To improve learning we need to challenge students’ thoughts. Through intuitive geometric statements, the human brain instantly poses smart questions and offers hypotheses while engaging in self-challenging explorations. Using geometry, which is too often neglected in our schools, engaging teaching can sometimes be achieved in contemplative pantomime settings almost without words. The aim of the workshop was to show the power of geometry in the development of the concept of proof. Several relatively easy geometric ideas were presented through simple thought-provoking questions and by the use of technology. The aim of these questions is not solely to motivate an answer; rather, it is much deeper and educationally wider. Namely, the aim is to motivate the understanding and the beauty of the resolved uncertainty brought by the certainty of a proof. In a way, a proof should be as much an emotional experience as a rational achievement. Participants were challenged by several thought-provoking questions followed by individual engagements in the form of short problem-solving sessions and concluded by joint discussions. Geometry was used to show that to learn and appreciate mathematics, it is necessary to understand the concept of proof. In order to understand the concept of proof, one also needs to experience the challenge of uncertainty that precedes the certainty of a proof.

At the workshop, several motivational techniques and ideas were actively presented. Participants were challenged using concrete, thought-provoking puzzles. In the introduction, some tricks were presented and participants were challenged to observe their own reasoning and motivation powered by cognitive puzzlement. Using thought-provoking questions, participants were guided to understanding (and feeling) the need for proof. Within the main part of the workshop, participants were presented with several problems that were mostly visual. Particular problems were given in “pantomime” fashion, i.e., without words, with the first challenge for participants being to formulate the problem as briefly and as elegantly as possible. By that, participants were invited to increase their sensitivity to “meaning” and provoked to consider the inflation of content that occurs when too many words are used (in explaining). Some challenges included “hands-on” tasks. During the workshop, geometric ideas and teaching accents involving the following subjects were discussed: Missing angle of a triangle, midpoints of a quadrilateral, apparent regular octagon, line through centroid, intersection of two squares, triangle on top of a square, geometric series formula, constructing parabola geometrically, ellipse by folding paper, parabola-ellipse analogy, geometric paradoxes, and sound technology geometrically.

All of the problems were very easy to formulate. Most of the problems were illustrated and communicated by the use of technology. Participants were challenged to formulate problems “formally” and to explore several “upgrades of (their) understanding”. Proofs and answers were obtained as cognitive solutions and conclusions to “provocative uncertainty.” In most cases, we showed that a proof is as much a rational conclusion as it is also an “emotional experience” (that comes as a solution to felt problem). Because of this, it is very important how the problem is introduced.

In the “sum-it-up” conclusion, participants’ feedback was discussed. Participants were also given access to the interactive dynamic presentations/visualisations (https://www.geogebra.org/m/mZpYbUmK, accessed 10 November 2016) that were used during the workshop.