Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS)

Daniel Otero – 11 Feb 2019

Daniel Otero is an Associate Professor at Xavier University, Cincinatti ( He is visiting the School of Mathematics and Statistics at the University of St Andrews until early April, as one of their Visiting Fellows in the History of Mathematics.

This talk outlined the approach of the TRIUMPHS programme ( to the teaching and learning of undergraduate mathematics. TRIUMPHS contributors design and deliver ‘Primary Source Projects’ in particular topics of mathematics, aiming to supplement or even partially replace, traditional teaching of those topics.

Projects are based around historical sources, including examples such as (an image of) a Bablyonian tablet illustrating their system of numeration (base 60). Students engage with the documents and take an active approach to discovering and learning about the topic. The aim is to encourage group work, moderated by the instructor.

Each project is designed to take anywhere between two or three class periods, and a whole semester, to deliver. The Bablyonian example covers two classes, while a unit on trigonometry is designed to take the first 4-6 weeks of a traditional module on the subject, then allow a faster track version of the same module to continue in the remainder of the semester.

These projects seem to be especially successful at engaging students who are not already strong mathematically, but are very challenging to ‘better’ students, who are sometimes uncomfortable with the active group style.

Further information about Danny:

Danny’s scholarship focuses on ways in which the history of mathematics can be used to improve the teaching of the subject at the tertiary level. He has designed three undergraduate modules for his own students that teach maths through engagement with primary sources. Since 2014 he has been co-PI on a National Science Foundation grant project titled Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS), and he is also a member of the MAA Spectrum series editorial board. While he is in St Andrews he is going to be working on completing his book, A Smoother Pebble, that aims to teach the basic concepts of calculus to students of the humanities through primary sources that exemplify the history of the development of its underlying ideas.


What makes examples exemplary for students? – John Mason

Mathematics Division Seminar

19 February 2018

Prof Mason has been teaching university mathematics for 40 years and during this time he has extensively investigated how to teach and learn mathematics effectively. He led the Open University’s Centre for Mathematics Education for fifteen years and has published numerous useful textbooks about maths educations that have become standard texts for students and lecturers.

This talk examined many of the ways in which the examples and ‘model solutions’ we provide to students might be more or less helpful to them, perhaps not working in the way that we expect! A number of techniques for engaging students within a session were discussed and even used during the talk.


Using Cognitive Science to Teach Mathematics in Higher Education – Carolina Kuepper-Tetzel

Mathematics Division Seminar

16 October 2017


Cognitive Science has revealed a number of learning and memory
phenomena that have direct, practical implications for learning and teaching.
I will highlight the most promising strategies alongside their original research
findings and will elaborate on ways on how to implement these in large
lectures and small seminars in mathematics. The talk will conclude with an
opportunity discuss the feasibility of implementation and ways to overcome
potential obstacles.

Automatic Assessment of Reasoning by Equivalence – Chris Sangwin

Mathematics Division seminar

Monday 27 February 2017


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.  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?