Chapter 4

Tutoring in Mathematics

4.1 The Purpose of Tutoring in Mathematics

Besides the lecture, the other main type of contact teaching in mathematics is the tutorial or exercise class.In this we include all such interactive teaching which is distinguished from the traditional lecture (Alwaysremembering that the lecture should be more interactive than just a monologue) covered in Chapter 3 bysome significant degree of interaction with the students. This may be through working with a numberof teaching assistants in a large exercise class, a small group tutorial or discussion class, demonstrationclasses, or individual one-to one tutoring.

In general the purpose of tutoring is to:

• provide supported practice in problem solving

• stimulate an environment for learning by doing

• provide training in communication/team skills

• facilitate learning by discussion.

In mathematics the major uses of tutoring are in problem solving or exercise classes, where the tutordoes not usually coordinate and manage the discussion process. Rather, students simply work throughproblems, and discussion amongst subgroups is generated spontaneously. Many mathematicians actu-ally prefer to work on their own, but there may be activities in which a coordinated team approach isuseful, say in modelling problems. The main point in such group teaching is that the focus is as muchon individual support for the student, to help them over the difficult parts of the course. The primaryfeatures of tutoring in mathematics are therefore:

• active participation

• face-to-face contact

• purposeful activity.

As Mason ([53], p.71) notes, the main advantage of tutoring in mathematics is that we can enter students’world, and can interrogate them, rather than trying to bring them into the lecturer’s world, which iswhat usually happens in the lecture. It is an opportunity to help the students through the most difficult

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conceptual reconstructions (Principle 7), such as abstraction. Also, small group teaching allows studentsto ‘teach’ each other and this takes advantage of the old adage that you only really begin to understandsomething when you come to teach it. Morss and Murray ([56], p. 50) give a useful list of the aims ofgroups and associated activities. They emphasise that we should explicitly share these with the students,so they know the purpose of the group activities. The tutoring environment is also a good place to helpstudents to learn how to learn (Principle 8). For example, when a student is stuck on a differentiationget them to write out everything they know about differentiation. Eventually they will find this is auseful ploy whenever they hit a sticking point. Eventually, they will get fed up of writing stuff out andwill start running through it in their mind - that is, thinking deeper. The tutorial also gives students theopportunity to watch experts in action, warts and all.

ExampleWhen extending the concept of a distance to any number of dimensions, we can get awaywith this in a lecture by generalising the sum of squares in a very natural way. The resultcan be expressed by a simple formula that is an obvious adaption of the two dimensionalcase. However, the axiomatic definition of a norm, extended to any vector space is muchmore of an abstraction, which most students find very difficult. Also they inherit a number ofmisconceptions in the abstraction. For example they may believe that the norm is linear in itsarguments. In a tutorial we can explore this much more interactively. We can help them thinkthrough the connections between the sum of squares and the axioms for the norm. We canpresent them with particular cases that show why the axioms are useful. We can show themcontradictions arising from the assumption of linearity of the norm, and the origin of thetriangle inequality. All these things require their active and prolonged engagement, whichcannot be achieved so easily in a lecture.

Much of what we need to say about tutoring has already been covered in Chapters 2 and 3 - so there willbe inevitable repetition, although here we go into more depth on engaging, enthusing and explaining,which are the key activities of the tutorial. There is also a great deal of generic literature on tutoring andgroup work, and you may meet this in your institutional staff development courses. While we do notwant to replicate this here, it is worth summarising the main points and give a few references for furtherreading if you feel the need. This is the subject of the next section. Then in the rest of the chapter we focusspecifically on tutoring in mathematics.

4.2 Summary of Generic Material on Tutoring

Most institutional staff development induction courses will contain something on what they might callsmall group teaching or tutoring. However, since this is the main mode of teaching in some subjectssuch as the arts and social sciences the slant of these generic courses is often towards small discussiongroups or seminars. While these have a role in mathematics, they are not the usual form of tutoring inour subject. Nevertheless, there is a great deal of tutoring that is generic (subject independent) and wecan learn from this. So we will briefly summarise this here. Most of it is common sense and self evident,but if further expansion is necessary there are ample references. The standard generic reference on smallgroup teaching is Jacques ([46]) You will see that although this contains a lot of good advice, it does notreally get to the core of the typical mathematics tutorial. Other useful generic references are [2], [10], [14],[36], [38], [52], [54], [57], [61], [70].

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Types of small group teaching methods

The following is a list of the different types of small group teaching that might be discussed in a genericcontext. Not all of them are common approaches in mathematics, but it gives an idea of the sorts ofpossibilities for group work with students.

Controlled discussion: strict control by tutorStep-by step discussion: planned sequence of issues/questionsSeminar: group discussion of a paper, a mathematical model or problem solution, presented by a studentProblem class: individuals working on problems and/or presenting solutionsTutorial: meeting with very small group, often based on an essayGroup tutorial: topic and direction from tutor, rest from groupFree discussion: topic and direction from tutor, tutor observesTutorless group: some direction from tutor, group may report backSelf-help group: run by and for students, tutor may be a resourceCross-over groups: brief discussions then transfers between groupsBuzz groups: very brief discussions generating ideas for follow-upSnowballing: pairs becoming small groups becoming larger groupsSyndicate: mini-project work reported to the full classBrain-storming: generation of ideas from group. No criticism until all the ideas are loggedSimulation/game: structured experience in real/imaginary roleRole-play: less structured activity in allocated or self-created rolesFishbowl: small groups within large, them discussion and reversalWorkshop: mixture of methods, usually directed at attitudes’ and skills’ developmentDemonstrations: illustrations of theoretical principles, solutions to problems, etcExercises: tightly structured experiments to provide dataStructured enquiries: lightly structured experiments, more student inputOpen-ended enquiries: students determine structure and report backProjects: student research - tutor provides supervisionPersonalized system of instruction (PSI): self-paced, tests on progressComputer assisted learning: often to simulate experiments, etcLecturing: a lecture to a small group - best done towards the end of the session, if at allVirtual learning environments: various electronic engagements such as discussion groups

The main types of small group teaching in mathematics are usually more limited, comprising essentially:

• exercise/problem classes

• board demonstrations

• discussions

• individual consultations.

However some mathematics lecturers use such things as project groups, role play, etc.

Aims of small group teaching

Morss and Murray [56] list the typical aims of group learning as:

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• understanding

• critical thinking

• personal growth

• communication skills

• group and team skills

• self-direction in learning.

They give examples of specific activities that can be used to achieve these aims. In the normal math-ematics tutorial we would be aiming for the first two and the last of these, although nowadays, withthe emphasis on transferable skills some people use small group discussions in mathematics to meet theother aims.

Group task and maintenance functions

The argument behind the range of group activities noted above relies on the wider learning opportunitiesand skills development that it stimulates. For example Jacques [46] summarises the sorts of tasks thateffective groups are intended to do as:

• Initiating - suggesting new ideas or a changed way of looking at the group problem or goal, propos-ing new activities

• Information seeking - asking for relevant facts or authoritative information

• Information giving - providing relevant facts or authoritative information or relating personal ex-perience pertinently to the group task

• Opinion giving - stating a pertinent belief or opinion about something the group is considering

• Clarifying - probing for meaning and understanding, restating something the group is considering

• Elaborating - building on a previous comment, enlarging on it, giving examples

• Co-ordinating - showing or clarifying the relationships among various ideas, trying to pull sugges-tions together

• Orienting - defining the progress of the discussion in terms of the group’s goals, raising questionsabout the direction the discussion is taking

• Testing - checking with the group to see if it is ready to make a decision or to take some action

• Summarizing - reviewing the content of the past discussion.

In fact opportunities for most of these will occur in a lively, well run mathematics exercise class or problemsession. Some are naturally less prominent because of the nature of mathematics - for example differencesof opinion can usually be resolved quite quickly! To keep the group on task and maintain an effectiveworking environment Jacques suggests that groups are supposed to be:

• Encouraging - being friendly, warm, responsive to others, praising others and their ideas, agreeingwith and accepting the contributions of others

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• Mediating - harmonizing, conciliating differences in points of view, making compromises

• Gate-keeping - trying to make it possible for everyone to make a contribution by, for example,suggesting limited talking time for all

• Standard-setting - expressing standards for the group to use in choosing its subject matter or pro-cedures, rules of conduct, ethical values

• Following - going along with the group, somewhat passively accepting the ideas of others, servingas an audience during group discussion, being a good listener

• Relieving tension - draining off negative feeling by humour or conciliation, diverting attentionfrom unpleasant to pleasant matters.

Main factors that may affect the dynamics operating within a tutorial group

Morss and Murray [56] list factors that may affect the dynamics of a tutorial group. Most of these applyto any sort of mathematics tutorial:

• commonality of purpose

• the interest and commitment of each participant to the aims of the group

• the relationship of members within the context of the group

• interaction of individual personalities

• personal agendas

• level of participation

• shared knowledge

• degree of cooperation

• group size

• setting and physical environment.

Morss and Murray also give a long list of activities one can use in groups, such as guided discussion, etc,along with a detailed sample plan for your first tutorial.

Facilitation skills required by tutors

Following Jacques, Morss and Murray [56] list the skills required by the effective tutorial supervisor:

• listening

• questioning

• explaining and clarifying

• encouraging participation

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• responding to students as individuals

• closing

• monitoring and evaluating

• provide grounding

• time management

• monitoring attendance.

We will treat most of these in the context of mathematics in this chapter.

Students also need to develop the skills above and effective tutors support such development in theirstudents. The success of small group teaching depends as much upon students - their knowledge andskills - as it does upon the tutor.

Managing and mismanaging the group

It is of course difficult to manage group work because we have to juggle the different abilities, attitudesand aptitudes of the individual students in a way that they all benefit, and that there is good progresstowards meeting the objectives of the session. This is not quite so bad in mathematics because one cankeep close tabs on any individual’s understanding by careful questioning (see below). In general, thecriteria for the successfully managed group include [42]:

• prevalence of a warm, accepting, non-threatening group climate

• learning approached as a co-operative rather than a competitive enterprise

• learning accepted as the major reason for the existence of the group

• active participation by all

• equal distribution of leadership functions

• group sessions and learning tasks are enjoyable

• content adequately and efficiently covered

• evaluation accepted as an integral part of their group’s activities

• students attend regularly

• students come prepared.

On the other hand it is easy to mismanage a group. Some common weaknesses that occur in small groupteaching are (HEFCE Specialist Assessor Training, UCoSDA):

• the goals of the session are unclear

• the structure of the class is unclear

• lack of preparation - by tutor and /or students

• tutor talks too much

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• lack of student participation

• low cognitive level of discussion

• questions rarely go beyond eliciting recall

• students are not involved in the topic

• discussion is unfocused for much of the time

• one or two students are allowed to dominate the discussion.

Skills for participating

The first thing most teachers notice when running a small group is the difficulty of getting full engage-ment and participation. This is particularly difficult in mathematics because of the high premium at-tached to precision. There is rarely room for opinion or waffle and most students are reluctant to sayanything unless they are sure it is right, in which case it is often not worth saying! One of the key skillsin running a small group, particularly in mathematics, is therefore to pose questions that are sufficientlyopen to encourage debate, yet lead to deep understanding and useful conclusions in the time available.In general terms Forster, et al, [33] list the sorts of participation skills the tutor needs to develop in thestudents as:

• listening attentively to others

• giving information to others

• asking others for information

• giving examples

• checking out what others have said

• giving reactions to the contributions of others

• asking for reactions to one’s own contributions

• initiating discussion by asking questions, giving ideas, making suggestions

• bringing together and summarising

• encouraging others to take part.

