Available Mathematics Education research projects:

*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.