| 1. |
- Bergman Ärlebäck, Jonas, et al.
(författare)
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A Modeling Perspective on Interpreting Rates of Change in Context
- 2013
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Ingår i: Mathematical Thinking and Learning. - : Routledge. - 1098-6065 .- 1532-7833. ; 15:4, s. 314-336
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Tidskriftsartikel (refereegranskat)abstract
- Functions provide powerful tools for describing change, but research has shown that students find difficulty in using functions to create and interpret models of changing phenomena. In this study, we drew on a models and modeling perspective to design an instructional approach to develop students abilities to describe and interpret rates of change in the context of exponential decay. In this article, we elaborate the characteristics of the model development sequence and we examine how students interpreted and described non-constant rates of change in context. We provide evidence for how a focus on the context made visible students reasoning about rates of change, including difficulties related to the use of language when describing changes in the negative direction. We argue that context and the use of language, forefronted in a modeling approach, should play an important role in supporting the development of students reasoning about changing phenomena.
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| 2. |
- Bergwall, Andreas, 1972-, et al.
(författare)
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The State of Proof in Finnish and Swedish Mathematics Textbooks : Capturing Differences in Approaches to Upper-Secondary Integral Calculus
- 2017
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Ingår i: Mathematical Thinking and Learning. - : Taylor & Francis. - 1098-6065 .- 1532-7833. ; 19:1, s. 1-18
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Tidskriftsartikel (refereegranskat)abstract
- Students’ difficulties with proof, scholars’ calls for proof to be a consistent part of K-12 mathematics, and the extensive use of textbooks in mathematics classrooms motivate investigations on how proof-related items are addressed in mathematics textbooks. We contribute to textbook research by focusing on opportunities to learn proof-related reasoning in integral calculus, a key subject in transitioning from secondary to tertiary education. We analyze expository sections and nearly 2000 students’ exercises in the four most frequently used Finnish and Swedish textbook series. Results indicate that Finnish textbooks offer more opportunities for learning proof than do Swedish textbooks. Proofs are also more visible in Finnish text-books than in Swedish materials, but the tasks in the latter reflect a higher variation in nature of proof-related reasoning. Our results are compared with methodologically similar U.S. studies. Consequences for learning and transition to university mathematics, as well as directions for future research, are discussed.
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| 3. |
- Björklund, Camilla, 1977, et al.
(författare)
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Implementing a structural approach in preschool number activities. Principles of an intervention program reflected in learning
- 2021
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Ingår i: Mathematical Thinking and Learning. - : Informa UK Limited. - 1098-6065 .- 1532-7833. ; 23:1, s. 72-94
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Tidskriftsartikel (refereegranskat)abstract
- We report here on an intervention implementing a structural approach to arithmetic problem-solving in relation to learning outcomes among preschoolers. Using the fundamental principles of the variation theory of learning for developing the intervention and as an analytical framework, we discuss teaching and learning in commensurable terms. The research question is how teaching grounded on a structural approach and designed based on principles of variation theory is reflected in children’s learning of numbers. To answer this, three analyses were conducted, addressing: i) how the children’s ways of experiencing numbers changed after participating in the intervention, ii) how the theoretical ideas were afforded in the intervention program, and iii) synthesizing how the affordance was associated with the children’s arithmetic learning. One group of eight children participating in the intervention program was chosen for thorough analysis. Progression was observed in how the children changed their ways of experiencing numbers during the intervention that allowed them to enact more advanced arithmetic strategies, which was associated with the structural approach in teaching. The results also show how analysis focusing on aspects discerned in learning and aspects afforded in teaching provides a way of describing arithmetic learning with significant implications for teaching practices.
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| 4. |
- Björklund, Camilla, 1977, et al.
(författare)
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Preschoolers’ reasoning about numbers in picture books
- 2020
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Ingår i: Mathematical Thinking and Learning. - : Informa UK Limited. - 1098-6065 .- 1532-7833. ; 22:3, s. 195-213
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Tidskriftsartikel (refereegranskat)abstract
- In this article we report on a study of children’s attention to numerical content in picture books. Specific research questions are 1) how the content in a designed picture book directs children’s attention to numbers, and 2) what kind of numerical reasoning the book reading entails. To answer these questions, we conducted an educational design research study in two cycles, with a specially designed picture book and two ways of reading: first without any prompting from the reading teacher, and second with teacher-child interaction. Nineteen preschool children (aged 3–5 years) and three teachers participated in video-observed individual reading sessions. The observations of the reading activities were analyzed with focus on the children’s attention to numbers as well as on their numerical reasoning during the book talks. Based on the observations, we suggest that preschool children do direct their attention to aspects of numbers, but that the design of the pictures and the framing of the reading sessions influence which numerical aspects are discerned. Furthermore, teachers’ support in transforming empirical clues into mathematical representations and comparisons seems critical for more advanced reasoning about numbers to occur.
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| 5. |
- Bråting, Kajsa, 1975-, et al.
