Keynote: The interplay between calculation and reasoning
Deductive reasoning and proof is one of the hallmarks of mathematics, and is an important factor in distinguishing mathematics from empirical sciences. Fluency in calculation, including symbolic manipulation in algebra and calculus, sit alongside deduction, reasoning and problem solving. “Core pure mathematics” is that essential amalgam which is universally studied by all mathematics, science and engineering students. It starts with traditional algebra, trigonometry and calculus, culminating with De Moivre’s theorem and its consequences while stopping short of real analysis.
Presentations of core pure mathematics often contain little explicit “proof” beyond formulaic proof by induction, but it is where proof starts for pure mathematicians. Furthermore, the work of core pure mathematics contains a key activity “reasoning by equivalence”. This is reasoning and is key in many of the deductions at this level, but it is very close to a calculation. Indeed, in many situations it can be treated formally as a calculation. This talk will look at the interplay between calculation and reasoning, with a focus on automatic assessment. To what extent can we automate the assessment of reasoning now, and where are the limits of automatic assessment in the future?