Paul Drijvers

• Utrecht University

Abstract The digitization of society impacts not only professional and private life, but also education. For the case of mathematics education, the question is how curricula may benefit from digital technology, and how digitization affects curricu- lum policies. To address these questions, we distinguish the intended, the implemented, and the attain...

Many students persistently misinterpret histograms. This calls for closer inspection of students’ strategies when interpreting histograms and case-value plots (which look similar but are different). Using students’ gaze data, we ask: How and how well do upper secondary pre-university school students estimate and compare arithmetic means of histogra...

Grasping mathematical objects as related to processes is often considered critical for mathematics understanding. Yet, the ontology of mathematical objects remains under debate. In this paper, we theoretically oppose internalist approaches that claim mental entities as the endpoints of process–object transitions and externalist approaches that stre...

In addressing the challenge of fostering functional thinking (FT) in secondary school students, our research centered on the question of how an embodied design can enhance FT’s different aspects, including input–output, covariation, and correspondence views. Drawing from embodied cognition theory and focusing on an action- and perception-based task...

Nowadays, mathematics teachers in K–12 strive to promote their students‘ mathematical knowledge and computational thinking (CT) skills. There is an increasing need for effective CT-embedded mathematics learning material and a better understanding of students’ views toward them. In this work, we present the results of a research study, which include...

In our data-driven society, it is essential for students to become statistically literate. A core domain within Statistical Literacy is Statistical Inference, the ability to draw inferences from sample data. Acquiring and applying inferences is difficult for students and, therefore, usually not included in the pre-10th-grade curriculum. However, re...

Interest is key to learning. Video is a promising tool for interest development in education, but professionals in education are in need of more theory-grounded guidance for production, selection, and use of videos. In previous studies, we developed and validated a model on film’s interest raising mechanisms in educational contexts, called the FIRM...

Over the last decades, digital technologies (DTs) have become ubiquitous in mathematics education. Still, their integration into classroom teaching and learning varies enormously. In this narrative overview, we focus on the different purposes for which DTs are used in mathematics education in order to study how the effectiveness of DTs depends on r...

When schools closed in March 2020 due to the COVID-19 pandemic, mathematics teachers and their students were confronted with emergency remote mathematics teaching (ERMT). To investigate how they experienced this in the first months, we set up an online questionnaire for mathematics teachers and their students in Flanders (the Dutch-speaking part of...

In addressing the challenge of fostering functional thinking (FT) among secondary school students, our research centered on the question of how an embodied design can enhance FT's different aspects, including input-output, covariation and correspondence views. Drawing from embodied cognition theory and focusing on action- and perception-based task...

As a first step toward automatic feedback based on students’ strategies for solving histogram tasks we investigated how strategy recognition can be automated based on students’ gazes. A previous study showed how students’ task-specific strategies can be inferred from their gazes. The research question addressed in the present article is how data sc...

Many students persistently misinterpret histograms. Literature suggests that having students solve dotplot items may prepare for interpreting histograms,as interpreting dotplots can help students realize that the statistical variable is presented on the horizontal axis. In this study, we explore a special case of this suggestion, namely, how studen...

Techno-mathematical Literacies (TmL), which are defined as a combination of mathematical, workplace and ICT knowledge, and communicative skills, are acknowledged as important learning goals in STEM education. Still, much remains unknown about ways to address them in teaching and to assess their development. To investigate this, we designed and impl...

To reach for abstraction is a major but challenging goal in mathematics education: teachers struggle with finding ways how to foster abstraction in their classes. To shed light on this issue for the case of geometry education, we align theoretical perspectives on embodied learning and abstraction with practical perspectives from in-service teachers...

Embodied cognition has recently gained increasing attention in mathematics education research. However, little is known about ways to use an embodied approach to reach for mathematical abstraction. In this study, we investigate this topic for the case of functional thinking (FT) using digital technology (DT) through a systematic literature study. W...

In March 2020, many schools worldwide were closed due to the COVID-19 pandemic. This closure confronted mathematics teachers with the challenging transition to emergency remote teaching (ERT). How did students experience ERT, and how did these experiences relate to context variables and to their teachers’ beliefs and practices? In particular, what...

