the condition?). [Note that all of these questions and the understanding all require thinking. Polya is listing how mathematicians think about problems!]
Second Devising plans. (Do you know a related problem? How was that solved? Do you know a problem with a similar unknown? Could you divide or restate the problem in ways you know how to solve? If you cannot solve all or part of the problem, can you think of a nearly similar problem that you can solve? Or can you think of theorems, rules or generalities that may help you think of a plan to address this problem?) Have you taken into consideration all of the essential notions involved in the problem? [Note that all of this is done in your head—thinking of potential plans.]
You may not find a single plan. You may have to devise, heuristically, and try a number of plans.
Third Carry out your plan. (Check each step. Is each step clearly correct? Can you prove it?)
Fourth Looking back; examining your solution. (Can you check your mental arguments, your thinking? Can you check your solution? Could you derive the same results differently? Can you see the problem and its solution at a glance? Could you use the methods for another problem?
From pages 37 to 233, the bulk of Polya’s book is a “Dictionary of Heuristics.” This is a detailed explanation of how a mathematician goes about solving problems—of how he or she thinks. Since math is considered a language, it is appropriate that Polya should call his main explanations of mathematical thinking a “dictionary.” Japanese teachers of math used his definitions as guides to teaching youths to think mathematically, and in solving life’s problems.
The final chapter, of less than ten pages, shows a variety of problems with hints and solutions. In teaching thinking, instead of the mere mechanics of solving set problems, Japanese math teachers found excitement in teaching that gave them self-satisfactions and self-respect that money or prestige could not match—even if the government would afford to give those to them.
The National Council of Teachers of Mathematics’ (NTCM) Manual is called: “Principles and Standards for School Mathematics.” It describes a future in which all youths have access to rigorous, high-quality math instructions and opportunity to learn important math concepts, ways of thinking and procedures with understanding. Youths also need to have access to technologies that deepen and broaden their understanding of math as a language for thinking.
Decisions made by educational professionals have important consequences for youths and for society. The Principles for school mathematics that guide them are:
Equity All children can learn math, if taught well; therefore all children should have equal access to learning math—without usual prejudices against females and minorities not being able.
Curriculum A curriculum is more than a collection of information, activities and tests. It must be coherent, focused on important mathematics, and well articulated across the grade levels.
Teaching Effective mathematics teaching requires understanding what students know (how they think) and need to learn and then challenging and supporting them to learn it well.
Learning Students must learn mathematics with understanding (thinking it through in their own mind), actively building new knowledge from experience (Dewey!) and prior knowledge.
Assessment Assessment is not primarily for ranking pupils. Assessment should support the learning of important mathematics and furnish useful information to both teachers and pupils.
Technology The use of technology is essential for teaching and learning math. It influences the math that is taught and enhances pupils’ learning of mathematical thinking by mechanizing math calculations and making it possible for pupils to work at higher levels of generalizations.
The Manual lists ten Standards for school mathematics to describe the mathematical skills, knowledge and understanding that youths should acquire--pre-kindergarten through 12th grade in four bands: pre-K-2ndgrade; 3rd-5th; 6th-8th, and 9th-12th grade. The ten are:
1.Number and Operations- Understand numbers, relationships among numbers and number systems; understand meanings of operations and how they relate to one another; make reasonable estimates;
2.Algebra Understand patterns, functions; analyze math situations and structures; use math models to represent, understand quantitative relationships; analyze change in various contexts;
3.Geometry Analyze 2- and 3-dimensional shapes, understand geometric relationships; describe spatial relations, using coordinates; apply transformations and use symmetry to analyze math situations; use visualization, spatial reasoning and geometric modeling to solve problems;
4.Measurement understand measurable attributes of objects, units, systems and processes of measurement; apply appropriate techniques, tools and formulas to determine measurements;
5.Data Analysis and Probability Formulate questions that use data, collect, organize, display data to answer them; use appropriate statistical methods to analyze data; understand inferences and predictions based on data; understand and apply basic concepts of probability;
6.Problem Solving Build math knowledge through solving problems in math and other contexts; adapt and apply strategies to solve problems; reflect on the processes of math problem solving;
7.Reasoning and Proof Reasoning and proof fundamental to math; investigate math conjectures and math arguments, proofs; use various types of reasoning, methods of proofs--in math and out;
8.Communication Pupils led to communicate math thinking coherently and clearly to peers etc.; analyze math thinking and strategies of others; use math language to express math ideas clearly;
9.Connections Use connections among math ideas; understand how math ideas build on each other to produce coherent whole; apply math thinking in contexts outside of math;
10.Representation Use representations to organize, record and communicate math ideas; apply and translate among math representations to solve problems; use representations to model math.
