Current questions

Brophy (2006) states that “social constructivist educators usually have much more to say about learning than about teaching” (p. 530). It is this gap in the literature that

interests me. It is all very well to ponder on whether the curriculum comes from a constructivist or a social constructivist epistemology, but the real issue affecting teachers is how to teach in a constructivist-compatible approach whatever the epistemological variations. Such variations have been discussed in chapter 2. The strength of the literature to date is that there has been a strong focus on how children learn. The weakness is that there has been little focus on how teachers should teach

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when they wish to adhere to a constructivist approach to teaching. Brophy (2006) ‘hits the nail on the head’ when he writes that “it is unrealistic to educate teachers to

implement social constructivist principles without systematizing them into operational models of teaching” (p. 530). When I conducted a “deep see” trawl of the

literature I was intrigued to find several authors who have tried to address the issue of constructivist-compatible teaching as opposed to learning. I now wish to give a brief summary of the work of such pragmatic authors.

Brophy (2006) outlines Graham Nuthall’s seven principles for effective

implementation of social constructivist teaching (See Appendix 1). What is interesting about his views is that they include aspects such as “ensure frequent repetition” and “repeat critical content” (p. 533). The interesting point here is that

Nuthall does not see teaching as transmission as being mutually exclusive to a social constructivist approach.

Gagnon Jr. and Collay (2001) spoke of a constructivist learning design (CLD) “composed of six basic parts flowing back and forth into one another in the actual

operation of classroom learning: situation, groupings, bridge, questions, exhibit and reflections” (p.xi). A more comprehensive outline is given as Appendix 2. The authors draw attention to the surface activation of students’ prior knowledge before

introducing them to new subject matter. They also stress the importance of the teacher providing questions, which instigate, inspire and integrate students’ thinking and sharing of information. Teachers’ questions usually fall into open or closed

categories. This can be a useful indicator of whether pupils are being allowed to construct their own knowledge or being funnelled into the teacher’s set knowledge.

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Furthermore, Gagnon Jr. and Collay (2001) highlight the use of groupwork more than Nuthall does (Brophy 2006).

A third author who grappled with what a constructivist approach to teaching might entail is Jaworski. Her work was to have a profound effect on how I analysed my own classroom observations. In the classroom study, which formed the bulk of her PhD thesis, she found it useful to analyse teachers’ modus operandi in terms of what she called “the Teaching Triad” (1996, p. 107). The three domains of the Teaching Triad were management of learning, sensitivity to students and mathematical challenge:

Management of learning is manifested in a set of teaching strategies and beliefs about teaching which influence the prevailing classroom atmosphere and the way in which lessons are conducted. Sensitivity to students in inherent in the teacher- student relationship and the teacher’s knowledge of individual students and influences the way in which the teacher interacts with, and challenges, students. Mathematical challenge arises from the teacher’s own epistemological standpoint and the way in which she offers mathematics to her students depending on their individual needs and levels of progress.

(Jaworski,1996, p. 108)

Jaworski has advanced my thinking in that she gives some guidelines as to what to look for in constructivist classrooms. It is useful to link her Teaching Triad categories with the advice given by Gagnon Jr. and Collay (2001). The following questions come to mind as regards observation of teachers:

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1. Does the teacher manage learning in such a way that the classroom atmosphere is conducive to optimum learning? In this regard Gagnon Jr. and Collay (2001) would suggest the use of groupwork.

2. Is the teacher sensitive to pupils’ needs? Gagnon and Collay (2001) would suggest helping the pupils to build mental bridges to enable them link their prior knowledge with the new subject matter.

3. Is there challenge for the pupils mentally in the work undertaken? It is here that Gagnon and Collay (2001) suggest that questions need to be inspiring for pupils and helpful to them in integrating their thinking.

A fourth set of authors endeavouring to shed some light on constructivist teaching approaches is Simon and Schifter (1991). They engaged in the Educational Leaders in Mathematics (ELM) Project (1989), which was conducted by the Summer Math for Teachers Programme at Mount Holyoke College, Massachusetts. The first aim of the project was “to create an innovative service program for precollege teachers of mathematics” (p. 309). A second aim was to study the effects of this program on teachers’ thinking and practice. What is interesting, from a research methodological viewpoint, is how the authors “evaluate teachers’ implementation of instructional

strategies learned in ELM and their use of a constructivist view of learning as a basis for their instructional decisions” (p. 323).

