Available Mathematics Education research projects:

Digital innovations and technologies (e.g. robotics, industrial automation, artificial intelligence, blockchain, virtual realities) play an essential role for Europe’s prosperity. An immense challenge about digitalization is that it transforms our labour markets with rapid pace (e.g. the labour demand of businesses and industry with regards to variety, types and amount of jobs in ICT sectors). However, a large majority of EU28 countries have recently experienced recruitment difficulties due to a substantial lack of suitable candidates (STEM Alliance, 2017) and only 15% of tech sector workers in the EU are women. Encouraging women into this [digital] field and building a gender balanced tech sector will play an important role in order to boost innovation and bring economic benefits to the European Economy. The European GEM-project investigates to what extent summer camps can contribute to girls’ attitudes and beliefs towards digital and entrepreneurial skills in the STEM domain. For more information see: https://icse.eu/international-projects/gem.

The EU-project will provide research instruments like questionnaires and interview schedules for combining quantitative and qualitative methods. The overarching question for your research project will be: How do summer camps impact the girls’ attitudes and beliefs towards STEM?

### Some references

Bielefeld, K. (2019). Female STEM Role Models: Increasing Girls in STEM Fields. https://blog.mimio.com/female-stem-role-models-increasing-girls-in-stem-...

Chapman, S. ., & Vivian, R. (2016). Engaging the future of STEM - A study of international best practice for promoting the participation of young people, particularly girls, in science, technology, engineering and maths (STEM). https://cew.org.au/wp-content/uploads/2017/03/Engaging-the-future-of-STE...

Milgram, D. (2011). How to Recruit Women and Girls to the Science, Technology, Engineering, and Math (STEM) Classroom. Technology and Engineering Teacher, 71(3), 4–11.

Rainey, K., Dancy, M., Mickelson, R., Stearns, E., & Moller, S. (2018). Race and gender differences in how sense of belonging influences decisions to major in STEM. International Journal of STEM Education, 5(1), 10. https://doi.org/10.1186/s40594-018-0115-6

Sjøberg, S., & Schreiner, C. (2010). The ROSE project. An overview and key findings. March, 1–31. http://roseproject.no/network/countries/norway/eng/nor-Sjoberg-Schreiner...

Tan, E., Calabrese Barton, A., Kang, H., & O’Neill, T. (2013). Desiring a career in STEM-related fields: How middle school girls articulate and negotiate identities-in-practice in science. Journal of Research in Science Teaching, 50(10), 1143–1179. https://doi.org/10.1002/tea.21123

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The corona crisis teaches us many things about disease, human behavior and about the importance of models. Measures such as social distancing, extending the number of IC units, and the closure of social life are taken mainly on predictions made by models of the way the disease will spread and develop.

As a consequence also the concept of models has been under scrutiny in the media and in politics. Responses vary from taking the models very or even too serious, to pointing out that models have nothing to do with reality.

For this reason it is interesting to investigate how the public, including secondary students, process the information on models and if and how they change their views on scientific models in general, and on the corona epidemic in particular. For this I propose a survey study, using an online questionnaire that includes questions on:

- Understanding of scientific models in general (based on work by Susanne Jansen)
- Understanding of the meaning of corona-related models
- Understanding of the basic concepts of the corona-related models (R0, exponential growth etc.
- Understanding of uncertainty in models.
- Awareness of the need for models.

The questionnaire will be spread broadly, using networks of teachers and students, social media, and other networks.

Rogier Bos and Ralph Meulenbroeks

The current pandemic of the coronavirus has led many governments to close school buildings as part of their containment effort. Teachers have responded at impressive pace by setting up new arrangements moving all their educational activity to online environments. But what will be the effects of this shift in the longer run? What is needed to make online education sustainable over longer periods of time? The Mathematics D Online project in the Netherlands provides a years-long experience providing online education at the secondary school level. Studying the project and the involved teachers and students might provide insight in the questions raised above.

The regular (not online) Mathematics D course is optional for students aged 15-18. Unfortunately, not many students opt for the course and several schools have stopped offering it because of the relatively high costs associated with teaching small groups. Mathematics D Online attempts to solve this issue by enabling schools to offer the course with reduced hours for teachers involved The course presently offers a complete online arrangement with schedules and exercises, an online course book, online videos (instructional and motivational), online hand-in exercises with feedback provided by online assistants, exams, and test exams. The teacher remains responsible for setting and marking the exams, and for providing one hour per week of face-to-face time with the student. The courses may thus rightly dubbed “blended” in nature, with a ratio of on- and offline work of about 80/20.

