Ann Kiefer

Mathematics is often perceived as an abstract discipline, disconnected from the everyday reality of students’ lives outside the school setting. To counter this perception, an increasingly adopted pedagogical approach is to incorporate more ‘real world’ examples into teaching (Coles, 2016).

To anchor learning in a concrete approach, we have chosen to approach solids and volumes through a creative project. Pupils are invited to design their own die by defining its shape, number of faces and the numbers, symbols or letters that will appear on it. The process continues with digital modelling of the die using Tinkercad software, followed by 3D printing in the school’s Makerspace. The learning is consolidated by a final presentation of their creation. This approach is fully in line with project-based learning (see Section 3.3 Teaching materials M2).

Among the many methods likely to improve student motivation, project-based learning has been frequently cited for several decades. It has become everyday practice in vocational education and higher education (Reverdy, 2013).

This pedagogical approach has the significant advantage of making the students autonomous, positioning them as true authors of their creation. By investing themselves in this project, they take charge of planning the stages of production and ensuring that they see their work through to completion (Reverdy, 2013).

The teacher’s role changes in this context, as he or she evolves from teacher to facilitator and motivator. They do not help the students directly, but make sure that they are on the right track and that their project is progressing.

One of the common criticisms of project-based learning is that it takes too long. Firstly, in this module we have ensured that the project covers exactly the elements of the official mathematics curriculum relating to solids and volumes. The final presentation of the die by the pupils forces them to confront the content, knowledge and skills relating to solids and volumes in the curriculum. As project-based learning does not work miracles, we have included mini-lessons (section 3.3. Teaching materials M3) relating to these different contents. These mini-lessons explain the content and ask students to complete exercises to check and verify their understanding. In this way, each pupil can progress independently and at their own pace: if a pupil knows how to calculate the volume of their die, they don’t need to do the mini-lesson and can work on their calculation straight away. However, research shows that discovery-based learning is only effective when learners receive timely feedback and are offered examples of detailed solutions (Alfieri et al., 2011). This is why we provide detailed solutions for all exercises (section 3.7 Solutions). In this way, students can consult them independently and self-monitor their learning. These mini-lessons also ensure that the knowledge required to complete the project does not exceed the student’s knowledge, a second criticism of project-based learning that we often encounter.

Project-based learning also has many advantages (Reverdy, 2013). In project-based learning, pupils and students learn by being active and maintaining a link with the real world, which enables them to nurture communication, cooperation, creativity and in-depth reflection. Attention to learning processes, not just content, is beneficial (Reverdy, 2013).

Educational researcher Robert DeHaan highlights the importance of promoting creative thinking in natural science education (DeHaan, 2009). According to DeHaan, natural science teaching is still often far from promoting this transferable knowledge in a sustainable way. The teaching is heavily influenced by facts and recipes, leaving learners with little knowledge and generating boredom. Studies have also shown that activities that encourage creativity significantly improve learning success in natural science teaching.

Research into project-based teaching also recommends cooperation between teachers from different disciplines. This is what we suggest in the fifth phase of the project, where the pupils have to invent their own game using the die created in the previous phases. We recommend doing this part of the project with the French or Digital Sciences teacher (see Section 3.4 Interdisciplinary ideas for more details).

We suggest starting the lesson with dice games organised into learning stations. Pupils explore six different stations in groups, at their own pace, without having to go through them all in the first two sessions.

A recent study (Abdelmalak, 2024) demonstrates the effectiveness of this model in developing pupils’ analytical, generative and evaluative skills. These cognitive improvements are probably the result of a combination of factors: involvement in stimulating tasks, collaboration between peers, alternation between individual and group learning, pedagogical support and a favourable learning environment. As several researchers have pointed out, intellectually stimulating activities contribute significantly to the development of fundamental mathematical skills (Abdelmalak, 2024).

At each station, students document their experience and answer reflective questions in writing. Dionne Cross (2009) points out that “writing is a powerful strategy for encouraging learning”, offering greater benefits than simple oral argumentation. This effectiveness is explained by metacognitive activation: by formulating their thoughts in writing, students diagnose the problem, plan their approach and constantly question their reasoning.

These games also introduce probability, a field with well-established educational challenges. The use of dice represents an accessible approach to introducing the concept of equiprobability. As Capaldi (2021) notes, discovery-based learning through a variety of games stimulates motivation and develops analytical reasoning, making up for teachers’ difficulty in creating engaging problems that illustrate mathematical applications in concrete terms. The questions proposed after each game focus on developing intuition rather than formal mastery of probability theory.

By the end of the module, students will have developed an intuition for probability while assimilating the essential concepts of solids and volumes through games and by designing their own die.

References

1. Abdelmalak, M. M. M. 2024. Promoting selected core thinking skills using math stations rotation. Research in Mathematics Education, 1–22. https://doi.org/10.1080/14794802.2024.2344209
2. Alfieri, L., Brooks, P. J., Aldrich, N. J. & Tenenbaum, H. R. 2011. Does Discovery-Based Instruction Enhance Learning? Journal of Educational Psychology, 103 (1), 1-18. doi: 10.1037/a0021017.
3. Capaldi, Mindy. 2021. Teaching Mathematics Through Games. 1st ed. Providence: American Mathematical Society.
4. Coles, Alf. 2016. Engaging in Mathematics in the Classroom : Symbols and Experiences. London ; Routledge.
5. Cross, D. I. 2009. Creating Optimal Mathematics Learning Environments: Combining argumentation and Writing to Enhance Achievement. International Journal of Science and Mathematics Education 7: 905-930.
6. DeHaan R. L. 2009. Teaching creativity and inventive problem solving in science. CBE Life Sci Educ. Fall;8(3):172-81. doi: 10.1187/cbe.08-12-0081.
7. Reverdy, C. 2013. Des projets pour mieux apprendre ? Dossier d’actualité veille et analyses n° 82 Février 2013. https://veille-et-analyses.ens-lyon.fr/DA-Veille/82-fevrier-2013.pdf?v=1361180601