Forster, et al, also note that students are encouraged to contribute in tutorials when:

• they feel comfortable with each other and the tutor

• trust and respect are displayed and support is given

• learning is seen as a co-operative exercise

• there is a clear understanding of what they have to learn

• they are aware of the importance of participation

• they are aware of the skills which they are expected to practices

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• students are set realistic and achievable tasks

• methods are used in early tutorials which foster students’ contributions

• ground rules have been agreed, for example:

– everyone prepares and attends

– everyone tries to contribute and helps others to do so.

So the tutor needs to provide the sort of group atmosphere that promotes such an environment.

Questioning strategies

While it is important to pose the right questions, it is equally important to pose them in the right way, theskill of questioning itself. This is clearly related to the cognitive skills we wish to encourage. In Section 2.6we looked at the classification of cognitive skills, noting that most taxonomies are too unwieldy for theaverage lecturer, and suggesting the MATHKIT for the practitioner. Either way, some such classificationcan be used to suggest different questioning strategies. For example, in generic terms, Forster, et al, [33],using Bloom’s taxonomy, classify the different sorts of questions students can be encouraged to ask:

Testing questions - used to elicit information and concerned with:

– checking knowledge: which limit test is most appropriate here?

– comprehension: what do you think is meant by...?

– application: what relevance would that have in ...?

– analysis: what qualities do they have in common?

– synthesis: could you summarise what you have said so far?

– evaluation: what do you feel is best?

Clarifying questions - used to ensure a shared understanding (often by elaborating a point previouslymade):

– what did you mean by...?

– can you give an example ...?

Elaborating questions - often provide a gentle way of encouraging students to say something morefully:

– can you tell me more about that?

– what does that make you feel?

It is clear how to interpret such questioning strategies in mathematics. There are also some commonerrors in questioning (HEFCE Specialist Assessor Training, UCoSDA):

• asking ambiguous/confused questions

• asking too many questions at once

• asking a question and answering it oneself

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• asking irrelevant questions

• asking a difficult question too early, so a student is deterred from answering

• asking questions in a threatening way

• asking confusing questions

• ignoring answers

• failing to see the implications of answers

• failing to build on answers obtained.

From this section on generic material on small group teaching we can see that much of it applies equallyto a mathematics tutorial. However it is perhaps a little arid and detached and somehow it lacks the feelof a buzzing mathematics exercise class, so we now turn to the specific issues of tutoring in mathematics.

H Exercise

Examine the typical mathematics tutorial in the light of the generic material of this section.

4.3 Tutoring in Mathematics

The main difference between tutorials and lectures is the closer relation one has with the students - wecan can now interrogate them, engage in discussion with them, we can start persuading and influencingthem and encouraging them to critically examine their own ideas. Hence, we define the mathematicstutorial by size - a mathematics tutorial can take place with a group of a hundred students, with a fewdemonstrators helping out. The distinguishing feature of a mathematics tutorial will be taken as anyenvironment where the object is to interact one to one with students, to be able to interrogate them,discuss issues with them, help them through their individual difficulties, and to encourage students towork together. This can of course be a large group or simply talking to a student in the corridor. So:

a mathematics tutorial is any interaction between the tutor and the student(s) in which there is theopportunity for in depth discussion of a particular mathematical problem or topic.

The way we do this also sets an example to them of how mathematicians work, how they think throughthings. You are not telling them something, you are now trying to get them to think about something sothat they can reconstruct their own ideas (Principle 7).

In many topics this might simply be an exercise class in which students work through problems togetherwith help on call from an expert such as yourself. Or it might be a one-to-one individual consultation- it might even be a telephone discussion, or video conference. It would also include discussion groupswith or without tutors (In the latter case the influence of the tutor would be felt through how they set upthe group and its tasks). It might be a demonstration class where students are given the opportunity topresent material to their fellow students -‘You really learn something when you are put in the position ofhaving to teach it’. So in a tutorial it is the interplay between students or tutors that provides a greaterrange of learning activity.

Tutoring can enhance the motivation of students, because they have more control over the proceedingsand more personal input and direct feedback. Tutoring has a wider range of learning activities includinglearning by doing and learning by trial and error in a safe environment, general transferable skills such ascommunication and teamwork. Also tutoring encourages learning through interaction and this is perhaps

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its prime strength. It involves all of the key gambits in a learning activity: initiating ideas and lines ofthought; explaining them; listening to new ideas; questioning with a genuine desire to know; respondingin a disciplined and organized way; discussing and playing with ideas. Each of these is a skill we shouldaim to develop to high levels at university.

There is one kind of interaction that is often forgotten in discussions of teaching, and which is possiblythe most important of all in mathematics - the interaction of the person with themselves. The working,thinking, mathematician is often like a self contained tutorial group, arguing with themselves, setting upconjectures to knock them down, critically examining one’s own ideas. One can probably recognise thosestudents who are the ‘real mathematicians’ as those that automatically adopt such ways of working. Intruth most people think and work like this, but not to the extent required in mathematics, and the tutorialis where we have to help them develop these skills (Principle 8).

Tutoring in mathematics is usually best restricted to particular areas of mathematics. For example, onewould not use it for routine topics such as matrix algebra calculations, where the intellectual skills em-ployed are limited. One would use it for topics that would benefit from discussion and debate. Forexample, a particularly long and involved proof; a tough mathematical modelling exercise; a sizeablepractical statistics exercise; harvesting a wide range of applications of a simple topic such as quadraticequations. All of these require a broad range of ideas and skills that an individual student might lack,but the group as a whole possesses. And members of the group will have to use their interaction skills toproduce the final product.

The particular objectives of a tutoring session will depend on the stage reached in the module, and wechoose the group activities to suit these. It might be small groups of four/five collaborating as a teamworking on a mini-project or a substantial modelling exercise, or it might simply be students working in-dividually through problems. But there are all sorts of imaginative activities one can try. For example onecould have pairs of students working through a complicated proof, one playing the role of the ‘prover’and the other the sceptic, criticizing the prover’s arguments.

H Exercise

Discuss the advantages and disadvantages of tutorials in mathematics. Which tutorials did you learnmost/least from as a student, and why? What do you think your students get from your tutorials?

4.4 MATHEMATICS for the Tutorial

In Section 2.2 we introduced the MATHEMATICS mnenomic to remind us of the sorts of things we needto consider when undertaking any teaching and learning activity. This comprises:

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.

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We might apply this to the typical tutorial as follows. Obviously the mathematical content needs thought.In a tutorial you are not going to labour through routine or peripheral material. Rather, you are going tofocus on the key ideas and also any issues that students find difficulty with. So your aims and objectivesare going to be higher level - I and T in the MATHKIT scheme. Put crudely your teaching and learningstrategy for the tutorial is going to be a mixture of students’ independent work, with you (or your as-sistants) intervening personally to help and support the students in developing their own solutions andtheir own approaches. The evaluation of the exercise can be by the usual student feedback mechanisms,with questions about how useful they found the tutorials. But also since you will be interacting directlywith the students almost on an individual basis, evaluation can take the form of minute by minute feed-back from the students. For learning materials you might simply issue problem sheets, or you mighthave a more interactive exercises, like a mini-project or role-play for which you will have to prepare moresuitable materials. So far as assessment is concerned, it may be that the tutorial is devoted to assessedcoursework, or you might decide a quick quiz at the end is useful to see what the students have absorbed.Time and scheduling issues are actually much more important than you may think at first sight. To getthe best out of the activity you have to keep the students to some sort of schedule - if there are three top-ics to consider then you cannot spend the whole session on just one of them. The initial position of thestudents now needs to be assessed on an individual basis in the tutorial - i.e. for each student you interactwith you always need to find out what their background knowledge is before you can move on and helpthem. The tutorial is one of the prime occasions for helping students with higher order ideas such aslinks between topics, the way they fit together, getting an overview, etc. In this way the coherence of thecurriculum can become clearer than is perhaps apparent from disjointed self contained lectures. Finally,one important role of the tutorial is for you and the students to get to know each other to produce a goodenvironment for learning. It is one of the few occasions on which you get the opportunity to meet thestudents as individuals and give them your undivided attention.

While MATHEMATICS provides a ready checklist for the things we have to think about in planning atutorial, the basic principles of Section 2.4 underpin everything that we do, as in all aspects of teaching.And just as we discussed suitable teaching and learning strategies for lectures (Section 3.3) this is easilyadapted for tutorials. However, in tutorials, because of the close interaction with individual students itseems appropriate to particularly emphasize three major features of group activities that are fundamentalto helping students learn mathematics, embodied in Principles 9-11:

• ENGAGE the students in productive mathematical work (Section 4.5)

• ENTHUSE the students about mathematics (Section 4.6)

• EXPLAIN mathematics to students with varied backgrounds (Section 4.7).

These skills are central to teaching mathematics by any means and they are difficult to master especiallyfor those new to teaching. They are particularly relevant in the tutorial situation, so we devote the nextthree sections to looking at these aspects of the mathematics tutorial. Following this we will look at thespecific practicalities of working with students and the different types of small group activity common inmathematics.

H Exercise

Work through the MATHEMATICS checklist for your next tutorial. Did it cover mosteventualities? If not make a note for next time.

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4.5 Engaging Students

In Section 2.9 we looked briefly at the issues around engaging students during the typical mathematicslecture. In that situation the opportunities are admittedly limited, but still allow some useful interactionthrough questions or short periods of activity such as problem solving. In the tutorial there are far moreopportunities for engaging the students, it is after all what the tutorial is all about. That doesn’t mean itis easy. In a tutorial of any kind it is easy to be lulled into thinking that the students are beavering awayproductively and can be left to get on with it. In fact, the likelihood is, particularly in the first year, thatthey will have difficulty finding the best way to work or take advantage of the tutorial. Often they areunable to plan their work, they will be unsure where to focus their efforts, they may not know the sortsof questions to ask, or even how to ask a ‘good’ question at all. Students may not be aware of the need tofollow up their tutorial work with independent work, consolidating the skills they have learnt, and theymay not know the best way to do this. Each student will have a different approach to learning, and maynot know how to adapt this to the tutorial situation. The tutor needs to consider all these aspects so thateach student gets the most out of the tutorials.

However, we should perhaps get one issue out of the way at this point. Tutorials are not always popular,particularly with mathematics students. Some students, for example, prefer to work on their own athome. So we have to accept that not all students will be interested in taking advantage of what we offerin tutorials. That is their choice, they are adults, and the teacher should not feel obliged to press hard toengage students who really prefer to fend for themselves. As long as we are always open to students’requests for help, so long as we offer them our support, then we have done our job and we can focus ourattention on those students who do wish to take advantage of it. So do not be afraid to ‘let go’ of studentsand leave those who wish it to their own devices. In this chapter we are looking at how to help thosestudents who want our help.

To engage students in a tutorial it is particularly important to ‘be everywhere’. This means movingquickly and efficiently from student to student, addressing each of their problems concisely and effec-tively. And if the issue seem to be of common interest, or it is raised a couple of times, open it up to thewhole group, maybe discussing it at the board. Give all students equal opportunity to use you, and ifthere are some who haven’t asked for your help, have a quick look at how they are getting on. Thereshould literally never be a dull moment. If it goes quiet and you are getting no questions, find out why. Ifthey have done everything give them something else. If they are all stuck and have given up address theblockage. Show by your attitude and demeanour that you expect everyone to be working and getting themost out of the session. Make it clear you want to help them, and that you welcome questions - and thatthere is no such thing as a silly question. Remind them that this is prime learning time. These days youcan remind them how much they are paying you to help them! You can soon sense in the first tutorialswhether or not they find it difficult to ask you questions. Find out why and make it as easy as possible.