According to a recent review by Boelens and colleagues (Boelens, De Wever, & Voet, 2017) there are four key challenges to the design of blended learning:

- incorporating flexibility
- facilitating interaction
- facilitating students’ learning processes
- fostering an affective learning climate

A recent internal report (Bos, 2019) describes the design choices within Mathematics D Online in the light of these challenges. A concrete major challenge is the decreasing engagement as time goes by. This raises questions on the motivational support during the course.

To our knowledge research into blended education addressing these challenges on the secondary level is very limited. The aim of the research proposed here is to fill that gap by studying the effects of the design choices made for the Mathematics D Online course.

The focus of your research, as a student, could be one or two of many questions, for example:

- What is the effect on motivation of the educational video clips? Do the educational videos provide enough autonomy and competence support?
- To what extent do the hand-in tasks, provided with feedback, contribute to the learning process or/and to the motivation of students, as reported by the students?
- To what extent is face-to-face education essential for motivating the students?
- How can community building between students and teachers be facilitated?
- How can self-regulation and self-assessment be stimulated and facilitated?

### References

Boelens, R., De Wever, B., & Voet, M. (2017). Four key challenges to the design of blended learning: A systematic literature review. *Educational Research Review,* Vol. 22, pp. 1–18. https://doi.org/10.1016/j.edurev.2017.06.001

Bos, R. D. (2019). Eindrapportage Project Blended Learning Wiskunde D. Intern rapport.

*Contact:* Anna Shvarts

Imagine a traditional classroom: a teacher is giving a lecture and the students are trying hard to follow the explanations. For some students, the material is boring, as they know everything ahead, and the others are barely able to follow as new terms are popping up all the time in the teacher’s speech without any obvious connection with drawings on the blackboard. New interactive technology promises to provide adaptive education by paving individual learning trajectory for any student. The problem is that technological tools are still much slower than the dynamics of individual tutors’ scaffolding of the students (Belland, 2017). This project aims to understand the complexity of effective student-tutor interaction in one-to-one settings with a distant aim of contributing to the technologically supported adaptive learning.

Joint visual attention between a student and a tutor means focusing on the same objects and similar understanding of their meanings. How do students come to see and name mathematical objects in a way that is similar to the teachers? Investigation of infants shows that naming the objects, which are in the focus of infants' attention is more effective for vocabulary learning than redirecting their attention to new objects (Tomasello & Farah, 1986). Theoretically, this project relies on an embodied approach to teaching and learning that stresses the complexity of student-teacher interaction. A student does not just follow the teacher, but actively participates in multimodal flow of teaching-learning interaction, where speech is combined with gestures to the visual display. It is crucial for the student to connect different modalities into one meaningful node (Radford, 2009). To understand the key aspect on this interaction, we introduce a notion of micro-zone of proximal development, which is a moment when a student and a tutor are coordinated in one modality (e.g., attending to the same visual display), but are discoordinated in other modality (e.g., they do not have a shared language to refer to it) (Shvarts & Abrahamson, 2019). In this project, we will investigate if micro-zone of proximal development is a moment when a teacher's intervention is particularly effective and lead to joint attention to the same mathematically meaningful object.

The methodology of the project lies on the crossroads of two technological innovations: touchscreens’ advantages for mathematical learning and dual eye-tracking’ advantages for investigation of the interaction between students and teachers. The tutors will support students’ problem-solving in technological activities for trigonometry (Shvarts, Alberto, Bakker, Doorman, Drijvers), that are designed in the Digital Mathematics Environment (DME) of the Freudenthal Institute. Dual eye-tracking, namely a synchronous tracking of two people's overt attention, will provide information about the visual perception of a student and a tutor's (see a video with a sample of data). At the same time, video and audio records will provide information about the participant's verbal and gestural utterances. Analysis of coordination and discoordination between modalities will shed light on the teaching-learning progress.

The study is a part of a broader experimental investigation of embodied processes in the introduction of mathematical tools. We invite the mastery students, who are interested in technology in mathematics education, in embodied processes and/or in acquiring skills of conducting research with innovative equipment such as dual eye-tracking.

Belland, B. R. (2017). *Instructional scaffolding in STEM education*. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-02565-0

Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. *Educational Studies in Mathematics*, 70(2), 111–126. https://doi.org/10.1007/s10649-008-9127-3

Shvarts, A., & Abrahamson, D. (2019). Dual-eye-tracking Vygotsky: A microgenetic account of a teaching/learning collaboration in an embodied-interaction technological tutorial for mathematics. *Learning, Culture and Social Interaction*, 22, 100316. https://doi.org/10.1016/J.LCSI.2019.05.003

Shvarts, A., Alberto, R., Bakker, A., Doorman M., Drijvers P., (in press) Embodied instrumentation: Reification of sensorimotor activity into a mathematical artifact. In *Proceedings of the 14th International Conference on Technology in Mathematics Teaching*

Tomasello, M., & Farrar, M. J. (1986). Joint attention and early language. *Child Development*, 57(6), 1454. https://doi.org/10.2307/1130423

*Contact:* Anna Shvarts

New technologies can help in achieving the ambition to make mathematics learning an exciting and memorable experience. Students have an opportunity to study mathematical content in a new form by manipulating mathematical objects on tablets or interactive whiteboards and smart tables, which become more and more accessible in ordinary classrooms. These technological enhancements require educational researchers to provide clear answers about the efficiency and limitations of interactive educational designs.

