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Abstract: Abstract We present here the lessons learned by iteratively designing a tutorial for first-year university students using computer programming to work with mathematical models. Alternating between design and implementation, we used video-taped task interviews and classroom observations to ensure that the design promoted student understanding. The final version of the tutorial we present here has students make their own logarithm function from scratch, using Taylor polynomials. To ensure that the resulting function is accurate and reasonably fast, the students had to understand and apply concepts both from computing and from mathematics. We identify three categories of such concepts and identify three design features that students attended to when demonstrating such understanding. Additionally, we describe four important take-aways from a teaching design point of view that resulted from this iterative design process. PubDate: 2022-04-26
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Abstract: This article presents the theoretical considerations leading to the design and development of a digital experience for teaching linear equations using a modified balance model for equation solving. We modified the balance to alter physics behaviour in a virtual reality (VR) experience, to strengthen students’ schemes for solving linear equations and help students to adapt their schemes to situations where negative numbers and mathematical negativity make equations abstract. We used the VR application in a small teaching experience with ten students and their mathematics teacher from a Danish grade 7 class (13–14 years of age). The exploratory study aimed to analyse and evaluate the effect of teaching with the modified balance in the VR application via a novel teaching experience. We report findings that show positive prospects for the use of VR in teaching linear equation solving including a new equation solving strategy enabled by the virtual environment. A majority of students gave a positive affective response to the experience, referred to, and were able to apply ideas from the VR experience to linear equation solving exercises on post-experience pen-and-paper exercises. Moreover, we report findings from a particular student case who showed interesting behaviour and reasoning, from which we provide in-depth analysis to understand future possibilities of teaching equation solving with VR. PubDate: 2022-04-07 DOI: 10.1007/s40751-022-00103-4
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Abstract: Abstract In this article, we conduct a qualitative systematic review of studies examining the use of digital games to promote students’ mathematical reasoning in primary and lower secondary schools. Digital games now have a prominent role in students’ leisure time, as has mathematical reasoning in curricula around the world. This study investigates how the affordances of digital game–based learning environments (DGBLEs) are used to support students’ mathematical reasoning. Through a thematic analysis, we construct five distinct themes that describe how mathematical reasoning is afforded in the DGBLEs in the reviewed studies: developing (winner) strategies, exploring an immersive environment, experimenting, designing learning games and solving tasks. By analysing the themes in relation to the reasoning and proof cycle, we found that DGBLEs primarily supported exploration, conjecturing and, to a lesser extent, justification. We conclude that students’ mathematical reasoning can be achieved through DGBLEs that specifically target exploration, conjecturing and justification, and by carefully structuring students’ interactions with and dialogues about the games played. PubDate: 2022-02-22 DOI: 10.1007/s40751-022-00100-7
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Abstract: Abstract This design-based study addresses the issue of how to digitally support students’ problem-solving by providing heuristics, in the absence of the teacher. The problem is that, so far, digital tutoring systems lack the ability to diagnose students’ needs in open problem situations. Our approach is based on students’ ability to self-diagnose and find help. To this purpose, we introduce a new type of digital, interactive, help-seeking tool called a heuristic tree. Students’ use of this tool is supported by a help-seeking flowchart. The design of heuristic trees is based on our reinterpretation of the notion of heuristic in terms of terms of compression. Our research question is: How do heuristic trees and the help-seeking flowchart influence students’ problem-solving behaviour' This question was studied in the context of a number theory course for in-service mathematics teachers. During five weeks, fifty students worked on fifty-five problems supported by heuristic trees. Our data consists of video observations of two small groups of students, a teacher log, interviews with these two groups, and a survey filled in by twenty-three students. The main results are that the support by heuristic trees and the help-seeking flowchart allows students to work in the absence of a teacher and to engage strongly with problems, maintaining ownership of the solution methods. Moreover, as intended by the tree structure, students learned to focus not just on the small steps of the solutions, but also on the general heuristic techniques, theorems, and concepts that should be learned in the process of finding those solutions. PubDate: 2022-02-12 DOI: 10.1007/s40751-022-00101-6
