An operational definition of learning

Harel, Guershon; Koichu, Boris

doi:10.1016/j.jmathb.2010.06.002

Cited by 33 publications

(67 citation statements)

References 12 publications

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“…Moments when students seem to be stuck in an inquiry context present research opportunities to explore student learning. Reflecting a constructivist perspective, Harel and Koichu (2010) provided an operational definition of learning as a continuum of disequilibrium-equilibrium phases which also supports this notion of 'getting stuck' in learning.…”

Section: The Importance Of Getting Stuckmentioning

confidence: 64%

“…Analyses of teaching and assessment were considered to show how alignment may take place. In this phase, theoretical analysis was informed using Piagetian frameworks that value learning as a journey or process that includes getting stuck (Harel & Koichu, 2010).…”

Section: Phase Three: Learning Mathematics In An Inquiry Classroommentioning

confidence: 99%

“…When a view of assessment, which focuses only on the attainment of particular goals, is used to describe learning in an inquiry setting, which does place importance on students coming to know, a mismatch occurs. A classroom culture of inquiry to learn mathematics places value on students; making accommodation for new ideas (Piaget, 1952); solving problems in reasonable and productive ways that are personal (Confrey, 1991); participating in a community of learning (Sfard, 1998a); and 'knowing' as an ongoing process or continua instead of a point in time (Harel & Koichu, 2010). Research undertaken by Black and his colleagues (Black et al, 2004) on how to improve formative assessment practices, highlighted the importance (to teachers particularly) of understanding models of how students learn.…”

Section: Assessment In Inquiry: a Mismatchmentioning

confidence: 99%

“…Vygotsky's (1978) zone of proximal development was the analytical tool in the second phase of study and was used to analyse how the classroom teacher adjusted her teaching based on feedback she received through formative assessment. The third and final phase looked closer at the mathematical learning revealed through formative assessment using the DNR framework (Harel & Koichu, 2010): a Piagetian-influenced framework. Duckworth's (2006) belief framework was also used to consider an interrelated synthesis of findings from this study, to assist in the development and refinement of assessment practices that align with using the inquiry pedagogy to teaching mathematics.…”

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confidence: 99%

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Measuring Messy Mathematics: Assessing learning in a mathematical inquiry context

Fry¹

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The pedagogy of inquiry to teach mathematics presents a seemingly messy yet busy classroom, where children are engaged in using purposeful mathematics to collaboratively generate effective and creative solutions to open-ended, ambiguous questions. Problems arise when describing student learning in inquiry settings, when assessment practices are chosen that do not align with or are not designed to capture learning in this context. This study will present an inquiry into mathematics inquiry classrooms, to analyse and interpret the relationships between three classroom elements of assessment, teaching and learning. Distinctive to this study, the researcher was also the classroom teacher which offers the reader close insight into school practices in these classrooms. The researcher aimed to understand if an alignment between these classroom elements could support teaching and learning in inquiry settings. Using design research as a methodology (Cobb et al., 2003), two primary, inquiry classrooms (Years three and six) are presented as three iterative phases of study, each using particular theoretical lenses and analytical tools. In the first phase of study, theoretical analysis of formative assessment practices was based on Dewey's (1891) conception of thinking as a process involving abstraction, comparison and synthesis. Vygotsky's (1978) zone of proximal development was the analytical tool in the second phase of study and was used to analyse how the classroom teacher adjusted her teaching based on feedback she received through formative assessment. The third and final phase looked closer at the mathematical learning revealed through formative assessment using the DNR framework (Harel & Koichu, 2010): a Piagetian-influenced framework. Duckworth's (2006) belief framework was also used to consider an interrelated synthesis of findings from this study, to assist in the development and refinement of assessment practices that align with using the inquiry pedagogy to teaching mathematics.Findings from three phases in this study revealed the inadequacy of summative assessment practices in capturing and describing student learning and thinking, fostered at higher levels through inquiry.In the first phase of study, analysis of assessment completed by students as part of their everyday, classroom curriculum reflected how such assessment only requires students to perform lower-level, reproductive thinking. In contrast, formative assessment opportunities encouraged students through inquiry to conceptualise their mathematical thinking in connected and abstract ways. The second phase of study focused on teaching in one inquiry classroom and characterised the difficulties classroom teachers face as they implement inquiry into their mathematics curriculum. Analysis of inquiry teaching and learning in this phase characterised how the teacher needed to be an engineer: able to interweave student ideas as potentialities, into the scaffolding of particular learning goals.ii Interweaving by the teacher, of students' connections t...