Examples

1. Legrand’s method of scientific debate In general students will be more inclined to par-ticipate in a tutorial situation if there is a conjecturing atmosphere ([53], p.72). That is, wehave to make the whole atmosphere of the tutorial clearly aimed at discussion, interac-tion, exchange, and so on. Mason describes the method of scientific debate used by MarcLegrand. This is designed specifically to generate the sort of environment we need. Inthis approach some sort of contradictory example or episode might be used to convincestudents that there is a need to adapt their ideas to meet new situations. This is usuallynecessary in the development of difficult concepts because students won’t switch on andexert mental energy unless they are convinced of the need to do so (Of course, studentsare not alone in this). For example when introducing the differentiability of functions the

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most stimulating examples are initially those of non-differentiable functions. Anotherfeature of Legrand’s debating approach is the inclusiveness designed to ensure that ev-eryone has the opportunity to contribute and think deeply about the topic. Also, theapproach aims to ensure that issues are resolved collectively in a spirit of respect for theviews of others. As mentioned above, another difficulty with engaging students is thatmany don’t like asking questions in class, and in fact don’t know how to ask non-trivialquestions. Mason has a number of ploys for encouraging this ([53], p.76). They need tomove from accepting that something works or is true because an expert says so to ask-ing why it works, and feeling the need to be convinced themselves when the occasiondemands it.

2. Krantz ([49], p. 119) considers the question of getting students participating so importantthat he devotes an appendix to encouraging class participation, advocating:

• get students to the blackboard, but only if they volunteer• have students prepare oral reports or min-lectures• have students take turns in writing and grading quizzes• if a student can’t do a problem, and openly admits this, invite him/her to come to

the board and explain where they are stuck• get students to regularly jot down things that are bothering them and then discuss

them in class• give regular reading assignments• have guest lecturers• mathematical POST-IT notes• make deliberate mistakes and ask the students to pick them up.

3. Sometimes, when working though a solution on the board or opening up a question tothe class generally you may get a long pause and silence. Persist! If necessary keep‘sharpening’ the question or giving clues. Mason ([53]) refers to this as ‘funnelling’ andurges caution in this because it can trivialize the question until it becomes effectivelyrhetorical. Of course we do not want this, and once you have actually got students an-swering and participating, start to work back and broadening the question. Think aboutthe sorts of questions you ask. Vague, open, questions are not necessarily the best meansof opening a debate - it might work better to go from specific to more general. Try tomake your questions stimulating, possibly provocative or intriguing. Only ask ques-tions worth waiting for. If students are inhibited about responding to questions, then getthem to write out their answer instead and hand it in.

Remember that in a tutorial environment you are actually doing more than just teaching mathematics.Part of the purpose of such classes is to teach the students to be able to interact fruitfully with others insolving problems and to learn how to learn. So they have to learn how to contribute positively, to be ableto take risks with ideas, to criticize other’s views, to ask penetrating and incisive questions. So you areentitled to press them to engage. Certainly, when they get jobs after graduation their boss will soon noticeif they sit there and don’t contribute.

You will get some ‘silly’ questions, and answers. But never ridicule an answer. Neither should you be bepatronising. Think about your general demeanor with students - be relaxed, welcoming, open, helpful.You may only be a few years older than the students - toss in the odd anecdote from your own studentdays - ‘I remember I found this very difficult myself, until ...’.

When a student asks you a question, seize upon that as an opening to get them engaged. So don’t ‘giveaway the store’ as Krantz puts it, but just give them a leg up to get started on finding their own answer.

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And often the question reveals a deeper problem that is the real issue to be addressed. Manoeuvre thestudent into considering this and check whether other students have a similar problem.

ExampleA first year student working on calculus problems asks ‘What is the derivative of 1/

√x ?’. We

don’t give the derivative or simply say ‘Look it up’. The real problem here is probably thatthis student is weak on indices. This is a very common problem, almost certainly shared byother students in the class. So you might ask the class generally ‘What is the closest standardderivative to this?’, or ‘What is another way of writing 1/

√x ?’. Either way, you are trying to

get them to see that they can rewrite it as x−1/2 and this is a standard derivative. To you suchsteps will seem trivial, but to the novice who is insecure with their indices they are difficultto work through and they need practice. And when you ‘answer’ their question in this waythe derivative they are after is not the issue, rather it is the thinking required to convert thefunction to a better form, the idea that you sometimes have to do this to make progress. Theywill of course meet such ploys time and again in integration, so you are preparing them forthat. And of course, once they have done this particular problem, get them to extend it andhence differentiate similar functions.

We also need to remember that working in a tutorial can be quite tiring and even stressful and frustratingfor students and tutors. You often find that students lose concentration halfway through an hour’s class,and are not really engaged with the material. Just remember that most people lose concentration afterhalf an hour! This is why it is often useful to have more than one topic or type of activity during thesession. You might have say two or three major points that you want to get across in the session, saythree big ideas. Then you can spend fifteen minutes on each, summarising and collecting ideas after eachone. In any case you can keep them on task by continually going back to each of them to see how theyare getting on. You might provide a quick relaxing break by a short story, joke, or anecdote, relevant tothe topic - or nothing at all to do with it! Just something to lighten things up. One lecturer slips in a bitof history - a famous mathematician. One invites all the students to get up and wave their arms about (!).Something topical - the cot death court fiasco for statisticians, what happened to Mars Beagle probe fordynamics, etc, etc. The key thing to remember is that not only might some students lose concentrationafter a while, but it is inevitable that all students will lose concentration eventually, and part of the jobof teaching is to anticipate and address this. And of course there is nothing wrong with controlled loss ofmomentum at strategic points - a fell runner does not maintain constant momentum over a range of hills.

H Exercise

For your next tutorial think about how you can engage students most effectively. How does thisinfluence the materials you prepare and how you structure the tutorial?

4.6 Enthusing Students

We touched on this in Section 3.9 and perhaps the issue of maintaining enthusiasm is more pressingin lectures, where students spend a significant proportion of the time fairly passive. In that case theproblem is more one of keeping the students interested and motivated to keep their interest up to followthe lecture. In the tutorial we can do more to inspire them about maths. We can spend the whole sessionmainly on using inspiring and motivational examples as a media for developing the mathematical skillsrequired. When covering a particular topic we are not confined to the strait-jacket of the objectives of

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the lecture. In a tutorial we can develop the beauty of a particular topic, we can get them to uncover forthemselves the fascination of some application.

It is however still worth thinking about motivation a little more deeply because in a tutorial we can makea big difference to students’ views of mathematics. To some people mathematics is a real demotivator,almost to the extent that they become anxious about mathematics (this is of course more prevalent inservice teaching). Skemp ([66], p.125) goes into detail on this. He describes the Yerkes-Dodson lawwhich essentially states that for simple tasks the stronger the motivation, the better the performance,whilst for complex tasks this is only true up to a point after which increasing motivation leads to poorerperformance. The consequences of this for the learning of mathematics are quite clear. While studentsare developing the basic skills through fairly routine tasks they will perform better, the more we canmotivate them. As they progress to higher order skills and more involved problems they need supportand a more relaxed pace to maintain their motivation, and this is precisely the environment we need toprovide in the tutorial. As mentioned in an example in Subsection 3.9.3, Skemp also suggests that as anintellectual activity mathematics satisfies an intrinsic need for mental growth. He argues that teachersshould proactively use this need for mental growth as an intrinsic motivation for learning mathematics.And for this one needs to provide lots of opportunities for success. This is the mastery as motivationargument, and the reason for providing lots of easy examples in the initial stages of any topic.

We have already mentioned the need for the teacher to be enthusiastic in the lecture (see Section 3.7). Thisis even more important in the tutorial. A well run tutorial can be exhausting because of the need to thinkon your feet and to keep everyone working hard. And you need to look as if you are enjoying it - even ifyou are not. Actually, if you really care for your students you will enjoy it, because you get a thrill eachtime a student overcomes a hurdle - particularly if you have been instrumental in the process.

So far as getting the students interested and motivated, we have already looked at this in Section 3.9.The suggestions there apply equally to tutorials. Make sure the basic teaching technicalities are satisfac-tory, and allow time for developing interest. As mentioned earlier, the tutorial should be addressing asmall, number of crucial points, so we need to ensure that the students appreciate what these are, andunderstand their importance.

ExampleIn the case of solving first order differential equations we might emphasize that there are es-sentially three types: Variables separable, linear and homogeneous. We can demystify therather strange form of these by telling them that these forms are chosen for no other reasonthan that they allow simple transformations or operations that lead easily to the methods ofsolution. Furthermore the classification of the types immediately suggests the means of solu-tion. Emphasize that this sort of tactic pervades mathematics - we are often classifying typesof mathematical objects precisely in forms in which they can be most easily dealt with andthat indeed suggest how to deal with them. This is one of the key ‘secrets’ of mathematics andappreciating it is just as useful as grappling with the details of solutions of a few differentialequations.

The importance and nature of examples was mentioned in Section 3.9, and as noted previously you cantake full advantage of this is in a tutorial. So make an effort to find inspiring, exciting examples, ratherthan only mundane applications from other subjects that they study. Nowadays there are plenty of popu-lar science and mathematics books which you can scour for such examples. Some useful names to look forare Devlin, Gribbin, Penrose, Singh, Stewart, Wells, as well as the mighty mathematical compendiums ofthe latest developments such as Engquist and Schmids’ Mathematics Unlimited - 2001 and Beyond [32].And of course there are almost unlimited resources on the web, with Wikipedia and open material andcourses produced by universities around the world we are spoilt for choice. For any undergraduate topic,at whatever level, it should be possible to dig out some stimulating motivational material that you can

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feed into a tutorial situation. And we need to find a range of such examples because different students aremotivated by different things - some prefer really practical commercial or industrial applications, someprefer more basic science, some are fascinated by the inherent beauty of the mathematics, some by itspower. See Section 3.9 for examples. And remember that beneath the most mundane topics can often befound fascinating depths.

ExampleA lump of coal is a rather dull and uninteresting object, but it is very useful for keeping uswarm. And it is dirty and something we only really want anything to do with at a distanceand when essential. But, if we study it carefully and delve down into its atomic structure wesee that the underlying element is Carbon. The atoms are Carbon atoms. So in principle, wecan rearrange these atoms to give us diamond, and in the formation of the universe Carbonwas the key element responsible for life. Diamond is clean, precious, sparkling and veryinteresting! And as a fossil, coal is evidence of changing climates in the past. So we see thatviewed in one superficial way coal is dull and boring, but looked at in other ways it is veryinteresting. So it is with most of mathematics.

The analogy in this example is useful in other ways. In general in explaining/ justifying/ motivatingmathematics we go from mental ‘atoms’ such as (a+ b)2, A− A = 0 in completing the square to ‘mentaluniverses’ such as maxima and minima of functions of many variables, where completing the squarebecomes the diagonalization of a quadratic form. This happens everywhere in mathematics. As anotherexample, quadratics and linear expressions are the ‘atoms’ of polynomials by the fundamental theoremof algebra. In a tutorial we can give the student the opportunity to search for such ‘atomic structures’themselves. What are the key atoms underlying a particular topic - when all the irrelevant detail hasbeen stripped away, what is the real underlying structure?

In a tutorial we can also encourage students to look behind the superficial nature of a question or problem.In fact, even the most mundane question contains the seeds of more interesting things. What could bemore mundane than: ‘Differentiate ex’? But you can point to the strange fact that the derivative is itself,and in fact this defines e. And it is the whole basis of the theory of linear differential equations - and it iswhy the exponential function is so important. And why is the derivative itself - how does this relate tothe limit definition? These are the sorts of things the students should be looking at in a tutorial, not justthe actual techniques of differentiation.