One of the promising new technologically enhanced activities in mathematics teaching is embodied design (Abrahamson, 2014). This design genre provides the students with an opportunity to develop new ways of seeing mathematical notations in sensory-motor tasks. The students manipulate elements on the large screen and receive immediate feedback, thus developing new perceptive strategies, which are called attentional anchors and can be traced in their gazes patterns (Duijzer, Shayan, Bakker, Van der Schaaf, & Abrahamson, 2017). These attentional anchors help students seeing the mathematical meaning behind visual inscriptions: for example, seeing a unit circle not as just a round shape, but also a way of approaching trigonometric functions.

However, the role of verbal communication with the teacher in these tasks is still under investigation. Are embodied manipulations with technological artifacts sufficient for perceiving mathematical meaning? Alternatively, do students necessarily need to talk about their embodied experience?

To answer these questions, we designed embodied tasks for trigonometry (Shvarts, Alberto, Bakker, Doorman, & Drijvers, 2019), and now we intend to conduct an experimental laboratory study with a learning table from Teaching and Learning Lab and Pupil-Labs eye-trackers. Three experimental groups will take part in the study. In the first group, the students will solo manipulate interactive elements. In the second group, a student will instruct another student on how to manipulate within the environment. And in the third group, a student will describe her experience to a tutor. After this embodied learning phase, the students will solve trigonometric problems with the support of technological tools.

We will record the learning process by eye-tracking of the students’ gazes synchronized with logging of their manipulations on the large smart table (see a video with sample of data). The analysis requires mixed qualitative and quantitative approaches. After coding the emerged sensory-motor patterns, we will conduct a between-group analysis of the learning gains and the activation of emerged patterns in the students’ problem-solving process.

We invite a master student with interest to develop skills in usage of contemporary educational technology and eye-tracking research equipment and with curiosity in a deep understanding of cognitive processes in teaching and learning.

Abrahamson, D. (2014). Building educational activities for understanding: An elaboration on the embodied-design framework and its epistemic grounds.* International Journal of Child-Computer Interaction*, 2(1), 1–16.

Duijzer, C. A. C. G., Shayan, S., Bakker, A., Van der Schaaf, M. F., & Abrahamson, D. (2017). Touchscreen Tablets: Coordinating Action and Perception for Mathematical Cognition. *Frontiers in Psychology*, 8, 144.

Shvarts, A., Alberto, R., Bakker, A., Doorman, M., & Drijvers, P. (2019). Embodied collaboration to foster instrumental genesis in mathematics. In K. Lund, G. P. Niccolai, E. Lavoué, C. Hmelo-Silver, G. Gweon, & M. Baker (Eds.), A Wide Lens: *Combining Embodied, Enactive, Extended, and Embedded Learning in Collaborative Settings, 13th International Conference on Computer Supported Collaborative Learning (CSCL) 2019, Volume 2* (pp. 660–663). Lyon, France: International Society of the Learning Sciences.

*Contact:* Paul Drijvers

In this research project, two recent developments will be integrated, one practical and one theoretical. From the practical perspective, digital tools for math and science education have become more sophisticated over the last decades and are getting widespread. The recent start of the Freudenthal Institute’s Science Teaching and Learning Lab (STLL) witnesses the relevance of this development in our institute.

One particular case of digital tools that may be used in mathematics education are motion sensors, that can be connected to handheld devices and laptops, and that can be used to measure motion, distance and speed. Such sensors may play a central role in exciting student activities (e.g., see Arzarello & Robutti, 2001). The effect of such activities on learning, however, is still unclear.

A theoretical perspective that may help here is the notion of embodied cognition. Core in embodied cognition is the awareness that mind and body are not separated entities, but that cognition is grounded in bodily experience. Perception and movement may drive cognitive development, and sensori-motor schemes form the basis for cognitive schemes (Lakoff & Núnez, 2000). The question is whether this view on cognition may help to exploit and explain the opportunities offered by the use of motion sensors in teaching the topics of graphs and functions to lower secondary students.

To investigate this question, introductory tasks will be designed for the topic of graphs and functions. These tasks will be field tested, either in the STLL, in local schools, or in a science museum (cf. Nemirovski, Kelton, & Rhodehamel, 2013). Both the design and the data analysis will be guided by notions from embodied cognition. The results should inform teachers, designers and researchers on the use of such digital tools in mathematics education.

