Figure - available from: Educational Studies in Mathematics

This content is subject to copyright. Terms and conditions apply.

Source publication

The transition from natural to rational numbers is difficult for most elementary school children. A major cause for these difficulties is assumed to be the “conceptual change” they need to undergo in order to see that several natural number properties do not apply to rational numbers. To appropriately handle pupils’ difficulties, teachers need well...

## Similar publications

Problematic learning situations (PLS) arise when students encounter learning difficulties and their teacher encounters difficulties assisting them. The current study looks at student and teacher difficulties revealed during PLS, in the course of instruction of basic geometrical concepts for average and below-average junior high school students, whe...

## Citations

... It has long been reported that many students have difficulty understanding rational number concepts (McMullen & Van Hoof, 2020;Obersteiner et al., 2019), despite their importance in school mathematics (Depaepe et al., 2018). Although many efforts have been spent on improving rational number instruction (McMullen & Van Hoof, 2020), students still experience difficulties in learning this topic (Wilson et al., 2013). ...

This study explored four middle-school mathematics teachers’ provision
of non-examples and instructional explanations in teaching
rational numbers to seventh-grade students. The data sources were
8 lesson hours of observations conducted in four different classrooms.
Yin’s [(2018). Case study research and applications: Design and
methods (6th ed.). Sage] cross-case synthesis technique was used to
code the four teachers’ non-examples. Charalambous et al.’s [(2011).
Prospective teachers’ learning to provide instructional explanations:
How does it look and what might it take? Journal of Mathematics
Teacher Education, 14(6), 441–463. https://doi.org/10.1007/s10857-
011-9182-z] criteria were used to code their instructional explanations.
The findings showed that the four teachers seldom used nonexamples
of rational numbers in their classrooms and provided lowquality
instructional explanations to their students. The implications
for improving teachers’ provision of non-examples and instructional
explanations are discussed.

... In line with the analysis of the results of the PASEC2019 tests of teachers' subject and teaching knowledge and skills and the international studies mentioned in the previous paragraph, it can be agreed that efforts to reinforce these two areas would benefit from being harmoniously combined in pre-service education and in-service training. By way of illustration, Depaepe et al. (2018) focused their education programme on the development of teaching knowledge and skills and found that these learning activities also had an impact (and in some cases even a greater impact) on the acquisition of subject knowledge and skills. This is undoubtedly an interesting avenue to explore with a view to reforming teacher education programmes. ...

The PASEC study is an African assessment in reading and mathematics, administered to primary school children in 14 countries in 2019. The study measures student and their teacher achievement in french and mathematics at the second and sixth grades.

... La investigación centrada en el aprendizaje de las operaciones con los números racionales en los estudiantes de educación primaria es abundante, señalando sobre todo las dificultades. Algunos de estos resultados, reflejan la complejidad para los estudiantes de educación primaria de la transición de los números naturales a los números racionales, por ejemplo, la debida al cambio conceptual, asociado principalmente a algunas de las propiedades, por ello los maestros necesitan de manera simultánea el CK y el PCK (Depaepe et al., 2018). ...

INTRODUCCIÓN. La formación en matemáticas de los futuros maestros debe fijarse en contenidos que tradicionalmente se han enseñado desde procedimientos poco comprensivos, como las operaciones con fracciones, para diseñar pautas formativas que se anticipen a los posibles errores que se han adquirido de manera previa. MÉTODO. En este trabajo, centramos nuestra atención en los registros de representación de las fracciones (pictórico, simbólico y lengua natural), con el objetivo de favorecer la comprensión y uso de las operaciones. Presentamos la investigación realizada con una muestra de 85 estudiantes que cursan el Grado en Educación Primaria. Los datos se recopilaron antes y después de una instrucción con material manipulativo. Se analizan muestras de trabajo desde la perspectiva de análisis del MTSK (Mathematics Teacher’s Specialised Knowledge). RESULTADOS. Los resultados del diagnóstico inicial señalan que utilizan números que les facilitan la representación, y que pocos de los futuros maestros utilizan el registro pictórico. El error principal es no reconocer que las partes en que se divide el todo son de igual tamaño. Los resultados tras la formación señalan un cambio en la elección de la representación y una mejora conceptual en los registros pictórico y simbólico, convirtiéndose en más habituales. DISCUSIÓN. El trabajo ha ayudado a identificar obstáculos en el aprendizaje de las operaciones con fracciones, que ponen de manifiesto la necesidad de trabajarlas desde una perspectiva que facilite potenciar la comprensión del contenido, y facilitar la enseñanza posterior desde el uso y conexión entre los distintos registros de representación.

