Fractions, Percentages, Decimals and Proportions

Galen, Frans van; Feijs, Els; Figueiredo, Nisa; Gravemeijer, K.P.E.; Herpen, Els van; Keijzer, Ronald

doi:10.1163/9789460911422

Cited by 37 publications

(12 citation statements)

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“…At the beginning of the basic session, the instructor stressed the importance of students’ conceptual understanding for teaching mathematics (e.g., Padberg, 2009; van Galen et al, 2008). Accordingly, the first block was dedicated mainly to conceptual aspects of fractions, such as the operator concept and the part–whole concept (e.g., Lamon, 2006; Wartha, 2009).…”

Section: Methodsmentioning

confidence: 99%

Teacher knowledge experiment: Testing mechanisms underlying the formation of preservice elementary school teachers’ pedagogical content knowledge concerning fractions and fractional arithmetic.

Tröbst

Kleickmann

Heinze

et al. 2018

Journal of Educational Psychology

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Pedagogical content knowledge forms the core of teachers’ professional knowledge; it refers to knowledge about making subject matter accessible to students. Thus, the formation of pedagogical content knowledge constitutes a crucial issue for educational research and practice. We investigated the contributions of content knowledge and pedagogical knowledge to the formation of pedagogical content knowledge about fractions and fractional arithmetic in 6th grade mathematics in a between-participants study with 100 German preservice teachers. The three experimental and two control groups received 7 hr of intervention spread out over two days. We assessed participants’ pedagogical content knowledge before intervention, between the two days, after intervention, as well as at 6-week follow-up. The control groups exclusively received instruction on either pedagogical knowledge or pedagogical content knowledge; each of the experimental groups embodied a specific hypothesis about the formation of pedagogical content knowledge. Providing support for a mechanism of amalgamation, a combination of instruction on content knowledge and pedagogical knowledge produced small but statistically significant growth in pedagogical content knowledge. Similarly, instruction on content knowledge exclusively was sufficient to cause small but statistically significant growth in pedagogical content knowledge. Prior instruction on content knowledge did not facilitate learning from instruction on pedagogical content knowledge. Nevertheless, direct instruction on pedagogical content knowledge caused medium and statistically significant growth in pedagogical content knowledge.

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Section: Methodsmentioning

confidence: 99%

Teacher knowledge experiment: Testing mechanisms underlying the formation of preservice elementary school teachers’ pedagogical content knowledge concerning fractions and fractional arithmetic.

Tröbst

Kleickmann

Heinze

et al. 2018

Journal of Educational Psychology

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“…Moreover, they should be able to do some (context-bound) calculations with those mathematical objects (CITO, 2013). During the teaching of proportion there is an emphasis on working with a ratio table, which either is given to students or they must recognize when it might be useful (Van Galen et al, 2005). As such, there is a large emphasis on applying learned rules and strategies instead of developing a deep understanding of proportion.…”

Section: Theoretical Backgroundmentioning

confidence: 99%

Touchscreen Tablets: Coordinating Action and Perception for Mathematical Cognition

et al. 2017

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Proportional reasoning is important and yet difficult for many students, who often use additive strategies, where multiplicative strategies are better suited. In our research we explore the potential of an interactive touchscreen tablet application to promote proportional reasoning by creating conditions that steer students toward multiplicative strategies. The design of this application (Mathematical Imagery Trainer) was inspired by arguments from embodied-cognition theory that mathematical understanding is grounded in sensorimotor schemes. This study draws on a corpus of previously treated data of 9–11 year-old students, who participated individually in semi-structured clinical interviews, in which they solved a manipulation task that required moving two vertical bars at a constant ratio of heights (1:2). Qualitative analyses revealed the frequent emergence of visual attention to the screen location halfway along the bar that was twice as high as the short bar. The hypothesis arose that students used so-called “attentional anchors” (AAs)—psychological constructions of new perceptual structures in the environment that people invent spontaneously as their heuristic means of guiding effective manual actions for managing an otherwise overwhelming task, in this case keeping vertical bars at the same proportion while moving them. We assumed that students’ AAs on the mathematically relevant points were crucial in progressing from additive to multiplicative strategies. Here we seek farther to promote this line of research by reanalyzing data from 38 students (aged 9–11). We ask: (1) What quantitative evidence is there for the emergence of AAs?; and (2) How does the transition from additive to multiplicative reasoning take place when solving embodied proportions tasks in interaction with the touchscreen tablet app? We found that: (a) AAs appeared for all students; (b) the AA-types were few across the students; (c) the AAs were mathematically relevant (top of the bars and halfway along the tall bar); (d) interacting with the tablet was crucial for the AAs’ emergence; and (e) the vast majority of students progressed from additive to multiplicative strategies (as corroborated with oral utterances). We conclude that touchscreen applications have the potential to create interaction conditions for coordinating action and perception into mathematical cognition.

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“…At an early stage of learning, representation models should be, according to van Galen et al (2008), very close to contextualised situations, but as schooling progresses, they should allow students to reason beyond a concrete situation. They suggest, therefore, the gradual replacement of "models of concrete situations by models for thinking" (p. 18), which they exemplify with the use and exploration of the double bar or the double number line with fractions, percentages, decimals, and proportions.…”

Section: Representations In Multiplicative Compa-risonmentioning

confidence: 99%

Model of Flexibility in the Conceptual Evolution of Multiplicative Comparison

Cebola

Brocardo

2021

Acta Sci.

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Background: Proportionality is a paramount topic in students' mathematical literacy. In this topic, it is fundamental to understand the multiplicative comparison and the flexible use of the concepts of ratio and proportion. Objectives: To analyse the articulation between concepts, resolution strategies, and representations used by students, supported by numerical relationships and properties of operations to characterise a model of the conceptual evolution of multiplicative comparison. Design: Research based design. Setting and participants: Two mathematics teachers teaching 6th grade (11-12-year-old students) and their 38 (18 + 20) students during the lessons in which five assignments were explored. Data collection and analysis: The data includes the transcription of the video recordings of the classes and the students' written productions. The data were successively revisited and condensed to identify and illustrate the aspects that make up the model of the conceptual evolution of multiplicative comparison. Results: The construction of the concept of multiplicative comparison prevails in the work in two spaces of measurement and in the exploration of the multiplicative relationship between the corresponding quantities within each one. Resolution strategies are first non-quantitative and become quantitative, initially with additive and then multiplicative characteristics. The representations are based on the ratio as a fraction, on the double number line, and on the table of ratios, being supported by numerical relationships and properties of operations. Conclusions: The results elucidate, adjust, and illustrate a theoretical model related to the evolution of the multiplicative comparison in two spaces of measurement.

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Fractions, Percentages, Decimals and Proportions

Cited by 37 publications

References 0 publications

Teacher knowledge experiment: Testing mechanisms underlying the formation of preservice elementary school teachers’ pedagogical content knowledge concerning fractions and fractional arithmetic.

Teacher knowledge experiment: Testing mechanisms underlying the formation of preservice elementary school teachers’ pedagogical content knowledge concerning fractions and fractional arithmetic.

Touchscreen Tablets: Coordinating Action and Perception for Mathematical Cognition

Model of Flexibility in the Conceptual Evolution of Multiplicative Comparison

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