Matematik problem solving and leaning mathematics PDF

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The general purpose of this dissertation is to define and explore what
mathematical problem solving entails. Seven criteria for rich problems will also
be formulated. Rich problems are defined as problems which are especially
constructed for mathematics education in a school con...

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Journal of Mathematical Behavior 24 (2005) 325–340

Mathematical problem solving and learning mathematics: What we expect students to obtain Kazuhiko Nunokawa ∗ Department of Learning Support, Joetsu University of Education, Joetsu 943-8512, Japan Available online 22 September 2005

Abstract The purpose of this paper is to re-examine the relationships between mathematical problem solving and learning mathematics. After introducing a diagram representing the notion of problem solving, four types of problem solving approaches used in mathematics classrooms will be distinguished according to which aspect of that diagram is attended to. In examining each type of problem solving approach, what is expected to be obtained by students and what can be possible support in that approach will be discussed on the basis of the research literature. Through such examination, it will be shown that a teacher’s choice of problem situations and ways of interventions is critical to enabling students to experience ‘authentic’ problem solving and the choice should reflect the teacher’s intention. In the last part, problem solving as a way to treat mathematical theories will be discussed and the limitation of problem solving approach raised by the issue of mathematical culture will be made explicit. © 2005 Elsevier Inc. All rights reserved. Keywords: Mathematical problem solving; Learning; Teaching; Problem situation; Mathematical knowledge; Mathematical culture

1. Introduction According to Lester and Kehle (2003), there is a “fruitful blurring of problem solving and other mathematical activity emerging from research on mathematical problem solving and constructivist thinking about learning” (pp. 515–516). They insisted that this blurring could lead to “a more authentic view of students’ cognitions as they exist in busy classrooms and in complex realistic settings” (p. 517). While we need to take students’ problem solving or thinking as such a complex activity, we might also need teaching opportunities where students can clearly experience a certain aspect of mathematical problem solving. When we intentionally incorporate problem solving activities into students’ learning of mathematics, it seems important for us to realize the relationship between the problem solving to be incorporated and the students’ learning to be realized through that problem solving, in order that students can experience what we expect them to experience. The purpose of this paper is to re-examine the relationships between mathematical problem solving and learning mathematics, referring to the body of research in this field. Specifically, the following three issues, which were suggested by one of the guest editors, will be examined in the rest of this paper.

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(a) What sort of learning do we expect students to obtain from problem solving experiences? (b) In order to maximize this learning, what kind of experiences should the students be exposed to, and with what pedagogical support? (c) Should this learning be aimed merely towards enhancing abilities in general problem solving, or should we use the problem solving perspective as a way to also treat mathematical theories? Here, as far as dealing with elementary and secondary school education, a mathematical theory can be interpreted as a systematic unity of mathematical contents to be learned, which is usually embodied by a unit or a series of related units. For example, the unit of multiplication of fractions can be seen a theory in this sense, since it consists of interrelated contents about multiplication of fractions, including what the multiplication of fractions means, how it can be represented on a number line, how to calculate it and what is the base of that calculation method, how the cancellation can be done in calculating it and how it can be validated, how it is related to the multiplication of whole numbers, and what kinds of situations it can apply to. Of course, this unit can also be considered a sub-theory of the theory of rational numbers. In the rest of this paper, after introducing a diagram representing a (simplified) conception of mathematical problem solving, I will treat question (a) and (b) in Sections 3 and 4 using that diagram. Extending a part of the discussion of Section 3, question (c) will be discussed in Section 5. 2. Various phases in mathematical problem solving To start discussion about the above question (a) and (b), let us consider a problem solving situation that can be found in textbooks. 2.1. Example 1 This example is the introduction problem of multiplication of two two-digit numbers in a Japanese 3rd grade textbook. The students have already learned the algorithm of multiplication of three-digit number and one-digit number (e.g., 128 × 8) and multiplication of multiples of 10 (e.g., 4 × 30 or 40 × 30). Problem. We will make small towers using plastic blocks. We will hand out 23 blocks to each child. How many blocks do we need if there are 12 children? One of the ways students can solve this is by adding the number of each child’s blocks (i.e., 23) 12 times, or by drawing the situation consisting of the blocks (Fig. 1) and counting the total number. In this case, the students can apply what they know to this problem situation. Some students may notice that they can use multiplication for this problem situation because there are some sets of the same number blocks. In this case, however, they cannot apply

Fig. 1. The original problem situation.