... Furthermore, the different types of ambiguity regarding the fraction concept, which were identified here, provide a theoretical contribution as an extension and refinement of Foster's categorization of mathematical ambiguities. Previous studies have advocated for the topic of fractions to be emphasized in teacher education programs, while giving pedagogical attention to the various models, interpretations, and representations of fractions (e.g., Ball et al., 2008;Depaepe et al., 2018;Marmur et al., 2020). In particular, Lee and Lee (2021) argued that PTs should be given opportunities to decompose content-and pedagogical knowledge of fractions; and observed that especially when presented with unconventional, ambiguous examples of fractions, PTs' levels of fraction-knowledge are easily discernable. ...

We explore the responses of 26 prospective elementary-school teachers to the claim “1/6.5 is not a fraction” asserted by a hypothetical classroom student. The data comprise scripted dialogues that depict how the participants envisioned a classroom discussion of this claim to evolve, as well as an accompanying commentary where they described their personal understanding of the notion of a fraction. The analysis is presented from the perspective of productive ambiguity, where different types of ambiguity highlight the prospective teachers’ mathematical interpretations and pedagogical choices. In particular, we focus on the ambiguity inherent in the aforementioned unconventional representation and how the teachers reconciled it by invoking various models and interpretations of a fraction. We conclude with a description of how the perspective of productive ambiguity can enrich teacher education and classroom discourse.

... Pedagogical Content Knowledge (PCK) is knowledge included "the way of representing and formulating the subject that makes it comprehensible to others" (Depaepe et al. 2018;Gasteiger et al. 2020;Loewenberg Ball et al. 2008;Ma'Rufi et al. 2019;Norton, 2019;Torbeyns et al. 2020). found categories of PCK consist of knowledge of content and students, knowledge of content and teaching, and knowledge of the curriculum. ...

... PCK has basically been implemented by teachers as knowledge about effective teaching content. The two main components of this knowledge are the teacher's knowledge of student misconceptions and the teacher's knowledge of learning strategies (Depaepe et al. 2018;Torbeyns et al. 2020). Ball et al. (2008) add that the PCK components are related to Knowledge of Mathematics Learning Standards (KMLS), where teachers need to be aware of curriculum specifications at each level. ...

... Gasteiger & Benz (2018) have found that teaching mathematics in early childhood (kindergarten) requires knowledge and skills for teaching mathematics in the classroom because sometimes there is a lack of vocabulary or concepts inherent in learning practices. In addition, PCK can be measured by assessing students' problem solving activities (Csíkos & Szitányi, 2020;Depaepe et al. 2018;Verschaffel et al. 2010). Verschaffel et al. (2010) emphasize that students' mathematics performance is considered to have an influence on learning practices, so it is better to find out how teachers assess students' solutions to mathematical problems. ...

In Indonesia, pre-service mathematics teacher education is strictly supervised so that
Pedagogical Content Knowledge (PCK) becomes an important aspect to build the
quality of learning. This study aimed to explore pre-service mathematics teachers’ PCK
based on gender and academic skills. To obtain rich and in-depth data, a qualitative
approach was used. A total of 70 subjects aged between 19 – 21 years old participated
in this study. There were two subjects selected based on their academic skills and
gender. Using a grounded theory approach, we conducted a preliminary analysis, open
coding, axial coding to obtain the three PCK components, namely Knowledge of
Subject Matter (KSM), Knowledge of Pedagogy (KP), and Knowledge of Student (KS).
Research findings revealed that the pre-service teachers’ pedagogical content
knowledge in terms of knowledge of subject matter was categorized as good in
mathematics learning. As for their knowledge of pedagogy, the male subjects presented
the concepts by employing the expository strategy, the female subjects with high skills
used the guided discovery, and the female subjects with average skill also employed the
strategy of expository. In the aspect of knowledge of students, the subjects with average
skills overcame students’ misconception by explaining the procedures and using the
strategy of asking, but the subjects with high academic skills did not only implement
the two previous strategies but also used their reasoning behind every procedure of
problem-solving that they carried out. These findings can be used as recommendations
for the development of mathematics learning.

... As an initial step, teachers should learn that students hold intuitive ideas about various mathematical topics, including the size and the type of the results of operations, since preservice teachers are insufficiently aware of such issues (Depaepe et al., 2015;Depaepe et al., 2018). Students should also learn about their intuitions, since these beliefs are often implicit and not under their conscious control (Fischbein, 1987). ...

When reasoning about numbers, students are susceptible to a natural number bias (NNB): When reasoning about non-natural numbers they use properties of natural numbers that do not apply. The present study examined the NNB when students are asked to evaluate the validity of algebraic equations involving multiplication and division, with an unknown, a given operand, and a given result; numbers were either small or large natural numbers, or decimal numbers (e.g., 3 × _ = 12, 6 × _ = 498, 6.1 × _ = 17.2). Equations varied on number congruency (unknown operands were either natural or rational numbers), and operation congruency (operations were either consistent – e.g., a product is larger than its operand – or inconsistent with natural number arithmetic). In a response-time paradigm, 77 adults viewed equations and determined whether a number could be found that would make the equation true. The results showed that the NNB affects evaluations in two main ways: a) the tendency to think that missing numbers are natural numbers; and b) the tendency to associate each operation with specific size of result, i.e., that multiplication makes bigger and division makes smaller. The effect was larger for items with small numbers, which is likely because these number combinations appear in the multiplication table, which is automatized through primary education. This suggests that students may count on the strategy of direct fact retrieval from memory when possible. Overall the findings suggest that the NNB led to decreased student performance on problems requiring rational number reasoning.