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Abstract: Computational thinking (CT) has acquired the status of a necessary 21st-century skill and is currently being introduced in school curricula around the world, despite a lack of consensus about what it entails. The aims of this review are to provide an overview of the existing literature on CT activities in primary mathematics education, and to articulate how it is integrated into the teaching and learning of primary mathematics. This systematic review presents and analyses the findings of 10 empirical studies, revealing a recent increased focus on the inclusion of CT in primary mathematics classrooms, as most studies are published around 2020. Our findings indicate two categories of such activities, one focusing on skills (such as mainly sequencing, looping, conditionals, debugging, decomposition, and abstraction) and one on process-oriented activities (communication, creativity, exploration, and engagement). Furthermore, we found that, while there are studies reporting on mathematics being taught directly through CT activities (full integration), in most studies, the mathematics content was emphasised, with CT built in as a way for students to demonstrate their understanding of mathematics concepts (partial integration). This review identifies current gaps in the field and the need to investigate further such process-oriented activities, the use of these activities in accelerated mathematics, and the need for different methodological approaches in primary mathematics. PubDate: 2022-02-11 DOI: 10.1007/s40751-022-00102-5
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Abstract: Abstract In this research, we explore the potential for computational thinking (CT) to benefit the spatial reasoning abilities of thirty-three middle-school students (aged 11–13) in grades 6 and 7. There is an increasing focus on the explicit development of spatial reasoning throughout the mathematics curriculum, such as geometry, as students’ success in mathematics is largely tied to their spatial reasoning abilities. CT provides a method through which numerous mathematics concepts can be taught or explored, but also offers an avenue through which spatial reasoning skill can be developed. We used the block programming website Scratch and programmable robotics in weekly ‘coding clubs’ run in two different elementary school classrooms for 10 weeks each. Lessons were structured to develop students’ CT abilities and mathematical reasoning, as well as being tied to their mathematics units such as geometry and number sense. Pre- and post-test data measured student spatial reasoning. Qualitative data in the form of student journal responses and teacher and researcher observations further illustrate incidents of learning which benefitted students’ spatial reasoning abilities. Our results found improvement over time. Implications for classroom instruction and further research will be discussed. PubDate: 2022-01-19 DOI: 10.1007/s40751-022-00099-x
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Abstract: Abstract The purpose of this study is to further our understanding of orchestrating math-talk with digital technology. The technology used is common in Swedish mathematics classrooms and involves personal computers, a projector directed towards a whiteboard at the front of the class and software programs for facilitating communication and collective exploration. We use the construct of instrumental orchestration to conceptualize a teacher’s intentional and systematic organization and use of digital technology to guide math-talk in terms of a collective instrumental genesis. We consider math-talk as a matter of inferential reasoning, taking place in the Game of Giving and Asking for Reasons (GoGAR).The combination of instrumental orchestration and inferential reasoning laid the foundation of a design experiment that addressed the research question: How can collective inferential reasoning be orchestrated in a technology-enhanced learning environment' The design experiment was conducted in lower-secondary school (students 14–16 years old) and consisted of three lessons on pattern generalization. Each lesson was tested and refined twice by the research team. The design experiment resulted in the emergence of the FlexTech orchestration, which provided teachers and students with opportunities to utilize the flexibility to construct, switch and mark in the orchestration of an instrumental math-GoGAR. PubDate: 2021-12-10 DOI: 10.1007/s40751-021-00098-4
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Abstract: Abstract We present results of a grounded analysis of individual interviews in which students play Vector Unknown — a digital game designed to introduce visualizing vectors, scaling vectors, vector addition, and vector equations, attending to the geometric and algebraic representations of vectors. The game was designed to be used at the beginning of a linear algebra course, and participants in the study were students who had not previously taken such a course. We categorize the strategies the students employed while playing the game and analyze how these strategies evolved throughout the students’ gameplaying. They range from less anticipatory button-pushing to more sophisticated strategies based on approximating solutions and choosing vectors based on their direction. We found that student focus alternated between numerical and geometric aspects of the game interface, which provided additional insight into their strategies. These results have provided revisions to the game and also informed our team’s plans for incorporating it into classroom instruction. PubDate: 2021-09-30 DOI: 10.1007/s40751-021-00093-9