For further details, please contact Paul Drijvers.

Arzarello, F. & Robutti, O. (2001). From Body Motion to Algebra through Graphing. In *Proceedings of the 12th ICMI Study Conference*, Vol. 1, 33-40.

Lakoff , G. & Núnez, R. (2000). *Where mathematics comes from: How the embodied mind brings mathematics into being*. New York: Basic Books.

Nemirovsky, R., Kelton, M.L., & Rhodehamel, B. (2013). Playing mathematical instruments: Emerging perceptuomotor integration with an interactive mathematics exhibit. *Journal for Research in Mathematics Education, 44*(2), 372-415.

*Contact:* Rogier Bos

In the new curricula for mathematics in HAVO-VWO in the Netherlands one finds a renewed attention for mathematical thinking (wiskundige denkactiviteiten). The curriculum designers want to move away from learning mathematics solely by memorizing sets of routines. Mathematical problem solving is a central constituent of mathematical thinking. To solve a problem, one needs to combine mathematical activities that one mastered before. So one needs to make strategic decisions, on a level that transcends the procedural.

A way to guide a student with these strategic decisions is by providing heuristics. A heuristic (Pólya, 1945) is a general strategy to attack a problem, e.g., investigate special cases. In this research we study how such hints and heuristics should be structured in a course. About heuristics Schoenfeld claims (1985, p. 73): “many heuristic labels subsume half a dozen strategies or more. Each of these more precisely defined strategies needs to be fully explicated before it can be used reliably by students”. So how should the use of these heuristics be built up? The idea under investigation is that hints in a course should be developed parallel to the way mathematical thought develops. Central in development of mathematical thought is a cognitive process called compression (Thurston, 1990, Barnhard & Tall, 2001). Compression is a transition in the mind from isolated procedural steps to more integrated processes, ending in cognitive units that Tall calls procepts. With a cognitive unit Tall means “a piece of cognitive structure that can be held in the focus of attention all at one time” (p.1). Hints and heuristics may represent such cognitive units, heuristics being the highly compressed ones.

The goal of this research project is to contribute to the teaching of mathematical thinking and in particular problem solving in HAVO-VWO. To this purpose, a course in the Digital Mathematics Environment (DME) of the Freudenthal Institute will be designed. The DME offers the possibility to build a structure of hints and heuristics and to monitor the students use of these.

The designed course will be field tested with secondary school students . The first result for these student should be that they develop compressed procepts in the domain of the course, and use these in their reasoning. A second result should be that students learn to ask for hints on the right level of their mathematical development, not continue to ask for low procedural hints, when they should be able to handle heuristics (cf. Roll et al., 2014).

Barnard, A. D. & Tall, D. O. (2001). A Comparative Study of Cognitive Units in Mathematical Thinking, *Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 2*, 89–96. Utrecht, The Netherlands.

Pólya, G. (1945). *How to solve it*. Princeton university press.

Roll, I., Baker, R.S.J.D., Aleven, V., & Koedinger, K. R. (2014). On the benefits of seeking (and avoiding) help in online problem solving environment. *Journal of the Learning Sciences*, 23(4), 537-560.

Schoenfeld, A. H. (1985). *Mathematical Problem Solving*. Orlando, FLA: Academic.

Thurston, W. P. (1990). Mathematical education, *Notices of the AMS*, 37, 844-850.

Contact: Lonneke Boels. Supervisors: Arthur Bakker and Paul Drijvers

### Short description of the general problem

Students have various conceptual difficulties with interpreting and making inferences from histograms. For example, students compute the mean of the frequencies instead of the mean of the data, see Figure 1. There are several hypotheses about the causes of these conceptual difficulties. For instance, students may not know that despite the two axes histograms represent univariate, not bivariate data (Cooper & Shore, 2010) and students may not know the required measurement level of data that can be represented in a histogram. The aim of this study is to reveal more precisely students’ conceptual difficulties and possible causes for their mistakes in interpreting histograms. To this end we will investigate the eye patterns of secondary school students (4 – 5 VWO) when they answer questions about histograms and other graphs with bars on a computer screen. We will also use a think-aloud protocol combined with video as input for qualitative analysis. Furthermore, areas of interest (AoI), fixation times, percentage of correct answers and time on task will be analysed (Holmqvist et al., 2011). The findings will be input for the design of teaching materials on this topic.

Figure 1 - Correct way of finding the mean in a bar graph of bivariate data (left) and a histogram of univariate data (right). Several students use the method that is correct for the left type of graph also when finding the mean in a histogram.

Figure 2. Histogram (a) and case-value plot (b) with heat map overlay. For the labels on the axis, see Figure 1. The arrow (a) points at the fixations around frequency 7 when students are asked to find the mean in this histogram (Boels, Ebbes, Van Dooren and Drijvers, in press).

