Mathematical problem solving and learning mathematics

1. Mathematical problem solving and learning mathematics Presenter: Chun-Yi Lee Advisor: Ming-Puu Chen Nunokawa, K. (2005). Mathematical problem solving and learning mathematics: what we expect students to obtain. Journal of Mathematical Behavior, 24, 325-340.

2. What do we expect students to obtain from problem solving experiences? Type A: emphasizing the application of mathematical knowledge students have Type B: emphasizing understanding of the problem situation Type C: emphasizing new mathematical methods or ideas for making sense of the situation Type D: emphasizing management of solving processes themselves

3. Emphasizing the application of mathematical knowledge students have • What we expect our students to obtain is an experience and/or knowledge of how and when to apply mathematical knowledge they have. • Teaching for problem solving

4. Emphasizing the application of mathematical knowledge students have • Support 1: enhancing students’ schemata • the students learn these schemata by induction through solving many problems that are related to the targeted mathematical knowledge. In order to support such students’ induction, one thing we can do is to deliberately select the problem situations they encounter. • Support 2: operating situations presented to students • operating problem situations so that they can elicit students’ mathematical knowledge more easily.

5. Emphasizing understanding of the problem situation • What we expect our students to obtain is the deepened understanding of the situation and/or the notion that new information about the situation can be induced by their mathematical knowledge. • Teaching in problem solving

6. Emphasizing understanding of the problem situation • Support 1: selecting and presenting a situation so that students want to get information about it • contradiction, uncertainty, and surprise. • Support 2: providing an empirical way of validating mathematical methods • there are students who are not sure whether the mathematical method used to generate new information is reliable or not. In such cases, we need to show that the used mathematical method is reliable and the information obtained through it really deepens their understanding of the situation. • prediction • Other supports: providing tools and calling attention to an appropriate interpretation • it can be a possible support to provide students with tools which help them implement their mathematical methods and obtain more information about the situation, like graphic calculators (Osawa, 1996) or computers. • In using realistic situations, it is also an important support to direct students’ attention to an appropriate interpretation of a mathematical result (Silver & Shapiro, 1992).

7. Emphasizing new mathematical methods or ideas for making sense of the situation • What we expect our students to obtain through problem solving is a mathematical content and how that new content is related to the mathematical knowledge they already have. • Teaching via problem solving

8. Emphasizing new mathematical methods or ideas for making sense of the situation • Support 1: selecting a situation that can bridge old and new knowledge • we should select appropriate situations that can bridge between the novel ideas or methods and the mathematical knowledge students have. • Support 2: promoting a shift toward mathematical knowledge • encourage students to reflect on their own activities (De Corte et al., 1996), to shift their attentions (Streefland, 1993), and to elicit mathematical knowledge from students’ results and formulate it. • Introducing notations (Streefland, 1988) or a shared vocabulary (Brown et al., 1989) may be helpful for such a support. • Support 3: bringing in appropriate values in the classroom • we need to select the situations where students can find out the necessity or the merit of new mathematical knowledge and the limit of the mathematical knowledge they already have for dealing with those situations. (Necessity, merit, limit)

9. Emphasizing management of solving processes themselves • What we expect our students to obtain is the wisdom concerning how to treat problematic situations, manage their solving processes, and put forward their thinking. • Teaching about problem solving.

10. Emphasizing management of solving processes themselves • Support 1: providing each component of problem solving activities • One is the explicit instruction of each component of the mathematical disposition (heuristic strategies). • Another is involving students in authentic mathematical problem solving, where we direct their attention to desirable behaviors at an appropriate time and/or scaffold their solving activities. • Support 2: scaffolding students’ activities in actual problem solving • metacognitive behaviors and appropriate beliefs have been instructed in the situated-learning manner. • heuristics and asking questions

11. Critical role of selection of situations • In the Type A approach, we expect our students to enrich their schemata of the targeted mathematical knowledge. • problem situations should be selected so that they can experience enough various kinds of situations. • Everyday or realistic situations should be selected if a mathematically educated person usually applies the targeted knowledge to those kinds of situations.

12. Critical role of selection of situations • In the Type B approach, we should select problem situations where students come to want new information about it and/or which can lead to new information that is interesting to them. • Problem situations should produce interesting or unexpected results that cannot be easily found without the use of their mathematical knowledge. • problem situations should include wondrous phenomena whose mechanism can be uncovered using students’ mathematical knowledge.

13. Critical role of selection of situations • In the Type C approach, we should select problem situations that are apt to elicit informal or na¨ıve approaches from students—some of which can be formulated into the targeted mathematical knowledge. • Problems situations should show the necessity and merit of the targeted new knowledge. • The support to be provided may depend upon the cultures of our mathematics classrooms.

14. Critical role of selection of situations • In the Type D approach, the authenticity lies in the experiences of thinking in mathematically valued ways. • We should select a problem situation that students can explore and operate upon in various ways that reflect important heuristics. • While scaffolding or coaching (Brown et al., 1989) their thinking, teachers need to design our support so that students can appreciate the power of the adopted ways of thinking or the metacognitive actions in achieving gradual understanding (e.g., students can find new and important information by themselves in investigating the situation in those ways)