This study aims to explore some of the necessary and sufficient conditions required, with a focus on the Singapore Bar Model approach, for pupils with autism to solve real life, mathematical word problems to enable them to make expected or better progress in mathematics. 

A significant driver for curriculum reform in the U.K. is that of international comparative assessments such as PISA and TIMSS. One such country, which leads the way in mathematical achievement and understanding, is Singapore. Consequently, the bar model approach has begun to emerge as a tool for supporting mathematical problem solving and conceptual understanding for primary school pupils within the U.K., and elsewhere.

Coupled with this, there has been a noticeable increase in the number of students being diagnosed with an autism spectrum disorder (ASD) over recent years (Lindsay, Proulx, Thomson, & Scott, 2013) with a steady increase over the last four decades (Baron-Cohen et al., 2009), and more than 70% of these pupils attend mainstream schools (APPGA, The National Autistic Society, 2017, p. 9). This increase in awareness and diagnosis has driven the need for a more ‘inclusive education’

When considering the underlying cognitive theories of autism, such as Theory of Mind, Executive Functioning and Weak Central Coherence (Levy, 2007), in line with various proposed models of solving mathematical word problems (Kintsch & Greeno, 1985; Polya, 1945; Skemp, 1978), we can begin to explore how a representational model, such as the bar model, may offer support for some pupils with autism, in terms of deconstructing the problem into detailed, focused parts. However, the global understanding and generalisation of the transfer of knowledge to new situations may be found to be problematic particularly for those individuals with weak central coherence, unless they have adopted strategies to overcome these challenges.

Consequently, as the bar model, along with other visual representational models, ‘consists of a series of rectangles in which the relationships of the rectangles are specified and presented globally’ (Ng & Lee, 2009, p. 285), it may be that those autistic individuals, particularly those demonstrating weak central coherence, may well find such an approach to problem solving less than supportive. 

The poster presentation outlines the key findings from the literature and a preliminary study, using qualitative comparative anlysis (QCA) to begin to establish the neceesary conditions needed for the bar model to be an effective tool to support mathematical problem solving amongst autistic pupils. Furthermore, the findings from the preliminary study are then used to guide the future conditions for analysis within the mains study.  

As an overarching research design, the theoretical foundations of QCA align firmly within the realist framework of causation. From a realist perspective, those observed behaviours and outcomes (the empirical) are determined by underlying mechanisms, which are not observable, and therefore may not be able to be measured, operating within specific contexts. Such realist research seeks to explore these mechanisms and the contexts within which they are activated, in order to provide an explanatory view of causation. Similarly, QCA, as a research approach, sets out to explore the multiple complex configurations of conditions (which may be contextual or mechanistic) necessary to give rise (or not) to a specific outcome. As a technique, QCA aims to explores set relations of conditions in terms of sufficiency and necessity for a particular outcome, and as such, can be either case- or condition-oriented. As a consequence of this ‘holistic approach to causal data analysis’, it is popular within research based upon theories which consider complex conjunctions of conditions and events (Thiem, 2014), as with the current study. Like any case-study approach, case-oriented QCA enables the in-depth case knowledge, in conjunction with cross-case inference, to strengthen the internal validity of the research. Additionally, it is best suited to purposively selected samples and explorative research designs (Rihoux & Ragin, 2009; Thomann & Maggetti, 2017) due to the focus on the interactions of specific conditions and outcomes of interest, as in the present study. Nevertheless, perhaps more importantly, as with the current study, justification for using QCA as a research approach and technique should lie within the focus on configurational analysis through a realist perspective, rather than the number of cases to be analysed (Thiem, 2014). As the ultimate aim of QCA, is to establish causal pathways, consisting of combinations of specific conditions giving rise to a particular outcome (CMO configurations as discussed earlier), the final number of cases analysed, is determined by the range of configurational pathways of conditions observed within the cases. Therefore, cases may be added, or dropped, throughout the study, in order to maximise the analysis on the specific configurations of interest. However, although still referred to as a ‘case-oriented approach’ by Thomann & Maggetti (2017), advocates for QCA have moved towards the use of the more focused term ‘configuration-oriented approach’ as a more fitting description.

As this study is still in the process of being carried out, the findings and conclusions are, as yet, unknown. At the time of the poster presentation, findings from the preliminary study will be available and therefore presented to the audience as a basis for the refinement of the next stages of research. Nevertheless, it is anticipated that the findings will begin to evidence some of the complex causal pathways involved in mathematical problem solving for autistic pupils. The findings will aim to consider the contextual factors necessary for successful application of the bar model approach for those pupils with autism.

APPGA, The National Autistic Society, A. A. A. (2017). Autism and Education In England. Baron-Cohen, S., Scott, F. J., Allison, C., Williams, J., Bolton, P., Matthews, F. E., & Brayne, C. (2009). Prevalence of autism-spectrum conditions: UK school-based population study. The British Journal of Psychiatry, 194(6), 500–509. Kintsch, W., & Greeno, J. G. (1985). Understanding and Solving Word Arithmetic Problems. Psychological Review, 92(1), 109–129. Levy, F. (2007). Theories of autism. Australian and New Zealand Journal of Psychiatry, 41(11), 859–868. Lindsay, S., Proulx, M., Thomson, N., & Scott, H. (2013). Educators’ Challenges of Including Children with Autism Spectrum Disorder in Mainstream Classrooms. International Journal of Disability, Development and Education, 60(4), 347–362. Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40(3), 282–313. Polya, G. (1945). Polya’s Problem Solving Techniques. In How To Solve It (pp. 1–4). Rihoux, B., & Ragin, C. C. (Eds.). (2009). Configuraitonal Comparative Methods: Qualitative Comparative Analysis (QCA) and Related Techniques. London: Sage. Skemp, R. (1978). Relational Understanding and Instrumental Understanding. The Arithmetic Teacher, 26(3), 9–15. Thiem, A. (2014). Navigating the Complexities of Qualitative Comparative Analysis: Case Numbers, Necessity Relations, and Model Ambiguities. Evaluation Review, 38(6), 487–513. Thomann, E., & Maggetti, M. (2017). Designing Research With Qualitative Comparative Analysis (QCA): Approaches, Challenges, and Tools. Sociological Methods & Research, 1–38.