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Section: The Importance Of Getting Stuckmentioning

confidence: 64%

Section: Phase Three: Learning Mathematics In An Inquiry Classroommentioning

confidence: 99%

Section: Assessment In Inquiry: a Mismatchmentioning

confidence: 99%

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confidence: 99%

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Measuring Messy Mathematics: Assessing learning in a mathematical inquiry context

Fry¹

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“…The public nature of students' thoughts, ideas and understandings paves the way for challenge to students' own and others' ideas and provides a means to develop the cognitive conflict to create the disequilibrium necessary to move students to seek new understanding (Harel, 2008;Harel & Koichu, 2010). Three specific provokers of cognitive conflict were identified as a result of this study: conflict brought about by the teacher; conflict brought about by students; and, conflict brought about by the task.…”

Section: Potential For Cognitive Conflict Increasesmentioning

confidence: 94%

Developing argumentation in mathematics: The role of evidence and context

Wells¹

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Multiple potential benefits to the introduction of argumentation into classroom environments have been identified and documented, including the potential to extend teaching goals to emphasise cognitive and metacognitive processes, epistemic criteria and reasoning, as well as the enculturation of students into the practices and discourses of a subject. Argumentation structures and practices offer the means to focus students on the need for quality evidence, potentially encouraging students to focus deeply on mathematical content. Much of the work with argumentation that has already occurred in mathematics is associated with justification of procedural choices to arrive at a correct answer. By contrast, mathematical inquiry offers the opportunity for students to engage in ill-structured, ambiguous problems that have neither a defined solution path nor a single correct answer. Thus there is great potential for argumentation to be effective in inquirybased learning environments. However, very little research has focused on argumentation practices of students undertaking inquiry of this nature.The study presented is thus exploratory; designed to both develop deeper understanding of Inquiry-Based Argument practices and possibilities, and to identify how students' developing use of evidence in argumentation could be understood and supported.Specifically, the study sought to address the following research questions: A design research methodology approach was utilised, with iterative cycles of inquiry and argumentation implemented in a single inquiry classroom of Year 4-5 students (n=27, aged 8-10). These cycles introduced the role of evidence, the structure of argument, and evidence and argument quality, to a class environment that embraced a knowledge building culture (Bereiter & Scardamalia, 1996). Berland and Reiser's (2009) Goals of Argument and McNeill and Martin's (2011)Conclusion-Evidence-Reasoning Framework were used to guide the instructional approach which was modified progressively to meet the developing needs of the students. 3Data were generated through videos of emerging classroom practices, interviews with students, student work artefacts, teaching notes, and observations over the course of ten months. A Grounded Theory Approach (Corbin & Strauss, 2008) was utilised for data analysis. The analysis identified several significant results of introducing Inquiry-Based Argument into the classroom:1. Introducing argument practices enabled the ‗visibilising' of student thinking, increasing opportunities to develop and utilise cognitive conflict.2. An increased focus by students on the need for quality evidence enables them to be able to put forth and negotiate arguments.In turn, these factors appear to enable students to:1. Develop complex appreciation for argument structure, progressing from intuitive responses to development of qualified arguments.2. Demonstrate understanding and appreciation of the critical role of evidence and evidence quality in the development of deepened mathematical understanding...

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Reflected Abstraction

Tallman,

O’Bryan

2024

Research in Mathematics Education

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An operational definition of learning

Cited by 33 publications

References 12 publications

Measuring Messy Mathematics: Assessing learning in a mathematical inquiry context

Measuring Messy Mathematics: Assessing learning in a mathematical inquiry context

Developing argumentation in mathematics: The role of evidence and context

Reflected Abstraction

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