Perhaps we should also think of the things that actually damage student motivation and that we shouldavoid (Most of this applies to lectures as well as tutorials of course). Being unapproachable or unavailablecertainly deters students. A tip might be useful here. There is actually not much danger in being asapproachable and available as possible - it is a fact of life that students will rarely come to see you anyway.That is, you can appear to be effusively available and helpful to students, but you won’t be flooded byvisits. Perhaps this aspect of human nature is a cynical application of Principle 2! Another turn off islack of clarity (Principle 3) for tutorial activities. Unclear instructions really frustrate students, who willsimply come back to you for clarification, which will just waste time.

As mentioned elsewhere, students like the freedom to pursue their own particular style of learning, andwill become demotivated if they are railroaded into one particular approach. So it will encourage studentinterest if we can offer choices in study methods, and this is possible in a tutorial situation in which thetasks are carefully designed. We have to recognise that we all learn in different ways. We don’t have tobuy into the learning styles industry, but we do need to be tolerant about how students approach somethings, even if it wouldn’t suit us. For example, as mentioned earlier, some students don’t actually like towork in tutorials, but prefer to work on their own, back at home. This probably accounts for occasionallypoor attendance at tutorials, which should not be taken too seriously so long as students are progressingsatisfactorily.

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ExampleThe following problem vividly shows up the different ways people might tackle mathematicalproblems. l is a large number and s a small number such that l+ s = 1. Which is larger, l2 + sor l + s2 ? Some people will try suitable values for l and s, soon discovering the result isalways the same, guess this is true generally and proceed to prove it, now they know wherethey are headed. Some on the other hand will simply jump straight in and substitute l = 1− sin both expressions to see what happens. Now it is tempting to regard the latter approach asthe best elegant method, and the former as a bit pedantic and routine. But in fact the formerapproach is very common - many good mathematicians like to play with things numerically,getting a feel for what is going on before committing themselves to symbolic approaches. Itis no good trying to discourage people from working this way if that is best for them, and iftheir approach is viewed as ‘wrong’ then that will only demoralise them.

Repetitive and dull material with no significant highlights or interesting features is of course demotivat-ing. The same goes for irrelevant material, or material whose significance is not made clear, or is unrelatedto experience or to rest of the course. Overload in the form of too much information or too much detailedcontent should be avoided. Arid conceptual terms and ideas, that are complex or abstract are of courseoff-putting, but they are unavoidable in mathematics so we have to make an effort to make them moreinteresting and palatable. Again, it will demotivate students if we fail to take sufficient account of theirbackground in mathematics - not only will the students not understand what we are saying, but it willbe clear to them that we are not even bothered to find out what they already know. Other off-puttingcharacteristics include verbosity, too much text to read, rambling explanations, lack of conciseness, etc.Students need to get to the key ideas quickly. No ‘road maps’, overall viewpoints and navigational aidsthat give students an overall appreciation of the topic will mean that they have no idea of the ‘magnitude’of the topic, and little motivation to take it seriously.

‘Boring’ theory lacking application and examples (like some lecturers’ experiences of generic teachertraining), particularly the classic ‘definition’, ‘theorem’, ‘proof’ of polished, finished pure mathematics,deters many people. While it may be eventually necessary, to tidy up ideas before moving on, it shouldnot be a first mode of presentation particularly at early undergraduate level. Unexplained jargon andtechnical terms are also off-putting (NB - students are not lazy or unusual in this respect, some academicshave been known to balk at a few simple terms such as ‘learning objectives’).

Students are put off by material that is too tutor specific, following the teacher’s interests for its own sake.Hobby horses of this kind are always risky, since one is requiring the students to share your interests andthis often alienates intelligent people because they have plenty of interests of their own. However, thereare positive benefits in giving examples of applications of the material in your area, as a way of illustratingits importance and usefulness. These days we are encouraged to bring our research into our teaching, butoverdoing this can seem like self-indulgence.

We of course face a particular problem with mathematics for non-specialists, service courses, as discussedfrequently elsewhere. Here you are already starting from a disadvantage, some of the students maypositively dislike mathematics. Service teaching of mathematics is some of the most difficult teachingthere is. Indeed one might say that the major problem in service teaching is getting the students interestedin the subject. Unfortunately, service classes are often given to novice teachers such as postgraduatesbecause it is regarded as elementary material. The topics might be elementary, the teaching is certainlynot. Apart from the many motivational ploys and suggestions for ‘relevant’ examples given elsewherethere is also a very important question of attitude of the tutor in service teaching. We need to make itclear that we know they are, as engineers, not as committed to mathematics, and tend to view the subjectas a tool. We need to make it clear that we respect this and we are adapting our teaching style and contentto suit this. We are going into their world to see what they need and not just presenting our mathematicalview of the world, expecting them to filter out what they need. We replace rigorous proof by ‘sensible’

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argument (but without abandoning rigorous logic). The tutorial is of course the ideal place for all this.(For ideas see http://mathstore.ac.uk).

Most people need time to get excited about things. So if we keep rushing through the material withoutallowing time for reflection and personal interests of the students then that won’t enthuse them. Similarly,if the material is at the wrong level, too hard, or indeed too easy. And of course obsolete or out of datematerial is very demotivating, and it is part of good teaching to keep material up to date.

Students also tend to lose motivation when we have to repeat material they have seen before (Or thinkthey have seen before). This is a particular problem in first year courses where, because of the wide rangeof student background, it may be necessary to go over basic pre-university material, which can reallydemotivate the well prepared students who want to get on with new stuff. In such cases we can perhapspresent the material in a way that few of the students will have seen before. This works well when weshow them short cuts, such as the cover-up rule in partial fractions. Or we might dig deeper than theyhave been used to. For example in integration by parts many might not have appreciated that it is simplythe reverse of the product rule - and then we can ask them to think about which integration technique isthe reverse of the chain rule. Here we have to find a delicate balance between losing the less preparedstudents while still challenging the well-prepared students.

In reading this section you will see that some of the demotivating effects are in fact unavoidable, with thebest will in the world. This just means that you have to work harder at minimizing the effects of suchthings. This can often be done by getting the students on-side. Warn them if something a bit uninspiringis coming up, but emphasize that it is essential for later work. Do your best to soften the blow - you mighttell them you also found it a pain as a student, but later came to realise how important it was (hopefully!).

H Exercise – Making mathematics and statistics interesting (If possible do this with agroup of colleagues)

The object of this exercise is to look at ways of making the most uninspiring material moreinteresting.

• Choose a mathematical or statistical topic that you find particularly boring, or that you thinkstudents find boring.

• Prepare a short presentation (less than five minutes) that makes the topic sound interesting.You are not presenting the topic, but the motivation for the topic.

• List the methods you used to make the topic interesting

4.7 Explaining to Students

Already discussed in Section 3.9, explaining is one of the key arts of teaching [12]. In any walk of life,being able to explain things clearly and efficiently is a great gift - and particularly so for the teacher. Ithas intellectual components (for example, knowing the topic well enough to adapt it to your listener),and emotional components (such as not becoming impatient). All of these must be marshalled when astudent asks you for help. When a student asks for help they are ideally primed to learn, so you cancapitalise on this. Explaining in the lecture is usually more difficult than in the tutorial, because onecannot easily interrogate the students to involve them in the dialogue so essential in the explaining ofdifficult ideas. In the tutorial we can of course discuss things in more detail and with better exchange ofideas and views, so explaining is easier. All the skills discussed in Section 3.9 can be brought to bear withgreater effectiveness.

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It is useful to bear in mind that all explanation is relative to the students’ capabilities and backgroundknowledge. So explanation is less a matter of the topic under discussion than of the present knowledgeof the student. We can keep questioning the student to check that they are getting the right idea or discernwhether they have any misconceptions. It also means that, depending on the listener, there are differentlevels of explanation. In mathematics we are often so concerned about rigour and precision that we jumpin too quickly and try to offer a deeper level of explanation than the listener can cope with. In general it isprobably better to start at a fairly crude level, and gradually dig deeper. And we shouldn’t be arrogant ordismissive about levels of explanation. There will always be someone who wants a more rigorous levelthan you can provide. For example the demands of Bourbaki, not to mention Russell and Whitehead,would embarrass most university lecturers!

So, in a tutorial we explain by directly interacting with students. We need to treat the student sympa-thetically and be tolerant of their difficulties. We must never use any sort of negative, derogatory ordemeaning response to a student’s question. At that particular instant your response to a student’s queryis very important to them. If you get it wrong you can spoil the relationship with the students for theduration of the course (and other students you haven’t met yet, because your reputation will get round).Be polite and helpful, but you don’t necessarily have to give the student what they may be asking for - aquick answer.

Note that there is a distinction between explaining how to tackle a particular problem, which is what weare often doing in exercise classes for example, and explaining a particularly difficult idea, concept ortechnique. In the problem we are ‘focusing down’, trying to find a pathway to a solution. In explaining adifficult concept or method we have to take a wider view, looking at the idea from a number of differentperspectives, justifying things not just because they work, but in a ‘natural way’ that is as sensible aslogical.

Examples

1. As an example of explaining how to tackle a particular problem, we might look back atan example in Section 4.5, where a first year student working through calculus problemsasks ‘What is the derivative of 1/

√x ?’. We described how to use this to engage the

students. In explaining the solution to the student it is quite clear where we need togo, but the important thing is where we need to take the student. As noted previously,their real problem may lie with indices. There is not much point us telling them that1/√x can be written as x−1/2. They will probably readily agree to this and continue, and

just as readily forget the lesson again, so that if they come across 1/√x− 1 later on they

are in no better position to proceed. So your explanation of how to tackle this problemmight focus on the laws of indices, encouraging the student to revise these before theyproceed. By interrogating the student you lead them to realize that they have to rewritethe function in indices form, xα, and then it is of course a standard derivative. And whenthey have done it, get them to do similar problems to consolidate the idea - and get themto integrate 1/

√x and such functions. Don’t be over concerned that this takes up some

time, the skills you are promoting are powerful and of more value to the student thanjust being told the derivative of 1/

√x.

2. As an example of explaining a difficult technique consider completing the square, whichwe have looked at before. This is actually quite mystifying at the elementary level, par-ticularly as it is sometimes presented in a quick logical algebraic sequence with littleexplanation as to why we are doing it. Sometimes it is justified by using it to solve aquadratic equation. This sows the seeds of future misconceptions because students oftenthen associate completing the square with a quadratic equation, rather than a quadraticfunction.

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So, in an explanation of completing the square the first thing to make clear is that it isapplied to a quadratic function, and has nothing to do with a quadratic equation. At theelementary level students often confuse functions and equations, so a bit of preliminarywork might be needed here. The difference can be vividly illustrated for example bydefining a function or expression that describes one’s current bank balance as a functionof time, let’s say B(t). Obviously this will be a useful function allowing us to calculateour bank balance at any time. However, the equation B(t) = 0 is a very different pieceof information! So this can be used to impress on the students the difference between afunction and an equation. Then we are applying completing the square to a quadraticfunction.As a preliminary to the explanation, it is also as well to explain why we would want tocomplete the square for a quadratic such as f(x) = ax2 + bx+ c. What does it do for us?The point is that the form given here for the quadratic tells us little about its properties.Because there is both an x and an x2 it is not clear how the function varies as x varies. Ifwe did have the quadratic equation ax2+bx+c = 0 to solve, it is not directly obvious howto do this because again we have the x and x2 mixed up. The problem is thus the ax2 +bxpart. Now provided the students have (x+ y)2 = x2 + 2xy+ y2 at their fingertips (and ifthey don’t then it has to be explained to them, carefully) then they can quickly appreciatethat we can replace the ax2 + bx by something like a(x + b/2a)2 − a(b/2a)2. For thebeginner in algebra this is actually quite a difficult step and the details may need to beexplained at length. In particular the inclusion of a(b/2a)2 − a(b/2a)2 = 0 needs to behighlighted and explained. We can tell them that this is a classic ploy in mathematics,where we introduce something and take it out again just to simplify an expression. Wecan tell them of other examples of this, as in finding a common denominator when wemight replace 1 by say (x− a)/(x− a), and ask them when this is valid. When we finallydisplay the completed square form we can then explicitly demonstrate how it is nowclear where the max/min occurs and what it is. And we could explicitly solve for x toget the usual quadratic solution formula. Finally we can emphasize the key points of thismethod – the ‘atoms’, namely the (x+ a)2 = x2 + 2ax+ a2 result and the common ployA−A = 0.