The aim of this closing chapter is to reflect on the book from outside the German mathematics education community. To do so, the introduction briefly sketches the scene for digital technology in mathematics education. Section 17.2 addresses the book and in particular addresses the straddle between the past and the future, between the practice and t...

Introduction Across the world, mathematical expressions are represented very differently in braille. The aim of this study was (1) to gain an overall insight in mathematical braille notations and (2) to investigate how mathematical braille notations support braille readers in reading and comprehending mathematical expressions. Method Twenty teache...

The forced online education during the COVID pandemic in 2020 and 2021 showed many teachers how valuable video can be as a teaching tool. Videos have the potential to raise pupils' interest in educational content. The mechanisms of raising interest, however, have hardly been studied in actual materials used in actual classrooms. This study aims to...

This paper comprises the results of a design study that aims at developing a theoretically and empirically based learning trajectory on statistical inference for 9th-grade students. Based on theories of informal statistical inference, an 8-step learning trajectory was designed. The trajectory consisted of two similar four step sequences: (1) experi...

Braille readers read and comprehend mathematical expressions while moving their fingertips over braille characters. The aim of this exploratory study was to investigate the effect of an intervention that teaches braille readers who use a braille display to use finger movements with a focus on the expression’s mathematical structure. The finger move...

We study the augmented reality sandbox (ARSB) as an embodied learning environment to foster meaning making in the context of bivariable calculus. We present the case of Tiago, a first-year bachelor chemistry student, performing a series of tasks based on embodied design, including perception-based, action-based and incorporation-based tasks. Tiago’...

Embodied cognition theory emphasizes that bodily interaction with the environment is important for all forms of learning, including mathematics. This theoretical trend coincides well with developments in motion responsive technology, and has resulted in numerous embodied technologies for mathematics learning. This review aims to contribute to clari...

The COVID-19 pandemic has confronted mathematics teachers with the challenge of developing alternative teaching practices—in many cases at a distance through digital technology—because schools were closed. To investigate what distance practices in secondary mathematics education have emerged and how teachers experienced them, we set out online ques...

Recent developments in cognitive and educational science highlight the role of the body in learning. Novel digital technologies increasingly facilitate bodily interaction. Aiming for understanding of the body’s role in learning mathematics with technology, we reconsider the instrumental approach from a radical embodied cognitive science perspective...

In this paper we focus on the link between the use of dynamic geometry software and student understanding for the solution of systems of linear equations from an instrumental genesis perspective. Three task-based interviews were conducted with an undergraduate linear algebra student majoring in mathematics education and proficient in using dynamic...

Digital technology is indispensable for doing and learning statistics. When technology is used in mathematics education, the learning of concepts and the development of techniques for using a digital tool are known to intertwine. So far, this intertwinement of techniques and conceptual understanding, known as instrumental genesis, has received litt...

A Correction to this paper has been published: https://doi.org/10.1007/s10649-021-10061-0

Braille readers use a braille display and text-to-speech synthesizer while reading and comprehending mathematical expressions and equations. Teachers need to have technological, pedagogical and content (TPACK) knowledge and skills for exploiting the potential of these devices in mathematics education. They have to understand how the use of assistiv...

Recently, computational thinking has attracted much research attention, especially within primary and secondary education settings. However, incorporating computational thinking (CT) in mathematics or other disciplines is not a straightforward process and introduces many challenges concerning the way disciplines are organised and taught in school....

Hypothesis testing involves a complex stepwise procedure that is challenging for many students in introductory university statistics courses. In this paper we assess how feedback from an Intelligent Tutoring System can address the logic of hypothesis testing and whether such feedback contributes to first-year social sciences students’ proficiency i...

Lay Description What is already known Intelligent tutoring systems often provide feedback at the level of steps within tasks (inner loop) and at the level of multiple tasks (outer loop). Inner loop feedback may support learning by guiding students through the solution process of a single task Outer loop feedback may support learning by advising or...

