The Problem with the “Problem Solving” Strand

When one thinks of problem solving in a mathematical sense, arithmetic word problems are typically considered to be the most sizeable part of the issue (e.g., Peter had eight sweets. He gave two sweets to his friend. How many sweets did Peter have left?). The typical explanation of such word problems is that they are linguistically presented, single- or multi-step problems requiring arithmetic solutions, but evidently that type of task is not all encompassing in the realms of mathematical problem solving.

The Problem Solving Strand in the UK curricular structure is primarily concerned with factors such as making decisions: deciding which operation and method of calculation to use (mental, mental with jottings, pencil and paper, calculator), reasoning about numbers or shapes and making general statements about them, solving problems involving numbers in context: ‘real life’, money, and measures, and not just arithmetical word based multistage problems (see Appendix E for learning outcomes for the Problem Solving Strand).

Given this dichotomy between the curricular strand of Problem Solving and the perception of mathematical problem solving as a measureable task, an attempt must be made to link Problem Solving (the curricular strand) with the previous psychological literature. It is probably most straightforward to focus on those more tangible aspects of problem solving in a mathematical

130 sense, which have been measured before. Therefore the focus is on arithmetical word problem solving.

8.1.1 Working memory and current thinking on Problem Solving as arithmetical word based problems

A significant body of work has concentrated on cognitive predictors of arithmetical word based problems solving tasks (Andersson, 2007; Fuchs et al., 2006; Fuchs et al., 2010; Hecht, 2002; Kail &

Hall, 1999; Lee, Ng, Ng, & Lim, 2004; Meyer et al., 2010; Passolunghi & Mammarella, 2010;

Swanson, 2006b; Swanson, 2011; Swanson & Beebe-Frankenberger, 2004; Swanson et al., 2008), but there has been considerably less research on the relationship between the UK curriculum, Problem Solving as a taught concept and the contribution of working memory (Holmes & Adams, 2006). Therefore the study is drawn towards research into working memory and arithmetic word problem solving.

Early research indicated that individuals with higher levels of working memory capacity have a tendency to perform better on learning tasks because they have more cognitive resources available to them (Daneman & Carpenter, 1980). Furthermore the effect of working memory on mathematical problem solving has been documented in a number of studies (Andersson, 2007;

Swanson, 2006b; Swanson & Beebe-Frankenberger, 2004; Swanson et al., 2008; Swanson &

Sachse-Lee, 2001; Zheng et al., 2011). Much of this research indicates that central executive functions seem to be among the key predictors of children’s performance in solving mathematical word problems as well as in written mathematical calculation (see Bull et al., 1999; Bull & Scerif, 2001; Gathercole & Pickering, 2000b; Gathercole, Pickering, Knight, et al., 2004; Lee et al., 2004;

Swanson & Beebe-Frankenberger, 2004). More recently Kyttälä and colleagues (Kyttälä, Aunio, Lepola, & Hautamäki, 2013) have specified that nonverbal CWM was having a direct effect upon performance in children’s arithmetic word problem solving. Relevant to this Chapter, Kyttälä et al also use the Odd One Out task and these data show nonverbal working memory made a

significant contribution to word problem solving above and beyond the role of general intelligence and age.

131 Turning briefly to deficits in arithmetical problem solving tasks, Passolunghi and Siegel (2001) suggested that when IQ was matched in peer groups, poor problem solvers were still performing at a lower level on working memory tasks than their IQ matched peers with better problem solving skills. This finding implies that working memory has an influence upon arithmetical word problem solving that is above that of IQ.

8.1.2 Calculation as a mediator between working memory and Problem Solving

Research consistently finds that basic calculation (addition and subtraction with sums less than 20) covaries with mathematics achievement (Durand, Hulme, Larkin, & Snowling, 2005; Geary &

Brown, 1991; Hecht, Torgesen, Wagner, & Rashotte, 2001) and Problem Solving is a measurable aspect of mathematics whereby it is theoretically appropriate to consider calculation as being an influential factor upon achievement in the Problem Solving Strand. In a cross -sectional study with a slightly older cohort than those children in the present study Andersson (2007) showed that three measures associated with central executive and one measure associated with verbal short-term memory contributed unique variance to mathematical problem solving when the influence of reading, age and IQ were controlled for (r² =.39, p=<.05). Andersson conducted a further regression model predicting arithmetic problem solving that included calculation as a predictor variable and he found that the amount of variance predicted increased to 63% (p=<.05)

suggesting that working memory and calculation both have a strong role in children’s problem solving performance .

Where Andersson (2007) suggests that the inclusion of calculation to the regression model strengthens the ability to predict arithmetical problem solving this study takes the view that calculation may actually mediate the relationship between working memory and problem solving.

Chapter 7 has already ascertained that working memory significantly influences performance on the Calculation Strand (between 20% and 32% of the variance in scores on Calculation, after controlling for age and IQ). It is expected that there will be a significant relationship between working memory and problem solving, but since a mathematical operation is frequently required

132 to find a solution to a typical arithmetical problem, it might be reasonably assumed that

calculation ability will mediate the relationship between working memory and Problem Solving.

8.2 Aims and Research Questions

There are three key aims regarding the relationship between Problem Solving in terms of a curricular strand, and working memory;

1. In line with previous chapters, the exploration of the relationship between working memory and Problem Solving is to be assessed using regression analyses. It is proposed that working memory as a whole will be predictive of Problem Solving.

2. It is also hypothesised that the CE-CWM component will be an independent predictor of Problem Solving in the cross-sectional analyses (similar to Kyttälä et al., 2013).

3. The study wanted to further explore the unique contribution of working memory to children’s mathematical problem solving when mediated by Calculation (identified by Andersson, 2007). Based upon Andersson (2007) it is anticipated that Calculation will mediate Problem Solving. This analysis is exploratory, and largely based on the premise that there are conflicting opinions as to the significance of the effects of working memory upon mathematical problem solving (Andersson, 2007; Kail & Hall, 1999; Swanson, 2006b;

Swanson & Beebe-Frankenberger, 2004; Swanson, Cooney, & Brock, 1993). To undertake these analyses modern statistical mediation methods were used (Hayes & Preacher, 2011).

The resulting data and analysis will be divided into two sections, primary regression analyses and secondary mediation modelling.