Diagrammatic Reasoning in Higher Education
A workshop on the teaching of mathematical concepts using diagrammatic reasoning in higher education. Part of the CARMA special semester on Mathematical Thinking.
9 November—11 November 2018 in Newcastle, NSW
Workshop delegates may also wish to consider extending their stay to add two further workshops within the CARMA Mathematical Thinking Special Semester: EVIMS on 13 Nov and Mathematical Thinking 14–16 Nov.
This workshop will investigate the potential for diagrammatic reasoning to be used to enhance the teaching of mathematics and abstract concepts across a range of disciplines (of business and economics, physics, engineering, and computer science) at the University of Newcastle. It will investigate the potential of diagrammatic reasoning as a novel pedagogical tool to enhance the learning of diverse student cohorts. Specifically, we will consider how it could afford a way forward for students for whom normal academic languages are a barrier, but who are more naturally pre-disposed towards visual and processual forms of understanding. Relevant diagrams to be considered include ologs and string diagrams, but could also include operads, props, and opetopes, which could be applied to engineering circuits, computational trees and networks, graphical linear algebra, and signal flow graphs.
We should emphasise the fact that any academic staff from disciplines other than mathematics, with an interest in this theme, are welcome to participate.
- Dr Daniel Barter (Australian National University, Australia)
- Professor Keith Devlin (Stanford University, USA)
- Dr Richard Garner (Macquarie University, Australia)
- Professor Rafael Núñez (University of California at San Diego, USA)
- Dr Judy-anne Osborn (University of Newcastle, Australia)
- Professor Ross Street (Macquarie University, Australia)
- Professor Dominic Verity (Macquarie University, Australia)
Registration has been closed.
Room X703 “Moot Court”, NeW Space building, Newcastle. See here for transport and parking information.
Ross Street: Calculating with string diagrams (Slides)
Motivating with linear algebra, I shall introduce symmetric monoidal categories (smc) and explain the string diagrams for which smc provide the environment. I shall show how familiar operations from vector calculus transport to smc where the properties can be expressed in terms of equalities between string diagrams. Geometrically appealing arguments will be used to prove a theorem with implications for multiplications on Euclidean space, a theorem of a type originally proved using higher powered methods.
Dominic Verity: An exploration of string diagrams in Computer Science (Slides)
When I was a teenager my Computer Science teacher tortured me with flowcharts!
At that time, received wisdom and accepted design practice demanded that all systems should have their logic and control flow explained diagrammatically, even before a single line of code could be written. Indeed my teacher made this a condition of access to precious computing resources; I quickly learnt to hate flowcharts, abandoning them as quickly as I could.
Ultimately the advance of ubiquitous and plentiful computing resources largely eliminated flowcharts from the computing curriculum. After all why bother carefully drawing up a flowchart, as a prelude to a brief session of code entry, when one can immediately open up a dialogue with the closest machine and hack up some code? That being said these pesky flowcharts live on, in hiding, and flourish like viruses, most notably in the internals of our programming language compilers (control flow graphs), in graphical environments targeted at programming pedagogy (Blockly, Scratch, Node Red, and friends), and in the papers of theoretical computer scientists.
Had someone told me that there was interesting mathematics to be had from the world of flowcharts I might have taken these diagrams more seriously.
In this talk I will try to illustrate some of the mathematics underpinning various graphical reasoning methods in computer science. Our discussion will touch on flowcharts, finite state automata, control flow graphs, electronic circuit diagrams, and their kin. We will build upon the graphical calculus introduced in Ross Street’s talk, extending it to explore structures called traced monoidal categories. This then will become a unifying frame for exploring the descriptive power of these diagrammatic methods and for understanding their utility in reasoning about and evolving computational systems.
Judy-anne Osborn: Education, Math-Pictures and the Modern World
I will discuss the use of diagrams and narrative in teaching mathematics. I will contrast a teaching area in which diagrams are currently used extensively: Combinatorics, with an area in which that is not so much the case: Calculus, and pose the question to what extent it needs to be this way?
