Visual Reprentation And Analytic Thinking In Solving Combinatorial Problems

By interviewing five students from senior Grade Three, the author of the essay made a study of the process on how high school students solve combinatorial problems. The study shows that visualization and analysis will naturally interact and influence each other in combinatorial problem-solving. For visual representation, consciously or unconsciously students are simplifying the visual representation so as to solve the problem successfully, but that is not to say, the problem is certain to be solved. Analysis can be broken down into four kinds of activities: inferring additional consequences, mathematical elaboration, imposing a new goal, monitoring statements.Though students are unconsciously making use of the combination of visualization and analysis in combinatorial problem-solving, this doesn't mean that students can solve problems successfully. There exist great differences when students apply visualization and analysis. It includes: (a) the differences in sifting those irrelevant hints; (b) the differences in transforming problem; (c) the differences in generalization of classification and fraction. Some differences determine whether they can succeed in solving combinatorial problems.In the process of combinatorial problem-solving, I find students represent various characteristics: whether they are pursuing preciseness or whether they do well in independent thinking and breaking through conventions.Finally, based on the research results, I offer some teaching implications. Firstly, seriously treat visualization in working out combinatorial problems. Secondly, tell the mistakes that students make of which kind of activities is involved in analysis. Thirdly, pay great attention to generalization of classification and fraction in combinatorial problem-solving. Fourthly, try to reduce the interferences to students from the problem circumstances.