In this master research you are involved in the whole process of this research. Together and in close contact with the researcher you will work with an eye-tracker and with secondary school students. You will also learn to analyse the data of an eye-tracker and how to use these data for qualitative and quantitative analyses. You will transcribe, and code the thinking aloud as a second researcher. You have the possibility to choose your own research focus that will be included in this study. The first pilots of this study have been done last winter. Refining pilots are foreseen for April after which the final study starts. Co-authorship of a journal article is an option.

**Data available:** Every Thursday and Friday after 11 am from 21 February onward.

**Key references:**

Cooper, L. L., & Shore, F. S. (2010). The effects of data and graph type on concepts and visualizations of variability. Journal of Statistics Education, 18(2).

Holmqvist, K., Nyström, M., Andersson, R., Dewhurst, R., Jarodzka, H., & Van de Weijer, J. (2011). Eye tracking: A comprehensive guide to methods and measures OUP Oxford.

*Contact:* Paul Drijvers. Co-supervisors Peter Boon and Sietske Tacoma.

Online resources for mathematics are widely available. The Freudenthal Institute’s Digital Mathematics Environment (DME) provides a rich set of digital activities for students, including teaching materials for geometry through the use of the software Geogebra, which is embedded in the DME.

Core issues in digital resources for mathematics education are automated scoring of student responses, adaptive feedback, and partial credit for relevant steps in the solution processes. To a limited extent, these facilities for geometry are available in the DME, but further refinement is needed. The aim of this research project, therefore, is to investigate how feedback and automized scoring including partial credit can be further developed in the DME and how students benefit from these features.

To do so, a theoretically underpinned student model for geometry in grade 9 will be designed. Based on this model, the automated scoring and feedback features of an existing module will be extended. The viability of the extensions will be assessed through a teaching experiment.

*Contact persons*: Annemiek van Leendert and Michiel Doorman.

In January 2015 the research project ‘ Improving the mathematical abilities of braille-dependent students’ started. In the first part of this project we will investigate how we can support braille-dependent students with reading and processing mathematical expressions. Our hypothesis is that insight in how sighted students perceive these expressions can give clues for support to braille-dependent students.

Let’s take, for example, the equation:

This equation has global characteristics, e.g. the ‘=’ sign and the sqrt-sign, and the equation has local characteristics, e.g. the 2 of x2 and the ‘+3’ part. A sighted person sees immediately that this is an equation, and almost immediately that this is an equation of two square roots. These insights provide direction to cognitive strategies for solving the problem.

How does a braille-dependent student perceive this equation? He reads this expression, with a braille-display connected to the laptop, in a Word-document. This is only possible when the expression is displayed in a linear form: sqrt(x^2/2) = sqrt((x+3)/2).

He reads the expression on a braille-display from left to right which makes it difficult to make a distinction between local and global characteristics. Therefore it is very hard to get an overview of the expression.

*Research questions*

The aim of this study is to better understand how braille-dependent and sighted students read and make sense of algebraic expressions and equations. We will perform an eye tracking sub study for the sighted students and a finger tracking sub study for the braille-dependent students.

The main questions of the eye tracking sub study and the finger tracking sub study are:

- What are the similarities and differences between the cognitive strategies of braille-dependent and sighted students?
- How are these similarities and differences related to tactile respectively visual strategies and to task characteristics?

For each sub study the following two research questions is addressed.

*Finger tracking sub study*

F1: What is the relation between the tactile strategies, cognitive strategies and the complexity of the task (in structure and in use of operations and numbers)?

*Eye tracking sub study*

E1: What is the relation between the visual strategies, cognitive strategies and the complexity of the task (in structure and in use of operations and numbers)?

We are looking for a master students to either contribute to the eye tracking study or to the finger tracking study.

The instruments for the studies are under development. A first set of tasks is designed. Methods for analyzing the eye-tracking data are based upon differences between number of fixation, duration, and switches and dwelling. Methods for analyzing the tracking data are under construction. The student will have to contribute to the optimization of these instruments, the data collection and the analysis.

*Contact:* Paul Drijvers

New mathematics curricula for havo and vwo were implemented in 2015. In these curricula, so-called Mathematical Thinking Activities are integrated, that should address problem solving, analytical thinking, and abstraction (cTWO 2012, Drijvers 2015, Van Streun & Kop 2012).

One of the important arguments behind this focus on mathematical thinking was the aim to make mathematics in school resemble more the mathematics as mathematicians perceive it, and to make students do mathematics more in a way that mathematicians do. The question, however, is: what is central in doing mathematics, according to mathematicians, what are the corresponding thought processes, and how well are these reflected in current didactical theories on the learning and teaching of mathematics?

