Chapter 3

The Mathematics Lecture

3.1 The Purpose of Lectures in Mathematics

In Chapter 2 we looked at curriculum design based on eleven basic principles of teaching underpinnedby accepted theories of learning of mathematics. We considered expression of content in terms of aimsand objectives, design of teaching strategies to meet the objectives, and finally the basics of preparation oflearning materials for a range of teaching activities. This really served to assemble the fundamental ideasthat we need to consider when engaging in any sort of teaching activity. In mathematics the primary formof teaching activity is currently the lecture supplemented by various types of tutorial and independentwork set for the student. In this chapter we look in particular at the mathematics lecture. Of course thegreater part of the effectiveness of a lecture resides in its planning and preparation. In Chapter 2, undercurriculum design we covered most of the general issues surrounding such preparation, whether it befor a whole module or a particular lecture. So we do not need to repeat this for planning and preparing alecture, but will simply summarize the main points in this chapter. This chapter concentrates mainly onthe delivery and evaluation of the lecture. This is underpinned by the same basic principles of Section 2.4and we will occasionally highlight where these are particularly relevant.

Definition of a lecture

The lecture is still the most common form of contact teaching in HE mathematics for anything but smallgroups (See Chapter 4). And of course there is a large range of approaches to such lectures. To accommo-date this range we will take a fairly precise definition of a lecture:

A mathematics lecture is a contact period during which the main activity is a carefully paced presen-tation, by the lecturer, of a particular mathematical topic, with the intention that when the studentsleave they will have the means, materials and incentive to study and to learn that topic to the desiredlevel.

Note that this definition does not imply a monologue from the lecturer, or absence of interaction with thestudents.

We are here concerned with how we can give good mathematics lectures that efficiently and effectivelysupport student learning. For this we will put together advice from experienced practitioners, results ofeducational research, commonly accepted good practice, and the views of students. Throughout, manyexamples are drawn from experienced practitioners, giving a range of perspectives.

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The purpose of lectures

Of course, the purpose of any lecture depends on the topic, the students and more besides. Properlydesigned a lecture can achieve a wide range of educational purposes. The really important thing is thatthe lecturer is clear about what they wish to achieve, and that they communicate this to the students.This is expressed in terms of the aims and objectives (i.e. learning objectives) of the lecture (Section 2.6).Normally the object of the lecture is to convey some important concept or technique, but it can also beused to good effect to inspire and motivate students, to impart values and change attitudes, encouragelearning skills and develop transferable skills.

The purpose of a particular lecture may perhaps be best appreciated when it is realised that lecturing timeis actually prime time. It is the one occasion on which the lecturer can convey ideas, concepts, attitudes,etc to all of the students at once and to get across the key messages in the subject. At other times studentswill usually be learning independently or in small groups, supported by materials that we provide forthem. Lecture time should therefore be quality time. Lectures may most productively be aimed at thekey messages in the subject, at those areas where students are likely to experience the most difficulty, atestablishing and encouraging higher order thinking skills, at developing attitudes and approaches thatcharacterise the subject, and at motivating the students. We might summarise this as:

The key purpose of a lecture is to get across the most important messages about our subject to themaximum number of students

The importance of lectures

Not only is the lecture prime time for conveying messages of all kinds to the students, but it is also a shopwindow for our subject and our community. So, lectures are important, because they can:

• be key learning experiences for the students

• motivate and inspire them

• influence students’ feelings about the subject

• show students what mathematicians do and how they work

• promote student confidence and self esteem

• convey the values of mathematics to students

• now cost students a great deal of money.

At the same time, we know that lectures are not absolutely essential in mathematics teaching, and manysuccessful mathematicians have managed very well without them. However, even the best and mostindependent student can benefit from a good lecture.

H Exercise

Discuss the advantages and disadvantages of lectures. Which lectures did you learn most/least fromas a student, and why? What do you think your students get from your lectures?

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3.2 Content and Objectives of the Lecture

In Sections 2.5, 2.6 we discussed content of the curriculum in general, and how it is expressed in terms oflearning objectives. The ideas and examples given there are applicable to planning and preparing for aspecific lecture, so here we will simply summarise the points as they would apply to giving a particularlecture, and for more detail you can refer back to Chapter 2. We have retained similar exercises, butapplied to the individual lecture, and these might be used as preparation for forthcoming lectures. Asdiscussed in Section 2.5 for the curriculum in general, the content and objectives of a lecture should:

• be made clear to the students (P3)

• fit coherently within the context of the course (P7)

• progress at a reasonable pace in both time scale and intellectual demands.(P5)

• take account of the mathematical background of the students (P5)

For more on these see Section 2.5, or work though the following exercises.

H Exercises

1. Think of a lecture you are soon to give and write a few lines that will inform the students ofthe content and purpose of the lecture.

2. Choose a particular topic from one of your lectures. What is the content and purpose of thattopic in the overall course/module? What does it rely on and where does it lead?

3. Break down one of your modules into say 25 lectures, each covering a specific topic(s)orobjectives. Write a concise description of the content and objective of each lecture, and sketchout how the different lectures are linked together.

4. Choose a particular lecture you are soon to give and analyse its pre-requisites. Whatknowledge and skills will the students need to cope with this, and what level of facility do theyneed in these skills? Are there likely to be any students who lack this facility, and how can youaddress this?

3.3 The Teaching and Learning Strategy for the Lecture

Again, designing teaching and learning strategies to meet the aims and objectives of a particular learningactivity are considered in Section 2.7. In designing the teaching and learning strategy and the learningactivities for a given lecture we might need to think about:

• the resources available (P1)

• the students’ abilities and background (P5)

• alignment with the learning objectives (P4)

• ways of engaging the students in active learning behaviour (P10).

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Refer to Section 2.7 for more details. The following example is perhaps an extreme example that at leastshows what is possible.

ExampleMillet [55] describes a very involved teaching and learning strategy designed for lectures ona Precalculus course for service students. In it activities were designed precisely to engagestudent activity in a way that would encourage learning. The 75-minute ‘lecture’ was notwhat we would normally regard as such! The format, outlined below, is very interactive.

1. An initial segment of administrative information followed by an invitation for studentquestions

2. A segment devoted to discussion of problems that had arisen from previous weeklycoursework

3. A group ‘surprise quiz’ came next, based on reading previously assigned for the lecture,culminating in a class discussion of results which moved the class onto new material(about 20 minutes into the lecture)

4. The fourth and longest segment concerned new concepts and methods growing out ofthe surprise quiz, and developed in a manner based on the previous discussion. Thiscomprised no more than one or two new key elements as the focus of the work

5. Time permitting any secondary new material was incorporated in the presentation anddiscussion of a closing issue

6. In closing students were reminded of homework, quizzes, material to be reviewed, etc.

Of course, we do not advocate that all lectures are planned in such meticulous detail, but in fact asyou develop experience you find yourself automatically thinking ahead about such things. Again, theexercises below can provide you with practice in this area.

H Exercises

1. Itemise the resources necessary for a few of your lectures. Include your contact time,preparation time (including thinking time!), costs of any materials, student time,accommodation facilities, equipment, etc. Now consider any efficiencies that can be made,without compromising the learning objectives.

2. Think of a particular lecture you are about to give. Are the students ‘ready’ for the proposedteaching method and associated activities? When you cover a particular point, will they beable to cope with your delivery method - will a straight lecturing mode suffice, or will theyneed more interactive and discursive explanation? Are they used to doing exercises unaided inclass?

3. Consider a number of lectures you might be giving soon and design appropriate teaching andlearning activities to meet the various objectives of the lectures.

4. Think about your next lecture and plan how you might introduce one or two short periods ofactivity to emphasize particular points. Aim to deepen the students’ understanding, withoutnecessarily sacrificing coverage of material.

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3.4 Preparation of the Learning Material for a Lecture

Media and message in the lecture

In Sections 2.8 and 2.9 we considered the issues behind preparing learning materials in general, splittingthe discussion into the media and the message. The application to a particular lecture is straightforward,and the exercises will provide practice if needed. In a typical lecture the main materials needed will be:

• the students’ notes, either transcribed from your presentation or in a handout, or electronic file

• problem/exercise sheets

• occasional administrative stuff such as timetables

• ongoing work such as coursework and associated solutions and feedback.

As emphasised in Chapter 2 remember that preparation takes longer than you expect, especially for thefirst delivery of a lecture, the writing style you use is important, and the volume of learning materialsshould be reasonable for the length of the lecture. Also the real bulk of preparation time lies in theintellectual construction of what you are going to say to the students, how you will express it, whatexercises you will give, how to phrase things, what depth of proof to use - the message.

Design of learning materials for the lecture - media

The list below summarises the sorts of things we have to think about when preparing teaching materialsfor a lecture. See Section 2.8 for details, or work through the exercises below.

• The alignment of the materials to the teaching and learning activities of the course

• The clarity and conciseness of the materials and the style of presentation

• The volume of learning materials.

H Exercises

1. Outline the learning materials you will use for a selection of your lectures and explain howthese will support the achievement of the learning objectives

2. Check your lecture materials for clarity and conciseness. Can they be improved? Can youreduce the number of pieces of paper?

3. Consider one of your lectures. At the end of the lecture, how much material will the studentshave? Does it seem about right? Compare with other modules.

Design of learning materials for a lecture - message

Again, this is simply an application of Section 2.9 to the case of a particular lecture. Also note that youmight not be able to incorporate all the suggestions here in your first lectures, but gradually they can beabsorbed until you do them automatically as the need arises. Based on the eleven principles of Section2.4 we suggest the following guidelines for the learning materials for a lecture. They should:

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• be appropriate to the students’ needs and level of understanding of the topic

• support the achievement of the objectives

• highlight/reinforce the key ideas

• provide hooks for students to hang ideas on - aide memoires, mnenomics, etc, to make it as easy aspossible to remember and internalize key results and ideas

• anticipate the difficulties students might encounter and support them in dealing with these

• highlight the essence of complicated ideas, theorems, methods, etc and support the development ofstudents’ intuitive views of formal arguments

• provide roadmaps and overviews of difficult sequences of arguments

• include devices for developing rapport with the students

For more details see Section 2.9, and if needed work through the following exercises.

H Exercises

1. Check your lecture material for each lecture. Do the notes, exercise sheets, etc take account ofthe students’ background - will the language and notation used be familiar to them, both interms of content and mathematical maturity?

2. Choose a typical first year topic such as partial fractions and prepare a succinct non-technical(i.e. accessible to those not yet familiar with the topic) summary of the topic and its relevance.

3. In lectures you are about to give what are some of the most difficult ideas to get across - wheredo students have the most difficulty? Why is it a difficulty and how can it be addressed?

4. Choosing examples from some of your own lectures, bring out the essential messages ofparticular ideas, expressing them in the simplest, most incisive way with a minimum oftechnical language and possible use of analogies.

5. Choose a number of key topics from some of your lectures and write succinct overviews anddirections through difficult sequences of arguments. Don’t forget that while they must beconcise and succinct, they must also be in terms that the students can understand.

3.5 MATHEMATICS for the Lecture

It is useful to have a checklist to run through before the lecture to avoid last minute hiccups and get youin the mood. If this is the first time you have given this particular lecture then you may spend sometime rehearsing it and getting your ideas together, even if everything is fully prepared. Most of us findit stressful in front of a class, and do not always function at our best, so we really should leave nothingto chance. Indeed a part of your last minute run through may be just to calm your nerves and bolsteryour confidence. For a run through of what you might consider when about to give a lecture, we can useMATHEMATICS (Section 2.2):

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Mathematical contentAims and objectives of the curriculumTeaching and learning activities to meet the aims and objectivesHelp to be provided to the students - support and guidanceEvaluation, management and administration of the curriculum and its deliveryMaterials to support the curriculumAssessment of the studentsTime considerations and schedulingInitial position of the students - where we are starting fromCoherence of the curriculum - how the different topics fit togetherStudents.

Obviously, the mathematical content of the lecture is important, and although we don’t usually havelesson plans we do have to think about the aims and objectives of the lecture. Read through your notesto remind yourself. You should of course know the subject matter inside out by now, but refresh yourthoughts on it. Usually the teaching and learning activity in a lecture consists of the students listeningto what you have to say, or watching you work, but of course there are many ways of engaging them infruitful activity and maybe today you might try something new. Practice your delivery, if only mentally.Help and support for the students may simply amount to directing them to a section of a text, or givinga handout on a difficult topic, or taking particular care in explaining a particular point. Maybe it is timeto give out a student questionnaire, but in any case you should be continually evaluating how the classis going. And what materials do you need for this lecture - a problem sheet, a few OHPs? Organisethem for easy access? And in materials include the accommodation and facilities of the lecture room -particularly chalk/pens/eraser! What about temperature, noise levels, over/under crowding, boards,screen, and other AV - order equipment needed in good time. In terms of assessment, do you need tohand out/collect in coursework or tell them something about the examination to come? Do you havetime to cover the next topic today, have you got to tell them the examination dates? Normally, if you arefollowing on from previous lectures, you will know the initial position of (most of) the students, but itis a good idea to remind them where you got to last time. And if it is the first class of the session thenyou may even need some sort of review. In fact does it follow on from the last lecture, or is it a newtopic altogether? If it is then you have to explain the links to the students, to ensure the coherence of themodule. Finally, remember why you are there - for the students. Think about what is best for them, seeany students that you need to, work to establish a rapport with them.

Of course, we may have to think about other things as well, but this provides at least a basic minimal set.In fact, if you construct a table with the MATHEMATICS letters as row headings then you can scribblein relevant notes for the lecture which effectively becomes your lesson plan, as well as a record of howthe lecture went for later. Of course, if none of this works for you, then devise your own checklist.

H Exercise

Work through the MATHEMATICS checklist for your next lecture. Did it cover most eventualities?If there is anything missing, make a note of it for next time.

3.6 Presenting the Lecture - an Overview

The next four sections are about actually giving the lecture. For many of us, particularly in our early years,this is something of a trial. While we hope this book will help you to prepare for this, it will probably bea long time, if ever, before you feel entirely relaxed when you go into the classroom. It will help in your

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early lectures if you can discuss your experiences and problems with an experienced colleague. Yourdepartment will normally provide a mentor for this purpose.

In giving a lecture we have a wide range of things to consider, even when we have prepared all thematerials we need. First there is the basic classroom technique including establishing a professionalpresence in the classroom, running the lecture to ensure the objectives are met, and structuring it formaximum impact, starting and finishing appropriately. There are the mundane technical issues of usingthe various resources in ways that will aid the learning process. Then there is the very important task ofproducing the right atmosphere in the classroom - creating a warm, relaxed, but industrious environmentin which people want to learn, enjoy the activity and feel free to get involved. Last, but most important,there is the job of ensuring that students are engaged in effective learning. So the next four sections coverthe following.

Basic classroom teaching technique (Section 3.7)

• professional and personal issues

• starting the lecture

• ensuring satisfactory delivery

• closing the lecture.

Use of resources (Section 3.8)

• use of accommodation - media and message

• use of black (white) board technique - media and message

• use of OHPs - media and message

• use of handouts - media and message.

Maintaining a conducive learning environment during the lecture (Section 3.9)

• establishing good relations with the students

• classroom management

• motivating and challenging students in the lecture

• encouraging student engagement and interaction in a lecture

• helping students to get the best from the lecture.

Lecturer’s facilitation and support for the students’ learning (Section 3.10)

• ensuring that students learn during lectures

• the lecturer’s appreciation of students’ understanding, needs and level

• careful explanation of complicated ideas.

A great deal of this material is generic, applicable to the teaching of any subject, so you might see someof it in your generic institutional staff development courses, and here we will focus on the mathematicalaspects. There is no harm in repeating such material in the context of mathematics, for mathematiciansof all people can appreciate the power of particular examples of general principles!

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3.7 Basic Classroom Teaching Technique

In this section we look at the basic mechanics of giving lectures, focusing on running the class in a pro-fessional, organised and interesting way, establishing effective communication and relations with thestudents. Specifically, we will be looking at:

• professional and personal issues

• starting the lecture

• ensuring satisfactory delivery

• closing the lecture.

