Mathematics in the Digital Age: The Case of Simulation-Based Proofs

Digital transformation has made possible the implementation of environments in which mathematics can be experienced in interplay with the computer. Examples are dynamic geometry environments or interactive computational environments, for example GeoGebra or Jupyter Notebook, respectively. We argue that a new possibility to construct and experience proofs arises alongside this development, as it enables the construction of environments capable of not only showing predefined animations, but actually allowing user interaction with mathematical objects and in this way supporting the construction of proofs. We precisely define such environments and call them “mathematical simulations.” Following a theoretical dissection of possible user interaction with these mathematical simulations, we categorize them in relation to other environments supporting the construction of mathematical proofs along the dimensions of “interactivity” and “formality.” Furthermore, we give an analysis of the functions of proofs that can be satisfied by simulation-based proofs. Finally, we provide examples of simulation-based proofs in Ariadne, a mathematical simulation for topology. The results of the analysis show that simulation-based proofs can in theory yield most functions of traditional symbolic proofs, showing promise for the consideration of simulation-based proofs as an alternative form of proof, as well as their use in this regard in education as well as in research. While a theoretical analysis can provide arguments for the possible functions of proof, they can fulfil their actual use and, in particular, their acceptance is of course subject to the sociomathematical norms of the respective communities and will be decided in the future.

Learners’ Functional Understandings of Proof (LFUP) in Mathematics: A Qualitative Approach

The purpose of this study was to present a revision and validation of the Learners’ Functional Understandings of Proof (LFUP) scale in mathematics using data collected from Grade 11 learners (n = 87) in a high school in South Africa. The LFUP scale was linked to the five-factor model (verification, explanation, communication, discovery, and systematisation) whose items were derived from existing literature on proof functions. Unlike the previous version of the scale, the new scale being validated here blends Likert-scale and constructed-response items to evaluate learners’ conceptions of the essence of the functions of an aspect central to mathematical knowledge development: proof. It is my contention that the LFUP instrument can be used as either a summative or a formative assessment tool, given the argument that learners often require motivation as to why they are required to write proofs. In short, this study provided an instrument to introduce learners to the concept of mathematical proof. Multiple regression analysis revealed that all five LFUP tenets correlated significantly with the total sum of all the functions of proof taken together. In the qualitative analysis, a substantial number of learners (52%) were found to hold hybrid beliefs about the functions of proof in mathematics.

Mathematical proof: from mathematics to school mathematics

Proof plays a central role in developing, establishing and communicating mathematical knowledge. Nevertheless, it is not such a central element in school mathematics. This article discusses some issues involving mathematical proof in school, intending to characterize the understanding of mathematical proof in school, its function and the meaning and relevance attributed to the notion of simple proof. The main conclusions suggest that the idea of addressing mathematical proof at all levels of school is a recent idea that is not yet fully implemented in schools. It requires an adaptation of the understanding of proof to the age of the students, reducing the level of formality and allowing the students to experience the different functions of proof and not only the function of verification. Among the different functions of proof, the function of explanation deserves special attention due to the illumination and empowerment that it can bring to the students and their learning. The way this function of proof relates to the notion of simple proof (and the related aesthetic issues) seems relevant enough to make it, in the future, a focus of attention for the teachers who address mathematical proof in the classroom. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

An illustration of the explanatory and discovery functions of proof

This article provides an illustration of the explanatory and discovery functions of proof with an original geometric conjecture made by a Grade 11 student. After logically explaining (proving) the result geometrically and algebraically, the result is generalised to other polygons by further reflection on the proof(s). Different proofs are given, each giving different insights that lead to further generalisations. The underlying heuristic reasoning is carefully described in order to provide an exemplar for designing learning trajectories to engage students with these functions of proof.