... Prospective teachers' knowledge of fractions has been a growing interest in mathematics education research in recent years, though unfortunately it has been shown that their fraction understanding is often weak (e.g. Castro-Rodríguez, Pitta-Pantazi, Rico, & Gómez, 2016;Depaepe et al., 2018;Park, Güçler, & McCrory, 2013). We suggest that focusing specifically on how prospective teachers deal with unfamiliar semiotic representations, may provide further understanding of how they conceptualise and conceive the notion of fractions, which in turn could be used to guide the teaching of the topic of fractions in teacher education programmes. ...

... In relation to these kinds of student difficulties, recent studies have focused on promoting teachers' pedagogical knowledge specifically in relation to the teaching of fractions and how it may support students' understanding of the topic (e.g. Depaepe et al., 2018;Jakobsen, Ribeiro, & Mellone, 2014). ...

We explore prospective elementary-school teachers’ attempts to provide signs and symbols with mathematical meaning in a case involving non-integers as numerator or denominator. The data comprises 33 responses to a task inquiring about the existence of numbers between 1/6 and 1/7, in which the participants were asked to compose a hypothetical classroom dialogue addressing this issue, and provide explanations on the mathematics involved. The analysis utilised the theoretical frameworks of “semiotic representation” and “concept image”. The findings indicate a strong reliance on prevalent fraction images and difficulties in assigning meaning to “6½” when it appears in the numerator or denominator of a fraction representation. These difficulties are examined via three approaches demonstrated in the scripts, relating to the symbolic representation of fractions, visual representation of fractions, and permissible operations on fractions. We suggest teacher-education courses may put further emphasis on the set model of fractions and atypical semiotic representations.

... Prospective teachers' knowledge of fractions has been a growing interest in mathematics education research in recent years, though unfortunately it has been shown that their fraction understanding is often weak (e.g. Castro-Rodríguez, Pitta-Pantazi, Rico, & Gómez, 2016;Depaepe et al., 2018;Park, Güçler, & McCrory, 2013). We suggest that focusing specifically on how prospective teachers deal with unfamiliar semiotic representations, may provide further understanding of how they conceptualise and conceive the notion of fractions, which in turn could be used to guide the teaching of the topic of fractions in teacher education programmes. ...

... In relation to these kinds of student difficulties, recent studies have focused on promoting teachers' pedagogical knowledge specifically in relation to the teaching of fractions and how it may support students' understanding of the topic (e.g. Depaepe et al., 2018;Jakobsen, Ribeiro, & Mellone, 2014). ...

We explore prospective teachers’ attempts to explain atypical fraction representations, particularly when non-integers appear in the numerator or denominator. As practical implication, we advocate the use of the set model to interpret atypical representations and expand learners’ understanding of fractions.

... Research has amply shown that a good understanding of rational numbers is an essential part of mathematical literacy, since it is required for learning more advanced mathematical contents such as algebra and calculus (Behr et al. 1983;Kieren 1992). However, the concept of rational number is one of the most complex to understand and master by students of primary and secondary education, and even adults show difficulties in understanding all its different aspects (Depaepe et al. 2018;Fischbein et al. 1985;Moss and Case 1999;Resnick et al. 1989;Torbeyns et al. 2015). ...

Understanding rational numbers is a complex task for primary and secondary school students. Previous research has shown that a possible reason is students’ tendency to apply the properties of natural numbers (inappropriately) when they are working with rational numbers (a phenomenon called natural number bias). Focusing on rational number comparison tasks, recent research has shown that other incorrect strategies such as gap thinking or reverse bias can also explain these difficulties. The present study aims to investigate students’ different ways of thinking when working on fraction and decimal comparison tasks. The participants were 1,262 primary and secondary school students. A TwoStep Cluster Analysis revealed six different student profiles according to their way of thinking. Results showed that while students’ reasoning based on the properties of natural numbers decreased along primary and secondary school, almost disappearing at the end of secondary school, students’ reasoning based on gap thinking increased along these grades. This result seems to indicate that when students overcome their reliance on natural numbers, they enter a stage of qualitatively different errors before finally reaching the stage of correct understanding.

... A lesson series was developed to evolve student-teachers´ content knowledge in rational numbers and pedagogical content knowledge (Depaepe, et al. 2018). In the intervention group, unlike in the comparison group, the teaching was codeveloped by researchers and focused on students´ misconceptions, as well as on strategies to overcome the same. ...