3.7.1 Professional and Personal Issues

In Section 2.4 Principle 2 emphasizes the fact that teaching is a matter of human interaction, and thatis the context in which ‘professional’ is used here. As well as the professional responsibility to do agood job technically, the teacher also has to cope with the extra demands of the human side of teaching- remaining detached in difficult situations, setting an inspiring example, treating all students equallydespite any personal feelings towards them, and so on. We will look at this generally later (Section 3.9),but for now we are concerned with the things that arise in the typical lecture, and how the needs of ourstudents might inform our professional judgement and behaviour. Essentially, we need to be enthusiasticabout our subject, confident and positive during the lecture, but alert to possible problems that mightarise. We have to conduct ourselves in a proper and considerate manner and be fair, firm but generouswith the students.

Enthusiasm

Time and again students put enthusiasm high on the list of qualities they like to see in their lecturers.But sometimes the topic we have to teach really is quite boring, and then perhaps we have to fake en-thusiasm! Or you might be open with the students and tell them that this part of the subject is in facta little uninspiring, but we will need it later on. In any event, you need the self awareness to recognisewhen you are feeling a bit jaded about a topic, and have the self-discipline to do something about it - youneed to ‘stay fresh’ ([53]). Then you have to find the interesting aspects of the topic - there must be some,otherwise you perhaps shouldn’t be wasting precious class contact time on it. You can stimulate yourown enthusiasm by asking yourself such questions as:

• history of the topic, where did it come from and why?

• where does it lead to, what areas of maths are built on it?

• what sort of applications does it have?

• can I derive any of the results involved by different methods?

• can it be generalized?

• how is it related to my own particular area of interest - if it appears not to be, why?

• why do the students need this anyway?

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• what concepts are at the core of the topic, and where else do these occur - can you imagine how theoriginal developers of the ideas found their way through them?

The point is to be imaginative about generating interest in the topic, and at the same time this might alsoprovide ideas about how to teach it.

Examples

1. Definitions and notation, particularly in something like abstract algebra or analysis, arenotoriously boring to put across. But there is usually a tale behind most definitions, areason for the name or notation and occasionally mentioning some of these can break upthe monotony. This might also help the students to understand why precise definitionsare so important, which is something they don’t always appreciate. And the sooner thedefinitions are used in subsequent material the better.

2. There is a lot of low level mathematics teaching, even in the most research intensiveuniversities, for example foundation or first year service teaching. If you are a keenresearcher in topology and you are assigned such a class to teach then it may be verydifficult to find something to enthuse about in solving quadratic equations, for example.But that is the nature of the teaching job - we have to try to be enthusiastic and conveythat to the students, even if the material is elementary. Of course, quadratics are veryinteresting to any keen mathematician. They lead us into complex numbers. For manygeometrical purposes they may be used to approximate other continuous functions. Theyhave a very rich history. Completing the square is an example of a magical trick usedeverywhere in mathematics - ‘getting something for nothing’ (adding and subtracting,multiplying and dividing, etc). Any polynomial with real coefficients, .... . The list goeson. Any elementary topic can be similarly elaborated.

3. Sometimes it is refreshing to approach a familiar task by a new approach or in a differentorder. However, this should not be at the expense of making it more difficult for the stu-dents to learn. If, for example, the lecture is devoted to a long proof, then the presentationcan be freshened up by simply bulleting the stages of the proof and then reconstructingthe details from scratch, with the students’ help, rather than by transcribing them fromprepared notes.

4. For such things as techniques of integration, matrix algebra you can simply choose dif-ferent examples each time, but be sure these are going to work out conveniently - forexample it would be a brave lecturer that chose to find the eigenvalues of a random 3by 3 matrix. It is amazing how unexpected complexities pop up when you try newlyinvented examples, and embarrassing when you have to try and rescue the situation. Sowhen you do manufacture new examples, test run them first.

5. Mason ([53]) suggests such things as changing the symbols in some piece of work (Whymust it always be x and y?). This may be a useful tactic, but again be careful becauseit requires intense concentration to keep this up (which is of course the point), and willconfuse students if you keep slipping between notations.

H Exercise

Think of the most boring piece of mathematics you can imagine. Now find something interestingabout it. What are the boring areas of mathematics, why are they boring, and how can they be madeinteresting?

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Confidence and awareness of the difficulties and stress that may arise

Good preparation will help your confidence levels, but it is still likely that problems will arise, you willmake mistakes, you will be uncomfortable with some of the students’ behaviour, and you may have tochange your plans. Such things may get to you and may be stressful - but usually things are not as badas they may feel. Try to maintain a sense of humour, and never lose your temper or control. Learn to notlet errors fluster you. Being nervous is OK, but learn to control it.

You may on occasions make mistakes in the class, everyone does ([7], P181). The real question is how youdeal with them. Some common difficulties that can arise include:

• losing your train of thought during a long sequence of arguments

• minor errors such as dropping a sign or writing the wrong symbol

• saying something different to what you actually write - as we will note later, in general any givenindividual will think, write and speak at different speeds

• omitting a crucial step because of your high familiarity with the material

• going off in the wrong direction.

Usually the students will pick you up on these, so be attentive to this. Be alert to advance warnings, suchas students suddenly starting to talk to their neighbours - sometimes they are just checking if you reallyhave made a mistake before raising it with you. You may sense a bit of discomfort in the class, fumblingwith their notes, looking at each other, etc. Sometimes a number of students shout out queries together,and you can’t figure out what they are saying. Take charge, quiet them down, scanning your work asyou do so, and if this doesn’t locate your error (It nearly always does), ask for one student to point it out,thanking them for that. When you reread your writing on the board you usually read what you think isthere, so if students have pointed out an error read more carefully! A few errors, corrected, at least showsthe students that you are human - but of course there must be no errors in key ideas and concepts.

H Exercise

In your next class make the odd deliberate mistake, and be alert for any student response. If nobodypicks it up go back and discuss it. Are they not paying sufficient attention, are you going too fast sothat it is all they can do to keep up with the writing, are they too frightened to question you? Thenreassure them that you are going to make real mistakes on occasions, and then you welcome theirintervention - they shouldn’t feel inhibited.

Setting an example of proper behaviour

To ensure that lectures run smoothly and encourage productive learning you will need to insist on good,responsible behaviour from the students. The best way to achieve this is to set a good example yourself,being punctual and proper in your behaviour. and treating students with respect and courtesy. Of course,students are people first, mathematicians second (or third, or ... ). So don’t let your feelings for them asmathematicians colour your treatment of them as people. This can be very difficult, even for experiencedlecturers. It is easy to favour the keen, capable student and to prefer to be in their company, but in fact itis often the weaker less interesting students who need the most attention. You will occasionally get somestudents that are a bit ‘bolshy’. This seems to be not uncommon amongst bright mathematicians. Usuallythey are well intentioned and genuine. Try to remember this when you encounter them and don’t fightfire with fire.

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Treating students fairly and generously

A common description applied to the good teacher is ‘Hard in the head, soft in the heart’. You have to sethigh standards of behaviour, thinking and debate, particularly in mathematics, but you also sometimesneed to be generous and give the benefit of the doubt. This does not mean compromising on quality orstandards, and it is perhaps not so much about being soft on the students, but being hard on yourself.All of us cannot help but let our own personalities and our own problems affect the way we deal withother people, and mostly, with our family, friends, working colleagues, we get away with this without toomany repercussions in everyday life. But we cannot afford this luxury when interacting with students -our position is more one of the priest or counsellor, and we have to set ourselves aside. There are manyoccasions on which one gets frustrated, irritated or impatient with some of the students and it is then youhave to guard against inappropriate behaviour - you should certainly never actually display any of theseemotions to the students.

Personality, disposition and attitude of the teacher

As Krantz ([49], page 42) argues, you are the most important thing in the lecture room - and you are aperson interacting with people. But you are not interacting in the ordinary sense of people interactingwith others say in a social or work context. Here, interacting to produce the best environment for learningis actually part of the job - in some sense you are actually acting. There are few professions in which ourpersonality and disposition are as crucial as in teaching. There are few professions that have such apowerful influence over their ‘customers’. A good teacher can influence young people in a positive wayfor the rest of their lives. A bad teacher can destroy confidence, alienating students from the teacher andpossibly the subject. It is part of our professionalism to be aware of this and to control our actions in thebest interests of an enlightened educational environment. Essentially you have to adapt your personalityand disposition to the needs of the job and continually monitor yourself to remain fresh, self-possessedand alert. You can, as much as possible be yourself, so you can relax, and help your students to get toknow you. Convince students that the material is doable, but it mustn’t look too easy. Avoid waffleand don’t be patronising. Be an objective honest broker. Take charge of ensuring that the students havemaximum freedom within an enlightened educational environment. Don’t abuse power. Both Wankatand Oreovicz ([72], p 36) and Baumslag ([7], p 141) emphasize that the most important attitude a lecturercan have is to be on the side of the students, to help them progress, and to keep encouraging them.

Examples

1. One of the most trying tasks in a lecture is to explain difficult ideas to the students witha genuine desire that they understand. This means that you have to keep questioningthem and insisting you get an answer, and treating answers in a kind and patient way. Ifyou are by nature impatient this is very difficult indeed, but it has to be done, and youneed to train yourself to cope with it. To show impatience or any other negative emotionis very counter productive and never advances the learning process.

2. Sometimes, when you are discussing an area of mathematics with your students, youmay inadvertently slip into the mathematical vernacular, forgetting who you are talkingto and thinking you are talking to a mathematical colleague, using all the familiar col-loquialisms. Try to avoid this. Also, don’t be unduly dismissive or opinionated about aparticular area of mathematics, whatever your views. Before the days of the RSA encryp-tion algorithm and other applications of abstract algebra one would often hear dismissivecomments from some lecturers about the pure mathematics parts of the curriculum and

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this might have influenced students against the subject. The students are not interestedin our prejudices, let’s allow them to develop their own!

H Exercise

Think about yourself as a person. Do you have any characteristics that might enhance, or possiblyimpede your abilities as a teacher? Are you naturally impatient, prone to arrogance, etc? Asktrusted friends and family. How can you best use these characteristics, or diminish any negativeeffect on your teaching?

3.7.2 Starting the Lecture

The way you start the lecture can set the mood for the whole class. You might settle them down with lightsmall talk, give time for the (slightly) late arrivals, or get the administrative stuff out of the way. When youdo actually start do it clearly and obviously - that is, command their attention and concentration. Thenstart with a brief outline of the purpose of the lecture, perhaps giving them a reminder of the previouslecture. See Wankat and Oreovicz ([72], p38) for a ‘start up list’ you can adapt to your own needs. Suchthings as mentioning a recent current event that relates to the subject of the class, or comments on pointsarising from recent coursework help students to settle into the new class.

A recap of where you got to in previous lectures can provide reinforcement of the previous material andset up a context for what you are about to do. Baumslag ([7], p143) for example emphasizes the needto continually introduce, summarise, review, and revise. He suggests the use of diagrams showing therelation of the current lecture to previous material. In the first lecture of a module you can give a summaryand overview of the whole module, but of course in layman’s terms.

Examples

1. When coming to discuss the factor theorem for polynomials, you can remind the studentsof previous examples of factorising simple polynomials such as quadratics and somecubics by inspection, and note the limitations of these methods. Then emphasise the factthat the factors indicate precisely where these polynomials vanish, and indeed to findsuch points is one of the purposes of factorisation. One can then point out to them that byreversing this process we can devise a way of factorising more complicated polynomials,and this leads naturally into the subject of this particular lecture. In doing this we havereinforced previous methods of factorisation, and established a context for the new workon the factor theorem.

2. Baumslag ([7], page 144) gives an example of an introductory overview for an elementarycalculus course. As he explains, the idea is to teach the main aspects of the topic conciselyusing little more than intuitive notions and ideas that the students will already recognise.

H Exercise

For your next few lectures, think ahead about what they will contain and draft a few brief lines ofintroduction that will serve to introduce the topic of the lecture and lay the foundations for the maincontent.

We can also spend a few minutes at the start of the lecture giving the students a clear vision of what thelecture will be about - essentially the learning objectives of the lecture. Then they can start to construct

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the context for assimilating the material and they can start to think about the nature of the intellectualdemands it might make on them. Some lecturers write out the learning objectives at a convenient spaceon the board or on OHPs and keep referring back to them. Note that there is a slight danger whenannouncing up front what you are going to do in the lecture. Some of the students might think they haveseen it before and mentally switch off, which is the last thing you want. So a bit of thought is neededabout how we describe what the lecture is to be about. This problem is particularly common in first yearclasses.

ExampleSuppose the lecture is to first year maths students with good A-levels and is devoted to in-tegration using partial fractions. Now many of the students will have seen this before, andwill tell you this, and may be impatient at you going over ‘A-level stuff’. They might not seethe purpose of the lecture. But the real purpose of the lecture may actually be to consolidateand extend their skills and speed up their performance, or maybe introduce them to someshort-cuts such as the cover-up rule for partial fractions. This is what should be emphasisedto the students from the start. They may have met partial fractions before, but it is unlikely(for the UK students anyway) that their skills will be as highly developed as they need to be.

3.7.3 Ensure Satisfactory Delivery

Speaking and writing clearly

While this really goes without saying, it is an area where new lecturers can have difficulties, and it in-volves a number of dimensions, from how you speak to how you write and how generally you ‘putyourself across’. Firstly, we have to ensure that what the students take away with them will enable themto achieve the learning objectives. In a mathematics lecture this usually translates into sending them offwith a good set of notes and instructions for future tasks. So organise what you write/say to make iteasy for them to get the message. Use sectioning, headings, etc skillfully ([7], p166). Krantz ([49]) advisesthat you write such things as conditions for theorems on separate lines, or as bullet points, not as a singlesentence with ‘and’ between them. Give each theorem/technique an easily identifiable name/number.Baumslag ([7]) also gives suggestions for dividing/arranging boards to good effect and using lines, ta-bles, arrows, and other diagrammatic ploys to link up associated ideas, equations, etc. Link what you sayand what you write considerately, to improve the quality of the students’ notes (It is a good idea fromtime to time to read through some students’ notes to see if your message really is getting across). Not allsuch ideas may work for you, but it illustrates the sorts of things we might do to help students get thebest from our lectures.

Pretty obviously, you need to ensure that the students can see and hear whatever you do. This is the sortof thing that a generic staff development course might help with. It may include such things as voicetraining, as for actors ([49] and [7], page 165). If you have concerns over your voice projection, use amicrophone. You can enliven your delivery with intonation, emphasis, body language - ‘fill the roomwith yourself’ as Krantz says, although in a dignified and reserved way. Face and talk to the students,establishing eye contact with them. To emphasise an important point pause, say and write it out clearly,tell them it is important, repeat it, ask the students if they have any questions, refer to the examinationand need for the ideas later in course. So letting students know when you are making a key point is animportant skill, which means continually thinking of the key ideas that you are trying to convey. Forthings such as use of boards, see Section 3.8.

ExampleSuch things as pronouncing and writing Greek symbols can cause students problems, and

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you need to be very emphatic about these things, particularly with first year classes. In fact,from experience at seminars, conferences, teaching observations it seems that this sort of thingis not uniform across mathematicians, especially of different nationalities. Even experiencedmathematicians sometimes find themselves having to double take at some strange symbol onthe board, or word uttered - so imagine what it is like for students. And of course mathematicsis so precise, sequential and fast moving that a second’s distraction trying to decipher a wordor symbol can lead to the student losing the thread altogether.

H Exercise

Make a list of all technical terms and symbols in your course that you think might be new to thestudents. Make sure you can write and say these clearly enough for them. Maybe issue the list to thestudents and keep referring to it when the terms and symbols first come up.

