Mathematical Problem Solving
Erwin Segal
Mathematical problem solving is an important domain to study. We have all had math courses of one kind or another. However, a large number of Americans do not appreciate mathematics. I do not think the situation, in general, is as dire in other cultures. People state outright, "I don't like math," or "I am bad in math," or they may make some other deprecating remark about mathematics. When this occurs, as is the case about making racist remarks in a racist society, the is likely to be a quick consensus from the hearers of the remark, that the ideas in the statement are shared. What a pity!
I do not know why the state of mathematical learning and problem solving has such negative feelings, nor why so many people are 'mathophobes'. I also do not believe that a large percentage of the American population are innately incapable of "doing" math. I do think that much of what passes for math education is not conducive to having people learn and to understand mathematical principles and concepts. Part of this is directly due to the mathematical community itself, and part of it is due to the fact that many of the teachers of math have become phobic and pass their phobia on to their students.
I believe that mathematical problem solving is an excellent model for representing schema theory, for illustrating the development of expertise, and for showing the value of scaffolding by the teacher or the textbook in the manner suggested by Vygotsky and the Zone of Proximal Development. It is an ideal domain for illustrating the value of analogic reasoning and the role of context. Furthermore, math is an ideal domain for showing the role of extending thought outside the brain and the use of sketches and diagrams to help solve problems. In addition to the above, mathematical problem soving is an ideal domain to illustrate the need of interpreting textual material (e.g., verbal problems) on some schema in order to see the relations between the parts and wholes so that the probems can first be understood, and then hopefully, solved.
In order for one to solve mathematical problems described in story form, one certainly needs some knowledge of the language, but I believe that problem solving difficulty is more often due to more mathematically oriented issues. One needs to know how to figure out what the problem is what is asked, one needs to know what the structure, or underlying schema, of the domain is, one needs to have some plan of action (either general problem solving strategy, or specific strategy based on the problem, and one needs to have the particular skills needed to apply the strategy.
Mathematical problem solving is a domain that requires the development of expertise, as does almost all other domains. Too many mathematicians and teachers, including to a great extent Mayer, seem to think that it is simply the application of some specific rules or strategies that one can learn on the spot. The most important underlying issue is to have some idea of the nature of mathematical schemata. Without the schema, solving problems becomes memorization and blind application of rules. This then must be followed by some strategic methods of using those schemata to solve the problems asked.
Elementary Mathematical Schemata
1. Numbers and the Arabic Numeral System. Ordinal and Cardinal Numbers
2. Geometric representations.
3. Linear, square and cubic units; other units
4. Areas
5. Searching for schemata in word problems.
6. Representing algebaic ideas geometrically and vice versa.
7. Proportional relations.
Mathematical problem solving is an important domain to study. We have all had math courses of one kind or another. However, a large number of Americans do not appreciate mathematics. I do not think the situation, in general, is as dire in other cultures. People state outright, "I don't like math," or "I am bad in math," or they may make some other deprecating remark about mathematics. When this occurs, as is the case about making racist remarks in a racist society, the is likely to be a quick consensus from the hearers of the remark, that the ideas in the statement are shared. What a pity!
I do not know why the state of mathematical learning and problem solving has such negative feelings, nor why so many people are 'mathophobes'. I also do not believe that a large percentage of the American population are innately incapable of "doing" math. I do think that much of what passes for math education is not conducive to having people learn and to understand mathematical principles and concepts. Part of this is directly due to the mathematical community itself, and part of it is due to the fact that many of the teachers of math have become phobic and pass their phobia on to their students.
I believe that mathematical problem solving is an excellent model for representing schema theory, for illustrating the development of expertise, and for showing the value of scaffolding by the teacher or the textbook in the manner suggested by Vygotsky and the Zone of Proximal Development. It is an ideal domain for illustrating the value of analogic reasoning and the role of context. Furthermore, math is an ideal domain for showing the role of extending thought outside the brain and the use of sketches and diagrams to help solve problems. In addition to the above, mathematical problem soving is an ideal domain to illustrate the need of interpreting textual material (e.g., verbal problems) on some schema in order to see the relations between the parts and wholes so that the probems can first be understood, and then hopefully, solved.
In order for one to solve mathematical problems described in story form, one certainly needs some knowledge of the language, but I believe that problem solving difficulty is more often due to more mathematically oriented issues. One needs to know how to figure out what the problem is what is asked, one needs to know what the structure, or underlying schema, of the domain is, one needs to have some plan of action (either general problem solving strategy, or specific strategy based on the problem, and one needs to have the particular skills needed to apply the strategy.
Mathematical problem solving is a domain that requires the development of expertise, as does almost all other domains. Too many mathematicians and teachers, including to a great extent Mayer, seem to think that it is simply the application of some specific rules or strategies that one can learn on the spot. The most important underlying issue is to have some idea of the nature of mathematical schemata. Without the schema, solving problems becomes memorization and blind application of rules. This then must be followed by some strategic methods of using those schemata to solve the problems asked.
Elementary Mathematical Schemata
1. Numbers and the Arabic Numeral System. Ordinal and Cardinal Numbers
2. Geometric representations.
3. Linear, square and cubic units; other units
4. Areas
5. Searching for schemata in word problems.
6. Representing algebaic ideas geometrically and vice versa.
7. Proportional relations.