1. A Problem Solving Approach to the teaching of Mathematics at the Secondary level Developing Habits of Mind Judith Sedi Davion Leslie
2. Objectives To explore the features of problems To establish a framework for developing problem solving skills in students. To explore the benefits of adopting a problem solving approach to teaching math.
3. Thinking-based curriculum Have you ever met students who can perform operations and algorithms but are unaware of what they are doing? slavishly follow algorithms regardless of what they are doing? cannot respond to context based questions – even though they can perform the operations implied in the questions? require an example before they can ‘solve a problem’ are not able to try different approaches in order to arrive at a correct answer?
4. Thinking based curriculum What is the thinking based curriculum? How can it address some of the problems mentioned before? What is the true purpose of teaching math in school? Does what exists now in schools qualify as the thinking based curriculum?
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6. “Problems that are truly problematic do not have clear or single solution paths” (Schoenfeld, 1985, p. 34). “Computational exercises for which students do not have a readily-accessible method or approach can be truly problematic” (Yackel, Cobb and Wood, 1988, p. 87). Problem solving
7. Problems are not to be seen as traditional word problems. Traditional word problems “provide contexts for using particular formulas or algorithms but do not offer opportunities for true problem solving” (NCTM, 1989, p. 76). Problems are, therefore, NOT contextualised algorithms According to NCTM
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9. It allows the student the opportunity to express their understanding of the concept
13. Developing habits of mind Introductory problem A tournament is being arranged among 22 teams. The competition will be on a league basis, where every team will play each other twice – once at home and once away. The organizer wants to know how many matches will be involved.
14. Habits of mind At first the problem appears difficult No known algorithms exist and students may not have an example of a similar problem. Students may not know how to approach the problem. How would you intervene at this stage? What would you say? Would you give an example? Would you model an approach? Would you do anything at all?
15. Habits of mind How do we start? How about simplifying the problem? Suppose instead of 22 teams, there were only 4? But still, how do we start? Well figure out a system for recording how many matches 4 teams will play.
22. Habits of mind By now you should realise that 4 teams will play 12 matches. Does this mean that the number of matches will be 3 times the number of teams? Will 22 teams play 66 matches? Perhaps we should try a few more cases to see. Which cases would you try and why?
23. Habits of mind You now know how many matches 3 – 6 teams will play. It’s perhaps best to make a table to capture your findings.
24. Habits of mind Now that you have a table, look for patterns. Write down the patterns that your are observing Look for horizontal (side to side) as well as vertical (top down) patterns Look for Differences (1st and 2nd) Relationships Rules Patterns such as symmetry, odd-even, number types, etc.
25. Habits of mind Use your patterns to solve the original problems with 22 teams. Which patterns/relationships are more helpful – vertical or horizontal? Can you find a general rule that tells you the relationship between the number of teams and the number of matches? Can you make it into an algebraic expression?
26. Habits of mind Try some simple cases Find a helpful diagram/creating models Organise systematically Examine results (make a table, etc) Spot patterns Explore/use the patterns Find a general rule
27. Some final thoughts The problem was stated with 22 teams; was this number too small, too large or challenging enough? What information in the question would require some explanations or background information? Is this best done as group work or as individual work? What could we change about the question to create an extension/variation?
28. Habits of mind Now, in your groups, attempt problems 1 – 3 on the activity sheet.