Refreshment breaks

Of course, even if the delivery is perfect few of us can sit through an hour or so of lecturing withoutneeding some sort of change of activity or break (But see below). And Principle 10 reminds us that,particularly in maths, most of us learn best if we are actively engaged in the process. So every 20 minutesor so provide some activity, change of activity or whatever to give the students a chance to rechargetheir batteries. This is very easy in mathematics - you can simply give them a short problem to do thateither emphasizes a point you have made, or primes them for something that you are about to say. Suchactivities can be directed at such things as the most difficult parts of the lecture material, the intellectualhurdles, or at some subtlety that they might otherwise miss. Maybe a perfectly routine piece of work,just to wake them up. Essentially, you have to think about what you want to get through in the lectureleaving say 10-20% ‘free time’, and then think of ways to get the students active to the best effect in thattime. There are plenty more structured activities you can ask the students to do for mini-breaks, some ofwhich are listed below ([35]):

1. rest/time out/silent reflection

2. read own or other’s notes as a prelude to the next stage

3. write down one or two questions that have occurred to you during the lecture

4. ask your questions

5. tackle a problem that will get you thinking about the next stage of the lecture

6. discuss a question of your own or the lecturer’s

7. apply a particular concept that has just been introduced

8. take a short test to check progress so far

9. planning and organising your notes and study of the subject so far.

Make sure that the instructions for such activities are clear and the students have any materials they needfor them.

An important point about such ‘refreshment breaks’ is that you have to build them in and keep a closeeye on the time - it can easily be frittered away in this way, Also, you have to develop the skills of bringing

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them back on task when you need to. Actually, this can have benefits elsewhere in your relation with thestudents. If they get used to the idea that you can let them go a bit - provide some playtime - but at theend of the day you call the shots, you control the activities, then that gives them rules and boundariesto work within. Another important point about this issue is that we should perhaps question the viewthat a proficient, able intellectual should not be expected to maintain attention for an hour on a topic ofrelevance to their interests. Certainly, one would not provide refreshment breaks in a research seminar,but these are not always very stimulating, and are usually more interactive than the average lecture,and in any case we rarely get past the first ten minutes ourselves! And our students are nothing likeso intellectually disciplined (yet), and your lectures are unlikely to be anywhere near as close to theirinterests as a research seminar is likely to be to yours. So the lecture is a continual balance betweenkeeping the attention of the students and trying to stretch their powers of concentration.

Examples

1. Try setting a small task for the students to do while the class is assembling for the start ofthe lecture - specified on an OHP, for example. Writing down a question they might haveabout the topic to be discussed, recalling a result that they will be using in this lecture,etc. Then for the first break refer back to that exercise and get them discussing what theywrote for a few minutes.

2. Working through particular cases of theorems often provides a convenient change ofpace or intellectual activity. Having spent a while working quite abstractly through theproof, requiring deep concentrated thought, going through a specific example of the the-orem provides a change of pace and level of difficulty, a change to more routine andundemanding thinking and is as good as a rest.

3. Extending a proof or result to a wider range of application, which may be non-examinableand therefore not so serious - more relaxing. A typical example here, for first year classes,is complexification of results and techniques in differential equations - just asking themto consider what happens if you allow your variables and functions to take complex val-ues can lead to lots of adventurous mathematics that may fascinate and enliven some ofthe students - it will at least encourage them to think around the subject without beingunder pressure to master it.

4. Discussing in pairs open questions such as ‘What is algebra?’, ‘What is the connectionbetween algebra and geometry?’. Such discussions provide a complete change for theclass from the meticulous attention to detail required for most of the lecture, and if chosencarefully may awaken interest in the topic of the class.

5. See the example provided by Millet [55] in Section 3.3.

H Exercise

Design a selection of ‘refreshment breaks’ for use in your lectures, some frivolous, some fun, somemerely a change of pace or intellectual demand. Remember to allow time for these in the schedulingof the class.

Pace

One of the most important aspects of good delivery is the management of the pace of the presentation.This is a frequent student complaint, particularly about new lecturers. Adopt a pace that the studentscan handle. Try to form a rough estimate of how long each part of your lecture will take, leaving a small

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amount of slack for questions and activities. Then try to maintain an unhurried even pace. If things don’tgo according to plan because maybe a topic took longer than you estimated, don’t accelerate to cramthings in. Shift things into your next lecture and readjust later. If you finish your plan early improvisesome additional material, or simply carry on into stuff planned for the next lecture - or finish early (See[72], p. 42). Another point about pace is that everyone thinks, speaks and writes at different speeds.Sometimes, when writing on the board, particularly during mathematical calculations, you are thinkingone thing, saying another and writing something else. You can for example find yourself missing outsome steps because your are thinking ahead too far! Slow down!

One way to control your pace is to keep asking yourself ‘Are they getting this?’, with a genuine desireto know. Keep looking for reactions in the class - puzzled looks, signs of boredom or disengagement,etc. And never let urgency to get through the material speed you up. You should plan and prepareyour delivery so that there is plenty of time to get across the main messages without having to rush. Butremember there is also such a thing as going too slow and labouring the obvious. As always try to getthe balance right.

H Exercise

Practice pacing your lectures, monitoring your speed of writing, talking, delivering generally. Getused to pausing occasionally to let students keep pace with you. Vary the pace depending ondifficulty of material.

Balance between theory and examples

Another component of satisfactory delivery is the balance between theory and examples. No one canlisten for very long to a monologue of information about any subject without very soon needing somesort of concrete example or application - something that they can play with to check that they have theright idea. Indeed most mathematicians usually commence a research problem by playing with examples.On the other hand too many examples and applications will simply slow down the delivery, so a balanceneeds to be struck. By an ‘example’ of course we mean any specific instance of a general principle, result,technique, etc. It doesn’t have to be ‘applied’ in the usual sense, but just something that illustrates ageneral statement. When explaining a long sequence of mathematical arguments one is often sparedthe need for examples because the logical development of the argument is self-explanatory. So one maylecture for some time before the need for an example arises, but sooner or later you will have to giveexamples of some particular point, if only to change pace and give the students opportunity to thinkabout what you have been saying. One thing is for sure, you will never give enough examples to satisfyall the students, because it is these that turn the material into things they can more easily grasp - thatgives it a more familiar context. As for the sorts of examples, these should not be too difficult nor tooroutine. See Mason [53] for a discussion of ‘what makes an example exemplary’.

Examples

1. Immediately following the presentation of any major new technique such as invertinga matrix or integrating by parts, we might present a range of worked examples show-ing the main features of the technique. For example for matrix inversion a couple ofnon-singular matrices, and then illustrate what happens for a singular matrix. Or forintegration by parts illustrate by a straightforward one such as xex and one that has tobe repeated such as x2ex or circulated, like ex sinx.

2. Following an important or subtle definition, give some specific cases, as for examplein the ε, δ definition of a limit, or the definition of a basis for a vector space. In this

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sometimes give examples, or get the students to devise their own, that illustrate whenthe definition breaks down, or stretches the extent of the definition to its limit. Mason([53], page 14) gives a useful tactic for this, in his ‘boundary examples’.

3. Another use of theory and examples in conjunction is the parallel treatment where sometheory is presented on one side and, in parallel stages a specific example on the other.Thus one may work through the proof of a theorem in the general case and in parallelpresent a particular example. Or, one might list the general stages in some technique suchas inverting a matrix, and in parallel do a particular example. See Burn et al ([16], Page53 for the solution of a differential equation by integrating factor, with the general casein parallel with a specific example. Or [21], p. 387 where the solution of linear equationsby determinants is illustrated in parallel with an actual numerical example.

H Exercise

Construct and rehearse examples at appropriate points in your module, making sure that these areillustrative of key ideas and results.

3.7.4 Closing the Lecture

The old adage ‘Tell them what you are going to do, do it, then tell them what you have done’ suggestswe close the lecture with quick summary of the main points of the lecture. We should also try to end ona high note, pointing to the (hopefully) exciting aspects of the lecture. Always end appropriately - not inmid-sentence or halfway through a point, better to finish a few minutes early. Don’t rush off at the endof the lecture, make yourself available, see any students that you need to. This is another mechanism fordeveloping good relations with the students.

One useful way to close a lecturer is to ask the students to spend a minute or two writing down what theyhave got out of the lecture - say three main points, and collect the results as the class finishes. When youclose, you can also say what you will move onto in the next lecture, which shows them where the work ofthis lecture leads and prepares them for new ideas they will be meeting next time. This is part of puttingthe lecture in context and developing an overview. In closing we may also need to remind the students ofany administrative matters, coursework submission dates, etc. Ask them to bring along anything specificthey will need next time. If it is actually important information, then mention it before they switch offand pack up.

3.8 Use of Resources

This section addresses issues of the use of the various available resources - board technique, use of OHPs,etc. Most of this will be generic and covered in your institutional staff development provision, so we willonly touch on some issues that arise specifically in mathematics. As noted in Sections 2.8,2.9 there aretwo aspects to the use of learning resources, which we can summarise in terms of media and message.

The media issues relate to actually using the resources to ensure that the communication of the message isefficient and effective. This includes such things as writing clearly on the board, large enough writing onthe OHP slides, not standing in front of the board or projector, talking loud enough or using a microphoneif necessary, and so on. Any good institutional staff development generic course will cover most of this,although they are unlikely to cover such issues as care needed in writing and pronouncing mathematicalsymbols, or in preparing mathematical OHPs.

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Using resources to actually get across the required message in the most effective and memorable way isas important, and far more difficult than dealing with the media issues. Try talking mathematics to acolleague, or over the phone, without being able to write things down. This will soon convince you of theprimacy of visual representation in conveying mathematical ideas and arguments, and should persuadeyou of the opportunity black/whiteboards and OHPs provide for both good and bad communicationof mathematics. Naturally, this is even more important in a teaching situation. When working throughcalculations with a colleague on a board you each have the opportunity to query what is written, toexplore different interpretations of what you see, to link closely with what is said. Most of this is deniedthe student in the typical lecture. Sometimes their main priority is just to get something down, that theycan decipher later, so they at least have some sort of notes.

As we have mentioned previously, Mason ([53]) makes a useful point about the use of resources. Henotes that we can treat teaching materials, such as handouts, OHP slides, etc as objects to be used inthe course of the lecture, to encourage ownership by the students and reinforce learning points of thelecture. For example, ask students to say what they see in a particular diagram or equation. Point toimportant features of an item. Ask students to add values to a table, or reproduce their own versionof a drawing. Get them to annotate the item with their own ideas and then discuss/exchange theirannotations with neighbours. Encourage them to customise their handouts, highlighting key learningpoints, incorporating additions, qualifications, etc in their own way. Hide what was shown and getstudents to construct it in their own way. Baumslag ([7], p160) also has some interesting ideas on theuse of props - using a torch to illustrate conic sections, students themselves to illustrate permutations andcombinations, an office spike to illustrate level surfaces, and so on. It is just a matter of being imaginative.

Based on the above, your use of resources might be developed in two stages: first the technical use ofthe equipment/environment then the application of those skills in getting across the key mathematicalideas you are teaching. These days there are of course many sorts of equipment and resources and ac-commodation. Material on some of the more specialist resources, such as the various forms of e-learningcan be found on the MSOR Network website (mathstore.ac.uk). In this book we will focus on black(white)boards, OHPs and handouts. These are probably what the majority of us use. Also, the principlesof the use and application of these will translate easily to other media. So, this section will look at thefollowing in the context of the lecture:

• accommodation

• black (white) board technique

• use of OHPs

• use of handouts.

Use of accommodation - media

The lecture room is actually quite important. For large classes certainly, the standard arrangement for thelecture is of a tiered lecture theatre. This is not always satisfactory, because you cannot easily get round tothe students (unless there is room for them to use alternate rows). It is useful if you can get close enoughto assist any individual. But even failing this, you can still walk up and down the aisles, which helps ingiving them the impression of being ‘amongst them’ rather than in front of them. Try to use the space inthe room constructively. If you can get a flat room then you can turn the lecture into a tutorial when youset the students some work.

Think about the conditions in the room. Too hot/cold? Poor lighting, poor acoustics, and so on. Whilesuch things are of course important in any discipline, they are particularly so in mathematics because it

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demands so much detailed concentration over prolonged periods, and any kind of distraction or discom-fort will get in the way of this. And few subjects require the same close attention to writing on the boardand OHP that are necessary in mathematics, so visibility has to be excellent for all students. So, as soonas you know the room and the number of students you have, check it out for all such issues and get anyproblems dealt with in good time. And remember that more experienced members of the departmentwill probably already know a lot about the quirks of the accommodation they use, so talk to them too.

Use of accommodation - message

The room and its fixtures can provide imaginative illustrations of some mathematical ideas. For a start itprovides a readymade three-dimensional coordinate system. The window blinds illustrate periodicity. Ora roller board is better if your have it because you can write on it. Once again it is a matter of imagination.

Black/whiteboard technique - media

Good board work is obviously important. There is a lot on this in the generic literature. For mathematicsspecific advice see Krantz ([49]). He exhorts us to write neatly in large letters in plain longhand or print.Work horizontally across the board, proceeding linearly, with not too much material. We can divide theblackboard into boxes to aid organisation by both the lecturer and the students. Label equations, usehorizontal and vertical lines. Draw sketches neatly and persist in getting students to do them ([49], Page40). Use sliding boards effectively. If right-handed write first on the right-hand board, then move to theleft, so as not to obscure what you have already written. Do not erase until you have to and then do itproperly. Don’t reuse stuff already written, write it out again. If your writing is poor, slow down, takespecial care. In truth, for most of us it is difficult to maintain such disciplined practice, but we have to try.

Black/whiteboard technique - message

When putting across the mathematical message by writing material on the board we have to think care-fully about the actual purpose of displaying it on the board. If it is fairly routine stuff that students canpick up pretty easily as they go along, then really all they have to do is copy it as a record. Note that thisis nothing like the waste of time and resources that some might think. The mere act of copying some-thing down can help the students to absorb it. This is particularly so with mathematics, because evenwhen copying it down verbatim it is usually necessary to think through the symbols and manipulations.Copying down the previous sentence verbatim would be ludicrous, but not so something like the powerseries of the exponential function.

On the other hand if the purpose of displaying material on the board is to develop deeper understanding(which most of it should be) then as Mason ([53]) points out we need to do more than just write thematerial on the board. We have to actually use it in some way. Once the material, say a proof of atheorem, or a worked example, has been written down and the students had time to transcribe it, thenwe need to study it in more detail. You can walk up to the back of the class and look at it with thestudents, discussing what you all see. Get them talking about it with you, see it from their point of view.Discuss together with them what are the important features - would it have been a good idea to highlightor underline some things - if so do it. Point out important features, or get the students to. Indicate to themthe logical structure, the steps made and the reasons for them. Can we improve the presentation in someway, is it clear enough? Interact with it, pointing things out, drawing connections. Repeat the key phrasesor points and keep coming back to them. Say things in different ways, and remind them of the meaningof key words and concepts. In all this you are using the board as a tool in developing understanding,

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rather than simply a record of information. If this serves no other purpose than providing a short breakthen it will be worthwhile.

Also of course the board should be used dynamically, as we go along, to illustrate mathematical think-ing and argumentation. The students can get a great deal from this, watching an expert develop theirarguments on the board, writing out their thinking (rather than just copying from their notes), maybebacktracking, making the odd mistake, asking questions, thinking asides, etc [71]. This is not to encour-age you to present messy boardwork, but to supplement the boardwork with a running commentary ofwhat you are trying to convey and why.

Of course, there are those of us who are not so demonstrative and might feel inhibited about some of thesuggestions above. No matter, the point is that you have to find your own way to use what you haveon the board, rather than simply writing the proof or whatever down and moving immediately on to thenext topic.

Another important aspect of getting the message across on the board is to make sure the message is allthere for everyone to see and ponder. So, for example don’t gloss over steps - put each one in alongwith words of explanation. When writing out mathematics steps on the board it is very easy to simplywrite the equations, and not write down the intermediate words and explanation you are saying at thesame time. This is actually storing up trouble, because what the students then get as a set of notes is alist of equations, and this is then how they tend to write out solutions of mathematical problems. We allknow this is not how good mathematics is presented, and that prose interjections are necessary in writtenmathematics such as books and articles. They articulate what the symbols and equations really mean. Ofcourse, writing out the prose also slows you down, which is nearly always a good thing!

