School of Mathematics Teaching

Principals and theory of teaching

A complete guide on the theory of learning and teaching.

These pages are maintained by Toby Bailey and are intended as a resource for those teaching in the School. What appears here is rather influenced by the lines my own (limited) reading has taken and I apologise for the errors and misinterpretations that surely exist, although there are contributions from others (thanks to LR and RF) and more would be welcome. 

This is a huge subject, but some familiarity with it is very useful:  at the very least it provides a bank of concepts to bring to bear when thinking about teaching and learning, and a vocabulary to discuss it with. 

The principal points it seems to me are these: 

  • There are different types of understanding ranging from procedural knowledge (e.g. the ability to carry out standard calculations) to the sort of deep understanding that we would like to instil. 

  • Each learner has to build their own understanding.  They do this by building in new information to what they know already. 

  • The idea that listening to (and/or reading) a beautifully organised account from an expert will of itself cause learning to happen is mistaken. Information transfer is not the main issue. 

  • Active learning is far more effective than passive learning. 

Types of Learning 

It is surely clear that whether even a small unit of mathematics has been learned is more than a yes/no question: there are degrees of learning ranging from the (probably short-lived) ability to reproduce a given form of words or carry out a very standard calculation through to what we might refer to as a "deep understanding" where the learner can apply the idea in different contexts and, say, vary it to suit different situations. So we distinguish surface learning and deep learning as part of a spectrum. For a well-known attempt to categorize parts of this spectrum in a subject-independent way, see Bloom's Taxonomy with its a scale of 1 (knowledge) to 5 (evaluation) and 6 (synthesis). Some keywords and types of evaluation are suggested for each. "Define" (as in "Define what it means for a function to be continuous") and "State" (as in "State the Fundamental Theorem of Algebra") appear at level 1?

APOS theory was an interesting attempt to categorize stages in understanding mathematics. It was perhaps over-sold as universally applicable and has fallen out of favour, but it can certainly illuminate the issues in learners getting a grasp on a lot of subjects. The four stages of understanding identified in APOS theory are as follows, illustrated by the popular example of "cosets": 

  • Action: the ability to carry out a concrete example. For cosets, the ability to compute a coset of a given subgroup of a concrete group. 

  • Process: the ability to consider and reason about the process. For cosets, the ability to imagine computing the all cosets of a subgroup of a general finite group (and hence understand the proof of Lagrange's theorem).

  • Object: the ability to think of the process or result of it as an entity in its own right. For cosets, the ability to think of the set of cosets of a normal subgroup (and hence to understand the quotient). 

  • Schema: A "schema" is a whole collection of processes, objects, and connections that constitute a full understanding of a mathematical thing. 

Even if one does not believe in this as a universal framework (and one probably should not) it does seem clear that "process-object difficulties" are often barriers for learners. Apart from cosets as above, think of the transition from thinking about a particular function to a space of functions (and perhaps the transition from "function as formula" to the mathematical idea is also one). Another example is learners who can take limits of sequences but find it hard to master the idea that a limit can be used to define the Riemann integral. 

Process-object difficulties also feature strongly in the excellent paper/booklet: 

Two other fascinating papers by the same authors are: 

  • Convergence of Sequences and Series Part 1 and Part 2. 

These appears chart students difficulties in learning basic real analysis and contains a lot of extracts from interviews with Year 1 students at a good UK university that makes interesting (and alarming) reading. The learners are divided into those who tend to reason visually and those who tend not to. They are also divided acording to whether they attended a traditional or more "active" course. Particularly interesting though is the relationship between success and the scale of whether the students could or did apply an internal authority to judge argeuments or whether they believed only in external authority: that maths being right or wrong can only be decided by a teacher. 

Expert performance 

Expert performance is quite a large subject. It is a good idea to understand the nature of our own expertise and surely too we should aim for our teaching to develop these characteristics to some extent. The first couple of chapters of "How People Learn" (ed. Bransford, Brown, Cocking; National Acadamy Press, 2000) are immensely informative on this and aspects of learning and teaching. 

  • Experts notice patterns in situations. They use chunking (perceiving a collection of features of a situation as a single pattern) to overcome the limitations of short-term memory. A classic example is that expert chess players can remember a chess position (but not a random arrangement of pieces) after just a short examination. 

  • Experts' subject knowledge is organised in ways that reflect deep understanding. Experiments show for instance that experts' schemas for mechanics problems organises thema round fundamental physical problems whereas beginners organise around superficial features (such as whether a spring is involved). 

  • Experts knowledge is "contextualised", meaning that methods and so on are connected to situations in which they can be employed, so that a situation will call up (with little effort) a collection of applicable techniques. 

  • Experts have "metacognitive skills": the ability to step back and think about their thinking. 

Links to Sites 

Some Important Themes 

  • The most important thing is the realisation that active learning is much more effective than passive.  

The best way to learn is to do; the worst way to teach is to talk.

Paul Halmos

Research has supported this view: passively listening to the material being "explained", however well, is not an efficient way of learning.  

