Abstracts - Earli

understand not only each individual representation but also the relationship betweenrepresentations. Passing from one representation to another is considered a fundamental processleading to mathematical understanding. This study intends to illustrate the importance ofconsidering the ability to construct and switch between multiple representations of a concept inanalyzing and assessing students’ learning outcomes in mathematics. The study explores thestructural organization of students’ abilities in carrying out conversions from one mode ofrepresentation to another in the context of functions and the alteration of this structure withstudents’ age. Students in grades 9 and 11 were given a test involving conversions of functions andother algebraic relations (inequalities) among verbal, symbolic and graphical representations. TheConfirmatory Factor Analysis and the Implicative Statistical Method were used to analyze thedata. The results show that despite the variation in students’ mean performance in the conversiontasks across the two grades, a common structural model is defined for capturing their use ofdifferent representations of functions in both grades. The model highlights the significant role ofthe initial representation of a conversion in students’ processes. Using the implicative statisticalanalysis, evidence is provided for the phenomenon of compartmentalization betweenrepresentations in students’ responses, which is indicative of their fragmentary understanding offunctions. These findings draw attention to the beneficial effect of including combinations ofrepresentations and conversion tasks from more than one source-representation in the assessmenttools of students’ mathematical learning.Using national assessment for getting a deeper knowledge of students’ mathematicalunderstandingMarja van den Heuvel-Panhuizen, Humboldt University / Utrecht University, GermanyAlexander Robitzsch, Humboldt University, GermanyDietlinde Granzer, Humboldt University, GermanyOlaf Köller, Humboldt University, GermanyIt is generally recognized that an assessment that supports learning requires an alignment betweenassessment and the goals of education. This alignment is a real challenge when education—like inthe prevailing approach to mathematics education—is aimed at bringing students to deepunderstanding. Therefore, assessment designers involved in mathematics education have put mucheffort in designing assessment tools that go beyond assessing basic skills and contain openproblems that draw on higher levels of mathematical understanding. In this contribution to thesymposium we like to share and discuss the results from a study into primary school students’mathematics achievements that focuses on higher-order competences such as reasoning, modeling,and smart calculating. The study is part of the ESMaG project that evaluates the national standardsfor mathematics in primary school in Germany. In order to assess the students’ mathematicalthinking the evaluation tool includes a number of open problems, which prompt the students to usehigher-order competences. In addition, we developed an analytic multidimensional codingframework through which we could handle the complexity of students’ responses to these openproblems. This new approach to coding in a large-scale assessment turned out to be an importantkey to get access to the students’ thinking. In our presentation we will argue that having elaboratecoding systems in a national assessment can result in a more informative assessment that gives adeeper view on the students’ mathematical understanding.– 166 –