Use of OHPs - media

Here we include most kinds of projection facilities, including such media as Powerpoint. While suchthings may differ in their technical facility they all basically project the finished material in toto on ascreen for viewing in polished form, rather than being put up sequentially by the lecturer with accom-panying explanation and commentary, as they might do on a board for example. There is a danger insuch projections being used simply as an exhibit of the final material, particularly in mathematics. Thestudents are denied seeing the material being developed, and simply reading off what the slide says isan insult to their intelligence. A compromise is to conceal the slide and reveal it sequentially, but that is apoor replacement for seeing the live development of the ideas that we talked about under the use of theblack/white board. So in consideration of such media the primary concern is how we use it to presentthe messages we want to send in the most effective way.

Much of the use of projectors as a media for transmission of information is standard generic stuff thatwill be covered in any institutional staff development course. For example, don’t stand between audienceand screen, don’t obscure projector with shoulder while writing on slide. In other words make sure everystudent can see everything they need to see when they need to see it. The best place for the screen is in thecorner of the room at a slight angle, if possible, and so on. Use a fine pen to better present mathematicalsymbols, etc. See Mason ([53], Page 44) for more on this is in the mathematical context.

Use of OHPs - message

Once projected of course the slide on the screen is no different to the board in terms of display (exceptof course they can be prepared in much better quality, it is easier to use colours, etc), so everything saidabove for boards applies equally for OHPs. However, slides are better for presenting summaries andoverviews of what you are saying. You can give students copies of the slides, whereas you can’t for the

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board, but don’t allow this to overload the students or effectively increase the pace of the lecture beyondthe ‘assimilation threshold’. Multiple slides can be used, one showing a summary, one more details. OHPand boards can be combined to good effect ([49], p. 40, [53]). And to reiterate the discussion above, thelive development of material, by the lecturer, on a board, or on an OHP, is far superior, particularly inmathematics, to projection of the final product on a screen.

Use of handouts - media

Again, you will get plenty of advice on preparation and use of handout materials in standard institutionalstaff development generic courses. So far as mathematics is concerned you can probably get all you needby looking at the various examples used by your colleagues. As mentioned elsewhere, don’t use handoutsto rush the course or overload students and, especially for large classes, think of the costs.

Use of handouts - message

Everything we said about boards and OHPs applies equally here, but in addition the students can nowtake these away and therefore have the opportunity to customize them and make them into referencematerials, revision aids, etc. See Mason ([53], P64) for different types of handouts and their use. Suchthings as handouts may vary in form from level to level to reflect the students’ developing independentlearning skills. Thus, in the first year students need quite a lot of support, with materials such as handoutsthat are pretty self explanatory and packed with examples. The level of guidance may be relatively highand demand little from the student in terms of their own original input. By the second year they needto be encouraged to become more independent and take more responsibility for their own learning. Thehandouts can be quite concise, brief and have some ‘white space’ that they are expected to complete withtheir own work, notes from the lectures, etc. In the third year their independent learning skills should bealmost fully developed, and they can be expected to make much more use of books and their own reading.One option here is to either use a set book, or provide book quality handouts, a little more advanced thanthey actually need. Then you help them through it in your lectures, covering the important parts, leavingthem to read up the details themselves. In the lecture your job is not so much to give them the material asto help them to assimilate it themselves. Incidentally, this move to getting students to rely on materialsto learn for themselves can be a bit worrying, because you may fear they won’t cope and will end uplearning too little, but in fact they invariably rise to the challenge, and often surprise you.

Putting it all together for maximum effect

It is (too) often said that lectures cannot develop ‘deep’ learning, by which is meant, presumably, theexercise of the full range of higher order cognitive skills such as synthesis, analysis, evaluation and dis-crimination (i.e. the full MATHKIT, Section 2.6). This is plain wrong. By combining boards, OHPs andhandouts and possibly demonstration models you can engage the students in all of these. For example,we can provide the students with book quality detailed handouts which they are expected to read eitherbefore or after the lecture. In the lecture we display the pages on OHPs, and work through them withthe students, just as we have previously described in studying a completed board. You pick out the keypoints and ideas, which if they want they can highlight on their handouts. You can go through any awk-ward steps or arguments in detail, going through extra examples or details on the board, which they cancopy or use to annotate their handouts. This way they are continually active and are having to think andorganise the material coming from different sources, putting it all together (with your help of course).They practice how to identify the ‘essence’ of things and not to be intimidated by complicated looking

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mathematical details. Much of mathematics is repetitive, we keep using the same arguments over andover. You can show this by getting them to scan though the handouts or book - pointing out how oftenthe same argument is used over and over. You can indicate what is important and not so important butmaybe interesting. Essentially, you and the students are working though a set of learning resources to-gether, and they are being trained in most of the cognitive skills, and even in the transferable skills suchas communication, team working, etc.

ExampleComplex analysis is a daunting subject for the novice, because there appears to be so muchintricate detail that is essential in actually implementing the methods of contour integration,for example. By using a range of resources we can help the students to understand the subjectwhile at the same time exercising all the skills required to read, assimilate, learn and useadvanced mathematics.

The students are issued with detailed handout notes, which they could probably use indepen-dently, without attending the lectures. Full proofs of the main theorems are given for example,but only a few key outline worked examples. Copious exercises are provided (With answers,but not complete solutions). As the course progresses they are expected to work through thehandouts and exercises, rereading as much as is necessary. In the lectures they are helped inthis by focused in-depth coverage of key points and key skills. The appropriate page is dis-played on the OHP (breaking all, the rules about density of words per slide, but of course theyhave the handout in front of them!), and the lecturer and students together work through theparticular topic(s) of the day.

In reality there are only a few major results in undergraduate complex analysis, one of thefirst of which they meet is the triangle inequality. The proof of this is given in the handout,but in outline. The details are worked out carefully and interactively on the board, constantlyreferring to the slide and handout. The students are free to copy the detailed proof from theboard as a supplementary note to the handout, or as most do, simply annotate their copy ofthe handout. By the end of the course most students’ handouts have been fully personalizedby detailed scribbles and annotations. As the course progresses the triangle inequality is ofcourse used time and time again, and this can be illustrated by going back through the notes.They soon realise the importance of the result, which further motivates them to understand it.They see for example that the integral estimation theorem is not a new complicated result, butjust the triangle inequality in disguise. They see lots of different methods applied to the sameor similar integrals and get used to judging which one to use in different situations becausethey can scan back through the annotated ‘book’ and see what has gone before.

A key subtle argument repeatedly used in complex analysis is the demonstration that a com-plex integral vanishes by showing that its modulus is less than an arbitrary positive quantity.The first time this occurs in the ‘book’ it is treated in great detail with careful explanation andexamples. The next time it is done more quickly, and they can flick back through the pagesof the handout to rework it and remind themselves. By the end of the course it passes with-out comment, except to point out to them how many times it has been used by flicking backthrough the notes. At appropriate points detailed examples are inserted and worked throughon the board, giving students the chance to practice them. This approach is independent ofthe size of the class, although students do need some space to work when trawling throughthe notes, for example.

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3.9 Maintaining a Conducive Learning Environment in the Lecture

Some environments are better than others for learning - and these can differ from person to person. Inthe lecture we have to ensure that the environment is optimum for learning for most of the students. Thismeans so many things - understanding which approaches and learning methods best suit your students,getting them relaxed and able to ask questions, keeping appropriate discipline so that the class runssmoothly. It is perhaps one of the most difficult things for the new lecturer to do, and even experiencedlecturers find difficulty, particularly if they are suddenly put in charge of a novel type of class for them(this effect has been very noticeable with widening participation).

Much of this section is really generic in nature, devoted to skills that all teachers need, in any subject,at any level. But the best place to learn about the issues involved here is in the department, with one’scolleagues and students. Your colleagues are likely to know the students and the particular challengesyou face. They can help mediate in difficult situations, and provide shoulders to cry on when the goinggets tough! In this book we have assembled a wide range of comment, advice, experience from across theHE mathematics sector, so there is some practitioner validity and relevance in what you read, but thatwon’t address all the situations you meet in your career. Every student in every one of your classes is anew experience who you may not have read about or come across before. In this section we will look at:

• establishing good relations with the students

• classroom management

• motivating and challenging students in the lecture

• encouraging student engagement and interaction in a lecture

• helping students to get the best from the lecture.

3.9.1 Establishing good Relations with the Students

Easier said than done. It may require qualities you might not have - a friendly nature, sense of humour,infinite patience, etc. But there is one overriding quality that will take you a very long way - you mustreally care for your students. Failing that, be able to pretend you care for them. Students will forgive alot if you have their interests at heart.

Krantz ([49]) holds as one of his key maxims for relations with students the need to respect them. This isabsolutely right. You have to treat them as people, not only as mathematicians - and as in most environ-ments until you are given cause to behave otherwise, it pays to be nice to people. While you are the onein charge, extend courtesy, consideration and respect to your students. Give value and do your best forthem. Be reliable, do what you say you are going to do. Be firm and fair and ultra-tolerant. Never loseyour temper or become impatient with your students, be utterly objective and detached when dealingwith them, especially in difficult situations (See 3.9.2 below). Find an ultimate deterrent or escape routefor those situations when things do start to get out of hand.

Baumslag ([7], p. 140) promotes a similarly student-friendly approach, advising the lecturer that the mostimportant attitude they can have is to be on the students’ side and to feel that it is important for them tosucceed. And let them know you want them to succeed. Encourage them and note improvements in theirunderstanding as the course progresses. Sprinkle the lecture with the occasional complement on theirwork - ‘I remember in the last coursework you did quite well on that point’. As Baumslag emphasizes,there is no need to pretend or be patronising about this - there will usually be improvements, but thesewon’t be so clear to the students.

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Examples

1. Particularly if you are teaching science or engineering students, it is highly likely thatsome will not be greatly interested in your subject - it is just something they have to do.It is all too easy to let this affect your relationship with the students, but it is part of thejob to avoid this - to keep good relationships with the students even though you havelittle of interest in common. You will be constantly asking the students to do things theydon’t really want to do - so if you have to also do some things you don’t want to do, thenyou at least have this in common!!

2. Sometimes a student will ask what you may consider a silly question. You have a veryimportant power relation with your students, and the way you respond to even sillyquestions can have a lasting demoralising effect on the student’s confidence and abilityto consult you or other lecturers again if you get it wrong. There is plenty of evidencethat even top academics can ask extremely silly questions in subjects outside their ex-pertise (and even within it), and when they do, they are no more immune to dismissiveresponses than anyone else.

H Exercise

Think back to your student, or school, days. Was there a teacher/lecturer who particularly inspiredor encouraged you, one you would wish to emulate? Why? Would they have the same effect on otherstudents? Note that it is not so much the individual reasons that are important here, but their longterm effect.

Another important aspect of building good staff-student relations is to treat them as individuals. Fromthe front of a class of 200 students it is difficult to see them as individuals (It may be difficult to see them atall!), but we have to try. Elsewhere you will see advice relevant to this issue, such as learning some studentnames, learning about their backgrounds, etc. Here we are more concerned with our overall attitude tothem. Never categorise students unjustly. It is tempting to fall for such generalizations as ‘These areengineers, they are not really interested in mathematics’, ‘These students are not interested in learning,they just want to pass the exam’, etc. But as Baumslag ([7], P 140) notes their motives might not be as youwould like, but as individuals they are entitled to their motives - if they can pass the examination withoutlearning that is your fault, not theirs! As we say elsewhere, the students are unlikely to be as dedicated tomathematics as you are, and much as you might wish it otherwise the nature of the job cannot be basedon that assumption - you have to work with what you have.

Examples

1. While writing on the board you may sometimes hear some chattering which you feelis beginning to escalate and become a nuisance. There is a tendency to think ‘This isa noisy class’ and to berate them generally (and therefore discomfort them all) with atelling off, which may in turn escalate into threats of removal (we will look at this inmore detail below). Now in fact you don’t have a noisy class - what you have is a fewindividuals who are chattering. You therefore must identify these - or some of them -and deal with them directly. You must not penalise the whole class because you havedifficulty identifying the individuals concerned.

2. If a student asks a question in class you may have a pre-formed impression of what theirdifficulty is, perhaps from similar experiences elsewhere. But this student may well havea different slant on things, may have their own specific difficulties. So, concentrate on

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them and try to determine precisely what their problem is. Their is no harm in establish-ing a mini-dialogue for a few minutes, because the other students will probably benefitfrom this, even if it is only to see that you are prepared to talk with students in a friendlyand considerate manner.

H Exercise

In a class you are currently teaching, how many students’ names do you know? What is thebackground of the students? Sketch out their likely mathematical knowledge. Is there likely to bewide variation about this? What are their likely motives for studying mathematics in your class?

3.9.2 Classroom Management

By this we mean maintaining an orderly environment, ensuring all the teaching and learning activitiesproceed expeditiously in a disciplined and productive manner. Of course, you should always be aimingto develop a productive, community working atmosphere within the class that is conducive to learning,and relies little on formal rules. However, circumstances do sometimes arise where it is necessary to dealwith some kind of troublesome behaviour, such as noiseness during lectures or disruptive late arrival.Advising lecturers how to deal with such instances is like advising parents how to bring up their children- the advice may be unwelcome, and they may not be in a position to take it anyway. But we will try.

Of course such problems can occur in any subject and are not specific to mathematics. They might there-fore be addressed within your institutional staff development department, or by departmental policies.Nevertheless, you might find the advice that follows useful, as it is distilled from a number of experiencedmathematics lecturers who have to deal with the issues on a daily basis, across a wide range of students.Also, there are actually some specifically mathematics aspects to these issues. For example a lot of mathe-matics teaching is to service students, whose main subject and interest may not be in mathematics. Theirmotivation and commitment to mathematics may therefore be less than that of mathematics students,and you may have to work harder to accommodate this. Another problem specific to mathematics is that,even for mathematics students, such a precise and efficient language can leave them behind faster thanmost subjects. Your students are therefore likely to be particularly sensitive to the pace of lecturing, andthe class can become very restless if this becomes too challenging, or too pedantic.

The first issue to address is when a problem actually becomes a problem. This is a delicate balancebetween your tolerance in encouraging a relaxed, friendly atmosphere and the students’ need for a con-ducive, controlled learning environment. It may be tempting, particularly in your first few lectures, toturn a blind eye to unwelcome behaviour. You may not think it is worth making a fuss. You may thinkthe behaviour is your fault, because you are doing something wrong. Or you simply may not know whatto do about it anyway. The way out of this is to ask yourself not whether it bothers you, but whether itinterferes with the other students, or the running of the class. If it does, then you must act to deal with it,the students will, quite rightly, expect that of you. In general it pays to be quite tough in the early stagesof a course, then gradually lighten up as you and the students all get to know each other.

So let us assume you have got to do something about the problem. Of course, prevention is better thancure. Set the ground rules in the very first class, clearly and concisely. Try to get the students on sideby involving them in maintaining a good learning environment. Students are less inclined to misbehaveif they feel you know something about them, so learn some names and get to know as much as youcan reasonably manage. Keep seeking feedback from them, informally and by frequent measures ofprogress. Keep them active. Be everywhere - wandering about the class rather than standing at thefront. Any cure of unwelcome behaviour must be systematic and fair - a first warning in pleasant mode,a second warning including statement of consequences, at the third offence impose the consequences

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without embarrassment or compromise. If the circumstances are extreme and it cannot be resolved byremoval of some students, leave (with dignity!!) rather than lose it. Note that leaving the classroom is anabsolute last resort. To do so penalizes the well behaved students not involved in the incident. In all mycareer as a university lecturer I have never found it necessary to abandon a class, although a few studentshave been removed along the way! This is probably the case for most lecturers.

