Faculty of Education - Research Publications
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ItemBeyond Accuracy: A Process for Analysis of Constructed Responses in Large(The Korean Society of Educational Studies in Mathematics, 2022-08-31)This article reports a detailed analysis of errors made by early algebra students who are learning equation solving. Whilst we aim to deepen the knowledge base that can be used by teachers and curriculum writers to plan instruction, the main goal of this article is to demonstrate a 5-step process that we have developed to analyse, summarize, visualize and grasp meaning from large numbers of short constructed responses to mathematics questions. This data-driven process is illustrated using responses from 3,010 students to an online test, which is intended to be highly diagnostic and used for formative purposes. The test consisted of 14 very carefully selected linear equations to solve. Students entered only the numerical answers. Although nearly all the errors identified have been previously reported in the literature, the study provides insights into their prevalence in this target population and demonstrates how they correspond to other characteristics of the students’ algebra, including the strategies that they use to solve these linear equations. It also provides a practical example of using short constructed responses as a window into students’ thinking.
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ItemSMARTA—online-diagnostic to reveal students’ algebraic thinking and enhance teachers’ diagnostic competencies(Danish School of Education, Aarhus Unviersity, 2022)Individual, diagnosis-guided support for learners is one of the most important factors in understanding mathematics and learning efficiently. Especially in the field of algebra many students often still lack basic competencies to handle variables, algebraic expressions and equations in a proper way. Digitally supported diagnostic systems offer the possibility to perform a deep understanding-oriented diagnosis with little time. This is the aim of the Australian SMART-system (Specific Mathematics Assessments that Reveal Thinking (Stacey et al., 2018)), which is currently being adapted for use in German-speaking countries. SMARTA is a twofold project to investigate the effects on students’ understanding of algebra and on teachers’ insight into formative assessment in the field of algebra.
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ItemSpecific Mathematics Assessments that Reveal Thinking: An Online Tool to Build Teachers’ Diagnostic Competence and Support Teaching(Springer, 2018)In this chapter, we describe the design of an online system for the formative assessment of students’ understanding of mathematics and discuss how it develops diagnostic competence and influences teaching. The smart-test system covers many mathematics topics studied by students between about 10 and 16 years of age. It is programmed to provide teachers with an automated diagnosis of their own students’ stages of development in specific topics and to report on an individual’s errors and misconceptions, in order to inform teaching. Our claim is that teachers’ diagnostic competence increases when they have easy access to information about their own students’ thinking. In turn, this can further improve teaching, and hence learning. By drawing together evaluative data from four sources, we highlight aspects of teachers’ initial responses to formative assessment and the effect of using this system on their knowledge for teaching and the subsequent changes to teaching practice. Overall, teachers report that using the smart-tests has improved their knowledge of the thinking of individual students as well as of students in general (i.e., their pedagogical content knowledge), and that they can use this information in several ways to adjust their teaching. Paradoxically, using smart-tests reduces the demand for teachers to have specific knowledge for diagnosis and at the same time increases this knowledge and so improves their diagnostic competence.
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ItemTeachers' views of using an on-line, formative assessment system for mathematics(COEX, 2012)In this paper we discuss some issues arising from teachers’ use of an on-line system for formative assessment of their students’ understanding of mathematics. The ‘smart tests’ cover many topics in secondary school mathematics, and are programmed to provide an automated diagnosis of students’ stages of learning in specific topics, to inform teaching. In particular, we highlight teachers’ views about: the desirability of formative assessment; the use that they make of this information; the provision of feedback; and the effect of using the system on their pedagogical content knowledge and subsequent changes to their practice .Overall, teachers report that use of the tests improves their knowledge of the achievement of individual students and students in general, and that they can use this information in several ways to improve their teaching.
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ItemEvaluating automated processes for revealing students’ mathematical thinking(AARE, 2010)Understanding student thinking is key to planning effective teaching. Research over three decades has established good knowledge of stages of development and common misunderstandings in many school topics. However, in order for teachers to make formative assessment a normal part of their practice, mapping of student understanding needs to be available for a greater proportion of the school curriculum. Online testing of students is beginning to create substantial databases which could provide relevant information. The aim of this symposium is to present and discuss methods which move towards automation of data analysis.
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ItemSMART Assessment for Learning(International Society for Design and Development in Education, 2009)“Specific Mathematics Assessments that Reveal Thinking,” which we abbreviate to “smart tests,” provide teachers with a quick and easy way to conduct assessment for learning. Using the internet, students in Years 7, 8, and 9 undertake a short test that is focused strongly on a topic selected by their teacher. Students’ stages of development are diagnosed, and sent to the teacher immediately. Where available, on-line teaching resources are linked to each diagnosis, to guide teachers in moving students to the next stage. Many smart tests are now being trialled in schools and their impact on students’ and teachers’ learning is being evaluated. Design issues are discussed.
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ItemGetting SMART about assessment for learning(The Mathematical Association of Victoria, 2009)“Specific Mathematics Assessments that Reveal Thinking”—or smart tests—provide teachers with a quick and easy way to conduct assessment for learning. Using the internet, students in years 7, 8, and 9 undertake a short test that is focussed strongly on a topic selected by their teacher. Students’ stages of development are diagnosed, and sent to the teacher within minutes. Many tests have been produced and are now being trialled in 7 Victorian schools. Where available, on-line teaching resources are linked to each diagnosis, to guide teachers in moving students to the next stage. This project is sponsored by the Australian Research Council and Victoria’s Department of Education and Early Childhood Development.
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ItemLearning arithmetic blocks: a concrete model for teaching decimals(Department of Science and Mathemtics Education, 2006)This booklet is an introduction to using the LAB model with your students. It outlines a number of activities using LAB to assist students in gaining an understanding of the decimal number system.
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ItemLesson ideas and activities for teaching decimals(Department of Science and Mathematics Education, 2006)The Department of Science and Mathematics Education has produced this booklet to assist teachers with students learning to work confidently with decimal numbers. It contains many classroom activities that will motivate and engage students making the teaching and learning of decimals both enjoyable and effective.
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ItemMisconceptions about density of decimals: insights from pre-service teachers’ work( 2008)Extensive studies have documented various difficulties with, and misconceptions about, decimal numeration across different levels of education. This paper reports on pre-service teachers’ misconceptions about the density of decimals. Written test data from 140 pre-service teachers, observation of group and classroom discussions provided evidence of pre-service teachers’ difficulties in grasping the density notion of decimals. Incorrect analogies resulting from over generalization of knowledge about whole numbers and fractions were identified. Teaching ideas to resolve these difficulties are discussed. Evidence from this research indicates that it is possible to remove misconceptions about density of decimals.