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Fig. 2. Transformed problem situation (decomposition of 12).

what they know directly because they have not yet learned how to multiply two-digit numbers by two-digit numbers. They need to explore the situation to find a contact point between their mathematical knowledge and the situation they are confronted with. If the students partition 12 children into 10 children and the rest (Fig. 2),1 the number of blocks for those 10 children can be calculated by 23 × 10. The remaining 2 children’s blocks can be calculated easily, 23 × 2. Both of these two multiplications are included in the mathematical knowledge the students already have. The total number of blocks can be figured out by adding these two products. There are also some children using other ways of decomposing 12 (e.g., 6 + 6) or transforming the problem situation. Through such reasoning, the students obtain the new information about this situation that the total number is 276. What sort of learning do we expect our students to obtain from the above problem solving experience? If we emphasize the fact that the students’ mathematical knowledge on multiplication of two- and one-digit numbers can be applied to this situation after it was transformed somehow, the students would learn that their knowledge can be used in a situation even if it seems different from the situations they encountered in learning that knowledge. In other words, the student would enrich their schemata of that knowledge. If we take over one of the students’ ideas (i.e., decomposing 12 into 10 and 2) and crystallize it into the vertical form of multiplication of two two-digit numbers, what we expect them to learn here is the vertical form of multiplication and the relationship between that vertical form and the mathematical knowledge they learned before. This examination of the above example shows that an answer to question (a) varies depending on our intention in using a particular problem solving experience at a particular teaching/learning situation. For investigating some types of learning we expect our students to experience through mathematical problem solving, here I would like to introduce one conception of mathematical problem solving and a diagram representing it. In this manuscript, it is assumed that mathematical problem solving is a thinking process as follows (see Nunokawa, 1994b, 1996, 2000): Mathematical problem solving is a thinking process in which a solver tries to make sense of a problem situation using mathematical knowledge she/he has and attempts to obtain new information about that situation till she/he can “resolve the tension or ambiguity” (Lester & Kehle, 2003). This image can be represented as in Fig. 3. In mathematical problem solving, since the mathematical knowledge a solver has cannot be directly applied to the situation at hand, the solver needs to transform the situation or find new perspectives on it so that her/his mathematical knowledge can be applied to it (Fig. 4). Basically, mathematical problem solving has two phases, trying to apply their mathematical knowledge through exploring the problem situation (1) and obtaining pieces of information about the problem situation (2). If the tension or ambiguity is not fully resolved, further exploration of the problem situation continues taking advantage of the obtained information (3). When the solver has identified some mathematical entities in the situation and finds new information about those mathematical entities, the 1

In Japan, the situation of Example 1 is usually translated into the number expression 23 × 12, instead of 12 × 23.

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Fig. 3. The conception of mathematical problem solving.

obtained information will sometimes be incorporated into the students’ mathematical knowledge (3′ ), as, for example, a theorem or a formula. In some cases, the reflection of used methods or the lack of effective methods may lead to constructing new mathematical methods or ideas (4). 3. Students’ learning and teachers’ support in four types of problem solving approach In this section, four types of problem solving approach in teaching mathematics will be distinguished depending on which phase in mathematical problem solving we intend to emphasize. Referring to the related research, what we can expect our students to learn and what kinds of support are available will be discussed for each type. 3.1. Type A: emphasizing the application of mathematical knowledge students have This is related mainly to the process (1) in Fig. 3 and corresponds to “teaching for problem solving” in Schroeder and Lester (1989). According to these authors, “the teacher concentrates on ways in which the mathematics being taught can be applied in the solution of both routine and nonroutine problems” (p. 32). Consequently, “students are given many instances of the mathematical concepts and structures they are studying and many opportunities to apply that mathematics in solving problems” (p. 32). Thus, when we adopt this type of problem solving approach, what we expect

Fig. 4. Matching a situation to mathematical knowledge.