However, whatever preventative action you take, you are bound to get some problems,. Perhaps themost common, particularly in a large lecture theatre, is talking and inattention. Usually the best time tohandle this is when it first occurs and nip it in the bud. While everyone must see that this behaviouris unacceptable, the class as a whole must not be chastised if disruptive talking is taking place, but theculprits must be identified and warned. If students are chatting, make direct eye contact with them so thatthey know you see them. Sometimes stopping the lecture, looking directly at the students, and resumingthe lecture when talking stops is enough to resolve the problem. However, this is allowing these studentsto dictate the progress of the class and should not be allowed to delay proceedings for too long. Physicallymove toward that part of the room, again making eye contact with the students. If you cannot identifythe source of the noise simply ask who is talking. If there is no answer, then their embarrassment shouldbe a sufficient lesson.

Having given fair warning, if you finally do identify someone talking then make an example of them andimpose the threatened sanction firmly, immediately and without malice - for example ask them to leave.Speak to the student (s) privately after class or before the next session. Explain that their behaviour dis-tracts the other students. Above all, make it clear that you bear no grudges, and that once the issue withthem is resolved it is forgotten completely and will have no further consequences. This is particularlyimportant, because you may shortly be marking their work, and you should not worry that your mark-ing will be affected by a spat over their behaviour. In fact students are not normally being deliberatelytroublesome, but have possibly simply not learnt the rules of reasonable behaviour, or have previouslybeen set a bad example by being allowed to talk in classes. Also, of course there are times when you wantthem to talk - during tutorial sessions or time out periods, for example, or to tell you that you have madea mistake, so you have to draw a balance between keeping a lid on them and allowing some chattering.

Another common problem is lateness and inattendance (of the students, hopefully not the lecturer!).Students shouldn’t skip lectures or be late/leave early, particularly if this is disruptive to others. It is bestto be firm on this, especially in the first few lectures. Many lecturers leave the question of attendance toindividual students. If you require attendance, be sure to have a system for reliably recording it and apolicy to follow-up on absences. If you feel that a student’s absences are excessive and are jeopardisingacademic performance, inform the student’s personal tutor and/or discuss it with the student. If a largeproportion of students don’t come to the lecture, consider the possibility that they do not find the sessionsuseful. On the other hand, students are adults and whether or not they attend is really their business.As for lateness, that is a different because it can disrupt the class, and should not be tolerated, exceptpossibly early in the first year when students are finding their feet, or when it really does not constitute adisruption.

These days there also seems to be a growing problem with eating and drinking in lectures and resultantlitter in classrooms. Usually there are regulations about this, and if so then they should be rigorouslyenforced - it is not unreasonable to expect a healthy youngster to last throughout an hour’s lecture withoutrefreshment, and it should be made clear that litter in the lecture room (or anywhere else) is unacceptable.Some of today’s students do not appear to have learnt this.

At some point in your career you may have to face a student who is resentful, hostile, or challenging in alecture. The following are a few suggestions for dealing with this:

• don’t become defensive and take the confrontation personally. Respond honestly to challenges,explaining - not defending - your objectives in the class.

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• at all costs, avoid arguments with students in class.

• when talking to a disruptive student, be objective about expressing your concerns.

• be honest when something doesn’t work as you had planned.

• in the highly unlikely event that a student becomes hostile or threatening, contact Security, informtheir personal tutor or head of department. Most campuses have disciplinary procedures that pro-tect staff as well as students.

• as a last resort leave and report the matter to the appropriate authority.

Despite the perhaps daunting nature of the issues discussed in this section take comfort from the factthat they are generally of an exceptional nature. If you genuinely care for your students and do yourbest to give them good value, then most of the time you will be able to create a lively but disciplinedenvironment. On the odd occasion when things go amiss then the bulk of the students will be behindyou in putting it right.

H Exercise

Discuss the issues raised above with as many experienced colleagues (not necessarily in mathematicsonly) as possible.

3.9.3 Motivating and Challenging Students in the Lecture

Mathematics is one of the most difficult subjects to concentrate on when you are simply listening tosomeone talking about it. If you are already very interested in mathematics then the last thing you wantto do is watch someone else doing it - you want to get on with it yourself, occasionally asking for helpwhen you need it. If you are not that interested in mathematics anyway, then it is even more difficult toconcentrate on lengthy mathematical arguments rolled out by someone else.

In a ’post-it’ exercise at the Plymouth SEDA (Staff and Educational Development Agency) Conference,April 1997 participants (probably very few of whom were mathematicians) were asked to identify sub-ject matter that damages students’ motivation. Some 80 items were produced, mainly generic such as‘too much information’. The only specific discipline related items were all related to mathematics orstatistics - ‘numbers, equations, graphs, mathematics, statistics and formulae, etc’. (10% of items). So inmathematics we seem to have something of a challenge!

In your generic staff development courses you may be told about such things as Maslow’s hierarchy ofneeds, and advised how to motivate students in general terms. Useful though this may be, the especiallydifficult task, for example, of motivating biologists in mathematics requires particularly imaginative tech-niques. Here we are focusing on the task of motivating and challenging students in the actual lecture.This is extremely difficult, because in order to get through the material you have limited time to spendwith motivational ploys. Also, you cannot get much feedback from the students to measure how effectiveyou are at motivating them. In a tutorial situation (See Chapter 4) you can use more time and ingenuityin motivating students, relating it more closely to the work they are doing.

Of course, the cynical amongst us will realise that there is one very potent extrinsic motivator - the as-sessment. And if all else fails you can remind them about that. Try this game (only once!). Pick on atopic, preferably a quite boring one. Wax lyrical about the interesting applications of this topic, it’s greatimportance in the history of mathematics, and so on. The students will listen politely, grateful for thechance to put their pens down and relax for a moment, and indulgent of your misguided interests. Then

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quietly mention that this topic is a favourite examination question, regularly cropping up in the past.Suddenly the room will erupt into activity as everyone scribbles reminders to that effect, annotates theirnotes, etc, and you will have their full attention. It is up to you whether or not you do actually haveany intention of setting the question this year! The sad truth is that although the examination is indeeda powerful motivator, it motivates to the wrong ends. It encourages students not to take an interest inthe course and its content but in the strategic task of passing the examination. Of course, we like to thinkthat students will be interested in what we are saying for its own intrinsic value and relevance, but asex-students ourselves we know the reality - and should use it. And of course, as it is such a fact of life,we have to ensure we set the right sort of examinations to capitalize on it. So, make (not too) frequentreference to the examination relevance of what you are doing, perhaps referring back to past examinationpapers. This may not score highly in the educational respectability domain, but it works.

So how do we motivate students and sustain their interest through a fifty minutes lecture? There is oneabsolute, invariable, necessary requirement, above all else, you yourself have to be infectiously enthusi-astic. We discussed the need for the lecturer to be enthusiastic in Section 3.7, here we will look at howto get the students interested. Of course, you have to get all the technicalities of teaching right first - thestudents have you be able to hear, see and follow what you are doing. And of course, you need to makeit easy for them to appreciate the key points, and then have mental time to play with them - they can onlyafford interest when they are spared the chore of trying to keep up. And the better your rapport withthem, the more likely they are to be infected by your enthusiasm.

We can motivate the students by giving examples, applications, asides etc that may be of direct relevanceto them. This does not necessarily mean venturing into their other subjects (For example, into electricitywhen teaching electrical engineers, etc). In fact, this sometimes puts students off. So try to think widelyabout relevance. It might be something topical, completely divorced from their academic subjects. Rela-tively recent examples might be:

• optimisation /A Beautiful Mind

• waves, sine functions/The Tsunami

• trajectories and rotation/England and the Ashes

• prime number theory/Recent developments in computer or business security

• Black-Scholes theory/The Credit Crunch

• Lorenz equations/Climate Change.

With Google anything is possible!

H Exercise

Think about some recent events from the world news that might have a mathematical context andcould fit into your teaching.

We can invoke their curiosity by asking the students stimulating questions (not rhetorical, and not rou-tine) at key points in the lecture. You are trying to unsettle them slightly, so they may feel a bit challengedto respond and engage. And always listen and respond positively to students’ comments and questions,taking every opportunity to open them out and draw in other students. Sometimes, you can ask fortheir help - you be the novice, they the experts (e.g., something from their own subject or experience asstudents, something relating to new technology, etc) - not teacher-learners, but learner-learners

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Justify results, theorems, etc ‘sensibly’, as well as logically. This is key to maintaining student interest. Nomatter how riveting the subject material, a few long sentences of dry impeccable logic (Not uncommon inmathematics!) are guaranteed to set eyes glazing over. Soften such things by simple, sensible explanationthat gives the gist in a nutshell. This is actually a difficult skill and requires that you have a thoroughknowledge of the topic yourself. There is no mathematical topic, especially at undergraduate level, forwhich such elementary heuristic explanations are not possible (discuss!). An occasional short amusingstory related to the topic in hand can often help them remember it and make it more palatable. If youthink back to your own student days you may well remember a particular result more because of somequip by the lecturer than the intrinsic interest of the result itself.

Above we have emphasised the need to motivate students. There is, however, a caveat with the desire tomotivate students. When at all possible it is much better to explain where and why you are going in acertain direction, or doing a particular step. But doing this to excess can sometimes slow things down andalso be very difficult in the context of the students’ current knowledge. Sometimes it may be necessaryto simply ask students to have a little patience or faith in what you are doing, to take a leap in the dark,assuring them that things will be clearer later on.

Examples

1. Teaching mathematics to non-specialists is notoriously difficult. They are unlikely tohave the same motivation as mathematics students, and indeed sometimes see mathe-matics as a boring or a daunting subject. Getting them interested in what you are tryingto say is sometimes an uphill task. There is however an abundance of advice on this inthe literature. For example in the context of teaching physicists and engineers Kummerer([50], p 330) recommends keeping in touch with parallel material in their other engi-neering courses, and occasionally taking a page from their notes in such a subject andillustrating the use of some mathematics you have recently covered with them. And asKummerer says students may sometimes ask you mathematical questions arising fromtheir other subjects - seize on these and explain how they can understand it using thematerial you have taught them. Baumslag ([7], p151) has similar suggestions for moti-vating engineers as well as ideas such as including quotations from famous engineers,or inviting guest engineering speakers along, or historical asides, etc. However, as withall such advice we must repeat a note of caution. An engineer or a biologist may notnecessarily thank you for doing an example from their own discipline. They may not bethat strong in the topic anyway, and it may simply mystify them further, as they try tosee the connections involved. Some may simply be happy to take your word for it that itis useful and just want to get on with the uncluttered mathematics of the topic.

2. Helen Keller (1880-1968) was a US writer who had become deaf and blind at 19 months.She was unable to communicate through language until she was seven, when she wasput under the tuition of Anne Sullivan. This may seem a strange example for mathemat-ics teaching, but her story illustrates an important point about motivating students. Afilm was made about Keller’s life in which Sullivan becomes increasingly frustrated withher failure to communicate with Keller. Suddenly, Keller imitates an action by Sullivan,who immediately recognizes this as an educational breakthrough and exclaims ‘That’sit - imitate first, understand later’ - and that is the point. In any form of teaching it issometimes best to simply require imitation from the student, knowing that with repeatedrepetition, understanding will follow later. In other words, the only motivation you cangive is to simply tell them to copy the process until they become familiar with it. Thisis very relevant to mathematics, which of course is full of complicated processes. Some-times the best motivation you can provide is that if they simply imitate enough times the

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inversion of a matrix by the adjoint matrix, eventually they will begin to understand it.The motivation is your assurance that it will make sense in the end.

3. The attainment of mastery is another powerful motivating force ([16] p. 83, 223). Virtu-ally everyone enjoys learning a new skill or getting on top of an idea - watch studentsin a tutorial, you can usually tell when they have mastered something. Skemp ([66],p. 134) argues that one of the prime desires fulfilled by successful learning is the desirefor growth. Regular success and the attainment of mastery feeds this desire and reassuresthe students that they are developing, so they gain confidence. We all know the boost weget when we finally crack something or master a new technique, suddenly understandsomething (the eureka moment). We not only feel pleased at the success, but it changesour attitude, propelling us forward more optimistically and energetically. So provide lotsof opportunities for students to succeed. In some circles there is currently a pedagogiccontempt for ‘drill’ exercises where students plough through lots of routine problems,but apart from establishing the basic skills this also provides long sequences of quicksuccesses that themselves are motivating influences. In the classroom there is no harm inasking the class a few rather straightforward questions (Not entirely trivial) that you aresure someone can and will answer. This gives you the opportunity to congratulate themon their progress.

4. Challenging to motivate. Most people, particularly students, like a challenge (HenceSuduko, Rubik’s Cube, etc), and this can be put to good use in helping them to learn.There is evidence that mathematics problems become the most interesting when they are‘just difficult enough’ - not too hard, not too easy. This is a matter of judgement, andyour knowledge of the class as a whole. And of course students will vary on what theyregard as the right level - a boring trivial problem for one student can be a fascinatingchallenge for another. So you have to have a range of problems to meet their needs. Ina lecture you are limited to the sorts of questions you can ask anyway, but you can stilltry to pitch the level right. It is judging this particular level for your class that is themain challenge you face! Kummerer ([50], p. 330) makes the point that teaching shouldbe done on (at least) three levels: for the minimum level cover all the absolutely essentialpoints, then 95% of the time should be spent addressing the average student (the mediumlevel), interspersed with some additional challenges for the gifted students (the upperlevel). In the same way we need to provide challenges in the lectures at these three levels,just one or two hard examples for the class as a whole won’t provide all students withsuitable challenge. They may keep the gifted students happy, but the weaker studentswill simply get nowhere and become demoralised. So, provide a range of challenges.But challenging students is not only about providing interesting questions, it is aboutthe whole teaching environment. When you explain something, make some of the stepsjust a little bit beyond the immediate reach of the bulk of the students and tease out theircontributions to help move the explanation along. Have sessions when you stretch them,interspersed with more easy-going periods (exactly like athletic training). The point isthat making the lecture environment just that bit challenging for the bulk of the studentsis a useful motivational tool.

5. Relevant examples - wonder and mystery. We have on occasions noted that apparently‘relevant’ examples may actually leave the intended audience cold. So what sort of ex-amples can be used to interest and motivate students? Often it may be nothing to dowith their degree studies, but some interesting, even mysterious, examples that all thestudents can appreciate whatever their background. An example I like to use with engi-neers is that of local realism arising in quantum mechanics. The students need no morethan A-level maths and don’t need to know anything about quantum mechanics. The

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importance and ‘correctness’ of quantum mechanics is evident to anyone who uses anyelectronic device. But few can get to grips with its apparent contradiction of commonsense in its violation of local reality - essentially the belief that things are ‘really there’when they are not being observed, and that no communication is possible at speedsgreater than that of light. Few can fail to be intrigued by this ‘paradox’, which has beenconfirmed by many experiments in one form of which the question of local reality [5]comes down to the value of a particular mathematical expression [45]

T = |1 + 2 cosx− cos 2x|.

For any theory satisfying local reality this quantity must be less than 2 for all anglesx. If there is any angle x for which T can exceed 2 then quantum mechanics cannotsatisfy local reality [25]. There can be few more conceptually perplexing and importantquestions in science or philosophy than this about local reality. And yet it is expressedin a simple mathematical form and can be answered by elementary mathematics (Nomore than A-level). The students are asked to determine all angles for which T exceeds2, which needs only a double angle formula and completing the square. This exampleis popular with the students because the problem of local reality has an almost mysticalappeal to anyone with any curiosity, it is easy to appreciate the issues, it is at the forefrontof modern science, the background is fairly easy to explain, and the mathematics is a niceexample of ideas that are normally a bit dry and uninspiring. Also, the solution illustratesthe necessity of having fluency with basic principles in order to be able to tackle multi-step problems.Another similar example in which an intriguing, easily understood, question is explainedby nice elementary mathematics is why it is colder in February than in December, whendays are shortest and we get less heat from the Sun? The resolution of this puzzle comesfrom the fact that the Newton’s law of cooling modelling the heating of the earth underthe periodic temperature variations from the Sun can be expressed as a linear first orderdifferential equation with a sinusoidal inhomogeneous part. Such an equation will havea steady state solution that is a linear combination of sines and cosines with the samefrequency as the sinusoidal ‘forcing function’. By the compound angle formula this isequivalent to a sinusoidal function with the same frequency as the forcing function, butwith a shifted phase - which accounts for the lag between December and February (ChrisBudd - private communication).The point about such examples is that they are ‘relevant’ to all of us and they challengeour common sense and impart a sense of wonder. Then to see them ’explained’ by rela-tively straightforward mathematics that the students can identify with generates a greatdeal of interest. A similar example at a more advanced level is the RSA encryption al-gorithm which can be used as an illustration in a first course on abstract algebra. Thetheory behind it is understandable at that level, but is rather dry, and by showing thestudents how they can use it to understand and use the most powerful encryption sys-tem known (Used millions of times a day at cash-point machines for example) intriguesand motivates them. It would be useful to compile a compendium of similar ‘inspiringexamples’ of elementary mathematics (See Exercise below).