The point of the second example is to illustrate the difference between ‘explaining’ completing the squareby working through the required sequence of algebraic steps, justifying each one ‘logically’ to producethe desired result, and really explaining what is actually going on in a way that conveys all the lessonsto be learned, and spells out why we are taking each step. Of course, the latter takes much longer, butthe likelihood is that the students will learn more from it and stand a better chance of understanding andretaining the ideas involved.

In mathematics we are often explaining a long proof or solution, the very length of which can be intimi-dating. Dealing with such things is a key aspect of explanation. Break it down into bite-sized chunks, butbe sure you also give the road map/overview enabling them to put it all together. Identify and emphasisethe key points - for example (a + b)2 and A − A in completing the square. Be economical with the truth- sometimes we have to ‘gloss over’ details or tricky bits in a first treatment, or at a certain level - but besure the students understand that this is what we are doing. Use memorable phrases that encapsulatethe topic. For example ‘Rings and sings’ in contour integration, referred to in Section 2.9, reminds thestudents that they are always having to think about the contour (ring) and the singularities. So, whenthey meet a contour integral ‘Rings and sings’ (hopefully) springs to mind!

In pure mathematics there are such things as theorems, lemmas, propositions, etc which are really ways ofbreaking up long pieces of work. But in fact this is not always helpful, and can impede fluidity. Sometimewe need to balance ‘sensible’ with logical explanation, and keep switching between the two. Keep asking

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the class what they think would be the next sensible step. Imagine how the person who first tackledthis topic might have approached it. Ask why each step is taken. Particularly in mathematics there is atendency to ‘pull rabbits out of a hat’. We say things like ‘Consider the function ...’, without saying whyand then proceed through extensive logical steps until the result we want suddenly pops out. This maybe elegant in the final polished presentation of the topic, but it is not the way most people think, and itis an unfair way to introduce difficult ideas to novices. Admit the ansatz for what it is - a device to giveus the result we want - the proof won’t work unless we take that particular type of function, or makea particular assumption. The person who first did it may well have started from what they wanted toprove and worked backwards to get the function they then ‘start’ with, or they might have used trial anderror to get a suitable function.

Sometimes students will ask how to do something that is on assessed coursework. It is easy, but shortsighted, to simply respond with ‘I can’t do that, it is coursework’. One of the purposes of coursework isto help students to learn, and if a student asks a question about it then they are in a good position to learn.You cannot of course show them how to do that particular question, because that is simply giving themmarks for nothing. But by interrogating them on the topic of the question you can narrow the problemdown to precisely what is getting in their way, their real lack of understanding, and this you can address.Get them to show you what they have done so far, work through all the parts they have done. Then,when you have identified the stumbling block (which you can normally predict in advance anyway, butinsist they find their own way to it), you have a number of options. Do a similar but different question,do it for everyone in the next class, re-teach that bit of material and send them off for another attempt,or simply tell them that you can’t answer that part - it would be unfair to the rest of the students. Ofcourse, sometimes you can’t really do anything at all because the the stumbling block is a clever trickthat the slightest hint will give away, so you have to use your common sense and judgement in suchcircumstances.

H Exercise

The object of this exercise is to look at how difficult mathematical or statistical ideas can be explainedin simple terms.

• Choose a particularly difficult piece of mathematics or statistics, at any level

• Prepare a short presentation (5-10 minutes) explaining this topic to a typical first year studentso that they would develop a reasonable interpretation of it in their own terms.

• List the methods you used to construct your presentation

4.8 General Points in Working with Students - Maintaining a Produc-tive Working Environment

As Principle 2 reminds us, teaching is an intensely human activity. Above all it is one human beinghelping another - having to help a lot of people all at the same time, as we do as lecturers, sometimesmasks this fact. If you have 100 students in your class and only interact by walking, talking and chalkingin the lecture room and marking their work, then it is difficult to view students as individuals, but that iswhat we must try to do.

In a lecture one is grateful if the students are quiet, well behaved, attentive, receptive, relaxed and sensibleenough to ask questions when they need to and engage with the limited activity that is on offer. In atutorial they have to be working, proactively producing, engaging enthusiastically with the activities,

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and we have to provide an atmosphere in which they can not only take part, but enjoy doing so. Thisis one of the occasions when interpersonal and emotional factors come into play in the teaching andlearning of mathematics.

4.8.1 Interpersonal and Emotional Factors in the Tutorial

Skemp [66] has a useful Chapter 7 on interpersonal and emotional factors in the learning of mathematics,which are important for the teacher to keep in mind in their interactions with students. One of the firstproblems that a tutor has is that some students may not actually like the topic you are teaching andindeed may have negative feelings towards it because of indifferent teaching in the past. Mathematics isparticularly dependent on good teaching, especially in the early stages at school, but also in the transitionfrom school to university where the change in learning is not just a matter of content, but also style, speedand levels of abstraction and rigour. This is not necessarily the students’ fault. You yourself will almostcertainly have some subject or topic that you don’t like simply because you had a poor teacher in the past.You cannot of course do anything about the students’ past teaching, but by being alert to such problemsyou can perhaps change or accommodate their attitudes to more positive ones towards mathematics.Skemp notes that the teacher has two tasks before even meeting the students - a conceptual analysisof the material followed by planning of the way in which the necessary concepts and schemas can bedeveloped, with particular attention to the stages at which accommodation of the learners’ schemas willbe necessary. Then when in contact with the students, the teacher has to set the general direction andguidance of the work, for explanation and correction of errors. Also, the teacher needs to create andmaintain interest and motivation.

So let us consider the face-to-face relation of the lecturer with the students and the effect that has onlearning based on the understanding of mathematics. While mathematics has a lot in common withother subjects, it is different in one crucial aspect. In the arts, for example, the authority in the subjectcomes from that of the teacher - it is largely a matter of their opinion. The only appeal in the event ofdisagreement is to a second opinion. In the natural sciences the authority is based on experiment and inthat case the students do not have to accept ‘because I say so’. But in mathematics it is even stronger - theauthority is based on internal consistency, and there is a great deal of agreement between mathematiciansand between teacher and learner on whether this internal consistency is present or not. If a teacher makesa mistake on the board and the student points it out, the teacher has to accept and correct it with goodgrace. So in teaching and learning mathematics the interaction is between intelligences, with mutualrespect for each others’. Skemp argues that this breaks down if the student is simply presented withmeaningless rules to accept uncritically - he calls this an insult to intelligence and compares it to actualphysical injury.

ExampleSkemp [66] gives the example of ‘shifting things from one side to the other’ in solving equa-tions. He compares this with teaching someone how to use the brakes in a car - telling themto depress the clutch at the same time as the brake, but with no reason. Most people will wantto know why they have to do this, rather than follow blindly the instructions. This soundsvery sensible, and normally one does provide a plausible reason for most things one does inclass. There is a caveat to this however. Returning to Skemp’s braking analogy, there are infact plenty of people who are quite happy to accept the instruction to depress the clutch atface value (me included) and can still make competent drivers. I know and accept that thereis some good reason, related to the engine, but I have more important things to worry aboutthan that, so will just get on with it - I just want to learn as quickly as possible to drive. Any-time I need to know more, I will consult an expert. This is exactly the same as, for example,teaching mathematics to engineers. In Skemp’s example of solving equations he admits that it

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takes a great deal of development to provide the basis of the change in sign when one moves aterm from one side to the other. The engineer does not have time for that. The real skill of theteacher comes in providing sufficiently plausible reasons to put the learner at relative ease.Thus in this case one might say ‘Adding two apples to this side of the balance is the sameas taking off two apples from the other’. Of course, this is not the whole story, but it makesit easier for the busy learner to accept the method into a useful and largely correct schema.What Skemp does in his justification is go through the study of algebra as a field, in whichone can talk about inverse operations such as subtraction being the inverse of addition. Butthis is only his level of explanation - Russell and Whitehead would regard it as rather superfi-cial. However, Skemp is right in that totally unjustified rules are bad for the learner and makelearning for understanding more difficult. The skill is to balance the level of plausibility withthe learner’s needs and background.

Another way in which interpersonal issues come into teaching is in the means by which the teachercommands authority. In general a teacher’s authority can be based on status and function, or on superiorknowledge. But there is confusion and indeed tension between these two forms of authority:

• exerting discipline and maintaining order

• attracting disciples because of superior knowledge.

In other circumstances these two roles are separated, as in the chairman and the members of a committee.Members cannot question the authority of the chair, but may argue amongst themselves. In the sameway, students should accept the controlling and discipline based role of the teacher, but should feel freeto question their intellectual and subject role if necessary. Some students and lecturers confuse the twoaspects. This is a particular problem in teaching mathematics, for which the intellectual demands are sogreat, and the need to maintain disciplined intellectual activity correspondingly so. And it is even moreso in service teaching where the students may not be there out of choice, but simply because they have todo the course to pass their examinations.

As Baumslag ([7], p.161) points out students are sometimes frightened of talking to the lecturer, or afraidof making fools of themselves, so we need to reassure them. We need to treat what they say with respect.We need personal qualities of respect, patience, ability to listen, ability to pitch conversation right forstudents, to encourage, be generous, admit our mistakes, and be in good spirits. It also helps to have areasonable personal interest in the students - ask how they getting on, etc. On the other hand some rude,challenging, and aggressive lecturers can still get the best out of students. If you do find yourself losingpatience with a student then politely draw the meeting to a close and rearrange it for a time when youcan compose yourself.

H Exercise

Think about your own personality and any effect this may have on the tutorial environment - areyou impatient, prone to ramble, etc?

4.8.2 Knowing how the Students Know the Subject

Just as in giving a lecture, not only must you know the material inside out yourself, but you must knowhow the students know it, or will come to know it. Also, you need to know as much about their back-ground interests and motivation as possible, so you can anticipate their difficulties, and quickly get ontotheir wavelength, to help you get your message across. This is particularly important in the case of service

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classes. Consult with colleagues about the students to find out what have they done in relevant classes.Read widely about the topic to get a range of perspectives - some students like a strictly logical accountof things while others like visual approaches. Rewrite notes on the topic in your own words - reassembleand restructure the ideas and content, make sure you can prove things ab initio yourself.

Design tutorial activity to meet the objectives including such things as organising the groups, assigningtasks and roles. For example, for a complicated modelling exercise you may split them into say groups of4-5. Don’t forget that all such tactics have implications for accommodation and possibly equipment. Andof course you will have to give them precise instructions. See Mason ([53], Page 105) for a wide range oftasks aimed at various types of mathematical objective.