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Abstract: Abstract While programming was introduced to mathematics classrooms in the 1980s, emerging robotic technologies have encouraged more widespread integration of these technologies to support the development of K–12 students’ mathematical reasoning. The recent emphasis of programming and computational thinking in K–12 education has highlighted the need to prepare future teachers appropriately to incorporate these technologies in their teaching. This study draws from the technology acceptance model and the theory of planned behavior to examine how pre-service teachers’ (PSTs) interaction with robots might influence their intent to use them in teaching. Two groups of such participants engaged in solving mathematical problems using robots in this quasi-experimental pre-/post-test study. Additionally, one of these groups had the chance to design and implement activities that integrated robots with first-grade students. The robots used in this study were Bee-Bots, simple programmable robots that can store up to 40 commands. At the beginning and end of a semester, both groups of participants completed a questionnaire about their perceptions, attitudes, and intentions towards using robots in teaching. In addition, the group who designed and implemented activities with robots provided qualitative reflections about their experience. The study’s quantitative and qualitative findings show that both groups of participants reported significant increases in their intention to use robots in teaching. These findings highlight that opportunities for PSTs to explore, ponder, and experience robotic technologies can promote the integration of these tools in their future teaching. PubDate: 2021-09-26 DOI: 10.1007/s40751-021-00096-6
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Abstract: Abstract When two third-graders collaboratively manipulated a multi-modal, digital learning device called TouchTimes (hereafter, TT), that introduces multiplication through visual, tangible and symbolic means, their thinking about quantity shifted from being additive to being multiplicative. In this study, I examine the children’s interactions around/with TT. My goal is two-fold: (1) to demonstrate the shift between the students’ additive and multiplicative thinking; (2) to explain how their multiplicative thinking emerged around/with TT. The emergence of multiplicative thinking does not refer to the students’ correct computations of multiplicative expressions as a response to verbal or number problems. Instead, drawing on an enactivist perspective, I identify the children’s thinking as their effective bodily reactions to a given unitizing task using TT—and I distinguish their multiplicative and additive thinking based on various researchers’ conceptions of multiplicative thinking. Data was created by video-recording the children’s interaction around/with TT. A retrospective analysis of the data reveals that the children’s effective action to solve the unitizing task developed through a history of recurrent interactions in this environment. PubDate: 2021-09-24 DOI: 10.1007/s40751-021-00094-8
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Abstract: Abstract This article queries how learning analytics systems can support content-specific group formation to develop students’ thinking about a specific mathematical concept. Automated group formation requires identifying personal characteristics, designing tasks to probe students’ perceptions, and grouping them to increase individual learning chances. Designers of automated group formation recommendation modules (GFRMs) rarely consider content-specific objectives. We draw on theories on conceptual learning in mathematics and dialogic thinking to emphasize the role of a dialogic gap between students’ voices to enhance individual learning. In an experiment, fifty 8th and 9th grade students solved three mathematical tasks in a pre-intervention-post-set-up: individually, then in dyads, and then individually again. We used a learning analytics system to collect fine-grained content-specific data on students’ responses based on four pre-defined aspects of the parabola concept. We compared students’ answers with those of their peers in order to identify interpersonal relations. The experiment results indicate that students’ thinking about the parabola concept was the most successfully developed when every group member had a different perception of this concept. We illustrate the learning trajectories of four students and elaborate on the learning sequence of one of these students in particular. This article suggests that the centrality of a dialogic gap in developing personal learning is probably content independent. We thus call for software engineers to think about GFRMs that can support content-specific learning and instruction. PubDate: 2021-09-22 DOI: 10.1007/s40751-021-00095-7
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Abstract: A Correction to this paper has been published: https://doi.org/10.1007/s40751-021-00089-5 PubDate: 2021-08-01 DOI: 10.1007/s40751-021-00089-5
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Abstract: Abstract Although geometric transformations are functions, few studies have examined students’ reasoning about these two important concepts. The purpose of this study was to examine the various ways students reasoned about functions in the context of pre-constructed, dynamic sketches of geometric transformations. We found that, regardless of prior experience, all students were able to reason about important aspects of the notion of function through dragging. Specifically, by using the idea of the semiotic potential of the artifact (the dragging tool), we were able to examine ways in which students with different backgrounds reasoned about functions and how the use of the dragging tool and semiotic mediation contributed to their descriptions of geometric transformations and functions. PubDate: 2021-08-01 DOI: 10.1007/s40751-021-00085-9