To address this issue, the research project includes a literature review and an interview study. In the literature review, knowledge about mathematical thinking will be synthesized. A provisional literature database is available as a starting point. In the interview study, a number of professional mathematicians will be interviewed. In this structured interview, the first part will address the work of the mathematicians themselves. The second part will address current theories on the learning of mathematics (e.g., Sfard 1991, Tall 2013) to see if the interviewees can connect these theories to their own experience. Three of such interviews have already taken place and are available. The interviews will be analyzed with software for qualitative data analysis (e.g., Atlas ti). The results will shed light on (1) what mathematicians consider core in their work of doing mathematics, and (2) if this matches with the notion of mathematical thinking and with the current theories in the field of mathematics didactics.

cTWO (2012). Denken & doen. Wiskunde op havo en vwo per 2015. www.ctwo.nl.

Drijvers, P. (2015). Denken over wiskunde, onderwijs en ICT. Inaugurele rede. Utrecht: Universiteit Utrecht. http://www.fisme.science.uu.nl/publicaties/literatuur/Oratie_Paul_Drijvers_facsimile_20150521.pdf.

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36.

Tall, D. (2013). How Humans Learn to Think Mathematically: Exploring the Three Worlds of Mathematics. Cambridge, MA: Cambridge University Press.

Van Streun, A., & Kop, P. (2012). Wiskundige denkactiviteiten. In Drijvers, P., Van Streun, A., & Zwaneveld, B., Handboek Wiskundedidactiek (pp. 339-368). Amsterdam: Epsilon.

### Year: 2018-2019

*Contact:* Arthur Bakker and Rosa Alberto

**The general problem:**

The theory known as Embodied Cognition argues that cognition is shaped by bodily experiences in interaction with the world, and is not just a matter of representation in the brain. The amounting evidence for this theory has led educational researchers to include data of action (hand movement) and perception (eye movements) in studies of learning, even in the domain of abstract mathematics. Moreover, educational researchers also think through the implications for embodied design (see https://embodieddesign.sites.uu.nl/). In the current project, titled The Digital Turn in Epistemology, we apply this theory to the domain of mathematics, in particular proportion and function. This project is funded by NWO and Noordhoff Publishers, and collaborates with the Digital Mathematics Environment (DME). We intend to integrate handwriting recognition with embodied activities.

Together with our international collaborator Dor Abrahamson from University of California, Berkeley, we have designed an embodied touch-screen application for students to learn interactively how to solve a simple proportion problem such as 2:4 = 4:? (Figure 1). See link (goo.gl/G8S1B9) for the task in the app store (IOS) or the web version (provided privately if you are interested). We have conducted eye-tracking studies with over a 100 students in primary and vocational schools (age 9-12), and recorded the hand movement, gaze behavior and thinking aloud conversation of the students.

We study how action (what students do), perception (what they see), and reasoning (what they say) interact to solve proportional tasks or tasks involving functions (say quadratic). We have found that students often are able to succeed in a proportion task before they can express verbally that it is about proportion.

We continue data collection and analysis to investigate how the app can stimulate and support proportional learning. Where we started with normal iPad-screens, we now use larger screens (iPad Pro) and intend to use large Ricoh multitouch screens in the Teaching and Learning Lab. We also intend to study pairs of students in interaction rather than individual students. We also intend to incorporate handwriting recognition software being developed within the DME project. The long-term goal of this project is to explore how mathematics education can benefit from embodied cognition research so that students can ultimately develop a better grip on mathematics.

*Figure 1. Sample screen shots from enacting the proportion app. The task is to move two cursors orthogonal directions in an interactive task and try to get a green feedback on the screen. The green feedback occurs only if one puts his/her fingers at locations that have the same pre-set proportional distance from a base point (for example 1:4)*

**Possible questions:**

This project challenges you to think about alternative, embodied ways in which students may acquire mathematical knowledge. It offers a methodologically challenging micro-study of mathematics learning. Eye-tracking data collection and analysis can be part of this project. You can define your research questions within the already existing data, or expand the project and collect more data.

The process will include a training session for coding and then the coding of a subset of existing data. This will help you generate your own research question. Within the embodied framework several directions can be thought of:

- Project 1: Our app enables students to solve proportional problems (for example 1:2 = 2:?) under different task conditions. In all conditions students are asked to find the green. In the original task students do this by moving two bars up-and-down (Figure 2a). In another variant of this task, students manipulate dimensions of a shape (in this case a rectangle) by moving up-and-down and left-to-right (Figure 2b). A yet untested task variant is for students to manipulate the slope of a line (Figure 2c). The main question is whether and how portraying the same proportion problem, under different representational conditions, lead to different solution strategies and conceptual development? What are the educational benefits and implications of various designs of the app? We also intend to design more complex functions for older students (e.g., tasks about quadratic equestions).

