Assessing Conceptual Understanding in Mathematics: Representations, Problem Solutions, Justifications, and Explanations

Niemi, David

doi:10.1080/00220671.1996.9941339

Cited by 56 publications

(44 citation statements)

References 20 publications

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“…Possibly, the arithmetical format is more difficult to use to construct a domain representation. Another possibility is that students failed to view mathematical symbols as reflections of principles and structures, but rather perceived them as indicators of which operations need to be performed (Atkinson et al 2003;Cheng 1999;Greenes 1995;Nathan et al 1992;Niemi 1996;Ohlsson and Rees 1991). This would mean that the textual and the conceptual format are more close to the code in which students can explain the domain to themselves, or maybe students consider those formats more suited to express their knowledge to the outside world.…”

Section: Discussionmentioning

confidence: 99%

Comparing the effects of representational tools in collaborative and individual inquiry learning

Kollöffel

Eysink

Jong

2011

Computer Supported Learning

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Constructing a representation in which students express their domain understanding can help them improve their knowledge. Many different representational formats can be used to express one's domain understanding (e.g., concept maps, textual summaries, mathematical equations). The format can direct students' attention to specific aspects of the subject matter. For example, creating a concept map can emphasize domain concepts and forming equations can stress arithmetical aspects. The focus of the current study was to examine the role of tools for constructing domain representations in collaborative inquiry learning. The study was driven by three questions. First, what are the effects of collaborative inquiry learning with representational tools on learning outcomes? Second, does format have differential effects on domain understanding? And third, does format have differential effects on students' inclination to construct a representation? A pre-test post-test design was applied with 61 dyads in a (face-to-face) collaborative learning setting and 95 students in an individual setting. The participants worked on a learning task in a simulationbased learning environment equipped with a representational tool. The format of the tool was either conceptual or arithmetical or textual. Our results show that collaborative learners outperform individuals, in particular with regard to intuitive knowledge and situational knowledge. In the case of individuals a positive relation was observed between constructing a representation and learning outcomes, in particular situational knowledge. In general, the effects of format could not be linked directly to learning outcomes, but marked differences were found regarding students' inclination to use or not use specific formats.

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Section: Discussionmentioning

confidence: 99%

Comparing the effects of representational tools in collaborative and individual inquiry learning

Kollöffel

Eysink

Jong

2011

Computer Supported Learning

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“…One needs to know, for example, the conceptual meaning of the multiplication sign. In arithmetical representations the underlying principle or concept is not as explicit as in diagrams and texts, and as a result most learners tend to view mathematical symbols (e.g., multiplication signs) purely as indicators of which operations to perform on adjacent numbers (Atkinson et al 2003;Cheng 1999;Greenes 1995;Nathan et al 1992;Niemi 1996;Ohlsson and Rees 1991).…”

Section: Representational Formatsmentioning

confidence: 99%

The effects of representational format on learning combinatorics from an interactive computer simulation

et al. 2008

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The current study investigated the effects of different external representational formats on learning combinatorics and probability theory in an inquiry based learning environment. Five conditions were compared in a pre-test post-test design: three conditions each using a single external representational format (Diagram, Arithmetic, or Text), and two conditions using multiple representations (Text + Arithmetic or Diagram + Arithmetic). The major finding of the study is that a format that combines text and arithmetics was most beneficial for learning, in particular with regard to procedural knowledge, that is the ability to execute action sequences to solve problems. Diagrams were found to negatively affect learning and to increase cognitive load. Combining diagrams with arithmetical representations reduced cognitive load, but did not improve learning outcomes.

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“…Thirdly, we can look at the wider understanding that children have of multiplication. Having adopted a clear theoretical basis for what we mean by understanding and reasoning in mathematics, and examined the implications for the definitions that we have used, it is clear that we need to use a variety of methods that encourage children"s reasoning in order to gain insight into their understanding of multiplication (we provided earlier example studies from Niemi, 1996, andLawson andChinnappan, 2000).…”

Section: Discussionmentioning

confidence: 99%

“…(Hiebert and Carpenter, 1992, p. 89) We might therefore provide a variety of opportunities for links to be demonstrated in order to get a broader picture of understanding. For example, Niemi (1996), in examining students" understanding of fractions, used a problem solving task where students justified their methods, an open-ended task where they explained fractions more broadly and a task examining the fluency with which students could access visual representations. In these tasks, the open-ended task provided the opportunity to see the possible links that students made, the problem solving task specifically looked at the reasoning used by the students when linking between the problems and their understanding of fractions, and the final task examined which links to visual representations the students could quickly access.…”

Section: Implications Of the Model Of Understandingmentioning

confidence: 99%

The array representation and primary children’s understanding and reasoning in multiplication

et al. 2008

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(2009) 'The array representation and primary children's understanding and reasoning in multiplication.', Educational studies in mathematics., 70 (3). pp. 217-241.Further information on publisher's website:http://dx.doi.org/10.1007/s10649-008-9145-1Publisher's copyright statement:The original publication is available at www.springerlink.com Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract We examine whether the array representation can support children"s understanding and reasoning in multiplication. To begin, we define what we mean by understanding and reasoning. We adopt a "representational-reasoning" model of understanding, where understanding is seen as connections being made between mental representations of concepts, with reasoning linking together the different parts of the understanding. We examine in detail the implications of this model, drawing upon the wider literature on assessing understanding, multiple representations, self explanations and key developmental understandings. Having also established theoretically why the array representation might support children"s understanding and reasoning, we describe the results of a study which looked at children using the array for multiplication calculations. Children worked in pairs on laptop computers, using Flash Macromedia programs with the array representation to carry out multiplication calculations. In using this approach, we were able to record all the actions carried out by children on the computer, using a recording program called Camtasia. The analysis of the obtained audiovisual data identified ways in which the array representation helped children, and also problems that children had with using the array. Based on these results, implications for using the array in the classroom are considered.

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Assessing Conceptual Understanding in Mathematics: Representations, Problem Solutions, Justifications, and Explanations

Cited by 56 publications

References 20 publications

Comparing the effects of representational tools in collaborative and individual inquiry learning

Comparing the effects of representational tools in collaborative and individual inquiry learning

The effects of representational format on learning combinatorics from an interactive computer simulation

The array representation and primary children’s understanding and reasoning in multiplication

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