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our students to obtain is an experience and/or knowledge of how and when to apply mathematical knowledge they have. This may be considered as enhancing students’ competencies of “recognizing” action (Hershkowitz, Schwarz, and Dreyfus, 2001). The diagram shown in Fig. 3 suggests that there can be two types of possible support we can adopt for students to experience phase (1) effectively: one is enhancing students’ mathematical knowledge so that it can be applied more easily and another is operating problem situations so that they can elicit students’ mathematical knowledge more easily. 3.1.1. Support 1: enhancing students’ schemata The former type of support can be found in the research and instruction based on the schema theory. It may be assumed here that students’ knowledge base should include not only conceptual and procedural knowledge, but also knowledge about typical situations that the targeted mathematical knowledge can be applied to. Such knowledge base or structure has been called a schema. For example, Owen and Sweller (1985) defined a schema as “a cognitive structure allowing a problem solver to categorize a problem and then to indicate the most appropriate moves for problems of that class” (p. 273). The teachers and researchers adopting the idea of schemata think that “problem-solving expertise is dependent on the acquisition of domain-specific schemata” and “problem difficulty is heavily related to the number of schemata that must be acquired” (Owen & Sweller, 1985, p. 274; see also De Corte, Greer, & Verschaffel, 1996, Greeno & Kintsch, 1985). They expect students to learn a rich body of schemata related to the mathematical knowledge from their problem solving experiences, so that they can solve as many various problems as possible using that knowledge. Usually, 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. For this deliberate selection, it may be necessary for us to categorize the situations where the targeted mathematical knowledge will be used (Greer, 1992; Marshall, Pribe, & Smith, 1987; Riley, Greeno, & Heller, 1983; Verschaffel & De Corte, 1997). Greer (1992) recommended that “students need to learn about a much wider range of situations modeled by the operations.” Reed (1999) recommended that “greater time should be spent in showing students how problems with different story content may share a common solution” (p. 115). He also pointed out that even when providing such support, “(e)ncouraging schema abstraction by requiring students to compare quantities and relations across isomorphic problems has been successful in helping them categorize word problems by common solutions,” but “less successful in enhancing their ability to use an example solution to solve an isomorphic problem” (pp. 114–115). A lack of “superordinate concepts that could describe the abstraction” (p. 115) may be one reason for this less success. Some researchers developed more direct instructional approaches that were designed to help students acquire schemata. The approaches introduced in Greeno (1987) adopted kinds of outer representations reflecting schemata and used those representations to instruct students. Owen and Sweller (1985) recommended providing problems with the specificity of their goal reduced. They changed, for example, the question “find the length of BD” into “find the lengths of all unknown sides.” They showed that working with goal-modified problems could diminish the use of conventional means-ends analysis and facilitate the acquisition of schemata. It is a way of enhancing students’ schemata to mitigate students’ feeling that their mathematical knowledge is unrelated to other situations than mathematical ones. Since students tend to demonstrate a great resistance to applying their mathematical knowledge to physics, Woolnough (2000) presented a program designed to help students build effective links between the mathematical equations used to solve problems in mechanics and the real world of moving objects. 3.1.2. Support 2: operating situations presented to students Another possible support we can adopt for students to experience phase (1) effectively is operating problem situations so that they can elicit students’ mathematical knowledge more easily. Hudson (1983) used the situation where birds tried to catch worms when he posed word problems about comparisons. He showed that adopting such situation raised the performances of students. This might occur because this situation could elicit the students’ knowledge needed to solve the problems, like one-to-one correspondence. De Corte, Verschaffel, and Win (1985) transformed problems into reworded ones so that students could comprehend the situation more easily. They added, for example, the sentence “the rest belong to Ann,” which highlighted the relationship between quantities in the problem and could elicit the appropriate schema more easily (see also Briars and Larkin, 1984).

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Using more realistic situations seems to make it easier for students to experience phase (1). Bottge, Heinrichs, Chan, and Serlin (2001) presented problem situations about running cars in more realistic fashion by using videos and experimental settings. After participating in their program, the students became able to apply their knowledge to the problem situations rather easily. The students understood “the nature of the problem almost immediately because they could see it in the video and in technology education classroom” (p. 311). Bottge et al. (2001) also pointed out that the students grew to understand the essential relationship in the problem situations “with repeated practice in a motivating activity” (p. 312). Using realistic situations is also expected to reduce math anxiety (Archambeault, 1993) and their familiarity would invite students to approaching and exploring those situations with, for example, everyday knowledge. 3.2. Type B: emphasizing understanding of the problem situation This type is related mainly with the process (2) in Fig. 3. What is important in this type is deepening students’ understanding of the situations that they are exploring using their mathematical knowledge. When a learner wants to know the total number of blocks in the situation of Example 1 for a certain purpose, this total number can be really important information for him and he experiences that he obtained new information using his mathematical knowledge. If a teacher adopts a problem solving approach in order that her students can appreciate an interesting proposition and the reason why it holds in that situation, her use of problem solving approach can be included in this type because such a proposition is the information about a certain kind of situations. Proofs of the propositions, especially “proofs that explain” (Hanna, 1995), show the mechanisms of the situation (Nunokawa & Fukuzawa, 2002) and which features of the situation make the properties possible. That is, proofs themselves comprise deeper understanding of the situation.2 In this type of problem solving approach, 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. This might be related to “building-with” action in Hershkowitz et al. (2001), in that students are expected to combine available mathematical knowledge to “build with it a viable solution to the problem at hand” (p. 215). However, it is important here that students feel that they can deepen their understanding of the situation using their mathematical knowledge, while such feeling is not required in Type A. Through such feeling, we also expect our students to obtain the belief in the power of mathematics that the mathematical knowledge they have can produce new useful or interesting information and uncover the mechanisms of the problem situations. Mathematical modeling of realistic situations is (or should be) included in this type, if learners have enough mathematical knowledge for making sense of the situation to be explored. Matsumiya and Yanagimoto (1995) designed mathematics lessons for 9th graders where the teacher asked the students to find out the distances, which could be seen from the skyscraper in their city and from Mt. Fuji, the highest mountain in Japan. This situation can be modeled by the Pythagorean Theorem (see Nunokawa, 2001b). After these lessons, some students wrote their impression like this: “I was very surprised because I noticed that we could see further than I had expected.” The students seem impressed with the unexpected information about the situation that was generated by their mathematical knowledge. This impression will strengthen their belief in the power of mathematics. 3.2.1. Support 1: selecting and presenting a situation so that students want to...