Communicating Mathematical Thinking?

Towards making Maths Understandable.Mathematicians as researchers need to better attend to communicating not just their definitions, theorems, and proofs, but also their ways of thinking:

Mathematicians who don t spell out the details of their work are like climbers who reach the top of a mountain without leaving hooks along the way. Someone with less training will have no way of following it without having to find the route for themselves, Katrin Wehrheim

This need for more clarity should be reflected in both their teaching of mathematics and in the communication of their research. Is it possible to reconcile into one mode of expression of understanding both the teaching method and the practice of doing Mathematics?

We need to appreciate the value of different ways of thinking about the same mathematical structure. We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics, with consequently less energy on the most recent results. This entails developing mathematical language that is effective for the radical purpose of conveying ideas to people who don t already know them. (On proof and progress in mathematics, William P. Thurston)

Consider in turn, issues related to teaching and then the expression of doing mathematics given that its apprehension is recognised as following one of two modes of understanding,(David Wees):

Relational mathematics ( knowing both what to do and why ) consists of building up a conceptual structure (schema) from which the student can produce an unlimited number of plans for getting from any starting point within that schema to any finishing point.

You can t plough a field by turning it over in your mind. Gordon B. Hinckley

In contrast, instrumental mathematics ( rules without reasons ) consists of the learning of an increasing number of fixed plans, by which the thinker can find their way from particular starting points to required finishing points deigned the answer to the questions. The plan is algorithmic enunciating what to do at any stage of the process.

Therefore teaching delivery may elicit two kinds of teacher-student mismatches, (Richard Skemp) :

A learner satsified to understand instrumentally, guided by a teacher (and book) who wants them to understand relationally.A meta-cognitive learner whose goal is to understand relationally, guided by a teacher who provides instrumental instruction.The first of these will cause fewer problems short-term to the student, but it will be frustrating to the teacher. The latter is more damaging to the student especially when it is effected from a written text.

A cautionary note from Cognitive Load theoryCognitive Load Theory, CLT tells us, why-students-make-silly-mistakes-in-class-and-what-can-be-done. Unlike learning styles theory, there seems to be a lot of research backing problems in overloading pupil s RAM! CLT suggests that the bottom-up incremental, instrumental approach is the best way to develop a pupil s appreciation of the methods of mathematics. At least to build up their confidence to motivate more self-driven enquiry. CLT suggest that any Big picture, relational teaching is perhaps an indulgence of the teacher yes they can connect a few dots and create some narratives to motivate example but for the most part this top-down view comes from within the learner over time.

Herein lies the conflict. Thurston s quote suggests that it is the relational way of understanding that mirrors the mathematical research method we as teachers should employ rather than the instrumental mode.

We look next at how either of the modes sit with the public perception of what actually is this business of doing mathematics.

What is the Study of Maths?There are two conceptions of mathematics, the currency of mathematics is:

ideasproofsThe latter mindset says that the study of mathematics involves an abstract deductive system consisting of:1. A set of primitive undefined terms2. Definitions evolved from the undefined terms3. Axioms or postulates4. Theorems and their proofs.

A public understanding caricature of mathematics is the Definition-Theorem-Proof (DTP) model which reads as follows:Define. mathematicians start from a few basic mathematical structures and a collection of axioms given about these structures,Theorise. there are various important questions to be answered about these structures that can be stated as formal mathematical propositions,Prove. the task of the mathematician is to seek a deductive pathway from the axioms to the propositions or to their denials.

The instrumental teaching mode involves the modelling of the solving of a problem and then the rolling out of problems with variations of minimal differences for the students to apply themselves and plays to this algorithmic theme.

A clear difficulty with the DTP model though is that it fails to beg the student to be curious of the source for the questions. A complete descri ption of what mathematics is at the practitioners level must include such Speculation: the making of conjectures, the raising of questions, intelligent guesses and heuristic arguments about what is most likely to be proven to be true.

Someone s sitting in the shade today because someone planted a tree a long time ago. Warren Buffett

A measure of increased success in the understanding of the field could thus be expressed by the extent to which successive researchers and interlopers apprehend and understand mathematics in its fully relational form.

Mathematical Objects by AssociationWhile students may approach a piece of mathematics from one tentative viewpoint, practising mathematicians will understand the concept in multiple ways.