Computational thinking (CT) has gained much attention over the last decades. Because of its strong focus on problem solving, CT has the potential to contribute to mathematical problem solving and mathematical thinking. Therefore, CT might fruitfully be addressed in mathematics education. How to do so in mathematics education practice, however, is l...

In electronic learning environments, information about a student's performance can be provided to the student in the form of an inspectable student model. While relatively easy to implement, little is known about whether students use the feedback provided by such models and whether they benefit from it. In this study, the use of inspectable student...

While various studies suggest that informal statistical inference (ISI) can be developed by young students, more research is needed to translate this claim into a well-founded learning trajectory (LT). As a contribution, this paper presents the results of a cycle of design research that focuses on the design, implementation, and evaluation of the f...

Nowadays, digital tools for mathematics education are sophisticated and widely available. These tools offer important opportunities, but also come with constraints. Some tools are hard to tailor by teachers, educational designers and researchers; their functionality has to be taken for granted. Other tools offer many possible educational applicatio...

There is increasing attention for the embodied and extended nature of mathematical cognition but the bodies of literature on embodied and on extended cognition have developed mostly separately. We propose a new step in the tradition of networking theories: design research from two theoretical perspectives to promote integration. Embodied design and...

This chapter reports on the work of Working Group 4 and focuses on the integration of digital resources into mathematics teaching and learning practices. There are five central sections, focusing on, instrumental genesis, instrumental orchestration, the documentational approach to didactics, digital resources and teacher education, and the design o...

As cognitive science reports joint action requiring tight intercorporeal coordination between two partners, we aim to evaluate the role of this coordination in computer-supported instrumental genesis for mathematics. In our dual eye-tracking design study we developed an embodied activity that potentially contributes to technologically extended prob...

Assessment is a crucial factor in the implementation of curriculum reform. Little is known, however, on how curriculum changes can be reflected adequately in assessment, particularly if the reform concerns process skills. This issue was investigated for the case of assessing mathematical thinking in a mathematics curriculum reform for 15–18-year-ol...

In technology-rich mathematics education, teachers nowadays experience the challenge to foster both mathematical thinking and computational thinking. To address this issue, the main research question is: How can a teaching-learning strategy, focusing on the use of digital tools, support 16-17 years old pre-university students in developing computat...

The workplace practices of engineers have changed due to the ubiquity of digital technology. So-called techno-mathematical literacies (TmL), seen as a domain specification of 21st-century skills, are essential for future engineers. How these TmL can be fostered in their education, however, is still unclear. To address this issue, we conducted a des...

Histograms are widely used and appear easy to understand. Research nevertheless indicates that students, teachers and researchers often misinterpret these graphical representations. Hence, the research question addressed in this paper is: What are the conceptual difficulties that become manifest in the common misinterpretations people have when con...

Students persistently misinterpret histograms. Based on a literature review we conjectured that the confusion of histograms with case-value plots was leading to inappropriate interpretation strategies. Hence, the question for this research is: what are the most common strategies for secondary school students when estimating the mean in histograms a...

The design and use of online materials for blended learning have been in the spotlight of educational development over the last decade. With respect to didactical courses, however, the potential of online and blended learning seems to be underexplored; little is known about its affordances for teacher education, and for domain specific didactical c...

Hypothesis testing is a challenging topic for many students in introductory university statistics courses. In this paper we explore how automated feedback in an Intelligent Tutoring System can foster students’ ability to carry out hypothesis tests. Students in an experimental group (N = 163) received elaborate feedback on the structure of the hypot...

Throughout the last three decades, mathematics educators have expressed high expectations of the benefits of using digital technology in mathematics education. In retrospect, however, we admit that this integration has not always been as successful and as smooth as we hoped for. What were the main phases in this period of drastic changes with respe...

The scarce research on teachers indicate that they often misinterpret histograms. We conjecture that the confusion of histograms with case-value plots is the source of many misinterpretations. Therefore, the question for this study is: what are the most common strategies of secondary school teachers when interpreting histograms and case-value plots...