Sometimes in a tutorial a student may ask a question that stumps you. You feel you ought to be able toanswer - in a lecture you can play for time and ask them to see you after class, but in a tutorial there is nohiding place! So, what do we do? Don’t blag it! Be openly yourself. If it is a question your are supposedto be able to answer - for example one of the tutorial problems - them apologize for falling down on thejob and sort it out as speedily as possible. If it is an ad hoc question that is peripheral to the main topicof the class, or not related to your course, then you might try it for a while. If you can’t do it relativelyquickly (you shouldn’t allow it to distract you from the main purpose of the class) admit it. If you thinkit appropriate say you will try it and get back later. Doing things you don’t really have to do is of coursegood PR.

There is no great shame in occasionally not being able to answer a question, especially if caught cold. Thestudents will benefit greatly from watching you struggle with it, even if you don’t succeed. They will seehow a mathematician works - see that it is OK to cross out, make mistakes and guesses. They will feelbetter about themselves and realise that ‘it isn’t just me’. It might even spur them on to try harder. If youthink the question is interesting and that it is appropriate, open it up to others in the class, ask if they cando it. Be careful you are not doing another lecturer’s coursework! The key point is not to be embarrassedif you can’t do a problem. We all have off days. It is the way you handle them that is important.

H Exercise

Choose a topic you expect the students to know and that you will be relying on in future work. Howdo the students ‘know it’? Do they see it in the same way as you, do they have the same facility, arethey used to your terminology and notation, do they have the necessary overview? Talk to colleaguesabout this.

4.8.3 Classroom Management in the Tutorial

We discussed classroom management in the lecture in Section 3.9. Most of what we said there appliesequally to tutorials so we will just summarize the main points and add anything that is particular totutorials. In general running a tutorial class is not so problematical in this respect - for a start there areusually fewer students to manage. However, it does have some special features that we will discusshere. Largely these amount to keeping the whole class on task, and curbing any disrupting or distractingbehaviour.

Be clear about your duties, responsibilities and status and make sure that the students are too. You maynot be very much older than some of the students, but don’t be afraid to exert authority if necessary.Some people believe that a ‘dress code’ is useful for establishing a distance between the students andthe lecturer. Certainly some young lecturers have reported that when they wore a suit the students paidmore attention to them! This may help to separate the two senses in which the lecturer needs authoritythat we talked about in Subsection 4.8.1. But in truth this is a minor issue. If some students are ‘messing

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about’ and not working on the problems, get in amongst them and get them down to work. That helpsthem, whether they appreciate it or not, and the rest of the class. Set ground rules early on and stick tothem. It is much better to be tough to begin with and ease up if necessary as the module progresses. It ismuch more difficult to tighten up after a lax start. You might negotiate some of the ground rules with thestudents. Maybe the start time is not convenient for all of them and a change is helpful to everyone. Butthen stick to the new start time agreed.

Keep order in the proceedings - when students (or anyone) are allowed the freedom to talk amongstthemselves they can easily stray off task and things can become unruly unless some order is imposed.Usually humour and an appeal to the students’ good sense and courtesy will settle things down. But ifit doesn’t, remain in control and politely but firmly insist that they do as they are told. Keep them ontask - that is, solving problems, or talking about solving problems. Don’t be too much of a slave driver,of course, a few minutes light banter can refresh everyone, but in the end, you and the students do havea job to do.

Learn as many names as possible, and try to build up knowledge of their personalities. Use any charactersin the class (politely and in a friendly way of course) to help develop rapport with students. Try to bring intheir interests - for example some might complain if working through a long list of tedious drill exercises- ask if there are any guitarists/pianists, etc in the group and ask them how they learn their chords -mathematics drill is the same thing. Be relaxed and friendly with them, while maintaining a respectableand professional stance.

Never be rude, sarcastic or derogatory, no matter what the provocation. Because if you are this willalienate most of the class, and in any case it is bad manners. It is not setting a good professional example.As a young lecturer, in a moment of frustration, I sarcastically announced ‘You lot are supposed to be thebrightest 10% of the population - I dread to think what becomes of the bottom 10%!’. Quick as a flash,from the back of the class, in a beautiful Irish lilt came the retort ‘They become university lecturers, surethey do!’. Fortunately that lightened things up considerably. I was never sarcastic with students again.

Examples

1. Wankat and Oreovicz ([72], p. 120) have some good advice on working with groups ofstudents, particularly on disagreement and conflict between students in a tutorial situ-ation. They advise that we set the climate from the start that conflict is to be resolvedtogether in the group. Argument should not be personalised. In conflict ensure thateveryone has the same accurate information and then help students to recognise simi-larities and differences. Use principles of debate and get them to switch sides and arguethe opposite case, to be Devil’s advocate. If conflict becomes heated, defer untill outsideclass. Nurture non-participants and arrange for better conditions under which they cancontribute - privacy, longer time, written submission, etc. It is found that women speakless in bigger groups, especially if their ideas are attacked. Over-participants or monop-olizers must be controlled. If bringing in other contributions doesn’t work, have a wordwith them. In time, other members of the group will tell them to shut up. Ask that eachstudent speak at least twice in the session.

2. Krantz ([49], p. 78) gives some useful advice on the nuts and bolts of running problemclasses, which are worth summarising. He points out that it is easy to fall into the trapof not taking such classes too seriously, but as we have repeatedly emphasised, they arein fact very important. But always remember to strike a balance between being wellprepared by doing all the problems beforehand, and spending too much time on prepa-ration. Use devices to liven things up. Think quickly, on your feet. Think how best topresent something or handle confusing points. Stick to problems of that week, i.e. limityour responsibilities. Ensure that you do the solutions the way the students need to do

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them. Don’t give away the store. Find ways to generate repartee. Set an example ofsharing, giving, caring so students are willing to muck in. Part of your job is to show thestudents how to do the problems, but also to show them that the problems are doable.

Try to be aware of any diversity issues that might affect the tutorial - disabilities, multicultural and lan-guage issues. Actually, in mathematics we are relatively lucky since it is something of a universal lan-guage. However, it is well known that ‘word problems’ in mathematics can present special difficulties forforeign students for example, and this can be a problem in the more discursive environment of a tutorialclass. But be careful about stepping outside of your responsibilities and expertise on such issues, someof which can be difficult even for the experienced lecturer. Consult an experienced member of academicstaff if in doubt.

On the rare occasion when we have a disruptive student, how do we deal with that? If someone isdisturbing the class it is not too hard to send them out of the class if they won’t toe the line. But keep itas a very last resort. Give a warning - ‘Three strikes and you are out’! Have a quiet word with them first.Remember that they are potentially disrupting the learning of other students, and these will invariablybe on your side in taking a strong line. If it gets that far, inform the appropriate authority. Don’t expressanger, malice or any other strong emotion. Stay in control. As noted elsewhere if they refuse to leave, youleave and report to the appropriate authority. You are not paid to handle personal confrontations!

Sometimes we do get students who simply will not participate. As noted in Section 4.5 there will bestudents who simply don’t want or need our help, but we are thinking here of behaviour that is not simplyindependence on the part of the student. Then you might try to find out why they won’t participate- it may be personal problems or illness. But don’t try to be a counsellor, refer it to the appropriatepeople. However, if a student does actually confide in you that shows a degree of trust on which you cancapitalize, so you might for example accompany them to the counsellor, if they want that.

H Exercise

Think about how the issues of this section relate to your own tutorials and discuss with colleagues.

4.9 Problem/Exercise Classes

4.9.1 Learning Mathematics through Problem Solving

By problem or exercise classes we mean the fairly standard form of mathematics tutorial in which stu-dents work through problems, seeking help when they need it. Problem solving is one of the mainactivities by which we learn mathematics. Ensuring that students get the best out of their problem classesis therefore important. It is also difficult, because we have the job of actually encouraging the students toengage fruitfully with the activity, and of assisting them in the most effective way when they get stuckand need help.

Note that one can think of problem sessions from (at least) two extremes. The most common use ofproblem sessions is that in which students work through problems whose main function is to supportthe learning of particular topics, maybe in some depth. In this case, while there may be some quite toughproblems, the bulk will be relatively routine and accessible to the average student in the class, more inthe nature of ‘practice’. Such sessions might be more commonly called ‘Exercise classes’. At the otherextreme there are problem sessions devoted to developing the skills of problem solving itself, sometimesto a very high level, almost involving mini-research projects. In this case of course the problems willbe far more difficult, some may not even have solutions. And the problems will take much longer, be

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more stressful for the students, require more sophisticated support from the tutor, and more attention topersonality and emotional issues. In fact this type of session is more technical and the new lecturer isless likely to be involved in a leading capacity. So in this book we tend to focus more on the first typeof problem session, which is more usual, although the skills developed will be also useful in the second‘problem solving methodology’ sessions.

On the face of it the sorts of activities involved in problem classes would seem to be pretty obvious - thestudents work though problems with your help. However, the tutor can be more proactive than this andthere are many things you can do to support and enhance the students’ learning during such tutorials.After all the student can plod though problems at home, making a note of any questions to raise withyou in the lecture or tutorial. In a tutorial they should be doing more fruitful things and working moreintensively.

Mason ([53], p.87, etc) gives a number of what he calls tactics that will help students in solving problems.You might show students how to approach a problem by simplifying the context to one that can betackled more easily, but still has the essential features. Thus, you might look at a special case, insertnumbers to replace symbols, choose more manageable functions, lower the dimension, etc. And of coursewe can reverse this and make an example more complex, so that it might lead to the solution of relatedproblems. In such tactics we might encourage the students to go from the particular to the general byextending results, or looking for examples with peculiar features. We have already mentioned the doingand undoing aspect of mathematics - factorising is the undoing of multiplying factors, partial fractions,integration, etc. Set problems and activities for students that look at this doing and undoing aspect ofmathematics. Mason gives some examples ([53], p. 89). The main objective is to emphasize to studentsthe principle that in order to undo a process you must be highly skilled at doing it. Another suggestionof Mason’s is to move from asking direct questions of the students, like ‘Give me an example of ...’ toprompting the students to ask the question themselves, as for example in asking ‘What question amI going to ask now?’, or ‘What question do I usually ask when you are stuck?’. Hopefully this willeventually lead them to ask such questions themselves, spontaneously.

Mason ([53], p. 90) also points out that when introducing new symbols or terms we load the studentwith a great deal of information to absorb. Not only do they have to understand for example what thesymbol stands for, but all the baggage that comes with that, all the properties and qualities of the thingrepresented. In fact, the mere act of assigning such a representative symbol can inhibit access to the ‘realthing’, and perpetuate any misconceptions or errors in a student’s concept image of the real thing. Masonsuggests that when the symbol, for example, is first introduced you should, for a while repeat the fulldefinition each time you use it, gradually withdrawing and paraphrasing the definition until eventuallyit can be abandoned altogether and the symbol alone can be used. This may be a little unwieldy in alecture situation, but can certainly be used in a tutorial situation. When you introduce new notation youcan also get students used to it by asking what they see, and then getting them to assemble the completedefinition or notation from the individual details. This capitalises on the fact that most people like first tobreak new things down into components, understand those and then reassemble the complete structurewhen the components are fully understood.

Mason has a great deal to say about the sort of examples one might use with students, asking what makesan example exemplary ([53], pp. 29, 80). Particularly powerful examples are those that reflect the waymathematics is actually done (Principle 6), that is those that require the student to look beyond the super-ficial details and irrelevancies of the situation, and simplify the problem until it is manageable. Masoncalls such examples generic ([53], p. 81). They provide practice in important skills such as specialising,generalising and finding counter-examples. Mason also emphasises the importance of getting students toconstruct worked examples for themselves. For example, impress upon them that it is better to be able toreconstruct a technique rather than relying on memorising meaningless steps in applying the techniqueto pre-set examples ([53], p. 79) (Principle 7). Related to this, Mason warns us to beware of students as-

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senting to a definition (or other result) that we give them, rather than having the confidence to assent forreal understanding. It is a familiar occurrence for students to parrot a definition remembered verbatimfrom the notes or book without really internalising what it means, or being able to use it in a new context.Of course, this is to be expected, and the remedy is to devote more time and practice to the definition andits use, if it is sufficiently important.

