Andreas Obersteiner | University of Education, Freiburg/Germany (original) (raw)

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Papers by Andreas Obersteiner

Describing and Studying Domain-Specific Serious Games, 2015

British journal of psychology (London, England : 1953), Jan 14, 2015

Many learners have difficulties with rational number tasks because they persistently rely on thei... more Many learners have difficulties with rational number tasks because they persistently rely on their natural number knowledge, which is not always applicable. Studies show that such a natural number bias can mislead not only children but also educated adults. It is still unclear whether and under what conditions mathematical expertise enables people to be completely unaffected by such a bias on tasks in which people with less expertise are clearly biased. We compared the performance of eighth-grade students and expert mathematicians on the same set of algebraic expression problems that addressed the effect of arithmetic operations (multiplication and division). Using accuracy and response time measures, we found clear evidence for a natural number bias in students but no traces of a bias in experts. The data suggested that whereas students based their answers on their intuitions about natural numbers, expert mathematicians relied on their skilled intuitions about algebraic expressions...

ZDM, 2015

ABSTRACT Understanding contingency table analysis is a facet of mathematical competence in the do... more ABSTRACT Understanding contingency table analysis is a facet of mathematical competence in the domain of data and probability. Previous studies have shown that even young children are able to solve specific contingency table problems, but apply a variety of strategies that are actually invalid. The purpose of this paper is to describe primary school children’s strategy use, and to explore the extent to which psychological theories of intuition and inhibition help better understand the cognitive mechanisms underlying these strategies. In an initial study, we investigated 231 second-graders’ performance on various types of contingency table problems in a paper-and-pencil test. In a second study, we asked 45 second- and fourth-graders to give reasons for their decisions on contingency table problems in an interview situation. Results of both studies suggest that ignoring relevant information and referring to additive rather than multiplicative relationships between cell frequencies were among the children’s primary strategies. These strategies can be explained by intuition, which the children were often not able to inhibit. We discuss the implications of this interpretation from a mathematics education perspective.

There has been a controversial debate if individuals solve fraction comparison tasks componential... more There has been a controversial debate if individuals solve fraction comparison tasks componentially by comparing the numerators and denominators, or holistically by considering the numerical magnitudes of both fractions. Recent research suggested that expert mathematicians predominantly use componential strategies for fraction pairs with common components and holistic strategies for pairs without common components. This study for the first time used eye movements to test if this method allows distinguishing strategy use on specific problem types in expert mathematicians. We found the expected fixation differences between numerators and denominators in problems with common components but not in problems without common components.

Competence models have been developed to describe levels of competence in mathematics and particu... more Competence models have been developed to describe levels of competence in mathematics and particularly in the domain of whole numbers. So far, only descriptions of what competence at different levels actually means are available, but current models do not describe how children can reach the next level. In this article, we propose a fine-grained description of the five levels of competence in the domain of numbers as proposed in a competence model for the primary school level that was based on theoretical and empirical research. Moreover, we discuss students􏰕 errors on three items to show how such a detailed analysis can provide additional information about how to support students in their development. We suggest that with such a combined empirical-psychological perspective, competence models can provide guidance for instruction.

The transition from high school to university mathematics causes many difficulties for students. ... more The transition from high school to university mathematics causes many difficulties for students. Reasons for that are often the overall view on mathematics (e.g., the axiomatic deductive character of university mathematics) and the learning culture that both differ greatly from secondary schools (for an overview, see . To bridge the gap between the secondary and tertiary level mathematics, it is particularly necessary that students have the opportunity to repeat basic mathematical theories and methods deeply, and learn how to adapt concept image and concept definition (Vinner, 1991, p. 68) appropriately. Furthermore, pre-service teachers of mathematics need to develop their pedagogical content knowledge that is important for their future profession as a high school teacher. For that reason they should learn to connect high school and university mathematics, and how to present and communicate mathematical contents. Therefore, the TUM School of Education in Munich (Germany) now offers innovative tutorials that aim at supporting students of mathematics education in the above-mentioned aspects of their mathematics educational training. These tutorials complement the basic lectures of linear algebra and calculus.

Die Ergebnisse der letzten PISA-Studie haben gezeigt, dass 15-Jährige in Deutschland im internati... more Die Ergebnisse der letzten PISA-Studie haben gezeigt, dass 15-Jährige in Deutschland im internationalen Vergleich über gute Leistungen in der Mathematik, im Lesen und in den Naturwissenschaft en verfügen. Doch über ein Länderranking von Schulleistungen hinaus gibt PISA beispielsweise auch Informationen darüber, wie Schülerinnen und Schüler ihren Unterricht wahrnehmen. Entsprechend beschreibt der Beitrag einerseits die Ergebnisse von PISA 2012 in Bezug auf die mathematische Kompetenz der Schülerinnen und Schüler und andererseits Merkmale des Mathematikunterrichts in Deutschland.

