Mathematics Education

Contact: Anna Shvarts, a.y.shvarts@uu.nl

New technologies transform the world of professionals: video conferences and collaboration on joint documents became a basis of everyday professional activity. The innovations now include bodily interactions with the technologies, starting from touch screens and continuing to virtual reality. Such technological innovations lead to major transformations in the professionals that include operating with 3D space: think of designers, architects, and engineers. These transformations call for updating the educational curriculum and developing new teaching methods that involve 3D space.

Virtual reality provides a unique opportunity to explore and actively construct geometrical objects in 3D. However, concrete opportunities and limitations of VR environments in learning about 3D space are still at the very beginning of their investigation (e.g., Price et al., 2020; Walkington, Gravell, & Huang, 2021).

Technological developments coincide with the development of new theoretical understandings of how cognition works: computer metaphor and information processing ideas give way to the new embodied cognition theories. Those theories highlight bodily interactions with the environment and talk about affordances—possibilities for actions—that the environment provides (e.g., Wilson & Golonka, 2013). Embodied perspective to learning with technology highlights that learners need to develop body-artifact systems so that they can fluently use technological tools (Shvarts et al., 2021).

In the master's project, you will work on design-based research (Bakker, 2018): you will design embodied environments (see Abrahamson et al., 2020) to teach the geometrical properties of 3D objects and explore the opportunities and limitations of constructing such objects in VR.

References:

Abrahamson, D., Nathan, M. J., Williams-Pierce, C., Walkington, C., Ottmar, E. R., Soto, H., & Alibali, M. W. (2020). The Future of Embodied Design for Mathematics Teaching and Learning. Frontiers in Education, 5, 147. https://doi.org/10.3389/feduc.2020.00147

Bakker, A. (2018). Design research in education. Routledge. https://doi.org/10.4324/9780203701010

Price, S., Yiannoutsou, N. & Vezzoli, Y. Making the Body Tangible: Elementary Geometry Learning through VR. Digit Exp Math Educ 6, 213–232 (2020). https://doi.org/10.1007/s40751-020-00071-7

Shvarts, A., Alberto, R., Bakker, A., Doorman, M., & Drijvers, P. (2021). Embodied instrumentation in learning mathematics as the genesis of a body-artifact functional system. Educational Studies in Mathematics, 107(3), 447–469. https://doi.org/10.1007/s10649-021-10053-0

Walkington, C., Gravell, J. & Huang, W. (2021). Using Virtual Reality During Remote Learning to Change the Way Teachers Think About Geometry, Collaboration, and Technology. Contemporary Issues in Technology and Teacher Education, 21(4), 713-743. Waynesville, NC USA: Society for Information Technology & Teacher Education. https://www.learntechlib.org/primary/p/219556/

Wilson, A., & Golonka, S. (2013). Embodied Cognition is Not What you Think it is. Frontiers in Psychology, 4, 58.

Contact: Anna Shvarts, a.y.shvarts@uu.nl

Gender differences in mathematics learning attract intensive attention due to the under-representation of women in STEM professions, including the fields tightly related to mathematics. This issue has a long history of investigation in mathematics education research, and now researchers question the mechanisms of how gender differences arise (Leder, 2019).

One of the theoretical visions of gender differences comes from theories of embodiment that assume different bodies are being treated differently in culture (de Freitas, 2008). Embodied cognition is a quickly developing approach that highlights that knowing anything, including very abstract notions, is grounded in the personal experiences of solving problems with their bodies, not just brains (Wilson & Golonka, 2013).

Applying this theory to practice, we can expect that embodied design—special activities designed to build mathematical knowledge from personal sensory-motor experience (Abrahamson et al., 2021)—can help create opportunities for people with various bodies to study mathematics fruitfully. Within the embodied design approach, we develop technological embodied activities to promote students’ deep understanding of mathematical concepts (Alberto et al., 2021). Those activities account for various idiosyncratic experiences and, theoretically, should open space for various bodies. However, we do not know yet how students of different genders approach embodied activities. More specifically, we need to know what kind of different support they might need when describing their experiences depending on the gender differences.

In the research project, you will work with one of the embodied activities developed in our previous studies. You may focus on investigating gender differences in experiencing and/or on teaching support needed to facilitate such diversity in embodied learning.

Sources:

Try out embodied activities at https://embodieddesign.sites.uu.nl/         

Abrahamson, D., Nathan, M. J., Williams-Pierce, C., Walkington, C., Ottmar, E. R., Soto, H., & Alibali, M. W. (2020). The Future of Embodied Design for Mathematics Teaching and Learning. Frontiers in Education, 5, 147. https://doi.org/10.3389/feduc.2020.00147

Alberto, R., Shvarts, A., Drijvers, P., & Bakker, A. (2021). Action-based embodied design for mathematics learning: A decade of variations on a theme. International Journal of Child-Computer Interaction, 100419. https://doi.org/https://doi.org/10.1016/j.ijcci.2021.100419

de Freitas, E. (2008). Mathematics and its other: (dis)locating the feminine. Gender and Education, 20(3), 281–290. https://doi.org/10.1080/09540250801964189

Leder, G. C. (2019). Gender and Mathematics Education: An Overview In G. Kaiser & N. Presmeg (eds.) Compendium for Early Career Researchers in Mathematics Education  (pp. 289–308). Springer International Publishing. https://doi.org/10.1007/978-3-030-15636-7_13

Wilson, A., & Golonka, S. (2013). Embodied Cognition is Not What you Think it is. Frontiers in Psychology, 4, 58.