Another activity is to involve students in the drawing of diagrams. Describe mentally what you havein mind and get them to try and picture it without drawing the diagram. This increases their ability togenerate and handle mental imagery. A similar type of activity is the scrambled proof, where studentshave to sort out the correct order of a number of steps in a particular proof. Or, they have to assemblea list of logical steps into an order that makes sense, makes a valid argument. Sometimes students (andyou) use ‘it’ in a mathematical sense, or some other pronoun. Often they do it without really knowingwhat ‘it’ is. Get them to explain what the ‘it’ is, get them to replace the pronoun by what it actually is.

When students work through a method or technique they have to think about each step, whereas thelecturer does not and so can attend to other things in parallel, such as checking answers. So we need todevelop students’ skills of diverting attention from a calculation. For this they have to internalise it ([53],p. 79), and when they have done this properly they will naturally think of other things simultaneously -just like we can solve mathematical problems as we drive along quiet roads, because the latter requiressuch little thinking. The learner keeps repeating a process until they have mastered it ([53], p. 99).

Sometimes a student will tackle a problem in a way we think correct but inefficient or inelegant. Whatdo we do? For example in evaluating the probability of an event it is in fact sometimes much easierto calculate the probability of the complement of the event and subtract from one. But what if one ofthe students uses the longer ‘direct’ method, calculating more probabilities than they have to? This isactually an interesting situation, typical of the sorts of sensitivities one has to balance in teaching. Whatwe do depends on the time available and cost/benefits analysis of the situation. It may be worth givingthe student their head if what they say/do will illustrate some point from which they and others mightlearn. But then the more elegant approach should be pointed out gently and without embarrassing them.Preferably the quicker method should emerge by discussion amongst the group, and not foisted on themby the teacher.

We hear many complaints about the way students write and present mathematics, and when you watchthem working through problems in the tutorial you will see lots of rough work that you wouldn’t everwant to read, much less mark yourself. But of course, we know this is how most of us work (See [39]) inpractice. But you also know that if you need to write a paper or book then you have to be much morecareful, and the written presentation of the mathematics is more comprehensive, tidier and readable. Thestudents are not so experienced however, and they write their mathematics the way the lecturer does onthe board. In a tutorial we have the opportunity to show the students how to write mathematics properly,so occasionally tell them that you want certain solutions written out carefully - give them guidelines ifnecessary at first. Also, sometimes go through problems on the board in full, setting it out in the way youexpect a final presentation to appear. Another reason students present mathematical work poorly is thatthey think they are writing either for themselves, so it doesn’t need to be thorough, or they are writingfor the teacher, who will ‘know what they mean’. Particularly for coursework, emphasize to them thatthey are actually writing for, say, someone who will be taking the module next year, who may not be ableto fill in the gaps, or easily deduce what they are doing. A further reason why students sometimes writemathematics badly is because they don’t really understand what they are trying to say. Explain to themthat the actual process of writing out the mathematics clearly, precisely and correctly actually helps themto learn.

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H Exercise

There is a vast literature on problem solving and problem solving methodology, in addition to thatcited here. In preparing for your own problem classes you might survey this more widely to gatherideas that can feed into your classes.

4.9.2 Writing Problem Sheets and other Tutorial Materials

The preparation of materials for teaching situations in general was discussed in Sections 2.8,2.9. Ideasand suggestions there can be applied just as well to preparing such things as problem or exercise sheets,so here we will simply summarise any additional points that might arise. Problem sheets are just asimportant as the actual lecture material. If written and constructed properly they are at the front line ofhelping students to learn. Having decided on the key objectives of the session, then the material mustbe designed to achieve those either during the tutorial or in subsequent independent work. We havealready said that the tutorial objectives will be high level, and so you will be trying to help your studentsto overcome some difficult coneptual hurdles, or possibly apply some ideas in new and subtle contexts.For this they are going to need high facility with the necessary elementary skills and ideas. If you have notalready ensured this previously then the materials must provide quick consolidation problems, firmingup the essential pre-requisites. This might take the form of lots of very simple problems that the studentsare expected to flash through in the first quarter of the session say. Then you need to build up slowlyto the higher order ideas, with problems and exercises steadily increasing in difficulty. This is the ‘step-laddering’ process (Section 3.10) where you are providing the individual rungs needed and helping thestudents up these.

Gradually you can progress to harder problems that miss out some of the steps, or switch them around.Only after this sort of ‘easing in’ material can you start to set tougher questions that require high facility.These might include not only problems that are difficult to solve, but questions that encourage students tothink about how they learn (Principle 8) - for example you might ask for them to write brief descriptionsof how all the steps fit together. You might ask them to generalise the results they have obtained, toanalyse the conditions under which the methods break down, etc. All this requires some thought if youwant to do more than just give a few problems without structuring them in any way. Remember thatthe object of the tutorial is not so much to have the students struggling with very difficult problems,but to help them to learn the subject matter, and to develop their learning skills. Also of course, in thetutorial you have to ensure that most of the students do actually get through all the steps and stages inthe problem sheet.

Problems set for the students need to be at ‘just the right level’ - not too hard, not too easy. This is amatter of judgement, and your knowledge of the class as a whole. And of course students will vary inwhat they regard as the right level - a boring trivial problem for one student can be a fascinating challengefor another. So you have to have a range of problems to meet their needs([50], p. 330). But usually thetwo extremes, lots of very straightforward problems, or one or two desperately hard problems, do notwork well. Also, try to make your problem and work sheets interesting. Explain the point of the differentquestions. You can call attention to connections with the rest of the course or with other subjects. You canask students to generalise the routine questions to more interesting contexts.

H Exercise

Survey as many examples of problem sheets as you can, particularly in topics related to yours andfor students at the same level. These can often be found on departmental websites. Harvest any goodideas!

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4.9.3 Running a Problem Class

Much that we said in Section 3.9 can be applied in problem classes, and there is no harm in repeatingsome of it here. Firstly, remember that you are the one in charge, and so it is you who have to run theclass. You have to make it clear that you expect them to work hard, that you are anxious that they learnthe material, and will go out of your way to help them in the process. Commence in a business-like way,making sure that the students fully understand the purpose of the particular activity of the session. Makesure they all fully understand what they have to do.

Ensure that all the students have the resources to do what is required. Many will not even have broughttheir class notes - tell them to do so next time, and if possible tell them what to bring before the session.Organise the students to suit the particular activity - possibly in groups, or in alternate rows in a tieredlecture theatre, so that you can reach all the students. Indicate some sort of schedule - for example manystudents start with the first question and work progressively throughout the whole session, not gettingvery far in terms of overview of the topic in the time available. So you may say something like ‘Spend 15minutes on the first three questions, then half an hour of questions 6- 9, and finally make sure you spenda quarter of an hour on the last three questions’. This way, they are wrestling with a range of issues, canget your help on each, and can consolidate the details in their own time. Emphasize that the idea is forthem to identify their sticking points quickly, while you are around to help them.

Set clear ground rules about orderly conduct of the class from the start (see below). Your priority is toestablish a conducive learning atmosphere, and most students will thank you for this, even if it meansyou being a little authoritarian. Keep things moving once you have started them off. And you will findthat you do have to maintain momentum and push them to work to best effect. That is just human nature(Principle 2!). You can keep reminding them that right now is the best time to sort out their difficulties,while the topic is still fresh in their minds, and while you are around to help them. It is much harderwhen they come to revise a few weeks before the examination - an hour spent now will save hours later.

Generate a sense of being everywhere in the classroom, move about from student to student, seeing howthey are doing. Be a pest, in the nicest way, continually asking how they are getting on, anyone stuck,how far have you got - anyone had a look at Question 4 yet? If similar questions crop up repeatedly tellstudents to help each other, go through with the whole class, or use something like Baumslag’s doublefile with outline and full solutions ([7], p. 161). If a student makes a small slip, and you are short of time,just check their overall solution and tell them to check for errors, come back to it later, etc. In problemclasses one way is to go through the solution on the board. Only issue model solutions when the studentshave made good attempt at it. Get the whole class to discuss how to find solutions.

Help the students with entry methods to problems, not specific hints. Starting a problem is very difficultfor some students, they sometimes have no idea how to start, and they just sit staring at blank paper -don’t show them how to start, because all that happens is the next step in the problem becomes a newstarting point where they get stuck again. In general the problem will contain a number of steps and totackle the problem one needs to understand each of the individual steps and also have an overview ofhow they all fit together. Ply the students with questions that tease out these different aspects, leadingthem hopefully to fill in the gaps themselves. Get them to write out the question in their own words.Or get them to write down everything they know about that topic - using their notes if necessary. Thinkabout yourself - how do you enter into a difficult problem, such as in your own research? Often thereason they can’t start is because they try to begin with steps that are too big - remember the proverb ‘Ajourney of a thousand ...’. Tell them to try examples, stick numbers in. Look for similar problems in theirnotes or books. You have to show then how to ‘enter’ a problem, any problem, and how to do this forthemselves. Get them to discuss it with fellow students. Get them guessing and assure them that this isa perfectly respectable tactic in mathematics, so if they are unsure how to start just take a guess at it andsee where that leads.

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Make sure that everyone is engaged - don’t concentrate on one particularly vocal group, gently bringin any shy isolated people, encourage them to help each other. Perhaps you might go through the firstquestion of a particular type at one or two points in the session. If things go quiet, and you find you arenot getting many questions, ask the class - ‘Anyone stuck? Has anyone done Question 3 yet? Anyonefinished - do you want some more problems?’.

Don’t let students spend too long on a question in the tutorial. It is better to give them an idea how toproceed and then move them onto other questions because they are then experiencing a range of problemsin the class, where they have the opportunity to get your help. Encourage students to use their notes orany other materials they have. Some students, particularly first years, find this difficult because in schoolthe problems they do are usually very closely related to what they have just done in class. In moreadvanced mathematics they may have to go back quite a long way, or interpret some of their material ina different way to how it is presented in the notes.

In the hothouse of a busy tutorial you might want to test a particular student’s knowledge about some-thing without making them feel uncomfortable in front of the group. Don’t put them on the spot. Have aquiet word without drawing attention to them. You could also give all students a short quiz and see whatthe response of the particular student is. You may have noticed that we have mentioned this tactic ofgetting the students to write something down a number of times. This is because people are more likelyto respond and contribute something if they can scribble it down rather than voice it in a crowd. Skemp([66], p. 124) gives a nice example of a teacher who elicited input from a student critiqued it, exposed itto examination from the class and yet did not in any way embarrass the student. Skemp makes the pointthat: ‘Those who really understand mathematics are not common; those who can communicate it, less so;those who are also excellent group leaders, fewer still; while those who can also communicate this lastability are rare indeed’. In problem classes, some students spontaneously group together and ask ques-tions amongst themselves and may not consult the tutor individually. They may even resent interferencefrom the tutor. If they learn better that way, encourage it! By talking to groups of people with similarproblems you at least save repeating yourself. Of course, if they still can’t resolve their issues, or if it istaking up too much time, then intervene. The question of who should initiate your involvement, you orthem, is a delicate balance. If they are arguing then at least one is probably wrong, and they may resentthe teacher being the one to point this out! But in arguing they are learning both the topic and debatingskills.

H Exercise

Prepare your next problem class in the light of issues discussed above and with discussion with yourcolleagues.

