Chapter 7 182 Cognitive diagnostic assessment (CDA) in mathematics designed for formative purposes aims to collect response behaviour that is indicative of students’ mathematical thinking. Mathematical thinking is defined as the collection of cognitive processes that are active during the problem-solving process (Breen & O’Shea, 2010). In these cognitive processes students use procedural, conceptual, and declarative mathematical knowledge to solve tasks (Rittle-Johnson, 2017; Voutsina, 2011). The focus in this dissertation is on students’ written response behaviour and associated mathematical thinking in third grade subtraction and addition. In Chapters 2, 3, and 4 the diagnostic value of response behaviour captured with the empty number line (ENL) was evaluated (path 1). Diagnosing students’ bridging errors in subtraction was the focus of Chapters 5 and 6 (path 2). With these studies I aimed to contribute to answering the following research questions: 1. What kind of response behaviour is considered diagnostically relevant for formative decision making in third grade mathematics? 2. What features should diagnostic tasks have to obtain response behaviour that is considered relevant for teachers’ formative decision making in third grade mathematics? In this final chapter, the main findings and limitations from each chapter are summarised. The summary is structured by the research questions above and by the two paths explored in this dissertation. Subsequently, directions for further research are discussed. This chapter ends with recommendations for teacher education about the design and use of diagnostic assessment for formative decision making. 1. Diagnosing Strategy Use with the ENL Students with different levels of mathematical ability have been found to differ in their strategy use (e.g. Baroody & Dowker, 2003; Beishuizen, 1993; Dowker, 2005; Kraemer, 2009; 2011). Hence, students with different ability levels use different conceptual and procedural knowledge in their mathematical thinking. The empty number line (ENL) is a frequently used didactical tool to support students’ progression in procedural progression in jumping strategies and conceptual understanding of subtraction and addition (Gravemeijer, 2004; Teppo & van den Heuvel-Panhuizen, 2013). When learning to add and subtract, students often learn to jump via the tens and hundreds (e.g. 73 - 16 = 73 - 3 = 70, 70 - 3 = 67, 67 - 10 = 57). When they gain more declarative knowledge about number facts, they will no longer need that intermediate step (e.g. 73 - 16 = 73 - 6 = 67,

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