Contact: Hang Wei (h.wei@uu.nl) or Rogier Bos (r.d.bos@uu.nl)

Developing functional thinking skills has been a central area of mathematics throughout primary, secondary, and tertiary education since the beginning of the twentieth century (Vollrath, 1986). Functional thinking is thinking in terms of relationships, interdependencies, and change. It is an essential step towards understanding the concepts of calculus. In this regard, functional thinking has received considerable attention because of its educational importance. Nevertheless, new insight from embodied design and technological advances allows for new approaches to design tasks that promote functional thinking.

Technology-enhanced instruction provides students with an opportunity to actively participate and reorganize the way in which they understand mathematics through higher-order thinking tasks (Lee & Hollebrands, 2008; Tanudjaya & Doorman, 2020), such as functional thinking tasks. While the new digital-embodied design (Abrahamson, 2014) in mathematics teaching shows high potentiality for developing functional thinking. The question becomes: how can digital-embodied designs be used to develop students’ functional thinking?

Your master research project will be part of the Erasmus+ FunThink project. You will co-design an innovative teaching-learning environment that aims at fostering students’ functional thinking in secondary education. The learning activities will be embodied designs and make use of up-to-date digital technologies, such as the DME’s GeoDefiner. There will be two testing cycles, one in the teaching and learning laboratory (TLL), and one in a real classroom. We invite master students who are interested in functional thinking and willing to develop skills in using educational technology.

For further details, please contact Hang Wei (h.wei@uu.nl) or Rogier Bos (r.d.bos@uu.nl).

Abrahamson, D., & Lindgren, R. (2014). Embodiment and embodied design. In The Cambridge Handbook of the Learning Sciences, Second Edition. https://doi.org/10.1017/CBO9781139519526.022

Lee, H., & Hollebrands, K. (2008). Preparing to teach mathematics with technology: An integrated approach to developing technological pedagogical content knowledge. Contemporary Issues in Technology and Teacher Education, 8(4), 326-341.

Tanudjaya, C. P., & Doorman, M. (2020). Examining Higher Order Thinking in Indonesian Lower Secondary Mathematics Classrooms. Journal on Mathematics Education, 11(2), 277-300.

Vollrath, H. J. (1989). Funktionales denken. Journal für Mathematik-Didaktik, 10(1), 3-37.

Contact: Sylvia van Borkulo (s.vanborkulo@uu.nl)

Context

Computational thinking (CT) is gaining interest in education, as it is a relevant skill in students' 21^st century workplaces and everyday lives. CT was initially defined by Wing (2006) as follows: "computational thinking involves solving problems, designing systems, and understanding human behavior, by drawing on the concepts fundamental to computer science" (p. 33). A broad range of aspects are involved in CT, as summarized in a framework by Kalelioglu, Gulbahar, and Kukul (2016) that includes the aspects abstraction, decomposition, data representation and analysis, mathematical reasoning, building algorithms, modelling, generalisation, testing and debugging. However, it is not clear how these aspects can be integrated in the secondary education curriculum.

Within the Erasmus+ project “Computational Thinking Learning Environment for Teachers in Europe” (<colette/>, project duration: 2020-2023, https://colette-project.eu/), seven organisations in the Netherlands, Germany, Slovakia, France, and Austria aim to address the need for accessible learning activities addressing different aspects of CT. The learning environment offers tasks related to upper and lower secondary education focussing on mathematics for different aspects of CT. Students use their smartphone to do the tasks. Part of the task set is related to block-based programming where the student evaluates the result in Augmented Reality (AR).

Research Project

The aim of this master research project is to use the Colette AVR smartphone environment to investigate how aspects of CT, such as algorithmic thinking, decomposition, and abstraction can be addressed in a series of programming tasks to design a cube building. In the programming task block-based coding is used and augmented reality to view the result of the code. A possible approach is to design a series of tasks in the colette learning environment and perform design research. Another approach might be to perform a teaching experiment.

Possible research questions are:

• How do programming tasks support the development of abstraction skills, for example in the use of variables and repeat blocks?
• How does AR and its embodied characteristics support the development of spatial skills in a cube building programming task?

More Information

For more information, please contact Sylvia van Borkulo s.vanborkulo@uu.nl.

References

Kalelioglu, F., Gulbahar, Y., & Kukul, V. (2016). A framework for computational thinking based on a systematic research review. Baltic Journal of Modern Computing, 4(3), 583–596. Retrieved from http://acikerisim.baskent.edu.tr/bitstream/handle/11727/3831/4_3_15_Kalelioglu.pdf?sequence=1

Milicic, G., Borkulo, S. P. Van, Medova, J., Wetzel, S., & Ludwig, M. (2021). Design and development of a learning environment for computational thinking: The Erasmus+ <colette/> project. Proceedings of EDULEARN21 Conference, (July), 7376–7383. Retrieved from https://www.researchgate.net/profile/Gregor-Milicic/publication/353339769_Design_and_Development_of_a_Learning_Environment_for_Computational_Thinking_The_Erasmus_COLETTE_Project/links/60f592a09541032c6d508428/Design-and-Development-of-a-Learning-Environment-for-Computational-Thinking-The-Erasmus-COLETTE-Project.pdf

Wing, J. M. (2006). Computational thinking. Communications of the ACM, 49(3), 33–35.

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-ST…

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-Schreine…

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.

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 (r.d.bos@uu.nl) and Ralph Meulenbroeks (r.f.g.Meulenbroeks@uu.nl)

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:
 

1. incorporating flexibility
2. facilitating interaction
3. facilitating students’ learning processes
4. 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 (a.y.shvarts@uu.nl)

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.

Contact: Rogier Bos (r.d.bos@uu.nl)

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: Michiel Doorman (M.Doorman@uu.nl)

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.