Grade Band Standards
All 10 Standards apply throughout from pre-K-12th grades; but some apply more in some grade bands than in others. The framework progressively forms the pupils’ math thinking.
[It is important for elementary school pupils to study math one hour every school day!]
Pre-K through 2nd Children intuitively begin to understand some math concepts (few vs. many, patterns, measurement, abstract values like two-ness, etc.) very early; but their understanding at school entry varies widely due to different early learning environments. Important to bring every pupil up to even level by second grade; BEST teachers needed in this most important band!
3rd to 5th Math must be kept interesting and understandable. Multiplicative thinking, equivalence and computational fluency are crucial, including fractions, division (with algebraic ideas introduced); this band too important to be left to “general” teachers—needs math specialists support, at least;
6th to 8th At this age pupils become students, able to reflect (think about thinking) and to deepen their understanding of math thinking, integrate algebra, geometry and a broad spectrum of math; able to deal with quantitative situations in their lives outside school; specialized math teachers;
9th through 12th[All students should take math through 12th grade!] Regardless of what students plan post 12th grade, all need to experience the interplay of thinking in algebra, geometry, statistics, probability and discrete math; they should be adept at visualization, describing and analyzing situations in math terms and to justify and prove math based ideas (not just problems);
Even with the challenges of the three American books, the Japanese may not have made such a remarkable change as in understanding mathematics as a language for teaching thinking if they did not already have the practice of Jugyokenkya (teachers mutually inspiring each other) while sharing ideas and methods for teaching thinking through mathematics. Can teachers in Jordan be convinced to allow other teachers into their classrooms, let alone seek to learn from each other?[One must also take into consideration the remarkable energy of the peoples (and teachers) of East Asia. Good teaching is extremely hard work, physically as well as mentally. Jordan should consider energy as a key qualification for recruiting teachers.]
Let us look at a larger view of how ordinary teachers can become very effective. For over a century “Normal” or Teacher Training courses have been based on the premise that all teachers can, and even should, teach by the same methods. Yet all research, and even common sense, tells us that every teacher has her or his own individual talents and skills; and each child is infinitely different from every other child. Also, each child varies from day to day. How then could one method fit all teachers, let alone all pupils, in all situations?
Teachers who try to teach as they were trained, only to follow someone else’s methods, often “burn out.” Whereas teachers who try to use methods that fit best with their abilities tend to keep on growing (i.e. teachers that are educated, and have deep culture, more than just trained) often get better, and even enjoy teaching more, each year.
What happened in Japan was that poorly paid, under-appreciated and little respected teachers learned to find joy and self-satisfaction—by teaching math as a language of thinking rather than as rote learning of formulae and premises to solve set problems. The math teachers in Japan did not get raises in pay nor recognition by the government, the public or even the parents of those they taught. Those teachers took for themselves deep satisfaction in teaching their pupils to think for themselves—hopefully to think better than they themselves could--the ultimate goal of a good teacher. They found real respect for themselves—not dependent on anyone else or any institution to confer it upon them.
How can the Japanese experience with Dewey, Polya and the NCTM’s (National Council of Teachers of Math) Principles/Standards be used in Jordan and the Arab World? What would the joy of teaching thinking, rather than the boredom of repeatedly having students memorize methods and theorems and ways to teach set problems, do for mathematics teachers in Jordan?