Developing Mathematical Understanding Through a Problem

traditional (teacher centered) approach and learning through a problem-solving approach in mathematics.

Table 2.1 Contrast between a Conventional (Traditional) Approach and Learning through a Problem-solving Approach

Approach to Learners Mathematics Instruction College Conventional (Traditional) Approach A Problem Solving Approach

Teacher’s Role  Lectures

 Assigns seats

 Dispenses knowledge

 Guides and facilitates Posses challenging questions

 Helps learners share knowledge Student’s Role

 Works individually 

 Learns passively

 Forms mainly “weak” constructions

 Works in a group (collaboration & cooperation)

 Learns actively

 Forms mainly “strong” constructions

Source:Masingila, Lester & Raymond (2011, p. 11)

The ultimate goal of a problem-solving approach in teaching mathematics is to enable learners to develop understanding of concepts and procedural skills in mathematics, and thereby improve their academic achievement in mathematics.

2.6.2 Developing Mathematical Understanding Through a Problem Solving Approach

It is important that all learners understand the mathematics they learn. Knowing how to memorise and execute a procedure is not enough. To understand something, according to Grossman (1986), means to assimilate it into an

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appropriate schema (cognitive structure).This explanation means the learner is expected to use his/her existing schema (or a network of connected ideas) to give meaning to new experiences and new ideas.

Deep understanding of mathematical concepts must therefore be the goal of all learners. According to Heibert and Wearne (1993), understanding something is one of the most enjoyable and satisfying intellectual experiences one can have. For Heibert and Wearne (1993) “understanding mathematics so well that one knows how it works confers an unparalleled sense of esteem and control” (p. 4- 5). How can learners develop understanding of the mathematics that they are learning?

A problem-solving approach, according to Davis (1992), leads to understanding. Although a problem-solving approach is time consuming, learners who actively engage in it develop, extend, and enrich their understanding (Heibert & Wearne 1993). For learners to develop understanding of mathematics through a problem-solving approach, the teacher‟s (facilitator‟s) role in ensuring a balance in engaging learners in solving challenging problems, examining increasing better solution methods and providing information for learners just at the right time is crucial (Hiebert, Carpenter, Fennema, Fuson, Human, Murray Oliver & Wearne, 1997). The need for learners to have a deep understanding of mathematics calls for a teacher using appropriate instructional approach and problem solving related tasks that will arouse and sustain the interest of the learners to develop understanding of concepts, procedure skills, and ability to

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synthesis, analysis skills and evaluate competencies (the higher levels of Bloom‟s cognitive domains). Selecting quality and interesting problem solving tasks is therefore required for development of understanding in learning mathematics via a problem-solving approach.

In “Selecting Quality Task for Problem Based Teaching”, Marcus, and Fey (2003), argue that “designing activities that will keep learners busy throughout the standard class period is relatively easy, but making sure such activities lead to learning important mathematics is much more difficult” (p. 55). They further argue that “finding and adapting problem tasks that engage learners and lead them to understanding fundamental mathematical concepts and principles and to acquiring skill in the use of basic mathematical techniques is itself a challenging task for teachers” (p. 55). To ensure selection of quality tasks for a problem- approach teaching, Marcus and Fey (2003) suggest four questions that need to be answered. These questions are:

 Will working on tasks foster teachers understanding of important mathematical ideas and techniques?

 Will selected tasks be engaging and problematic yet accessible for many learners target classes?

 Will works on tasks help learners to develop their mathematical thinking?

 Will working on the task in a curriculum builds coherent understanding and connections among important mathematical topics? (p. 55-56)

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These questions suggested by Macus and Fey (2003) seem to be calling for a mathematics curriculum that is problem-based (task driven) and that will enable learners to develop understanding of mathematics through the use of a problem- solving approach.

Studying the mathematics curricula, Wu and Zhang (2006) note that international trends in mathematics curriculum development indicate an increase focus on problem solving and modeling in countries from the West as well as the East (the extent that this approach has been embraced in most African countries including Ghana is not yet known). Reflecting on curricula development in mathematics, Anderson (2007) suggests to mathematics curriculum developers to include problem-solving experiences in the mathematics curriculum. Anderson‟s convincing reasons are that problem- solving experiences will make learners be able to use and apply mathematical knowledge meaningfully, develop deeper understanding of mathematical ideas, become more engaged and enthused in lessons, and finally, learners will appreciate the relevance and usefulness of mathematics.

A good use of a problem-solving approach curricula calls for efficiently using problems in the context that make sense to the learner: “if a learner does not have a good sense of what he or she knows, he or she may find it difficult to be an efficient problem solver” (Schoenfeld, 1987, p. 190). A problem-solving skill entails more than drawing on one‟s background knowledge; instead,

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information must be effectively applied to new problem situations (Fosnot, 2005; Pirie & Kieren, 1994; Thompson, 2000).

Mathematics topics are interwoven. For example, knowledge about a procedure for adding common fractions may be needed when it comes to the addition of rational functions. It is therefore important that learners understand the topics in mathematics relationally and not by rote. This view of why it is important for learners to develop understanding in mathematics topics is supported by Heibert and Carpenter (1992). They explain that understanding a mathematics topic ensures that everything one knows about the topic will be useful. When solving mathematical problems, learners develop a deeper understanding of mathematics because it helps them to conceptualise the mathematics being learnt (Schoenfeld, 1992). To sum it up, for a student to develop understanding of mathematics through a problem-solving approach, the teacher‟s role of selecting quality and interesting problem-solving task is important. Also the mathematics syllabus, which mostly drives the teaching and learning of mathematics, should contain quality and interesting problem-solving activities. The student should also be prepared to learning mathematics actively rather than passively by doing, recording, and communicating mathematics.

In UCC-CCE, distance learners like most pre-service teachers learners are most often engaged in routine mathematical activities and are not exposed to meaningful problem-solving tasks (Boaler, 1998). Such an approach to teaching

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fails to promote critical thinking skills (Brown-Lopez & Lopez, 2009). Therefore, if learners are not exposed to effective problem-solving skills, through a problem-solving approach, they are unlikely to demonstrate understanding of concepts and procedural skills in the subject content knowledge. This study will determine if through the use of a problem-solving approach, UCC-CCE distance learners‟ understanding of mathematical concepts and critical thinking skills may improve (Hines, 2008) and thereby may improve their achievement in mathematics.

2.6.3 Impact of a Problem-Solving Approach on Achievement of Learners