6. The motivating power of the abstract. One of the great powers and beauties of mathe-matics is that of abstraction. In schools there has been a move away from abstraction inmathematics and a move towards the concrete. And many reputable teachers of math-ematics, for good reasons, argue for concrete over abstract. But of course at universitylevel abstraction is the very essence of mathematics. For example, at school and in earlyteaching at university matrices are a compact means of handling large amounts of data

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or large systems of equations. Matrices are in that case a manipulative tool, comfortinglyconcrete and numerical. But in more advanced mathematics they become conceptualsymbols in which the concrete properties are abstracted in order that we can, for exam-ple, view a system of autonomous homogenous linear differential equations in exactlythe same way as we do the single equation y’ = y. This graduation from the concrete tothe abstract is one of the most difficult stages in intellectual development [69], and wecannot expect students to achieve it overnight when they come to university. It has tobe gradually and proactively developed. Given sufficient time the students are perfectlycapable of appreciating the power of abstraction, and many of them will delight in thefreedom it gives them. If they fully understand it, the realisation that a system of differ-ential equations can be treated analogously to the simple first order equation in a singledependent variable is liberating and exciting, and motivates and challenges the students.

H Exercise

Compile a compendium of motivational methods/resources in mathematics.

3.9.4 Encouraging Student Engagement and Interaction in a Lecture

In a lecture you are limited in what you can do to get the students engaged, but it is still possible. Weneed to respond sympathetically to queries and show that we expect them to interact and contributeat appropriate times in the lecture. Concentrate not only on the academic content of the lecture, butdevices for developing rapport with the students. Most people learn more easily when they are (nottoo) relaxed about showing their ignorance, and feel able to ask questions when needed. So your modeand attitude of delivery needs to incorporate this if possible. It is not easy to do this - some peopleexude empathy, warmth, helpfulness without even trying, some simply leave others cold no matter howhelpful and sympathetic they are. Being relaxed yourself, a little light-heartedness, and interest in thestudents can help. Perhaps it’s the things you shouldn’t do that make the difference - no sarcasm, noimpatience, no ‘go and look it up’, etc. Baumslag ([7], p. 149) makes the point that students will learn howto ask ‘sensible questions’ by experiencing you asking questions, so pepper the lecture with your ownconsidered questions. But remember that such activities can eat up time, and not always productively, souse sparingly and strategically.

When students do pluck up the courage to ask questions in class respond to them in such a way as to helpthem as much as possible, and others in the class. When a student asks a question they are at their mostready to learn, so we need to capitalize on this. Build an environment for interaction, and build students’confidence. Ask the students questions, but insist that you get answers. Question and answer in largelectures can be particularly difficult. If you want students to answer and ask questions you need to:

• explain the question clearly, possibly displaying it visually

• give them time to get their thoughts together

• provide privacy so that asking and answering do not feel so threatening

• allow students the opportunity to prepare questions and answers before ‘going public’

• never criticise or ridicule an answer - and indeed build on it

• persist.

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Some lecturers ask particular students questions, by name. So long as you have developed a good rapportwith the students, this is fine, but be careful not to embarrass a student by doing this.

Examples

1. If covering techniques of integration, for example, ask students to design a flow chart oralgorithm for tackling integration of rational functions. Encourage them to devise theirown mnemonics, for example. These days, many UK school pupils are used to beingtold which method to use in a problem and to concentrate largely on how to performthe method. So, having learned the different methods of integrating rational functions(Substitution, partial fractions, completing the square) they need considerable practicein dealing with unseen problems without being given the method. They need to de-velop skills of deciding on the appropriate methods, then discriminating between differ-ent methods that are equally valid but of varying practicality. Such flow chart construc-tion can be applied to any number of mathematics topics that involve consideration of anumber of different cases or methods - solving polynomials, solving first order differen-tial equations, testing convergence of series, etc.

2. If you have just worked through a proof or argument with an involved structure, getthe students to work through particular cases in small groups - noting down the mainstages and how they link together. Then get them to generalise by for example replacingparticular numbers by symbols. We can do the same thing with a complicated techniqueor algorithm.

3. Baumslag ([7], p. 149) suggests some questions the lecturer might have for the class con-cerning a theorem that is currently being studied:

(a) Does the theorem seem reasonable?(b) What does it mean geometrically or algebraically?(c) Can you give an example(d) Why does it have such peculiar conditions?(e) Which theorem is it related to?(f) Could we generalize the theorem?(g) What is the converse? Is the converse true?(h) Where can I use this in subject S?(i) Is there an example that basically illustrates the whole theorem?

Treating the occasional theorem in this way, and encouraging the students to do so can getthem used to discussing mathematics with others.

H Exercises

1. Think of a recent lecture. Could you have engaged the students more, involved them in moreinteraction? How? Would you still have got through the same material, but possiblydifferently? Think of a lecture to come. Can you use any of the lessons learned to make it moreinteractive, without sacrificing coverage?

2. Choose a theorem or standard result and frame questions such as those in Example 3. Do a fewexamples where you provide answers to the questions.

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3.9.5 Helping students to get the best from the lecture

Principle 8 (Section 2.4) reminds us that students may also need advice and training in using lecturesand other learning experiences, that is in learning how to learn. The real place to do this is the tutorial,but there are some things we can do in the lecture to help the student to get the best out of the lecture.One thing we can do is set a good example as we work though the material on the board. It is easy toplough through the meticulous logical development of the material, simply vocalizing the board workor the slides or whatever. Then all the students get is essentially a copy of your notes and what you say.But instead, you can develop the arguments as if on the hoof, noting down the main points and actionson the board/slides, showing how you might reason your way through the material as if your had neverseen it before. ‘Now, I need something which when I differentiate it is going to give me this on the LHS,any ideas? Didn’t we see something like this last week?’ And so on. This way, you not only presentthe material in a more honest and understandable way, but you are demonstrating to the students how amathematician actually thinks and works (As opposed to how most textbook writers write). As well ascopying just what you say, students will learn to copy what you do and how you think, they will get afeel for the traditions and ethos of the subject.

Baumslag ([7], p. 77) compares lectures to a sandwich - the preparation by the students is the bottomlayer of bread, without which the filling falls out in a mess, the top layer the student’s work after thelecture without which the filling won’t be retained. So, you can advise students to prepare before thelecture by consolidating the previous material, think about where it is leading. Then of course tell thenthey should attend all lectures, and then after each they should work through the notes, test themselveson the material, write out a summary, and work though a few problems, any work set, and so on. Ofcourse, as an ex-student yourself, you do not need reminding that ‘You can take a horse ...’, and manystudents will ignore this excellent advice - but then you have at least done your bit. Mason ([53], page 95)also gives some advice on helping students to study generally. We will return to this topic in Chapter 4,in the context of tutoring.

As mentioned previously, a ’minute test’ at the end, getting them to identify the main points of the lecture,can aid their learning and also tell you how well they have assimilated the material. You can then brieflytell them what the key points are, and advise them to spend some time before the next lecture ensuringthat they understand each of the points, and producing an example of each, and raising any questionsthat still bother them at the next lecture. This is a good ploy because identifying and exemplifying themain points usually carries the heaviest intellectual demand, and they are being focused on it and helpedto start it by you, while it is still fresh in their mind. They also know that if any problems arise you willdeal with them next time.

Don’t forget that many departments have booklets such as ‘How to study mathematics’, or your univer-sity may put on general study skills courses. So you don’t want to duplicate these, but concentrate onhow students can get the best out of mathematics lectures and yours in particular.

H Exercise

Does your department have such booklets of advice for students on how to study maths, how to getthe best out of lectures, etc? Do you know of anything useful in the literature on this?

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3.10 Lecturer’s Facilitation and Support for the Students’ Learning

3.10.1 Helping Students to Learn

This is really what teaching is all about - to educate our students to the best of our (and their) ability, andreally help them to learn. In Chapter 2 we emphasised this, we summarised ideas about how studentslearn and used them to underpin the eleven basic principles (Section 2.4) that guide us in the job of help-ing students to learn. Also in Chapter 2 we based the design of teaching strategies and the preparationof learning materials on these principles. Now we want to turn this into concrete ideas we can use in theclassroom to facilitate and support student learning. How for example do we help students in making thereconstructions necessary to develop new concepts and ideas? In this section we consider such things,and look at:

• ensuring that students learn during lectures

• the lecturer’s appreciation of students’ understanding, needs and level

• careful explanation of complicated ideas

• getting feedback from and to students.

3.10.2 Ensuring that Students Learn during Lectures

The first thing to recognise is that making a real effort to ensure that the students learn, and that they takethe initiative in learning, usually takes a lot more time in lectures than if you simply tell them about thetopic, assuming it is ‘going in’. It also of course takes longer preparation, as discussed in Section 2.9, andas a result of this you may find that you are not able to ‘cover the syllabus’. This is a serious concern - ifyou slow down so that the attentive students really learn, then you may not get through all the materialon which subsequent lecturers may rely. The priority has to be getting through the syllabus, until you cannegotiate a revision of the syllabus agreed by all your colleagues. Also, remember that the lecture contacttime is only a proportion of the student learning time for the course - you might for example have 30hours lectures, but it is assumed that the students will put in twice that in their own learning time. If themodule is designed properly, then if the students really do put in that time, they have a good opportunityto assimilate what you cover. In the lecture you provide the raw material on which they must work - it isnot the job of the lecturers to do the learning for the students, but to make it as easy as possible for themto learn.

It requires sustained concentration to focus what you say and do in the lecture on helping the studentsto learn rather than just reciting what it is you want them to know. As well as most of the skills andtraining advocated in this book, it also requires a genuine desire on the part of the lecturer to interactwith the students in a range of ways that help their learning. If we are describing a technique such asintegration by parts it is so easy just to write down the formula and work though a few examples, withhardly any interaction with the students. However, if we really want them to understand then we haveto work with them through the origin of the formula, emphasize that it is the reverse of the product rule,repeat it in words time and again, and then work though examples drawing in the students at each step,constantly referring back to the formula, making them do the work, while you write out the solutions asthey construct them. This is so much harder, but is essential if we want to help students to learn, ratherthan just copy our notes down. The lecturer actually talks less, but then what they say has to really hithome, and again, this takes thought and concentration.

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3.10.3 The Lecturer’s Appreciation of Students’ Understanding, Needs and Level

In Sections 2.5 and 2.9 we emphasized the need to pitch materials at the right level for the students, and alot of that is equally relevant in the lecture itself. You should certainly have a good insight into the back-ground knowledge of your students, and their ‘facility level’ and be able to talk to them in that language,but there is a little bit more to it than that. You need to empathize with your students and ‘get on theirwavelength’. This is one of the most important aspects of any form of teaching, but particularly applica-ble in a lecture. A lecture is not about talking to quick thinking stenographers. It is about explaining towidely varying human beings. One of the biggest problems here is getting used to the fact that few of thestudents are really that interested in what you have to say, particularly in the case of service classes. Thisdoesn’t necessarily mean that they are lazy or disrespectful, but maybe that they are busy and have otherequally important priorities. You may ask why you should make extra efforts to explain things to peoplewho are not interested? To illustrate this we will go through a scenario that you may be familiar with.

Most of us have had to learn some software in recent years. In this we are in a similar position to, forexample, an engineer learning mathematics. We want to be able to use it reasonably well, and will investsome effort in learning it, but it is really a very small part of what we do and we have so many otherthings crying out for priority. If we get stuck we may ask a ‘techy’ for help. He talks way above yourhead, berates your lack of expertise then wearily sorts it out himself. However good he is as a techy, he isno good as a teacher.

You call in techy2. He listens patiently to your problem, reflects a moment, then leans across, taps a fewkeys, presto done, and lets you follow suit to check you can do it yourself - job done. But is it? He hasjust got you to imitate him - what do you do if a similar but different problem arises again? This guy isvery helpful and has solved your problem by doing it for you, but he has not helped you to learn, he isnot a good teacher.

You call in techy3. She listens to your problem, probably sees the solution straight away, but doesn’tlet on immediately. Instead, she asks questions to find out what you already know, how motivated youare, how much time you can devote to this task, what your precise needs are. Then she explains, usinglanguage that she now thinks you will understand, and in a depth that she thinks will benefit you most.Your problem is solved - only now your understanding will be much more permanent and portable. Sheis a good teacher. She appreciates that although you are willing to learn, this topic does not have a highpriority, so she has to explain in the most efficient and effective way in the context of what you alreadyknow and your possible future needs. You don’t want to be an expert, and you are not interested in thefiner details - you are not lazy and stupid, you are just busy and in an unfamiliar environment. Techy3has had to work harder to get onto your wavelength, but she has done the job required. She has alsoreduced the probability that you will have to go to her again!

The analogy with teaching mathematics to less than avid students is obvious. Being able to explain thingsclearly, at the level of the student, is one of the most important skills of a teacher [12]. In a lecture to say100 students it is of course even more important and more difficult. They are probably all at differentlevels. Each may understand different fragments of what you say. You can’t easily interrogate them tofind out what they know or whether they have understood you. For this reason it is best to work near tothe lowest common denominator of what you think they know, and must continually be on the lookoutfor signs that you are losing them. Lewis and White ([51]) give some ideas of how to adapt your languageto that of the students.

Examples

1. Particularly when taking a first year class, we sometimes adopt a level of language thatreally expects too much of the bulk of the students. For example we might automaticallyuse the term ‘polynomial’ without comment. But although it is mentioned in most A-

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level syllabuses, few first year students are really familiar with it - the same certainlyapplies also for ‘rational function’. What’s more, few students appreciate the value ofclassifying the functions in this way. So, particularly in a first year class, we need to usesuch terms considerately. If we say something like ‘Unlike polynomials a typical rationalfunction is not continuous everywhere’ to first year engineers (or even mathematicians),the likelihood is that we will simply not be talking the same language as the students.They may have seen all the words before, but few of them will recall and string togetherthe meanings quickly enough to get the message before we move on. Instead we willprobably help the students more if we say something like ‘A typical polynomial (Sketchone) has no breaks in its curve because it yields a definite number for every value youinsert, whereas a typical rational function (Sketch one - 1

x will do) will have breaks whereits denominator becomes zero, and at such points it is not defined’. Most students willget more out of this form than the previous one, which we can start to use later whenthey really do understand it.

2. Another problematical object for many first year students is the exponential function -again it is in the A-level syllabus, but not always taken seriously in schools. Conse-quently some students are apprehensive about it. In particular they don’t always see theconnection with indices. You can help them by emphasizing that algebraically that is allthere is to the exponential function - the laws of indices. There is nothing special aboutit in manipulative terms. And the apparently mysterious value of e comes about simplybecause the exponential function is its own derivative, and therein lies its importance.This is far more enlightening than starting off with the series or limit definition, or theexample of compound interest.

H Exercise

Practice explaining difficult mathematical ideas to a non-mathematician, a friend, partner, relative -anyone who will make an honest attempt to understand what you are saying. Notice the sort ofquestions they ask and think about whether your students might be worried about similar questions,but feel unable to raise them.