People have amazing facilities for sensing something without knowing where it comes from (intuition) for sensing that some phenomenon or situation or object is like something else (association) and for building and testing connections and comparisons, holding two things in mind at the same time (metaphor). These facilities are quite important for mathematics. ( William P. Thurston)

Consider the notion of a vector. To communicate the idea to students you might consider giving examples of vectors in the abstract and then instances of how they are realised kinematically in nature. A vector is then:

a directed line element/segment usually represented by an arrow whose head type indicates its kinematical form in the field of mechanics,has as archetype the position vector an orientated directed distance from the origin of some arbitrary co-ordinate axis that enables the location of the head of the arrow to be determined by the tuple (x,y) in which the base of the arrow is at (0,0)a displacement vector of relative positionsthe (absolute/relative), (instantaneous/average) velocity vector, being the rate of change of those various types of displacement.the absolute acceleration vector as the rate of change of velocity or indeed the snap of the relative acceleration.That these are vectors objects and not scalar ones can be made clear by saying that non-examples of vectors are:

distance, speedTemperature, Pressure as Real number Field valuesAnd then you might want to consider why there is no scalar equivalent of acceleration. Such mapping of physical quantities to their algebraic symbols then helps uncover common student errors such as setting vectors equal to scalars, or comparing finite quantities with infinitesimals. Interpreting Derivatives, Dray et al. Of the latter consider then the notion of a derivative:

Physicists: A ratio of (very) small changes in quantities?Mathematicians would consider the slope of the tangent line that is the limit of the slopes of secant lines?The average rate of change, no matter how small the domain, is different from an instantaneous rate of change.The physicist would ask: How does one take the limit of discrete, numerical data, such as that measured during an experiment and Calculus then becomes less about the formal limits of analysis , rather more the art of infinitesimal reasoning: quantities that are small enough for the purpose at hand.

Students do not distinguish between the graph of the function and the graph of the function s rate of change which might be attributable to students not conceiving of rate of change as a quotient of two quantities, discussing the rate of change as a slope but not speaking of slope as a quotient (the change in a function s value being so many times as large as the corresponding change in its argument). Instead, they talk about slope as the function s steepness. As another example, students often use a tangent line and rely on visual judgments to sketch the derivative function. Interpreting Derivatives, Dray et al

In increasingly more developed mathematical language the derivative can be thought of as ( William P. Thurston):

This is not just a list of different logical definitions but rather a list of different ways of thinking about the derivative. Only by embracing the Relational mathematics viewpoint, making contact to tangible instantiations of otherwise increasingly more abstract conceptions will the student be able to apprehend higher order abstractions by building on longer term retained and bedded down comprehensions.

Relationism is the Categorising of EquivalencesAs you learn more and more math, your brain will run out of room if you don t use category theory to organize your knowledge. And once they get deep enough into category theory, most mathematicians realize it s a lot of fun. It s very clean, very conceptual, and impossible to forget once you understand it. John Baez

At a higher level, an appreciation of the interplay between the objects of mathematics and their physical realisations as a route to understanding may be gained by considering the primary differential geometric object of Einstein s General Relativity (GR). Within the freely falling (elevator) framework of GR we have,

Physical coordinates in different reference frames are related by Lorentz transformations (as in special relativity) even though those frames are accelerating or exist in strong gravitational fields. (Physical time and physical space in general relativity, Richard J. Cook)

The defining object of Einstein s GR, then is an observational platform consisting of a network of laser-based stopwatches and rulers measuring Einstein s falling elevator in an ambient gravitational field. Specifically the frame field is defined as a collection of observers distributed over space and moving in some prescribed manner. Each observer, ( local witness ), O is assigned space coordinates that do not change. The Os are at rest ( ) in these coordinates. Each O carries a standard measuring rod and a standard clock that measure proper length and proper time at his/her location. The basic data of GR are the results of local measurements made by the Os.

This object can be traced alternatively from Gauge theories in which an equivalence class of bosons fields gives rise to a uniquely realised (vector) gauge boson excitation through the Gauge Principle. The underlying Gauge Equivalence maps to Einsteins Equivalence Principle. This later developed Gauge Principle itself applied by Yang-Mills rather pertained to the three non-gravitational forces effectively realises a soldering form object that knits together the local flat Lorentzian space of Special Relativity with the global co-ordinate diffeomorphism invariance principle of GR.

Relationism is everything in mathematics, category theory is telling us here that the soldering form is a functor mapping the locally equivalent gravitational and inertial viewpoints as espoused by the Equivalence principle. By employing this gauge object rather than the metric object of Einstein, Mathematicians can then see further equivalence between gravitational and non-gravitational interactions that would otherwise appear distinct in their nature.

What is the Plan Stan?From those that appear to know, on the virtues of planning your means of communicating:

Plans are of little importance, but planning is essential. Winston Churchill

Give me six hours to chop down a tree and I will spend the first four sharpening the axe. Abraham Lincoln. lt;/p>Both recognise that drawing together relationships rather than collating a bunch of mere blunt instruments is more fruitful in the long run. I have not achieved this here by any stretch, but the template from How to give a great research talk seems like a reasonable structure to follow:

Abstract (4 sentences)Introduction (1 page)The problem (1 page)My idea (2 pages)The details (5 pages)Related work (1-2 pages)Conclusions and further work (0.5 pages)Note here that related work comes as number 6 not number 3 in the list, a feature of some many educational articles that in trying too hard to justify the earnest of their efforts, lose the reader in a myriad of cross-referential tag teaming.

Above all endeavour to plan to make your mathematical expositions relational in content no matter how intrinsically abstract or abstruse that may be.