To assess implementation of strategies, ELM adapted the Levels of Use (LoU) measure, developed by Hall et al. (1975), which consists of a structured interview and a five level classification scheme for rating teachers’ responses. The five levels

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were named: non-use, mechanical use, routine, refinement and integration. What is more interesting, from my viewpoint, is that ELM developed a new instrument, the Assessment of Constructivism in Mathematics Instruction (ACMI), which has a parallel format to the LoU. ACMI data were obtained during the same interview session and were rated according to the classification scheme shown as Appendix 3. Although the classification scheme is quite general, it does show that in following a constructivist epistemology a shift is required, whereby the focus moves from how the teacher is behaving in the classroom to how the pupil is learning. At the highest level (level V) collaboration among teachers is advocated as a way of advancing learning. This is certainly a challenge for me in this study as I seek to investigate teachers moving towards adopting constructivist-compatible pedagogies; firstly as individuals, and then as part of a community of practice.

LoU ratings were based on nine strategies, which were modelled during ELM instruction. The strategies are included as Appendix 4. As could be expected Simon and Schifter (1991) found that “changes in teaching strategies were more easily and more rapidly made than changes in teachers’ views of learning with its concomitant effect on instruction” (p. 327). In other words, it is easier to bring about instructional

shifts than philosophical ones. This is allied to the proviso that Fosnot (2005) brings to the discussion. She argues that “reform-based pedagogical strategies can be used without the desired learning necessarily resulting” (p. 279). This is because

constructivism is a theory of learning, not a theory of teaching, and many educators who attempt to use such pedagogical strategies confuse discovery learning and “hands-on” approaches with constructivism. For instance, children may be observed

engaging with a mathematical problem using manipulatives. However, this does not necessarily mean that they are operating at their zone of proximal development.

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Holt-Reynolds (2000) gives another illustration of this misconception when she describes a prospective teacher named Taylor. In her classroom Taylor used active learning methods such as encouraging pupils to offer their opinions during English literature class. However, such opinions were not challenged by the teacher or other students. Taylor made the mistake of equating participation with learning. She needed to “see constructivist pedagogies as techniques for teaching, not merely as strategies for activating kids” (Holt-Reynolds (2000, p. 30). In other words,

activation is an essential aspect of extrinsic motivation but it does not follow that activation will ensure pupils are cognitively challenged. Such challenge is at the heart of reform mathematics which seeks to move pupils beyond the mundane problems inherent in school mathematics textbooks. The primary curriculum and Project Maths programmes aspire to a ‘minds-on’ and not just a ‘hands-on’ approach

to mathematics. Gagnon and Collay (2001) use the phrase ‘mental bridges’ to refer to the linking of prior knowledge with new subject matter. However, this needs to be an ongoing process so that pupils are constantly reinventing and reinterpreting their knowledge. Certainly, the use of sociocultural tools, like calculators, computers and the internet have a role to play in helping pupils to expand their ‘mathematical horizons’, to quote Ball’s (1993) phrase. It can be seen that I am trying to weave several authors’ writings into a constructivist framework which would enable me look at current practitioners’ classroom teaching in terms of its relevance or

otherwise to constructivist theory.

3.14 Summary

In summary, this chapter commenced with an overview of constructivist research on the teaching of mathematics internationally and in Ireland from a problem solving perspective. I looked at Piagetian and Vygotskian epistemologies in the old and revised primary mathematics curricula respectively. I outlined the serious lack of

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guidance in the curriculum documents for teachers on how to implement and assess constructivist pedagogies. The contribution of several authors to such implementation has been considered. These authors include Brophy (2006), Gagnon Jr. and Collay (2001), Jaworski (1996b), Simon and Schifter (1991) and Fosnot (2005). In the next chapter I focus on the research design and methods required to investigate constructivist approaches in the classroom.

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Chapter 4: Choosing the research