Chris Sangwin

1. We begin with the premise that making mistakes is an inherent aspect of all human activity. While most errors occur unintentionally, some are deliberate – crafted to deceive, manipulate, exploi...

For over a quarter of a century, computer algebra systems have been utilised in assessment systems to transition away from multiple-choice questions towards enabling automatic assessment of students’ answers which are student-constructed mathematical expressions. Objective properties of these expressions are then established automatically, and the software generates outcomes including numerical marks, formative feedback, and detailed statistical information. The advantages of these systems include providing students with immediate, personalised, and adaptive learning experiences, which help them identify and potentially correct mistakes more quickly. Despite these benefits, such software still has significant limitations compared to human tutors, including the inability to offer subjective feedback or engage in meaningful dialogue with students. However, it excels in providing consistency, scalability, and efficiency, allowing for assessments to be conducted on a large scale. Overall, the implementation of automatic assessment, supported by computer algebra, has started to influence the field of mathematics education, offering new possibilities for personalised and adaptive learning experiences. This includes high-stakes examinations. This chapter provides a review of a quarter of a century of use and contemporary examples and discusses the future of this field in the context of emerging generative AI.

What is it that makes us judge two proofs of the same theorem to be the same or different? This is not an idle question: one central aspect of judging mathematics is the novelty of the mathematics presented. This is important everywhere, from the peer-review system, to assigning international prestige, to funding agencies’ grant decisions. It even matters to some extent in examinations, to avoid accusations of collusion. Surprisingly, philosophers of mathematics have not paid the question of novelty much attention. In this Article, we will consider the appealing conjecture that the main ideas that make up the proof, the essence of a proof, can indeed be identified and that very different styles of proofs can share common main ideas. Further, that a particular theorem can be proved using quite different, independent main ideas. As a means of exploring whether this is plausible, we will present a number of novel proofs of the following theorem.

There is a long tradition of mathematicians valuing collections of proofs, e.g. [1] is a famous recent collection. In the past, professional mathematicians have also written many different proofs of single important theorems, e.g. [2]. Proofs of the Pythagorean Theorem were collected by [3], and [4] discusses mathematical style through proofs of the irrationality of . The recent work of [5] is a discussion of mathematical style via a comparison of 99 different ‘proofs’ the following ‘theorem’.