A problem that we often encounter in our teaching, is that through abstraction the intuition behind mathematical results get lost. Of course, for mathematician abstraction helps to push intuitive ideas to their limit, but for a student abstractions can become a notational burden. Rigorous mathematics runs the risk of turning cute ideas into conceptual monsters fully detached from what students understand. If this bridge between abstract notation and intuition is not built, mathematics can turn into a science om memorizing formulas; something only the smartest can do.

Guided reinvention is a dialectical approach that can help build the bridge between abstraction and intuition. The key idea is that students discover mathematical concepts themselves through exercises by starting from something they know, and using it to build new theory or techniques. This way, the students actively navigate the landscape of mathematical abstraction themselves. The notation is not something they have to memorize, but something they actively create. The ideal result is that students understand where mathematical theory comes from, but also feel empowered that this mathematical theory was something they could invent themselves.

We implemented guided reinvention techniques in two courses about (algorithms on) networks. The courses teach several topics, and each topic starts with a so-called “orientation session”. In these sessions guided reinvention takes place. Students work or Jupyter notebooks that combine theory with coding exercises. The theory is there to introduce key notation or problems, while the coding exercises are there to help students discover algorithms or ideas. For example, we applied this concept to teach students the basic ideas behind link prediction (i.e., finding missing edges in a network). During the session students are guided through examples to think about principles that can help to find missing edges, and they even invent naive algorithms themselves for specific example networks. At the end of such a session, students can share ideas they had with each other and the teacher, such that they can form a reference point when mathematical abstraction and notation is introduced during later lessons.

Even though in these orientation sessions students learn through coding, it must be specified that coding is not necessarily the point. Jupyter notebooks proved a natural way to combine theory and discovery exercises, and since our student come in knowing how to code, it is also a good way to make mathematics feel tangible. However, for students that do not know how to code there are other ways to facilitate discovery through guided reinvention. Teachers can for example make apps for students to play with to discover a theorem or simply create pen-and-paper exercises that guide students towards discovering something new.

When implementing guided reinvention in your course, there are a few key principles to keep in mind. First, the teacher should not simply let students work on guided reinvention problems without help. Instead, the teacher should walk around, help students that are clearly stuck by nudging them in the right direction without robbing them of the discovery, and encourage students to share the knowledge they obtained.  Second, there should by a discussion at the end where students actively share their result. During this discussion, the teacher should synthesize key discoveries and make sure all students understand the key ideas. Otherwise, these key ideas cannot be used in subsequent lessons. Finally, the teacher should create a safe atmosphere where students can try different things without feeling scared to make mistakes. The goal of guided reinvention is to empower students through mathematical discovery, not to break them down by calling their ideas “dumb”.

All in all, guided reinvention can be a great tool to help students understand why mathematical ideas are true and how they came to be. It gives a reference point for subsequent lessons and helps bridge the gap between cute ideas that lie at the foundation of mathematics and the abstract notation that has been developed around these ideas. Guided reinvention can show students that mathematics is not for the smartest kids of the class. It is something that everyone can participate in. And the best thing is: guided reinvention shows the student this by letting *them* do the work!