(författare)
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Exploring the intersection of algebraic and computational thinking
- 2021
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Ingår i: Mathematical Thinking and Learning. - : Informa UK Limited. - 1098-6065 .- 1532-7833. ; 23:2, s. 170-185
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Tidskriftsartikel (refereegranskat)abstract
- This article investigates how the recent implementation of programming in school mathematics interacts with algebraic thinking and learning. Based on Duval's theory of semiotic representations, we analyze in what ways syntax and semantics of programming languages are aligned with or divert from corresponding algebraic symbolism. Three examples of programming activities suggested for school mathematics are discussed in detail. We argue that although the semiotic representations of programming languages are similar to algebraic notation the meanings of several concepts in these two domains differ. In a learning perspective these differences must be taken into account, especially considering that students have to convert between registers with both overlapping and specific meanings.
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| 6. |
- Gardesten, Malin, et al.
(författare)
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Students’ participation in mathematics in inclusive classrooms : a study of the enacted mathematical and relational knowing of teachers
- 2023
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Ingår i: Mathematical Thinking and Learning. - : Taylor & Francis Group. - 1098-6065 .- 1532-7833. ; , s. 1-22
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Tidskriftsartikel (refereegranskat)abstract
- Given the need to increase equity in mathematics education, this study draws on earlier research connecting the mathematical and relational know-ing of teachers to determine how such combined knowing enables elemen-tary school students’ inclusion in mathematics. Through a participatory perspective on learning based on social practice theory, empirical examples in the results illustrate how the mathematical and relational knowing of teachers enables diverse participation in communities of classroom mathe-matics. For students’ spatial, social, and mathematical inclusion in the class-room, the enactment of both mathematical and relational knowing of teachers is important. Further, the results indicate that this enactment of both mathematical and relational knowing can be made by one or two co- teachers. Thus, when advocating for inclusive and quality mathematics education for all students, the mathematical as well as the relational knowing of teachers ought to be considered.
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| 7. |
- Hemmi, Kirsti, et al.
(författare)
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Curricular approaches to algebra in Estonia, Finland and Sweden – : a comparative study
- 2021
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Ingår i: Mathematical Thinking and Learning. - : Taylor & Francis. - 1098-6065 .- 1532-7833. ; 23:1, s. 49-71
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Tidskriftsartikel (refereegranskat)abstract
- The aim of the study is to investigate and compare approaches to algebra in the Estonian, Finnish, and Swedish national core curricula (Grades 1–9). Despite the similarities in the school systems of these neighboring countries, the analysis reveals three quite different curricular approaches. The Estonian approach shows influences of the Russian Davydov School. The Finnish approach to some extent resembles the traditional one whereby algebra is addressed first at the lower secondary level and then in a formal manner. However, there are also characteristics typical of the functional view, which dominates the Swedish curriculum. Here, as opposed to the documents from the other two countries, a transition to more formal sophisticated methods at the secondary level is not visible at all. We discuss the results in relation to earlier research and students’ different learning outcomes in light of international evaluations.
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| 8. |
- Holmlund, Anna, 1987
(författare)
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How numbers influence students when solving linear equations
- 2024
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Ingår i: MATHEMATICAL THINKING AND LEARNING. - 1098-6065 .- 1532-7833.
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Tidskriftsartikel (refereegranskat)abstract
- Most students need to learn to solve equations as a part of their education, but far from all succeed. In this study we investigate the potential impact of variation in the choice of numbers as coefficients when students in upper secondary school solve linear equations. A test with linear equations was distributed to 111 16-year-old vocational students and phenomenographic interviews were performed with 23 of them. The findings show that exchanging coefficients (e.g., changing a and b in $a = b\cdot x$a=b & sdot;x) from natural numbers to other numbers significantly changes how equations with the same structure are handled, despite access to a calculator. A framework of five conceptions is described explaining how numbers are experienced in the foreground or background of attention, partly depending on what meaning the equation is given. The results indicate that number types need to be considered, not only in teaching arithmetic, but also in algebra.
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| 9. |
- Juter, Kristina, 1971-
(författare)
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Limits of functions as they developed through time and as students learn them today
- 2006
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Ingår i: Mathematical Thinking and Learning. - 1098-6065 .- 1532-7833. ; 8:4, s. 407-431
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Tidskriftsartikel (refereegranskat)abstract
- This article compares first-year university students' development of the concept of limits to mathematicians' historical development of the concept. The aim was to find out if students perceive the notion as mathematicians of the past did, as understandings of the concept evolved. The results imply that there are some similarities—for example, the struggle with rigor and attainability. Knowledge of such critical areas can be used to improve students' opportunities of learning limits of functions. Some teaching aspects related to the study are also discussed.
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| 10. |
- Juter, Kristina
(författare)
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Limits of functions as they developed through time and as students learn them today
- 2006
-
Ingår i: Mathematical Thinking and Learning. - : Routledge. - 1098-6065 .- 1532-7833. ; 8:4, s. 407-431
-
Tidskriftsartikel (refereegranskat)abstract
- This article compares first-year university students' development of the concept of limits to mathematicians' historical development of the concept. The aim was to find out if students perceive the notion as mathematicians of the past did, as understandings of the concept evolved. The results imply that there are some similarities—for example, the struggle with rigor and attainability. Knowledge of such critical areas can be used to improve students' opportunities of learning limits of functions. Some teaching aspects related to the study are also discussed.
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