Cognition and Instruction, 2014

ABSTRACT External number representations are commonly used throughout the first years of instruct... more ABSTRACT External number representations are commonly used throughout the first years of instruction. The twenty-frame is a grid that contains two rows of 10 dots each, and within each row, dots are organized in two groups of five. The assumption is that children can make use of these structures for enumerating the dots, rather than relying on one-by-one counting. We compared first-grade children's performance on two types of computerized enumeration tasks, in which between one and 20 dots were presented in random arrangements or on a twenty-frame. The number of dots was a strong predictor of response times and accuracy rates in the enumeration task with random arrangements but not in the twenty-frame task. Performance on the twenty-frame task was correlated with performance on a number and arithmetic test, even when other cognitive variables were statistically controlled. We discuss these findings in the light of theories on utilizing external representations to support numerical learning. Free Full-text Download: http://www.tandfonline.com/eprint/3w4tZMyzPWb7CdsHEZz3/full

ZDM, 2010

Scientific collaboration between neuroscience and mathematics education has mostly focused on bra... more Scientific collaboration between neuroscience and mathematics education has mostly focused on brain imaging trying to inform education. This study aims at meeting expectations of both research fields, thus enhancing the ecological validity. We investigated the influence of age, mathematical competency, and task characteristics (format, complexity) on students’ arithmetic performance. Based on behavioral data from a pilot study (n = 73), arithmetic tasks

Learning and Instruction, 2013

ABSTRACT Theories of psychology and mathematics education recommend two instructional approaches ... more ABSTRACT Theories of psychology and mathematics education recommend two instructional approaches to develop students’ mental representations of number: The “exact” approach focuses on the development of exact representations of organized dot patterns; the “approximate” approach focuses on the approximate representation of analogue magnitudes. This study provides for the first time empirical evidence for the specific effects of these approaches by implementing them in a highly controlled learning environment. 147 first-graders were randomly assigned to one of three intervention groups that used an “exact”, an “approximate”, or both versions of the same computer game, or to a control group. Performance on tasks requiring exact or approximate number processing as well as achievement in arithmetic were measured before and after the intervention. Results show that performance improved on tasks related to the exact or approximate number aspect trained, but there was no crossover effect. Achievement in arithmetic increased for the experimental groups and tended to be higher after only exact or only approximate training. Implications for teaching and learning in the classroom are discussed.

Describing and Studying Domain-Specific Serious Games, 2015

British journal of psychology (London, England : 1953), Jan 14, 2015

Many learners have difficulties with rational number tasks because they persistently rely on thei... more Many learners have difficulties with rational number tasks because they persistently rely on their natural number knowledge, which is not always applicable. Studies show that such a natural number bias can mislead not only children but also educated adults. It is still unclear whether and under what conditions mathematical expertise enables people to be completely unaffected by such a bias on tasks in which people with less expertise are clearly biased. We compared the performance of eighth-grade students and expert mathematicians on the same set of algebraic expression problems that addressed the effect of arithmetic operations (multiplication and division). Using accuracy and response time measures, we found clear evidence for a natural number bias in students but no traces of a bias in experts. The data suggested that whereas students based their answers on their intuitions about natural numbers, expert mathematicians relied on their skilled intuitions about algebraic expressions...

ZDM, 2015

ABSTRACT Understanding contingency table analysis is a facet of mathematical competence in the do... more ABSTRACT Understanding contingency table analysis is a facet of mathematical competence in the domain of data and probability. Previous studies have shown that even young children are able to solve specific contingency table problems, but apply a variety of strategies that are actually invalid. The purpose of this paper is to describe primary school children’s strategy use, and to explore the extent to which psychological theories of intuition and inhibition help better understand the cognitive mechanisms underlying these strategies. In an initial study, we investigated 231 second-graders’ performance on various types of contingency table problems in a paper-and-pencil test. In a second study, we asked 45 second- and fourth-graders to give reasons for their decisions on contingency table problems in an interview situation. Results of both studies suggest that ignoring relevant information and referring to additive rather than multiplicative relationships between cell frequencies were among the children’s primary strategies. These strategies can be explained by intuition, which the children were often not able to inhibit. We discuss the implications of this interpretation from a mathematics education perspective.

There has been a controversial debate if individuals solve fraction comparison tasks componential... more There has been a controversial debate if individuals solve fraction comparison tasks componentially by comparing the numerators and denominators, or holistically by considering the numerical magnitudes of both fractions. Recent research suggested that expert mathematicians predominantly use componential strategies for fraction pairs with common components and holistic strategies for pairs without common components. This study for the first time used eye movements to test if this method allows distinguishing strategy use on specific problem types in expert mathematicians. We found the expected fixation differences between numerators and denominators in problems with common components but not in problems without common components.

