The Strategy Teaching Model

The STM guides the explicit teaching of number strategies (Hughes, 2002). Three ways students need to interact with new mathematical concepts are recommended. These are using materials, using imaging, and the abstract stage of using number properties to represent ideas (MoE, 2007d). The STM begins by acknowledging the existing knowledge and strategies that students bring to their learning.

The STM was influenced by, and appropriated from the Pirie-Kieren (P-K) Theory, the seminal work of Pirie and Kieren (1989, 1992, 1994) from Canada. The power of the P-K Theory is that it interprets the growth of mathematical understanding, not the understanding of mathematical growth (Martin, Towers, & Pirie, 2000). The dynamic relationship between the phases of using materials, using imaging, and using number properties is illustrated by the double-ended arrows in Figure 2.1.

Figure 2.1: The Strategy Teaching Model (MoE, 2007d, p. 5).

Using materials enables students to see and manipulate representations, equipment or diagrams. The use of materials in the STM differs from the more experiential or “hands-on” orientation where equipment was used to “keep students actively engaged” (Higgins, 2005, p. 89). In New Zealand, teachers of students in their early years of schooling have traditionally used materials to teach mathematics, and there has been an expectation that older students would

experience more book-based studies (Higgins, 2005; Hughes, 2002). The STM anticipates that students of all ages would be accessing materials, thus reflecting the sociocultural influence cultural tools and artefacts have in mediating learning.

The using imaging phase is an attempt to bridge students’ conceptual construction from materials to abstraction and assist them to make the connection between materials and generalisations or concrete and abstract cognition (Hughes, 2007). Teachers can provoke the use of imaging by moving between materials and imaging and imaging and abstracting, shielding materials from students, allowing students to see but not manipulate materials, and asking them to imagine the materials (Higgins & Parsons, 2009; Hughes & Peterson, 2003; Wright, 1991; Wright et al., 2000).

According to Hughes (2002), if children are having trouble imaging it can be assumed that manipulation of materials has not led to successful learning so the teacher should provide the materials again and fold-back to the using materials phase of the STM. Folding-back means returning to a previous phase of the STM (Pirie & Kieren, 1992). For example, if students are experiencing difficulty imaging addition problems to 10, the teacher may re-introduce materials to support students. A return to a previous phase does not indicate a return to the original activity but rather prompts a new activity stimulated and influenced by outer level knowing. By folding-back, a deeper understanding is achieved because the student has the opportunity to extend, reflect on, and reorganise their thinking before returning to the outer layer (Pirie, 2002; Pirie & Kieren, 1994).

Success at the using imaging phase indicates readiness for the final phase of using number properties. Students at the using number properties phase reason directly with the numbers, make generalisations, and do not need to use materials or imaging. Progression to using number properties is promoted by increasing the complexity or size of the numbers involved (MoE, 2007d). With larger and more complex numbers a reliance on materials or imaging becomes too onerous. At the using number properties phase students are also expected to look at the numbers they are working with and to apply the most efficient strategy for those numbers (Cobb, 2012).

Hughes (2007) identified a problem with the way some teachers were using the STM. He noted they were “reducing the model to a step-by-step set of rules that they delivered … not listening, observing, understanding, and acting in response to students’ actions and words” (p. 2). The MoE (2007d) also noted a “serious misunderstanding of the teaching model [that] should never be encouraged” (p. 6). Students should not be practising on materials, imaging, and/or number properties through teacher-provided worksheets, independent of guidance and observation from the teacher. It was never intended to be the teacher’s responsibility to lead the students through each phase of the STM and to the solution (Hughes, 2007). The teacher’s duty is to provide tasks at suitable phases and stages, observe and appropriate students' actions and discussions about the tasks, and ask questions that support students to derive their own mathematical understandings at each phase. New knowledge and strategy learning occurs when students shift from “an externalised representation to a visualised idea and then to an internalised representation” (Higgins, 2005, p. 89).

An essential component of the STM is the expectation that students are able to illustrate and articulate their strategies at each phase (MoE, 2007d). Students must be able to clearly explain their thinking before moving to the next phase. If the thinking is not clear then more experiences are required at the same or an earlier phase. There is no designated time frame in which students move through the three phases. With some concepts students may move through all three phases in one lesson or they may spend several lessons exploring thinking at any phase of the model.

Each pedagogical tool of the NDP - the Number Framework, Diagnostic Interview, and STM, emphasises the need for students to be able to illustrate and articulate their thinking and listen to others’ explanations. More important than the answer is the mathematical thinking and reflection that led to it (Anthony & Walshaw, 2009). The teacher’s guide to the Number Framework (MoE, 2007b) provides illustrations of students verbalising the strategies they could use at each stage. Opportunities for students to think, communicate, make connections, and reflect through pictures, diagrams, words, and symbols are emphasised. The Diagnostic Interview (MoE, 2007c) requires students to share what they know and explain the strategies they used on specific tasks. As students’ progress

through the phases of the STM they are expected to explain, reason, and justify their mathematical thinking when using materials, imaging, and number properties. The ideas and values of the Number Framework, Diagnostic Interview, and STM are encapsulated in the NDP teacher resource books, which are examined in the following section.