3.10.4 Careful Explanation of Complicated Ideas

This is one of the key skills of good teaching. How well students learn depends on how effectively weexplain what we are trying to get across. There are good explainers, and there are bad, and skills in thisarea can be improved by education and training ([12]). We will look at this in great detail, and breakdown the task of effectively explaining ideas into the following stages. All of these derive either fromour eleven principles or the underlying educational research (Section 2.3), and some have already beendiscussed in the context of producing learning materials (Section 2.9):

• assist students in ‘chunking’ information

• provide hooks for students to hang ideas on - aide memoires, mnemonics, etc, to make it as easy aspossible to remember key ideas

• provide appropriate step-laddering and lubricate arguments

• anticipate the difficulties students might encounter and support them in dealing with these

• provide regular overviews

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• continually highlight/reinforce the (small number of) key ideas conveyed in the lecture

• bring out and highlight the essence of a complicated idea, theorem, method, etc and support thedevelopment of students’ intuitive views of formal arguments

• emphasize the respectability of guessing and the necessity of checking

• alert the students to patterns that may assist their understanding and learning

• help students to build up connections by providing roadmaps and overviews of difficult sequencesof arguments - try to express proofs ‘sensibly’, as much as ‘logically’

• alert students to the fact that some teaching and learning is provisional, and may have to be revisedat a later date.

Assist students in ‘chunking’ information

The concept of chunking of information, ideas and processes in cognitive development is fairly wellestablished ([58]). But it is only a first step on the path to deeper understanding - we all break complicatedideas into smaller more manageable units (After all, that is what theorems, lemmas, propositions, etc arefor!). What is not so good of course is to stop there, without subsequently internalising the chunks andassembling the whole picture from them. In the lecture environment the ‘bite-sized chunks’ translatesinto breaking down the topic or result into smaller more easily memorised results, and emphasisingthese, continually revisiting them, hammering them home, illustrating them, etc. And from time to timeyou need to test that the students have actually swallowed them!

Examples

1. In teaching introductory calculus one might regard the standard derivatives as bit-sizedchunks. The important cases of these can be thoroughly learned and understood inde-pendently as separate entities. For mathematics students they would each be proved indetail, for engineers maybe just hand-waving, but for all students they must be knownthoroughly, available to instant recall (absolutely not on a formula sheet!), which the lec-turer can easily check with short verbal or written quizzes. These bite-sized chunks arethen readily available to the students as they move on to more complicated functionssuch as products and compositions. Then, when they are learning the product rule, theycan concentrate on its structure without being distracted by the need to think hard or re-fer to the formula sheet for the standard derivatives involved. Ditto, standard integrals.

2. When teaching matrix inversion by the adjoint matrix, the bite-sized chunks might com-prise evaluating the determinant (thereby checking that the matrix is non-singular), eval-uating each cofactor, using these to construct the adjoint matrix, dividing by the deter-minant. In this case the bite-sized chunks are actually steps in a sequential process, andeventually the student will have to get used to putting them together in the right order,but initially it may help if they can focus on them independently.

H Exercise

Think of a few lectures you have to give. Can you break each down into more manageable chunks insome way. How would you do this. How would you ensure that the students appreciate the fact thatyou are ‘chunking’ the information? Think about how you would put them together at the end of thelecture to form a coherent whole.

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Provide hooks for students to hang ideas on - aide memoires, mnemonics, etc, tomake it as easy as possible to remember key ideas

Preferably funny, or otherwise memorable. Also, link them to existing knowledge and practical applica-tions. But emphasize that eventually the student should normally internalize the idea and dispense withthe hook (See Section 2.9). This is of course a key device in mathematics, employed all the time. Acronymsare widely used in mathematics (Yet we pour scorn on them if they are used by university administra-tors and managers!). In Section 2.9 we have already referred to some examples in this area: BODMAS,CAST, ‘rings and sings’, Sketch GRAPH. And indeed MATHEMATICS is a useful aide memoire forteaching itself. Below we give some more examples. But remember the warning about over reliance onsuch ‘tricks’ (See Section 2.9). Also, there is a tendency to criticize such aide memoires saying they aremindless devices that do not reflect real understanding. Of course, that might be true if they are simplyleft as labels. But they should not be. They should be similar to the name of a friend, where the veryname (a label!) conjures up all sorts of memories, knowledge, attitudes and feelings. A related ploy is toname rather than number important equations and theorems whenever possible. It is usually easier torecall something with a name than with a number!

Examples

1. Hooks can sometimes take the form of brief memorable anecdotes, maybe a little his-torical tale, or a recent news story. For example Gauss’s summation of the integers, thePythagorean’s dismay at the irrationality of

√2. Many such stories can be found in the

books by David Wells, for example [74], [73].

2. Much mathematical notation and terminology can be made more memorable with theright form of expression or explanation. Baumslag ([7], p. 171) gives some examples, suchas noting that ‘there Exists’ is backward E, ‘for All’, upside down A, Z for the integerscomes from the German zahlen, etc. Or using ‘An r × c matrix’ instead of ‘An m × nmatrix’. However, this last example serves to signal caution in adapting widely acceptedmathematical notation to make things more palatable for the student. Clearly, such anotation will become cumbersome and inappropriate when one starts doing summationsover subscripts, and we will then have to get the student used to a new notation. Andwhat sort of understanding will a student have of matrices if they have to keep beingreminded of ‘rows - columns’. There are points at which one simply has to impress uponthe students that they have to get use to the standard mathematics conventions becausethat is simply what they are, popular conventions. And in this particular example theroot convention ‘i, j, k, l,m, n, ...’ for integers should not be compromised to cater for anew situation.

H Exercise

Think of some topics covered in your lectures. Can you find some ‘hook’ that will make it easier forthe students to remember and retain the material? If it is a technique, use your imagination to findsome easily remembered acronym or mnemonic. If it is an overview of a sizeable portion of work findsome sort of analogy that might be suggestive and easily remembered.

Provide appropriate step-laddering and lubricate arguments

In sequences of arguments, as for example involved in putting together the bite-sized chunks, one im-portant factor is the ‘height’ of the steps in the arguments. Obviously, if the logical or intellectual jump

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from one step to another is too large then most of the students will simply not be able to follow you.This is particularly important in a lecture, where the students might not be able to go over the argumentagain, and not only that but they have to keep up with subsequent arguments. On the other hand thereis clearly a limit to how many steps to put in to help the student follow. Judging the ‘step-laddering’needed is a difficult task. It is however better to get it wrong in the student’s favour. See [16], p. 19.It is not only the intellectual steps that you expect of students that can affect learning, but the way youpresent the argument or explain the steps. You can lubricate the argumentation by, for example, givinganalogies, repeating a key phrase that ‘unlocks’ a mental block, scribbling a quick example, asking theclass to consider a leading question.

Examples

1. The product rule in differentiation. Having ensured that the students know their stan-dard derivatives thoroughly (see above), you can move on to the product rule. You mightprove it for mathematics students (by the way, be careful with notation - you might usethe usual ‘u-v notation’, but if your writing is sloppy then the u and v may look the same- perhaps use f -g!) and even for engineers, because it is so important. But for engineerstry and lubricate the argument by giving the analogy of the expansion in area of a heatedrectangular plate with sides u, v. When it comes to actually applying the product for-mula you will have to think about the steps you include. Eventually of course you don’treally want the students to use the formula, but rather to differentiate the product on thefly without introducing u and v. But for complete beginners this is probably a step toofar initially, so we include all the steps, ‘Put u = ...’ etc, write down and substitute intothe formula, collect and simplify terms, etc. This level of step-laddering would certainlyhave sufficed for the 10% participation of a couple of decades ago, but not always fortoday’s 40-50%. Many students come to university clinging to the full use of the formula,and our task in the first year is to wean them off this, which can only be done by lots ofpractice. Ditto integration methods.

2. For the inversion of a matrix example the steps will of course depend on their previouseducation. For example if they haven’t met cofactors before then you will have to takesteps to deal with them before forming the adjoint matrix.

H Exercise

Analyse some of your recent lectures. When you work though a sequence of arguments on the boardfor example, is the logical spacing about right for the bulk of your students? Are you sure that theyhave a reasonable chance of fitting the steps together comfortably, and when you are explaining thesteps in class do you ease the logical progression by what you say and do?

Anticipate the difficulties students might encounter and support them in dealingwith these

As you are lecturing you will know that some things that you are going to say/do will be quite subtleand challenging for the average student. Let the students know this - and modify pace and presentationaccordingly. Tell them - ‘Now we come to a very difficult step, most students have problems with this- I remember when I saw it for the first time I had a lot of trouble’. Then prepare them for somethingthat is hard by waking them up, ask them pointed questions, get them to discuss with their neighbours,map out how we are going to approach it and why, emphasis that it is hard, but it is also very important.

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Sneak up on it, keep referring back to it, etc. Make them see that you are anxious for them to get this -you know it is hard, but you will do all you can to help them.

An excellent source of the sorts of difficulties students have is Mason ([53]). Most of his first chapter isdevoted to this subject. However, there is a danger of seeing the students as being unequal to the task oflearning mathematics until these difficulties are ‘diagnosed’ and dealt with. In fact this sells students fartoo short and rather defeats the object of supporting them. Much of what are characterised as ‘studentdifficulties’ could equally be attributed to their lecturers! Denial of having seen something before, un-certainty about precise definitions (It wouldn’t take long to find a mathematician who is insecure aboutthe definition of a function), calling to mind things you are confident about in preference to those thatare new to you. These and many other characteristics are common to all learners, whether they be a firstyear undergraduate or an experienced researcher - move the latter out of their specialist area and theycan err as well as the next person. So, provided one views the students’ difficulties as simply stickingpoints that anyone can encounter, and one is alert to these and willing to do something about them in anon-judgmental way, then Mason will give you lots of food for thought.

Examples

1. As mentioned earlier, when meeting the integrating factor method in linear differentialequations students usually find difficulty in the reversal of the product rule used to con-vert the left-hand side of the equation to the derivative of a product after multiplyingby the integrating factor. You can help considerably by revisiting the product rule anddoing a few examples of converting expressions to derivatives - i.e. using the derivativerule backwards.

2. General angles greater than 360 degrees, especially for inverse trigonometric functions,also give most students some problems, so it will help them if you spell out in detailthe multiple angles that can result in finding inverse trigonometric functions, or findinggeneral solutions to trigonometric equations. Here the step that they often find difficultis expressing the general result in terms of an arbitrary integer. This is the problem ofexpressing a sequence of numbers in a single formula, and we can help the students tograsp the general result by methodically working through the angles produced as we gothrough multiples of 360 degrees.

3. Concepts such as convexity, linear dependence, equivalence relations, etc always causeproblems - why? Because they are difficult! Baumslag ([7], p. 172-177), provides someuseful advice on minimizing the difficulties, such as only using one term rather than thetwo that confuses the students (e.g. convex not concave, dependence not independence,etc). But this masks the real problem, as we said, that these topics are intrinsically diffi-cult. What is perhaps most surprising is that some lecturers, having thoroughly mastereda topic, are not able to appreciate that the topic is difficult for the newcomer. It is a fineskill to be able to assess what a novice will find difficult and then to make it easier forthem. In fact, concepts like linear independence are exceedingly difficult at first blush.In truth, one cannot usually say one comes to ‘understand’ them. Rather, one just getsused to them! They require extended and lengthy treatment (And that is probably thereal cause of the problem, we think we don’t have time for this in our over-crowded syl-labuses), lots of practice and discussion, lots of revisiting and lots of time to absorb andassimilate the complex ideas involved.

4. Sometimes you may have to rethink how you would phrase things to make them morepalatable for the student. For example in partial fractions we are often tempted to saysomething like ‘Equating coefficients ...’. Even the best students would be hard put toreally understand this move, and they are not likely to see immediately the connection

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with linear independence ([53], p 27). It is probably kinder, and probably more truthfulto emphasize the fact that since the result is an identity, it must therefore be true for allvalues of x, which can only be the case for a polynomial if the corresponding coefficientsare equal. Even if this is presented in a vague hand waving way, it will probably go downbetter than ‘Equating coefficients...’

5. Abbreviations such as ‘Iff’ take a long time to get used to, as does the actual idea of‘necessary and sufficient’. They need to be carefully explained ([53], p 27), with lotsof examples. In fact, schools do not appear to develop pupils’ logical skills to a highdegree. One often finds that not only do first year students sometimes have problemswith mathematical knowledge and facility, but they may have problems with logicalargument, especially the precise and sophisticated levels found in mathematics. This isnot the students’ fault and there is no point bemoaning the situation and pressing on as ifit doesn’t exist. One has to face the problem and develop the required skills from scratch.And a module in propositional calculus is not necessarily the best way to develop theseskills - perhaps better to give copious examples.

H Exercise

We have given a large number of common ‘sticking points’ for students. For all your teaching, scanthrough the material and identify such points where you think the students might have particulardifficulty. Now devise ways to ease their path!

Provide regular overviews

Even when the bite-sized chunks have been put together and students understand the connections, andcan execute techniques and proofs with some facility, they still need to get an overview of the material,to see what the general message is, to get an intuitive feel for what is going on. This can be developed byensuring from the start that the students understand what the main objective is, that they are fully awareof the purpose of the lecture. You might for example state this at the beginning, repeat it at appropriateintervals in the lecture, and at the end, in a summary. Wankat and Oreovicz ([72], p. 42) make the pointthat this presents one of the major problems to new staff, because they are often overspecialized and maynot yet have developed overviews themselves. This is why teaching, especially at the elementary level,can be much more difficult than we might imagine. There may be many approaches to a particular topicand the good teacher needs to know them all, to be able to judge which is most appropriate in any giventeaching situation.

Examples

1. Having developed a lot of techniques of integration - by parts, substitution, partial frac-tions, trig identities, and so on, you can point out to the students that this plethora ofmethods can broadly be summarized under three main approaches:• Manipulating the integrand into a more helpful form (partial fractions, trig identities,

etc)• Reversing the product rule (by parts)• Reversing the chain rule (substitution)

Then ask them to notice one obvious omission (reversing the quotient rule) and thusto discover for themselves a new and unusual integration method. This gives them acompact oversight of what integration is about. It packages it and makes it look moremanageable (hopefully!).

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2. Laurent series are a traditionally difficult area for students, with the problems of choos-ing the right series for the different regions of convergence, and then algebraically ma-nipulating the expansion to give the appropriate series. When lots of examples, withdifferent cases of singular parts and non-singular parts, have been presented, bring thewhole lot together with an overview using diagrams. Link the different regions with theconvergence regions of the different forms of the series involved. Show how the dia-grams can be deduced from the singularities of the function to be expanded, and howthey dictate the forms of the series in each region. It is difficult to make this sensibleto the students until they have done a few mechanical examples, but after this, such anoverview can crystallise the key ideas.

H Exercise

Work through your own modules and identify where such overviews and synopses would be useful.You will probably only need one every few lectures, but allow time for them when they are needed.

Continually highlight/reinforce the (few) key ideas conveyed in the lecture

Mathematics is particularly concept efficient, which can make it either easy or difficult to learn. Any givenundergraduate course contains only a relatively small number of really basic ideas, concepts, techniques,proof methods, etc. But by the time these have been recycled, applied, tweaked, the resulting set ofnotes or book can run to many pages of complicated looking mathematics that can be intimidating to thenovice. The job of the lecturer is to reverse this process, to root out the important core ideas and help thestudent to absorb these and to recognise them in different guises throughout the course. That is, to helpthem through the wood, and also to help them in developing the skills to find their own way throughother woods. You can do this by listing the key points at the beginning of the course, flagging up everytime we meet them, summarising them at the end of the course. ‘If you learn nothing else, learn this!’

Examples

1. In a one hour lecture on partial fractions to first year engineers, say, the key points maybe: i) the reason we might need to do it, ii) what a typical resolution into partial fractionslooks like, iii) what can be done with the resolution that cannot with the compoundfraction, iv) the use and properties of an identity, v) the process of decomposition, vi)checking by putting back over a common denominator.

2. In a lecture on separating the variables in partial differential equations the key points arei) the form of the solution z = f(x)g(y) ii) a function of x can only be equal to a functionof an independent variable y if both are a constant iii) the boundary/initial conditionsalso need to be separated iv) solution of the resulting ordinary differential equations.