*Figure 2. Different conditions of the app with the same proportional problem.*

- Project 2: Many embodied tasks are focused on individuals learning in interaction with the app. In this project we aim to explore possibilities for collaborative embodied learning. We envision students working together on proportion task on larger tablets or the interactive multi-touch learning table (Ricoh) available in the Teaching and Learning Lab (TLL: https://teachinglearninglab.sites.uu.nl/).

Collaborative learning also means we aim to track movements and gaze of multiple students (dual and multi-eye tracking). Questions that are of importance are: How do students learn when they are collaborating? What are advantages and drawbacks in exploring embodied design apps collaboratively?

### References:

Abrahamson, D., Shayan, S., Bakker, A., & Van der Schaaf, M. (2016). Eye-tracking Piaget: Capturing the emergence of attentional anchors in the coordination of proportional motor action. Human Development, 58(4-5), 218–224. doi:10.1159/000443153.

Duijzer, C. A. C. G., Shayan, S., Bakker, A., Van der Schaaf, M. F., & Abrahamson, D. (2017). Touchscreen tablets: Coordinating action and perception for mathematical cognition. Frontiers in Psychology, 8, 1–19. doi:10.3389/fpsyg.2017.00144.

### Master Theses from previous students in the project:

Duijzer, Carolien. (2015). How Perception Guides Cognition: Insights from Embodied Interaction with a Tablet Application for Proportions – An Eye-Tracking Study. (Unpublished Master Thesis). Utrecht University.

Cuiper, Anne-Ciska. (2015). Vocational students’ search patterns while solving a digital proportion task of the Mathematical Imagery Trainer For Proportions application. (Unpublished Master Thesis). Utrecht University.

Veugen, Marijke. (2016). Defining Mathematical Proportional Embodied Learning Within Eye Tracking Measurements. (Unpublished Master Thesis). Utrecht University.

Boven, Loes. (2017). Coordination of Action and Perception Processes in an Orthogonal Proportion Tablet Task. (Unpublished Master Thesis). Utrecht University.

*Contact*: Paul Drijvers and Sylvia van Borkulo

The new curricula for pre-university upper secondary mathematics education stress the importance of mathematical thinking as a learning goal. In the meantime, a plea is made for a better integration of computer science, and computational thinking in particular, in other topics. As mathematical and computational thinking clearly have a lot in common (Angeli et al., 2016; Drijvers, 2015; Tall, 2013; Weintrop et al., 2016), it makes sense to investigate how the two can be addressed in mathematics education. To address this challenge, a research project entitled ‘Computational and Mathematical Thinking’ was granted by NWO. The main research question of this study is: How can a teaching-learning strategy, focusing on the use of digital tools, support 16-17 years old pre-university students in developing computational thinking skills related to mathematical thinking in pure and applied mathematics courses? The consortium, led by the Freudenthal Institute, consists of five schools, Utrecht University, Radboud University and SLO. The three-year project will start on January 1, 2019.

The research project that we propose here concerns a first design cycle, including a pilot, of a 5-lesson teaching unit for vwo-5 addressing mathematical and computational thinking. In collaboration with the schools in the consortium, a suitable subject and target group will be identified, as well as appropriate ICT-tools. The student will set up a theoretically underpinned design and will closely monitor the pilot. The results will shed light on the interplay between mathematical thinking and computational thinking, and will inform the continuation of the bigger NWO study.

For further details, please contact Paul Drijvers.

Angeli, C., Voogt, J., Fluck, A., Webb, M., Cox, M., Malyn-Smith, J., & Zagami, J. (2016). A K-6 computational thinking curriculum framework: implications for teacher knowledge. Journal of Educational Technology & Society, 19(3), 47–57.

Drijvers, P. (2015). *Denken over wiskunde, onderwijs en ICT*. [Thinking about mathematics, education and ICT.] Inaugural lecture. Utrecht: Universiteit Utrecht. http://www.fisme.science.uu.nl/publicaties/literatuur/Oratie_Paul_Drijvers_facsimile_20150521.pdf

Tall, D. (2013). *How Humans Learn to Think Mathematically: Exploring the Three Worlds of Mathematics*. Cambridge, MA: Cambridge University Press.

Weintrop, D., Beheshti, E., Horn, M., Orton, K., Jona, K., Trouille, L., & Wilensky, U. (2016). Defining computational thinking for mathematics and science classrooms. *Journal of Science Education and Technology, 25*(1), 127–147.

