This section discusses internal representations and specifically visualisation. A widely used reading comprehension strategy, and a part of many problem solving and teaching models
(Sections 2.1 and 2.2), there appears to be less research on visualisation, especially in
mathematics, and there are calls for more exploration into how it can be more effectively taught and used in mathematics pedagogy.
Visualising, imaging, or mental imagery is the ability to create, use, interpret, and reflect upon a picture, image, or diagram in one’s head (Arcavi, 2003) and is identified as an effective strategy for increasing comprehension (De Koning & van der Schoot, 2013; Ministry of
Education, 2006). Sadoski and Paivio (2013) describe its importance as a strategy for developing students’ literacy: “The research literature can be summarised conclusively; spontaneously-
26 occurring mental imagery is a natural and important part of literacy, and educating students in the strategic use of mental imagery is a successful practice in various aspects of literacy learning” (p. 115).
To be able to produce a representation learners need to be able to read the information and access their prior knowledge related to the context, firstly ‘seeing’ what they know in their head, that is to visualise what they understand the content to be. Creating visual images in their mind enables readers to make links to their prior knowledge and experiences, and to have deeper comprehension of what they read including at an inferential level (Davis, 2007). Sadoski and Paivio (2013) add to this when they state “Without the activation of mental representations, no meaning can be present” (p. 50).
When summarising the literature on internal visualisations in reading, De Koning and van der Schoot (2013) surmised that students who are trained in the use of mental imagery improve their reading comprehension when compared to students who do not receive any training in a comprehension strategy, though when compared with other reading comprehension strategies the results are less convincing. De Koning and van der Schoot (2013) surmised that creating a mental representation of what is described in a text helps students to better understand the text, and when instruction in how to create mental representations is included the chances are
increased that an accurate mental representation can be constructed. The authors also concluded that even though either an external or an internal representation can improve comprehension, the end goal is the automatic construction of internal mental representations. The authors noted that internal and external representations have usually been used and tested separately and a
hypothesis from the authors was that the two forms could have complementary roles and this was an area that required further investigation.
27 Creating visual images is an important part of literacy and especially reading
comprehension; therefore, it seems logical that this step should be useful in the comprehension and solving of mathematics word problems. With comprehending word problems identified as a barrier for many students (Fuchs & Fuchs, 2007; Hegarty et al., 1995; Kajimes et al., 2010; Kenney, Hancewicz, Heuer, Metsisto, & Tuttle, 2005; Sousa, 2011), then the use of a reading strategy to assist students when solving mathematics word problems merits investigation. Visualising, whether it be internal or external, is an effective strategy for improving a reader’s comprehension of a text (Davis, 2007; De Koning & van der Schoot, 2013; Frey, Fisher, & Berkin, 2009; Graham, 1990; Ministry of Education, 2006; Park, 2012; Zimmerman & Hutchins, 2003) and therefore it is worth investigating if visualising can help students solve mathematics word problems.
Representations are a way of showing mathematics concepts, be it a graph, table, equation, diagram, or drawing and to be able to produce a representation students need to understand what the problem is. An external representation (such as a drawing) can be produced if first an
internal, non-verbal representation, a mental image, can be produced (Van Meter et al., 2006). Creating external representations enables students to share what their understanding is and to discuss what they have produced (Goldin & Kaput, 1996) as well allowing others to ‘see’ their thinking (Diezmann & McKosker, 2011).
Visualising, or using imaging, is an important part of the Strategy Teaching Model in the Numeracy Project (Ministry of Education, 2008) (section 2.1). This teaching strategy places using imaging as a crucial transitory stage when moving children from using manipulatives to using number properties. Being able to produce representations of their mental images is also identified as a way that students can show a deep understanding of a concept. Sousa (2008)
28 learning concepts. The pictorial stage involves providing or creating visual representations that will help students to visualise mathematical operations during problem solving.
As noted earlier, low achievers in mathematics tend to have poorly developed visual memory and visualisation skills (Mulligan & Mitchelmore, 2009, as cited in Mulligan, 2011). Without the ability to visualise the problems and mathematics content involved, students are going to struggle to solve word problems.
In their summary of the literature around visualisation and mathematics learning, Booth and Thomas (2000) found that results on the effectiveness of visualisation were inconsistent and there were opposing views on the effectiveness of the strategy. They surmised that mental imagery can be valuable as part of a process but if there is too much reliance on visualisation it can affect performance. In a review of papers presented at the Annual Conference of the International Group of the Psychology of Mathematics Education, Presmeg’s (2006) final
recommendations included the need to find what makes mental imagery effective in mathematics and to investigate effective pedagogy utilising visualisation in mathematics education. Similarly, Arcavi (2003) concluded his report on the role of visual representation in mathematics with comments that visualisation is a central issue in mathematics education, and though warning about thinking visualisation will be a cure all, he stresses the need to better understand visualisation and how it can help in the teaching and learning of mathematics.
While seen as an important, and successful, strategy for helping with reading
comprehension, the literature on visualisation is mixed as to the effectiveness of it as part of teaching and learning in mathematics education. However, there are calls for more research into visualisation to help gain a deeper understanding of it and any possible role visualisation can have in mathematics education.
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