PRINCE HAMIDU ARMAH School of Education

Beyond Show and Tell to Problem Solving: Exploring the Discrepancy between Mathematics Teachers’ Problem Solving Beliefs and Practices

Fifth Asian Conference on Education, Osaka-Japan, 23-27, October, 2013

Outline of this presentation1. Introduction

2. A historical glimpse on teaching mathematics

3. Problem Solving: Current reform-oriented vision for teaching mathematics

4. Two major impediments of adopting problem solving approach

5. What are beliefs/ Problem Solving beliefs

6. Impacts of beliefs on teaching mathematics: Ernest’s (2004) Schematic Model

7. Beliefs and practices: Azjen's(1991) theoretical perspective

8. Conclusion and Possible further research

Introduction Social psychological influences play a significant role in

teachers’ pedagogical practices.

Mathematics teachers' actions and classroom decisions are driven by their views and beliefs.

To understand the nature of teaching mathematics, one must understand teachers' processes of thinking about teaching, and the theories and belief systems that drive these processes.

In this presentation, I examine the influences of teachers’ beliefs on pedagogical practices within the context of the theoretical formulations of Azjen (1991) and of Ernest (2004).

A historical glimpse on teaching mathematics

During the mid 20th century success in learning mathematics usually meant ones competence in the use of arithmetic algorithms.

1960s- the New Math movement in the USA (or modern maths in UK) whose aim was to help learners understand the structure of mathematics together with its unifying ideas, and not just as computational skill. – Pedagogical failure

1970’s -Back to Basics movement reinstated the earlier view that success in mathematics was concomitant to one’s ability to compute accurately and swiftly. - Drill and practice approach.

1980’s and 1990’s – Reform movement swung the curricular pendulum toward what is now called the development of ‘mathematical power,’ which involves investigation and problems solving.

Current reform-oriented vision for teaching mathematics In today’s world – mathematics teaching and learning

emphasis problem solving approach(OECD, 2012; MOESS, 2007).

What is a problem solving? A working definition: you are engaged in problem solving when you are trying to achieve something and you don’t know a straightforward way to do so.

A problem solving teaching approach(philosophy) in mathematics suggests significant changes in teaching practices such as:

The use of context, concrete materials, calculator graphics, tables, etc as a means to help students grasp abstract concepts.

Group work, including students sharing and justifying their ideas.

Teacher as orchestrator of classroom discourse and facilitator of learning experiences. (Herrera & Owens, 2001, p. 89)

Two major impediments of adopting problem solving approach

Interpretation of Problem SolvingProblem solving as a lesson goal, a process, a basic skills, a teaching approach, learning theory and philosophy of mathematics. Each understanding has implication for classroom practices.

Investigational work in school mathematics could be subverted by the view that there is a unique correct outcome

Implementation of Problem Solving approachAdoption of PS approach is underpinned by teachers beliefs about problem solving and its place in mathematics(Schoenfeld,1985).

Teachers more easily accept teaching innovations that are in accordance with their educational beliefs (Fang, 1996, Ertmer, 2005 ).

Any shift towards adopting problem solving approach requires not only curriculum change but also change in teachers deeply held beliefs( Ernest, 1991).

What are beliefs/ Problem Solving beliefs

(Educational) Beliefs: Personally held mental constructs, understandings, premises or propositions about the world which is

thought to be true (Wong, 2013) regardless of rational inconsistency (Roulet,1998).

Beliefs in a mathematics teaching context refers to the epistemological views of the nature of mathematics, the conception of teaching and learning as well as the principles of education that teachers hold to be true (Ernest, 1991).

This description of education beliefs also known elsewhere as ‘personal epistemological beliefs’ (Roulet, 1998; Pajares, 1992), that is, beliefs about the nature of knowledge and teaching and learning of mathematics, is the focus of this paper

Beliefs are so powerful that two teachers could have similar knowledge but adopt different teaching approaches (Ernest, 1988, Bishop, 1996).

Problem Solving beliefs: The belief that mathematics is a social construction which develops and changes. Mathematical

knowledge is a cultural product which may be adulterated with errors and that errors play substantial role in mathematics in respect of the process of conjectures, proofs and refutations.

Beliefs – Practices Relation

How does beliefs influence teachers’ practices and what account for the discrepancy between teachers’ problem solving beliefs and pedagogical practices?

Impact of Belief on Practice: Ernest’s(1991) theoretical Perspective

• The teacher’s philosophy of mathematics underpins two secondary components relating to espoused-enacted theories of teaching and learning.

• The translation of these espoused theories of teaching and learning into classroom practices, subject to the constraints and opportunities of the school/classroom context, are noted as the enacted models of teaching and learning.

• The way mathematical texts and curriculum materials in the classroom are used is a determinant the extent of the implemented curriculum.

Beliefs and practices: Ajzen's(1991) theoretical perspective

Theory of Planned Behavior(TPB): Human behaviour is predicated by the intention (I) to adopt or perform the behaviour (B). The intention on the other hand is underpinned and predicted by three factors: Attitude toward the behaviour (A), Subjective Norms (SN), and Perceived Behavioral Control (PBC)

Predicting intention-one needs to know: whether the person is in favour of doing it (attitude), how much the person feels socially pressured to do it (subjective norm) and whether the person feels in control of the action in question (perceived behavioural control).

These intentions together with perceptions of behavioural control, account for considerable discrepancy in actual behaviour (Azjen, 1991, p.179).

Conclusion “A person’s understanding of the nature of mathematics

predicates that person’s view of how teaching should take place in the classroom” (Hersh, 1986, p.13)

Belief is a function of teachers classroom actions; stimulate teachers’ conception of content knowledge and guide their instructional practices (Wong, 2013; Schoenfeld, 1999).

Changing instructional approach towards problem solving is not only a subject of curricula reform or of providing materials, but also a matter of challenging the traditional personal philosophies of teachers.

How can teachers willingness to adopt problem solving teaching approach be increased?

Within the confines of constraints and opportunities provided by context, do teachers enact pedagogical practices that are instigated by their mathematical problem solving beliefs?

As beliefs shape practices, how can teachers’ educational beliefs be incorporated in pre-service teacher education programmes?

Further research questions

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