Contact: Marianne van Dijke and Paul Drijvers

Statistical literacy is essential for informed citizenship in our data-driven society (Gal, 2002). Statistical literacy is considered as one of the 21st century skills that students should acquire (Thijs, Fisser, & Hoeven, 2014). Therefore, our statistics education in secondary school should prepare students to understand and interpret statistical information. Statistical inference (SI) is about making statements concerning an unknown population based on observed sample results. In contrast to descriptive statistics, which concerns describing the data under investigation, SI includes the handling of sampling variation, due to the interaction between data and chance. However, learning SI is difficult for students (Castro Sotos, Vanhoof, Noortgate, & Onghena, 2007). Therefore, we focus on the design, implementation and evaluation of a learning trajectory for SI for students in VWO 3.

This study comprises the third cycle of a design based research that started September 2016. In this phase, we intend to conduct a teaching experiment that consists of 12 lessons which will be carried out in two periods, the first in March and the second in May 2019, at 10 – 20 schools in the Netherlands, to test the designed and revised materials from the previous cycles (Van Dijke-Droogers, Bakker & Drijvers, 2018). The implementation of innovative educational materials by different teachers, is a challenge. For this reason, we invite master students who want to deepen their knowledge and expertise in the implementation of renewing teaching materials in educational practice, to participate in our research. The student will focus on the role of the teacher(s) in the implementation process, for example by developing a teacher observation instrument.

Castro Sotos, A. E., Vanhoof, S., Noortgate, W. van den, Onghena, P. (2007). Students’ misconceptions of statistical inference: A review of the empirical evidence from research on statistics education. *Educational Research Review, 1(2)*, 90–112.

Gal, I. (2002). Adults’ statistical literacy, meanings, components, responsibility. *International Statistical Review, 70*(1), 1–51.

Thijs, A., Fisser, P., & Van der Hoeven, M. (2014). *21e-eeuwse vaardigheden in het curriculum van het funderend onderwijs* [21st century skills in the curriculum of basic education]. Enschede, The Netherlands: SLO.

Van Dijke-Droogers, M.J.S. van, Drijvers, P.H.M., & Bakker, A. (2018). Repeated Sampling as a step towards Informal Statistical Inference. In M. A. Sorto, A. White, & L. Guyot (Eds.),* Looking back, looking forward. Proceedings of the Tenth International Conference on Teaching Statistics* (ICOTS10, July, 2018), Kyoto, Japan. Voorburg, The Netherlands: International Statistical Institute.

Contact: Michiel Doorman

Inquiry-based learning (IBL) has been advocated by science and mathematics educators as a means to make students be actively engaged in content-related problem solving processes and in reflecting on the nature of science. Several reasons created the need for IBL varying from improving content-learning, fostering motivation, creating opportunities for learning 21st century skills like creativity, critical thinking and working collaboratively. However, a discrepancy can be found between the need to make IBL accessible to students and teachers’ current classroom practices. Three research projects are proposed in close cooperation with these running projects on this issue.

The first research project accompanies the implementation process of IBL by selected teachers within a case study design. The teachers will be guided in this process and data is retrieved on their beliefs and implementation strategies (e.g., necessary aids and evaluation tools), classroom implementation is observed and students’ responses during and after the implemented unit are considered (e.g., creativity and levels of inquiry). A sample collection of students’ responses is available for an initial explorative study.

In the second project, the central focus is on the redesign of traditional tasks within teacher professional development units. Teachers will be supported and observed during the process of redesigning a closed textbook task into an IBL-oriented task and interviewed afterwards. The aim is to extract the determinants for successful redesigning processes to be able to enhance a research-based tool kit containing redesigning aids to guarantee successful implementation of tasks. The tasks can be selected from subjects within the science or mathematics domain.

The third project concerns a textbook analysis in one or more science domains. You will analyze to what extent textbooks provide opportunities for inquiry-based learning and opportunities for experiencing how scientists think and work. This textbook analysis can be performed in close collaboration with a PhD student who investigates similarities and differences between China and the Netherlands on this issue in lower secondary mathematics education.

References:

Capps, D. K., & Crawford, B. A. (2013). Inquiry-based instruction and teaching about nature of science: Are they happening?. Journal of Science Teacher Education, 24(3), 497-526.

Minner, D. D., Levy, A. J., & Century, J. (2010). Inquiry‐based science instruction—what is it and does it matter? Results from a research synthesis years 1984 to 2002. Journal of research in science teaching, 47(4), 474-496.

Swan, M., Pead, D., Doorman, L.M. & Mooldijk, A.H. (2013). Designing and using professional development resources for inquiry based learning. ZDM - International Journal on Mathematics Education, 45 (7), (pp. 945-957) (13 p.).

Master Theses from previous students in the project:

Nur Rahmah Sangkala (2018). The influence of inquiry based learning on Indonesian students’ attitude towards science.

Lysanne Smit (2016). A better understanding of 21st century skills in mathematics education and a view on these skills in current practice.