Abstract Introduction: Braille readers encounter difficulties when reading mathematical expressions. In this exploratory study, we created a setting to investigate these difficulties. Method: Using a motion-capturing system, we analyzed the tactile strategies of three braille readers while they read mathematical expressions. To compare tactile with...

Teachers are increasingly using video in their lessons, with various aims (eg, to raise students’ levels of conceptual knowledge or interest). Videos that can be used for educational purposes are numerous, ranging from instruction videos to fiction films. Such videos have different characteristics, for example regarding the amount and structure of...

Statistics is a challenging subject for many university students. In addition to dedicated methods of didactics of statistics, adaptive educational technologies can also offer a promising approach to target this challenge. Inspectable student models provide students with information about their mastery of the domain, thus triggering reflection and...

In this response paper to Hoyles’ contribution “Transforming the mathematical practices of learners and teachers through digital technology” focuses on three points. First, more knowledge is needed on why teaching and learning practices should transform, into what will they transform, and by what or by whom will they be transformed. Second, a sugge...

Many people misinterpret histograms. The conjecture is that some of these misinterpretations emerge from the application of interpretation strategies associated with case-value plots. To investigate this, eye-movement data were collected from six university students solving questions on histograms and case-value plots. Analysis of gaze data and cue...

Students’ strategies solving problems about statistical graphs Histograms appear to be easy to interpret. However, research indicates many persistent misconceptions. One common misconception becomes manifest when students estimate the mean from histograms. Students often perceive the bars in a histogram as representing one single case each, instea...

The aim of this closing chapter is to reflect on the content of this book and on its overall focus on the development of mathematical proficiencies through the design and use of digital technology and of teaching and learning with and through these tools. As such, rather than making an attempt to provide an overview of the field as a whole, or tryi...

The benefit of using digital tools in education, and in mathematics education in particular, is subject to debate. To investigate this benefit, we focus on effect sizes on student achievement reported in reviews of experimental and quantitative studies. The results show significant positive effects with modest effect sizes. Possible causes for this...

Digital assessment of mathematics is becoming widespread, but still comes with limitations and constraints. A central question is how to design digital tests that assess mathematical knowledge in a valid way. Based on literature on validity and on assessment with and through technology, we identify arguments for and opportunities of digital assessm...

The use of technology and other resources for mathematical learning is a current issue in the field of mathematics education and lags behind the rapid advances in Information and Communication Technology. Technological developments offer opportunities, which are not straightforward to exploit in regular teaching. In CERME10 TWG16, the recent resear...

Due to increased use of technology, the workplace practices of engineers have changed. So-called techno-mathematical literacies (TmL) are necessary for engineers of the 21st century. Because it is still unknown which TmL engineers actually use in their professional practices, the purpose of this study was to identify these TmL. Fourteen semi-struct...

An instantly graphable formula (IGF) is a formula that a person can instantly visualize using a graph. These IGFs are personal and serve as building blocks for graphing formulas by hand. The questions addressed in this paper are what experts’ repertoires of IGFs are and what experts attend to while recognizing these formulas. Three tasks were desig...

The use of technology and other resources for mathematical learning is a current issue in the field of mathematics education and lags behind the rapid advances in Information and Communication Technology. Technological developments offer opportunities, which are not straightforward to exploit in regular teaching. In CERME10 TWG16, the recent resear...

Springer open access from https://www.springer.com/gp/book/9783319336657

In an increasing number of mathematics classes throughout the world, technology is being used for the teaching and learning of mathematics. But knowledge is limited about the long-term development of students’ mathematical thinking when learning mathematics with the use of technology. This article reports on the development of a student and the rol...

To analyse the use of a dynamic geometry environment for learning the notion of parameter in linear algebra by two students, we combine a semiotic mediation approach with a lens of students’ thinking modes. This manner of approaching the data promises a detailed understanding of the observed phenomena.

Due to increased use of technology, the workplace practices of engineers have changed. So-called Techno-mathematical Literacies (TmL) are necessary for engineers of the 21 st century. Because it is still unknown which TmL engineers actually use in their professional practice, the purpose of this study is to identify these TmL. Fourteen semi-structu...