Richard Garner: Rigour and diagrams
Mathematics is often considered to have a privileged epistemological status, in that true mathematical statements are not just true, but uncontroversially true. It may seem a reasonable assumption that this lack of controversy stems from the unique rigour with which mathematical arguments are invested. I will argue that diagrammatic and other non-standard reasonings provide a useful tool to interrogate this assumption, and that mathematical rigour as commonly understood is mainly a product of historical happenstance. I hope to illustrate this with examples from topology, 2-category theory and game semantics.
Daniel Barter: Diagrammatic methods for computing defect fusion in topological phases
My research is focused on (2+1)D topological quantum field theory. Recently I have been computing defect fusions in these models using diagrammatic methods. At this point, there is no other way that these computations can be done. I will explain a little bit what all this means, and demonstrate some example computations.
Heath Jones: Diagrammatic Reasoning and the Teaching of Introductory Physics
Recent decades have seen significant increases in both the number of Australian higher education providers and their student populations. At the same time, the gap between student preparedness and the demands of tertiary study has widened. In particular, a lack of mathematical readiness represents a major pedagogical challenge for tertiary educators of quantitative disciplines.
This talk will explore the ways in which diagrams can provide an alternative analytical framework for teaching a quantitative discipline (introductory physics) in a context where prior mathematical experience cannot be automatically assumed (pre-university foundation studies). The extent to which diagrams can be used as an adjunct (or substitute) to traditional approaches (algebra) will be discussed.
James Juniper: Thoughts on Diagrammatic Reasoning by Practitioners of Diagrammatic Reasoning
The paper discusses the pedagogical motivation for the Workshop before considering some philosophical aspects of Diagrammatic Reasoning (DR) in the Digital Economy. The next section of the paper defines DR and provides a brief genealogy before moving on to examine some subsequent developments in the field. One feature of string diagrams—their power of abbreviation—is then discussed. The views of DR practitioners, as expressed “in their own words”, are then reviewed across a variety of application domains. The paper concludes after a brief review of some opportunities for the future development of DR.
Rafael Núñez: Inductive reasoning or mathematical induction? Investigating mathematical thinking through ‘visual proofs’
Formal deductive proofs are essential for the validation of bodies of knowledge (i.e., theorems) in modern mathematics. The elaboration of proofs involves reasoning with concepts that are often intertwined with complex supporting notations and algorithms associated with them. How to investigate the principles underlying these concepts without getting over-entangled with notational features? Here, using ‘visual proofs’ we investigate fundamental differences between standard inductive reasoning and reasoning by mathematical induction. While the former is acceptable means of discovery (but not justification), the latter —a formal, deductive proof strategy— can be used to show that a theorem is necessarily true for all natural numbers. I will present a study conducted in our lab by Josephine Relaford-Doyle in which we examine the nature of the conclusions drawn from a visual proof by induction. We find that, while most university-educated viewers demonstrate a willingness to generalize the statement to nearby cases not depicted in the image, only viewers who have been trained in formal proof strategies show significantly higher resistance to the suggestion of large-magnitude counterexamples to the theorem. We conclude that for most university-educated adults without proof-training the image serves as the basis of a standard inductive generalization and does not provide the degree of certainty (necessity) required for mathematical proof. With the help of ‘visual proofs’ as a methodological tool, it is possible to empirically show that, despite its name, mathematical induction is not standard induction, but a deductive proof strategy that requires technical training.
Keith Devlin: The Versatile Power of Simple Diagrams in Mathematics
I’ll focus on one particular diagram that has proved to be powerful from both the most abstract realms of pure mathematics to the application of mathematical thinking to messy, day-to-day phenomena.
CARMA Special Semester on Mathematical Thinking 2018
Attendees might want to consider the following sister events to this workshop.