  • So what can we do instead in a large class in a lecture theatre? See Student-active Learning and Peer Instruction: for some ideas. Student-active learning embraces a number of themes around the idea of promoting engaged active learning. One component is Eric Mazur's "Peer Instruction" technique used very successfully in Physics and elsewhere. (At the least, everyone should watch the video of Mazur's talk "Confessions of a Converted Lecturer".) For a mathematician's view on this see the excellent article by David Bressoud

  • Threshold concepts are important concepts in a subject that are necessary for progress in the subject generally. The idea is that one should concentrate on these more rather than stuffing curricula with as much knowledge as possible. See http://www.ee.ucl.ac.uk/~mflanaga/thresholds.html. For a few more remarks see DoTblog. 

Other Resources 

  • "Mathematics teaching practice", a guide for university and college lecturers by John H. Mason is a standard reference. There is a copy in the JCM library.  

  • "53 ways to ask questions in mathematics and statistics" by Ruth Hubbard has lots of interesting ideas for different sorts of questions to address typical concerns of lecturers and students. There is a copy on loan in the Main Office. 

  • Taking Learning Seriously by Lee Shulman. His analysis of three pathologies in learning are well worth reading: amnesia (the rapid forgetting of material not well integrated into the learner's understanding); fantasia (clinging to or returning to inappropriate views) and inertia (knowledge present but apparently not able to be usefully applied). 

  • Self-explanation is the process of asking yourself what you are understanding sentence by sentence as you read. Studies show that reading once with this process is more effective than using a similar amount of time to read twice. 

  • Understanding by Design is a Curriculum design approach that proceeds backward:  (1) decide on outcomes - what do we want students to be able to do at the end; (2) Assessment - how can students demonstrate those outcomes; (3) design a teaching approach.  This is in contrast to the traditional university approach of spending great effort on determining a syllabus and only a little on outcomes and the nature of assessing those. The book by Wiggins and McTighe referenced there contains a lot of thought-provoking material. 

  • Richard Skemp's book The Psychology of Learning Mathematics is directed primarily at school teaching, but is well worth some attention.  Chapter 2 on the formation of mathematical concepts draws useful attention to the importance of related concepts in understanding new mathematics.  Chapter 3 on Schemas explains how learning something "properly" (as opposed to memorising a procedure" depends on fitting it into appropriate conceptual frameworks ("schemas").  Facilitating this should be a major aim of teaching.  At times though a new generalisation or mode of understanding requires the learner to abandon an old schema and adopt a new one or at least make substantial changes, both of which can be very hard to do (see "threshold concepts" above).  An example might be the shift from an understanding of "function" in terms of formula to one based on the usual definition.  Chapter 12 on relational versus instrumental understanding (which seems to equate roughly to the distinction between "conceptual understanding" and memorised procedures) is also worth study.  This chapter is available also here.  These sections are used in our Y4 Maths Ed class - thanks to RF for pointing them out. Also used there is an article by M Swan which contains a useful discussion (around page 33) on running group work (in e.g. tutorials). Gillian Hatch's article Maximising energy in the learning of mathematics is also worth a read - I was struck by her analysis of the dangers in "explaining" mathematics to students (pages 133-135). 

Papers 

Just a list of things we have encountered and that seem to be important: 

  • "Do third-year mathematics undergraduates know what they are supposed to know?" by J. Anderson et al, International Journal of Mathematical Education in Science and Technology, Volume 29 (1998), 401--420: A sample of final year students were tested on what one might think of as core Year 1 material. The results are rather frightening. 

  • More Maths Grads Sheffield Hallam's "Reflections" site. 

  • Everyone should get an A by David MacKay (the "without the hot air" man). A rather interesting analysis of conventional assessment. 

Misc 

  • The University of Minnesota Physics department has a very comprehensive 2-week training programme for TAs. The materials are all online  - a lot are particular to Physics (labs, etc) but a lot also translates to Maths. I love the picture below, explained as follows: 

"Learning physics through problem-solving is a difficult, time consuming, and frustrating process -- like climbing a steep mountain. Many students try to run around this mountain by using their novice problem-solving strategies. Some of them give up (drop the course). A course structure must include scaffolding (ladders) that help students learn how to solve problem solving and barriers (fences) to keep students from succeeding in using their novice strategies." 

 

learning physics through problem solving using mountain example

 

  • The Torch or the Firehose? An experienced MIT mathematician's reflections on teaching. This is mainly directed at giving "recitations" which is not a form of teaching we do, but much of the advice applies to lectures and tutorials. I like the advice (on blackboards) that "standing in front of what they write is a method many teachers use to cover the material". 

  • Berkeley's Office of Educational Development

  • A great little note on "Content Tyrrany" - allowing the "I/we have to cover this that and the other" argument to dominate the more important issue of real student learning outcomes. 

  • There was an interesting experiment in Warwick some years ago, teaching a first course in Analysis in a problem-based way. Much of the material came from the book "Numbers and Functions: Steps to Analysis" by RP Burn. This book is a rare example of a textbook written from a pedagogical point of view: mosty of it is problems. The preface and a few pages is available on Google Books.