Students have difficulties in analysing statistical data and drawing correct inferences from data (Boels et al., 2015). According to Garfield and Ben-Zvi students should know “How to graph the data as a first step in analysing data” which can be used for a quick and global answer “to the question of interest” (2008, p.15). Students make several mis...

Students make several mistakes when interpreting graphs with statistical data even with seemingly simple graphs such as histograms, for example when comparing two graphs (Friel, Curcio, & Bright, 2001; Lem et al., 2013). This review is a first step in revealing possible causes of students’ difficulties with histograms. Note that we now have an art...

This book addresses the problématique of the teaching and learning of school algebra. The problématique addressed by this book can be posed in general terms as follows. On the one hand, algebra is considered a central subject to be studied in junior high and high schools in almost all educational systems around the world. The reasons for this centr...

This book addresses the problématique of the teaching and learning of school algebra.‘ Problématique’ is a word borrowed from French and it describes not just problems in isolation, but comprehensive challenges posed by a certain large-scale theme. In other words, it consists of articulating questions and dilemmas arising in a certain area. The pro...

Available from Springer Open : http://link.springer.com/book/10.1007%2F978-3-319-33666-4

In this paper we investigated the relationship between the use of a digital tool for algebra and students' algebraic understanding from an instrumentation theory perspective. In particular, we considered the schemes that students developed for algebraic substitution using an applet called Cover-up. The data included video registrations of three sev...

This topical survey establishes an overview of the current state of the art in technology use in mathematics education from an international perspective, including both practice-oriented experiences and research-based evidence. Three themes are discussed: Evidence for effect; Digital assessment; Communication and collaboration. Exploiting the oppor...

In this chapter, we revisit Michèle Artigue’s classic 2002 IJCML article and draw out what we consider to be the core theoretical ideas and key dimensions of the body of work on tools and tool use that Michèle not only elaborated but also inspired others to further develop. We trace the evolutionary path of these core ideas, noting the ways in whic...

The use of digital tools in algebra education is expected to not only contribute to master skill, but also to acquire conceptual understanding. The question is how digital tools affect students” thinking and understanding. This paper presents an analysis of data of one group of three grade seventh students (12-13 year-old) on the use of a digital t...

It is still largely unknown what are effective and efficient strategies for graphing formulas with paper and pencil without the help of graphing tools. We here propose a two-dimensional framework to describe the various strategies for graphing formulas with recognition and heuristics as dimensions. Five experts and three secondary-school math teach...

Digital technology plays an increasingly important role in daily life, mathematics education and algebra education in particular. To investigate the effect of a technology-rich intervention related to initial algebra on the achievement of 12–13 year old Indonesian students, we set up an experiment. The experimental group’s intervention focused on e...

The integration of digital technology confronts teachers, educators and researchers with many questions. What is the potential of ICT for learning and teaching, and which factors are decisive in making it work in the mathematics classroom? To investigate these questions, six cases from leading studies in the field are described, and decisive succes...

The notion of didactical engineering has influenced and characterized contemporary research in mathematics education in France to an important extent. In this paper, we address the following from an insider’s and an outsider’s perspective: (1) the way this notion is theoretically grounded, (2) the kinds of design research practices has it led to an...

The integration of digital technology into secondary mathematics education is not yet a widespread success. As teachers are crucial players in this integration, an important challenge is not only to attract early adopters, but also to support mid-adopting teachers in their professional development on this point. The questions addressed in this Chap...

The integration of digital tools in mathematics education is considered both promising and problematic. To deal with this issue, notions of webbing and instrumental orchestration are developed. However, the two seemed to be disconnected, and having different cultural and theoretical roots. In this article, we investigate the distinct and joint jour...

Online resources are widely used for educational purposes, such as the training of skills. For algebra education in particular, online resources are expected to contribute to skill mastery in an efficient and effective way. However, studies that underpin these claims through a randomized experiment are scarce. To experimentally investigate the effe...