3. Baumslag, ([7], p.168), gives useful guidance on how to discuss the statement of a the-orem, which is relevant here if a key theorem occupies the bulk of the lecture. The keyideas can often be emphasized of course with an example, and Baumslag recommendsworking though this in parallel with discussing the theorem, to illustrate the variouspoints. Amongst other things he lists the following key points to refer to when discussingtheorems:

• Comment on the conditions of the theorem, and any peculiarities• Comment on the conclusions of the theorem, and any peculiarities• Indicate the geometrical and analytical significance of the theorem

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• Explain the intuitive content of the theorem• Discuss the applications of the theorem• Demonstrate that the theorem holds under the given conditions for an example• Show how the theorem fails for an example where the conditions do not hold

The last entry is included on the grounds that it draws students’ attention to the con-clusions, and it is, as a general rule salutary to see when theorems don’t hold as well aswhen they do ([53]).

4. Completing the square is difficult conceptually for many students at the elementary level,for a number of reasons - they don’t always realise how important it is and so perhapsdo not take it that seriously. However, they also have difficulties because amongst allthe trickery involved in completing the square it is not always clear to them what arethe absolutely essential key ideas involved. These are simply thorough facility with theexpansion (x+ a)2 and use of the 0 = A− A ploy. These are the two building blocks forcompleting the square. If the students are totally familiar with them, then they shouldhave little trouble with completing the square. If either is shaky or unfamiliar to them,then they are likely to struggle with the method. So before embarking on the methodsram home these ideas, keep referring to them in the course of the process, and list themas the key ideas at the end.

H Exercise

Think about your next few lectures. In each case write down the three most significant ideas in thelecture, the ‘foundations’ of the topic. Will the students be able to pick these out and appreciate them?How can you ensure that these key ideas really do come across? If there are more than three such ‘bigideas’, are you sure that the students will be able to cope with them all in the time available?

Bring out and highlight the essence of a complicated idea, theorem, method, etc andsupport the development of students’ intuitive views of formal arguments

Mathematics is full of complicated ideas which can be overwhelming to the novice, but most of them,certainly at undergraduate level, can be explained in quite simple ways, almost accessible to the educatedlayman (See Devlin [30], for convincing evidence of this). Try to give the student an intuitive feel for anidea - what are the core points? One way to do this is to treat it as a challenge for yourself to explaina difficult bit of mathematics to a non-mathematical friend. Mason ([53], p25) refers to a descriptionthat captures the essence of an idea as an ‘intensive definition’. The formal definition is referred to as‘extensive’.

Tall ([69], p17) emphasizes the importance of helping the student, in their development from pre-formalmathematical thinking to a more formalised approach, by aiming at developing a preliminary insight intowhat is going on. This is often the reverse of the natural approach of the experienced mathematician whotends to break down difficult and extended ideas into palatable chunks, develops these and then puts thewhole together. But when the novice sees these chunks, they are ignorant of their motivation and theireventual destination, or the whole picture into which they have to fit. At first sight of these chunks theymay in fact, in the absence of guidance, come to think of them in ways that inhibit later inclusion in awider picture. By bringing out the essence of the idea from the start, and continually fitting the ‘chunks’into it, this can be avoided.

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Examples

1. As an example of the dangers of ‘chunking’ without a guiding insight into an idea Tall([69]) considers the traditional teaching of the definition of a derivative by the followingsteps:

(a) define notion of a limit

(b) for fixed x consider the limit of (f(x+ h)− f(x))/h as h tends to zero

(c) call the limit f ′(x), then allow x to vary to give the derivative as a function

As Tall notes, the student is often stumped at step (a), with the idea of a limit ‘plucked outof the air’ for no apparent reason. Other possible cognitive obstacles in this developmentare discussed, but the main point is clear. The lecturer fully understands the ‘big picture’and knows where they are going with these steps. They know that essentially all they aredoing is approximating a rate of increase at a point by the average over a small distancefrom the point to a neighbouring point. They are then letting the neighbouring pointapproach as close as we wish to the initial point and then assuming that this will give usa finite value that gives the derivative at the point. They also know that they can chooseany point x on the curve for this process and so the derivative will itself be a function ofx just like f(x). The last three sentences constitute all that is needed, with sketches on theboard, to convey the essence of the idea of the derivative to the students, as a preliminaryto working through the steps (a)-(c). To the uninitiated student however all this insightis hidden behind the jargon and so they have no overall idea of where they are going.

2. The adjoint matrix method of inversion of matrices can seem daunting when displayedas an algorithm. But with the use of one or two simple examples it is easy to give apreliminary insight by showing how for example the cofactors simply come from thesystematic elimination of each variable, as in Cramer’s rule. It is essentially a notationdesigned to facilitate this recipe.

3. In the solution of second order differential equations by the Frobenius method ([19]) thedetails of choosing the form of the series and the strange emergence of the log term canbe mysterious to the beginner. In this case a preliminary feel for the essence of these ideascan be obtained very simply by working through the analytical solution of the Cauchy-Euler equation. In this case the general features that arise in the Frobenius method comeout explicitly and the log term is no surprise. So in this case the essence of a complicatedidea is conveyed by a much simpler analogy.

4. Mason ([53], p. 52) suggests crystallizing the essence of a theorem by going back over theproof of a theorem, illustrating the structure and describing the salient features, alongwith an overview of the role of the theorem in the module as a whole. After doing thisa few times one may then be able to encourage the students to do it themselves, withoutprompting from the lecturer.

H Exercise

Actually, there are not that many ideas per module that either need their essence exposing, or forwhich it is very difficult to do so. However, there will probably be some in your modules. Identifythese and find ways to convey the essence, using verbal explanation, board-work, analogies,examples, etc.

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Emphasize the respectability of guessing and the necessity of checking

Often the formal, finished, polished presentation of mathematics gives an impression that it alwaysmoves with inexorable logic from A to B with no intervention from the imagination. Of course we knowthis is not the case - but the students may not. In mathematics we have phrases for making guessing andleaps of faith seem respectable - ‘conjecture’, ‘by inspection’, ‘trial and error’ ‘taking an ansatz’, ‘methodof undetermined coefficients’, etc. All are grand clothes for nothing other than glorified guesswork -educated though it may be. Share this secret with the students. Encourage, indeed insist, that they getused to guessing and help them to lose the fear of being wrong. ‘Make mistakes - but quickly!’. In suchguessing, get them used to asking questions such as ‘What would be the simplest thing to do in these cir-cumstances?’. And of course, most importantly, they must realize the imperative of checking. Baumslag([7]) refers to this as the ‘Do the mathematical two-step, guess-check, guess-check’. In some ways thisprovides a reason for not giving answers or solutions to those exercises that can be checked - for examplethe solution to a differential equation does not need to be given, since the students can (should) checkit themselves by substituting back into the equation. This not only gives them practice in differentiationand hammers home the concept of the solution to a differential equation, but also gives them confidencethat they can find their own way about mathematics.

Examples

1. The method of undetermined coefficients in finding particular integrals for inhomoge-neous linear differential equations is a classic example of ‘guessing’ made respectable.Of course, it is so familiar to us now that it hardly seems like guessing - but that is whatit actually is, and this should be pointed out to the students. By inviting them to makeguesses themselves, before showing them the ‘correct’ choice, you can give them a feelfor the whole idea of progressing in mathematics by trying out the obvious avenues. Andof course in the method of undetermined coefficients we are forced to ‘check’ in order tofind the coefficients.

2. A similar example to Example 1 is the repeated roots case of the auxiliary equationin solving second order differential equations with constant coefficients. The auxiliaryequation gives us only one solution so we are forced to look elsewhere for another. Atthe elementary level we can’t justify the correct choice by for example the Jordan NormalForm, so it just has to be treated as the ‘obvious’ guess - which we later check by insertionin the equation.

H Exercise

In most areas of mathematics there are some critical stages or techniques that lend themselves toeducated guesswork - ‘by inspection’, ‘assume a solution of the form...’, etc. Identify some of these inyour course and prepare the corresponding lectures so as to highlight these to the students. Usethem, to emphasize that mathematicians often proceed in this way, by trying out simple guesses - butof course they immediately check them.

Alert the students to patterns that may assist their understanding and learning

Some people define mathematics as the science of patterns, and certainly patterns pervade the subject.Students should always be alerted to patterns in whatever they are doing and they should be encouragedto search for them themselves. Of course, the patterns can be of all sorts of different types - algebraic,

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geometric, logical, conceptual, numerical, etc. Fortunately this is one topic that is now given considerableprominence in UK school mathematics teaching, so students should appreciate the idea quite readily.

Examples

1. See the pattern used to summarise binomial theorem pattern in Section 2.8

2. Baumslag ([7], p. 155) notes how the right picture can illustrate the structure of an idea,giving the well-known example of the proof of the countability of the rational numbers.

3. Another type of pattern in mathematics is the generalisation - ‘perpetuating the presentpattern’. The simplest examples are such things as sequences - like 1, 4, 9, ..., but there areothers much more powerful. For example generalizing the Pythagorean length

√x2 + y2

from the plane to Rn√x2

1 + ...+ x2n. Or tensor calculus as a generalisation of vector

calculus. All these are generalizations of a pattern. In this case one can carry the patternfurther into abstractions such as metric spaces or groups. See Tall ([69], p. 11) for thedistinction between generalization (easy extension of a pattern) and abstraction (hardreconstruction of a pattern).

4. Mason ([53], p15) discusses a tactic that encourages students to develop and expressgeneralities.

H Exercise

Identify useful patterns in your own teaching that may illustrate important points in your lecture.

Help students to build up connections by providing roadmaps and overviews of dif-ficult sequences of arguments - try to express proofs ‘sensibly’, as much as ‘logically’

This is related to the previous item about patterns, but particularly relevant with a long proof. You canwrite down a summary of the steps to begin with, or provide a handout. Try to make each step ‘sensible’(not necessarily the same as logical) - why would you do that, why did the originator do that? Sometimesthe standard proof is presented in a sanitized, ‘perfect’, logically most efficient way that often prompts thequestion ‘Why?!’ from any curious student. Sometimes it may be best to use a different form of the proofthat does make each step fairly obvious. And the classic ‘rabbit out of a hat’ trick of pure mathematicswhere a form of expression is selected, or a particular condition is stipulated, can be exposed for what itoften is - a device to make the proof work.

Examples

1. The proof of a major theorem such as Cauchy’s theorem cries out for some sort of road-map linking all the steps and showing how they build up into the final picture. Youcan treat it as a group exercise within the lecture, only going through it for the studentsyourself when they have had a good try. Here the basic idea is to build up the proof byfirst deriving it for triangles, using a reduction process on smaller and smaller trianglesto provide estimations for the integrals required. This can then be used to derive theanti-derivative theorem and hence Cauchy’s theorem.

2. A similar example is the proof of Lagrange’s theorem in Group Theory. In this case thestructure is one of continual searching for cosets by a repetitive process until the group isexhausted. The connections and structure of the proof can be lost in the technical detailsif attention is not drawn to it.

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3. Mason ([53], p. 54) advocates using tree diagrams to illustrate the structure of a proofand to bring together the various ideas and steps.

H Exercise

Choose a few of your lectures and study the major ideas in each of them. Can the complexities ofthese be illustrated by appropriate roadmaps and overviews? Devise exercises that might help thestudents to construct these themselves.

Alert students to the fact that some teaching and learning is provisional, and mayhave to be revised at a later date

As higher levels of mathematics are reached it becomes increasingly necessary to compromise betweenexactitude and economy of explanation. Due to time or other limitations we sometimes have to be a littleeconomical with the truth in the interests of progress through the material. We perhaps avoid a subtletythat the students will revisit in a later course. On such occasions at least warn them, as a matter of intel-lectual honesty, that that is what you are doing, and that you cannot at the moment give them the wholepicture. The students don’t have to believe that everything they learn is the absolute last word - indeed,informed scepticism is an essential asset for any educated person. The first two examples below are caseswhere at some stage in a child’s mathematical education they are taught firmly established ‘facts’ usuallywithout qualification, which are later overturned to reveal whole new realms of mathematics. Rules inmathematics are made to be broken! John Bell - ‘All no-go theorems show is a lack of imagination’.

Examples

1. Students do not always realise for themselves that such things as Pythagoras, 180 degreesin a triangle, etc are restricted to plane geometry and do not for example hold in generalon a sphere. But then, when alerted to this fact, they may protest that one would nothesitate to calculate the diagonal of a field using Pythagoras, despite the fact that it ison a ‘spherical’ earth. This is an example where their previous teachers have had to be‘incomplete’ in their explanation to avoid muddying the waters. Point out that this willoften happen, because we are continually adapting to new circumstances in mathematics.We cannot at every stage go into all the detailed restrictions on our statements.

2. Ditto for the oft-quoted assertion in early school mathematics that we can’t take thesquare root of a negative number.

3. Tall [69] makes the point that we teach students a great deal about solving differentialequations analytically, and this might give them the impression that most equations canbe solved in this way. Of course, the number of differential equations that can be solvedanalytically is negligible and they are the exception - that’s why we need numerical meth-ods. We can explain this to the students and perhaps show them a few differential equa-tions that can’t be solved in analytical form.

H Exercise

When are you being economical with the truth in your lectures - do you need to give any healthwarnings? You often feel this when you make some statement and then immediately think toyourself ‘Ah, but, ..’. At this point it is probably best to be open with the students and add a qualifier‘There are occasions when this breaks down, but these cases don’t concern us here’.

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We have deliberately gone into great detail in this subsection on explanation, because it is so fundamentalto teaching. Good skills in explanation will not only help the students, but also save you time by enablingyou to progress more rapidly and reduce the likelihood of students coming to you for clarification.

3.11 Evaluating and Developing the Lecture

Even the most experienced lecturer sometimes finds something to improve in a lecture they may havegiven a number of times. Certainly, in your first years of teaching, every lecture is actually a learningexperience as much for you as for the students, and so each one needs to be evaluated so that you candevelop it further and improve your teaching. The Higher Education Academy UK Professional Stan-dards Framework (http://www.heacademy.ac.uk) holds as one of its professional values a willingnesson the part of practitioners to regularly evaluate and develop their performance and teaching practice.Your institution may have mechanisms for this as its own quality assurance and enhancement policies.So it is likely that you will have some sort of formal evaluation of your lecturing in your first few years ofteaching at least. This is NOT the subject of this section, although its results may of course feed into yourdevelopment. Such formal arrangements will usually follow some specified protocol that will normallybe explained to you through your institutional staff development unit. What we are referring to here isthe continual day to day process of learning and self-improvement that most academics take as a matterof course.

In fact this area is important, because it is how we measure the effectiveness of the impact of any trainingwe receive. Baumslag ([7]) gives a delightful and memorable analogy to illustrate this - the differencebetween a good and an indifferent carpenter. The good carpenter sharpens his tools before putting themaway after finishing a job. So the good lecturer, on leaving a lecture, spends a few minutes jotting (ortapping!) down the main topics of the lecture, good and bad points, ideas for doing things differentlynext time. This takes a little time, but is invaluable. You may put off doing this, because you think thethings are so clear that you will remember them - but you probably won’t. Occasionally give the studentsa very quick feedback form to see how things are going. By now you will be getting a fairly good ideaof what makes for a good lecture in theoretical terms, so construct for yourself a checklist of the sort ofthings you might want to think about after the lecture.

Wankat and Oreovicz [72, p 321] recommend setting ‘minute papers’ at the end of a lecture, containingsuch questions as ‘What is the most important thing you learned today?’ or ‘What questions do you stillhave’.

And, of course there is the minute by minute ‘micro-evaluation’ we do throughout the lecture, in whichwe watch the students to get a feel for how the class is going. It helps if you can cultivate a couple of thestudents who you can rely upon to give you feedback on progress. If students make comments, good orbad, don’t take it personally. During the lecture is by far the easiest time for a student to sort out problemsand misconceptions, and almost certainly other students will benefit from the issues addressed. This isone reason why you should identify in advance the difficult areas where you know most students willhave a problem, then you can address them there and then.

H Exercise

Design a short checklist for use in evaluating how a lecture has gone, including questions for thestudents and for yourself.