Competence models have been developed to describe levels of competence in mathematics and particu... more Competence models have been developed to describe levels of competence in mathematics and particularly in the domain of whole numbers. So far, only descriptions of what competence at different levels actually means are available, but current models do not describe how children can reach the next level. In this article, we propose a fine-grained description of the five levels of competence in the domain of numbers as proposed in a competence model for the primary school level that was based on theoretical and empirical research. Moreover, we discuss students􏰕 errors on three items to show how such a detailed analysis can provide additional information about how to support students in their development. We suggest that with such a combined empirical-psychological perspective, competence models can provide guidance for instruction.

The transition from high school to university mathematics causes many difficulties for students. ... more The transition from high school to university mathematics causes many difficulties for students. Reasons for that are often the overall view on mathematics (e.g., the axiomatic deductive character of university mathematics) and the learning culture that both differ greatly from secondary schools (for an overview, see . To bridge the gap between the secondary and tertiary level mathematics, it is particularly necessary that students have the opportunity to repeat basic mathematical theories and methods deeply, and learn how to adapt concept image and concept definition (Vinner, 1991, p. 68) appropriately. Furthermore, pre-service teachers of mathematics need to develop their pedagogical content knowledge that is important for their future profession as a high school teacher. For that reason they should learn to connect high school and university mathematics, and how to present and communicate mathematical contents. Therefore, the TUM School of Education in Munich (Germany) now offers innovative tutorials that aim at supporting students of mathematics education in the above-mentioned aspects of their mathematics educational training. These tutorials complement the basic lectures of linear algebra and calculus.

Die Ergebnisse der letzten PISA-Studie haben gezeigt, dass 15-Jährige in Deutschland im internati... more Die Ergebnisse der letzten PISA-Studie haben gezeigt, dass 15-Jährige in Deutschland im internationalen Vergleich über gute Leistungen in der Mathematik, im Lesen und in den Naturwissenschaft en verfügen. Doch über ein Länderranking von Schulleistungen hinaus gibt PISA beispielsweise auch Informationen darüber, wie Schülerinnen und Schüler ihren Unterricht wahrnehmen. Entsprechend beschreibt der Beitrag einerseits die Ergebnisse von PISA 2012 in Bezug auf die mathematische Kompetenz der Schülerinnen und Schüler und andererseits Merkmale des Mathematikunterrichts in Deutschland.

Cognition and Instruction, 2014

ABSTRACT External number representations are commonly used throughout the first years of instruct... more ABSTRACT External number representations are commonly used throughout the first years of instruction. The twenty-frame is a grid that contains two rows of 10 dots each, and within each row, dots are organized in two groups of five. The assumption is that children can make use of these structures for enumerating the dots, rather than relying on one-by-one counting. We compared first-grade children's performance on two types of computerized enumeration tasks, in which between one and 20 dots were presented in random arrangements or on a twenty-frame. The number of dots was a strong predictor of response times and accuracy rates in the enumeration task with random arrangements but not in the twenty-frame task. Performance on the twenty-frame task was correlated with performance on a number and arithmetic test, even when other cognitive variables were statistically controlled. We discuss these findings in the light of theories on utilizing external representations to support numerical learning. Free Full-text Download: http://www.tandfonline.com/eprint/3w4tZMyzPWb7CdsHEZz3/full

ZDM, 2010

Scientific collaboration between neuroscience and mathematics education has mostly focused on bra... more Scientific collaboration between neuroscience and mathematics education has mostly focused on brain imaging trying to inform education. This study aims at meeting expectations of both research fields, thus enhancing the ecological validity. We investigated the influence of age, mathematical competency, and task characteristics (format, complexity) on students’ arithmetic performance. Based on behavioral data from a pilot study (n = 73), arithmetic tasks

Learning and Instruction, 2013

ABSTRACT Theories of psychology and mathematics education recommend two instructional approaches ... more ABSTRACT Theories of psychology and mathematics education recommend two instructional approaches to develop students’ mental representations of number: The “exact” approach focuses on the development of exact representations of organized dot patterns; the “approximate” approach focuses on the approximate representation of analogue magnitudes. This study provides for the first time empirical evidence for the specific effects of these approaches by implementing them in a highly controlled learning environment. 147 first-graders were randomly assigned to one of three intervention groups that used an “exact”, an “approximate”, or both versions of the same computer game, or to a control group. Performance on tasks requiring exact or approximate number processing as well as achievement in arithmetic were measured before and after the intervention. Results show that performance improved on tasks related to the exact or approximate number aspect trained, but there was no crossover effect. Achievement in arithmetic increased for the experimental groups and tended to be higher after only exact or only approximate training. Implications for teaching and learning in the classroom are discussed.