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Abstract: Abstract In this article, we provide an empirical example of how digital technology; in this case, GeoGebra may assist students in uncovering—or whiteboxing—the content of a mathematical proof, in this case that of Proposition 41 from Euclid’s Elements. In the discussion of the example, we look into the impact of GeoGebra’s “dragging” functionality on students’ interactions and the possession and development of students’ proof schemes. The study and accompanying analysis illustrate that, despite the positive whiteboxing effects in relation to the mathematical content of the proposition, whiteboxing through dragging calls for caution in relation to students’ work with proof and proving—in particular, in relation to students seeing the necessity for formal proof. Moreover, caution must be paid, e.g., by teachers, so that students do not jump to conclusions and in the process develop inexpedient mathematical proof schemes upon which they may stumble in their future mathematical work. PubDate: 2021-08-01 DOI: 10.1007/s40751-021-00088-6
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Abstract: Abstract Spatial ability is considered a major factor of intelligence and is increasingly important in times of digitization. This article explores the fostering of spatial ability through computer-aided design software. Different notions of spatial ability will be discussed, and, finally, a concept consisting of five aspects will be described. In addition, literature reviews on the connection between the use of computers and the fostering of spatial ability, as well as on the use of 3D printing technology in mathematics education, are given. Building on this, a case study is presented which examines the work of two middle-school students using computer-aided design software within a workshop at the University of Siegen. From the data material, basic possible actions within such software are derived. These are, based on theory, connected with the five aspects of the specific concept of spatial ability used. The results show various perspectives for the fostering of spatial ability with computer-aided design software. PubDate: 2021-08-01 DOI: 10.1007/s40751-021-00084-w
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Abstract: Abstract Digital media have become increasingly important in recent years and can offer new possibilities for mathematics education in elementary schools. From our point of view, geometry and geometric objects seem to be suitable for the use of computer-aided design software in mathematics classes. Based on the example of Tinkercad, the use of CAD software — a new and challenging context in elementary schools — is discussed within the approach of domains of subjective experience and the Toulmin model. An empirical study examined the influence of Tinkercad on fourth-graders’ development of a model of a geometric solid and related reasoning processes in mathematics classes. PubDate: 2021-07-12 DOI: 10.1007/s40751-021-00092-w
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Abstract: Abstract In this article, we examine secondary mathematics teachers’ work with resources using the Documentational Approach to Didactics lens. Specifically, we look at the resources and a teacher’s scheme of use (aims, rules of actions, operational invariants, and inferences) of these resources across a set of lessons (macro-level analysis) that aim towards students’ preparation for the examinations and how this use emerges in a set of three lessons on the same topic (micro-level analysis) as a response to contingent moments. We propose the terms scheming—a teacher’s emerging scheme of use related to the same set of resources used for the same aim—and re-scheming, namely, shifts in such scheming. Our analysis of lesson observations and the teacher’s reflections on his actions from a post-observation interview demonstrate the interplay between the stable characteristics of the scheme of use and the scheming and re-scheming in individual lessons. We conclude this article with a discussion on the methodological potential of using both macro- and micro-level analyses in the investigation of teachers’ use of resources. PubDate: 2021-06-25 DOI: 10.1007/s40751-021-00091-x
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Abstract: Abstract This article is part of a wider research project that has the educational goal of supporting students in the production of conjectures, arguments and proofs, as well as promoting a move from the production of arguments expressed in colloquial registers to arguments expressed in literate registers. In this regard, we Giovannina Albano, Umberto Dello Iacono and Maria Alessandra Mariotti designed and implemented a digital educational environment that allows students to formulate and prove conjectures; three different working areas are available where students can work on a geometrical open problem sometimes individually, sometimes in collaboration. In this article, I report on an empirical study aimed at investigating the functioning of one of these areas, the ‘Working with others’ area, where small groups of students are expected to discuss and formulate a shared solution to a problem. The research question concerns if and to what extent the communication tools, specifically designed to foster students’ collaboration, can promote the production of mathematically acceptable arguments. The qualitative data analysis shows that the ‘Working with others’ area seems to foster discussion within the group and can make students aware of their mistakes. Moreover, it can bring out some students’ misconceptions and can provide useful information upon which the teacher can trigger fruitful discussions. However, this working area does not appear to foster a significant improvement of the production of mathematically acceptable arguments, produced by students in a collaborative and sharing mode. The integration of specific components within this working area seems to be necessary to support the student in moving from argumentation to proof. PubDate: 2021-06-06 DOI: 10.1007/s40751-021-00090-y
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