4.9.4 Dealing with Large Differences in Ability within a Small Group

It is often said that if you get any two mathematicians together, one will be extremely good and theother rubbish! Certainly, in a tutorial you will get a wide range of abilities and understanding. In suchcircumstances we must resist the temptation to focus on either the very weak or the very good student -both pay the same, both are entitled to equal ‘added value’! We need a range of problems/activities/workto challenge and to help at all levels. Because all students should have the course pre-requisites, there is alimit to the variation one would expect. If all students will eventually have the opportunity to consolidatethe work in their own time (for example if full worked solutions will be issued), then one can perhapsafford to let the better students sort their problems out themselves later. Or, we can set some very hardproblems to keep these students occupied.

When you are addressing the group as a whole, aim for a middling level, but emphasize any points that

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are particularly important and the weaker students might miss. On the other hand, you can occasionallypresent the most elementary material from an advanced standpoint that even the better students mightnot have seen, and that will not leave the weaker ones too perplexed. Always be specific about what thestudents will have to know when it comes to assessment (Principle 3). For example, they will be keen toknow whether the proof of a particular result is necessary. Here you can reassure the weaker students,but you can also challenge the stronger students to fully understand the proof, regardless of whether itwill be examined or not.

Often, people are at different levels because they have different levels of motivation and interest, so try tofind presentations and explanations that will interest or intrigue a wide range of students. For ideas seeSection 4.6. A point perhaps not sufficiently emphasized there is that there is one very strong motivatingstimulus for many students in HE and that comes out strongly in tutorials - their self esteem, whichis generally high. If challenged with a problem such students often feel compelled to solve it for noother reason than they should be able to - you can capitalize on this by demonstrating (reasonably) highexpectations of them and showing that you expect them to be able to make progress on even difficultproblems.

You may occasionally find that one student in the class is much less or much more able than everyone else.First find out why they are so exceptional. Use this as an excuse to devote more time and effort to them,so that this does not seem like favouritism. See them out of class if this is appropriate. More able is not somuch a problem because you can always give them some really challenging problems (maybe somethingfrom your own research you have been struggling with!). Invite them to help less able students. Muchless able is more of a problem. Assuming that they are in the right class/group and therefore have theappropriate pre-requisites, then the simple fact is they will have to work harder to ‘catch up’. Get themto think about how they can do this - maybe they have other subjects that they are strong in, so they cancoast through those and borrow some of that time? How much work are they likely to need? Be veryefficient and targeted about how you assist such students - don’t bother them with things you think theyare less likely to need and focus on those that are absolutely essential. Be strategic about advice, andstructure it so that the help it gives impinges on a number of areas.

Examples

1. If you are teaching rules of differentiation to first year engineers, concentrate on just theproduct rule and function of a function rule with the weaker students. They are (very)less likely to need the quotient rule anyway, and if they have a thorough understandingof the product rule and function of a function then they can muddle through a quotient.

2. When proving a theorem in, say, analysis encourage the weaker students to concentratemore on the actual effect of the theorem rather than on the conditions under which thetheorem holds. For example, if proving Rolle’s theorem you might bother the weakerstudents less with the conditions of continuity and differentiability imposed, and con-centrate on convincing them that the derivative must vanish somewhere, whereas youmight expect the stronger students to examine carefully how these conditions fit intothe proofs. This might sound like heresy to the pure mathematician, but teaching is acontinual balancing of such compromises.

H Exercise

Think about what you know about the students in your class, their abilities and interests. Deviseways of giving added value to the full range of ability.

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4.10 Demonstration Classes

In demonstration classes the teacher may be literally demonstrating how to tackle mathematical prob-lems. This requires all the skills of lecturing, and so most of what was covered in Chapter 3 is relevanthere. But you are not really there to merely go through the problems for the students on the board, whilethey imitate you. In fact, you are simply the scribe when going through problems on the board. Encour-age the students themselves to develop the solution by asking pertinent leading questions and givingcareful hints that still require them to think. You simply write down what they come up with, see wherethat leads and backtrack if necessary. You are demonstrating mathematical thinking, not mathematicalwriting. And when you are writing out solutions, ‘be mathematical’ in the sense that you display yourthinking as you work through. Think aloud, spell out options, maybe try a few dead ends, some guess-work, etc. The idea is to develop the solution as we might do it in rough, in practice, ourselves and notjust presented in the final polished form. When you have done it in this rough way, get them to writeup the final solution tidily and carefully. When completed, leave the solution up on display and as theywork through a similar problem keep asking them to refer to the solution when they get stuck again.

If you want students to come up and work through problems on the board (Only if they are happy withthis and volunteer - remember that public speaking is one of peoples’ greatest fears, and they are thereto learn, not to be embarrassed), then the best situation to aim for is one where the environment is sorelaxed and the rapport so good that students will spontaneously go to the board to make a point. If thegroup is small enough you may indeed hold the session in front of the board, so you are all chipping in -more a remonstration class than demonstration class! But remember that there is value in just watchingan expert at work, and occasionally a demonstration class might be just that. It is basically a highlyinteractive lecture, using all the skills mentioned in Chapter 3 for the lecture as well as the tutorial skillsof this chapter.

ExampleIn demonstrating how one might derive the series for the exponential function from its def-inition as the function f(x) that is its own derivative and satisfies f(0) = 1 we might startwith an assumed series with unknown coefficients. We then insist that the students now tellus how to find these coefficients. If this is their first sight of such a situation it is quite difficultfor them. Normally, when faced with finding coefficients in this way they will have had onlya finite number to find, as in breaking into partial fractions for example, but now they havean infinite number of unknowns and so are not looking for a closed system of equations. Theclass will have to be pressed quite hard to come up with suggestions, but we have to resistthe temptation to do it for them. If we keep harping back to the definition someone will even-tually suggest substituting into the equation f ′(x) = f(x), and so we dutifully do this on theboard, but go no further, just ask ‘What now?’. Thinking back to similar problems as in partialfractions one student may suggest equating coefficients, in which case, go with that and de-rive the sequence a1 = a0, 2a2 = a1, 3a3 = a2, 4a4 = a3 and so on. Again, ‘What now?’. Andso we continue, only a few hints will be needed to entice a general recurrence relation out ofthe class and hence determine all coefficients in terms of a0, which they know is one from theinitial condition.

On the other hand one of the students may have initially suggested repeated differentiationof the series and the equation f ′(x) = f(x) to determine the coefficients one at a time, as wedo when finding Taylor series. If so, no matter, we simply follow that line to its conclusion,even if we wanted to go via the previous route to illustrate recurrence relations. At the endwe can lead them through that form of the solution as an alternative. The point is that weare demonstrating the solution using prompts from the students, wherever they lead. Theskills they learn in doing this are far deeper and more useful than simply watching us work

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through the solution by a method of our choosing, and it is these skills that we should bedemonstrating.

4.11 Group Discussion/Project Classes

4.11.1 It Helps to Talk!

Group discussion is not so common in mathematics. Similarly project classes are more common in en-gineering. But there is certainly room for such things in mathematics. Skemp ([66], p. 121) looks at thebenefits of discussions in mathematics - mainly between students rather than with the teacher. The advan-tages are based on the enlightenment that we all feel when we start to tell someone else about a problem -sometimes, halfway through, we actually solve it. Discussion provides opportunities for ‘thinking aloud’.There is also the opportunity of interrelating your ideas with those of others. Discussion also stimulatesnew ideas, and reveals connections between ideas.

Below are some examples of the sorts of things that might lend themselves to discussion in mathematics.

• Modelling a ‘real-life’ situation in say engineering, biology, etc where there is are a number of pos-sible methods and one has to achieve a compromise between practical solution and representingreality. Here of course the students have first to know the various methods one might use and thenecessary physics, biology, whatever, on which to base their discussion.

• The examination of a relatively long and subtle argument, as in proving that every n-dimensionalvector space over a field is isomorphic to the vector space of n-tuples of elements from the field.Here the objective would be for the discussion group to convince themselves that all of them reallydo understand the concepts and logical steps involved and could present and exemplify it them-selves in their own words. Of course, this is only worthwhile for results such as this, of majorimportance.

• Discussion of the ‘best’ way to introduce a difficult concept (That of isomorphism arising in theprevious example would for example be ideal, or the notion of an equivalence relation), perhapsdevising guidelines for the lecturer on how to introduce the topic.

Wankat and Oreovicz ([72], p. 116) discuss the advantages of discussion groups in engineering subjects,but they apply equally to mathematics. Basically they are:

• promotes higher order thinking skills

• retention of material usually improved

• can change attitudes (affective objective)

• greater intellectual development

• students more actively involved

• can improve students’ group interaction and communication skills

• students can be leaders and teach other students

• more likely to lead to commitment to the subject

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• can be fitted in anywhere, say to break up lectures.

However, as Wankat and Oreovicz warn, discussion also has disadvantages:

• harder for tutor to conduct properly

• time consuming and rate of transfer of information low

• students don’t show improved learning of knowledge, comprehension and application objectives

• may be difficult to obtain student participation, especially in mathematics

• students must know something before an intelligent discussion can take place

• only practicable for small groups - but can always break larger groups up

• may be less acceptable to students such as mathematics students who want to learn from an expert

• meaningful discussion may be difficult with immature students

• mathematics students sometimes think that the transferable skills such as communication and teamskills should be taught separately, not in mathematics classes.

Wankat and Oreovicz also advise that the tasks set for small group discussions should:

• have several possible solutions

• be intrinsically interesting

• be challenging but doable

• require a variety of skills

• allow all group members to contribute.

4.11.2 Running Discussion Groups

Again, Section 3.9 covers most of what we will need for running discussion groups. We need to ensurethat the accommodation and resources are suitable for the session, and to decide how to use them. Infact, this is quite a technical matter and may be out of your hands, but don’t be afraid to raise the issue iffor example the room is not suitable for what you have in mind.

Skemp ([66]) talks about the advantages of discussion in mathematics. He believes the best number ina discussion group is two or three, but this is somewhat unrealistic these days and things should workfine with say half a dozen students. Skemp emphasizes the importance of allowing time for thinking ina group, which doesn’t necessarily have to be continuous discussion. And, particularly in mathematics,members of the group may need to engage in some sort of activity like doing calculations in order toinvestigate points made. So a discussion in mathematics may be somewhat more varied than in a subjectsuch as say history. In mathematics, time-out may be needed to raise questions, provide responses, etc.Good personal relations are of course needed between members of the group, and good facilitation bythe tutor.

Even in small groups, participants contribute unevenly and the problem increases with the size of thegroup. To increase participation, try breaking larger groups into smaller ones for some time, and perhapswithdraw from the discussion for some of the time, leaving the students to themselves. As in any kind

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of teaching situation, give the objectives of the session and explain briefly what you are going to do, andwhy. Set rules/agenda from the beginning and make sure everyone understands the requirements forthe session to work well. This includes timekeeping, disciplined working, etc. Give some motivating in-troduction - a bit of relevant history, an example, reference to the examination or coursework, somethingrelevant to them, etc. (Section 3.9). Give clear instructions on a handout or, better, displayed on the boardor OHP so they can easily refer to it.

Start with easier tasks to get the students used to working together. Give them ‘entry methods’ to topics,not ‘hints’. That is, general principles they can adopt to get started on any discussion or project. Havethey seen a similar thing before? How does it differ from other examples they have seen, and how canthey generalise what they already know to suit this new case? Can anyone suggest an approach that therest of the group can check out? If you find it is essential to give a hint to actually get a discussion off theground, then perhaps that means the task is not expressed well in the first place and next time you mightrewrite it. The point about discussion group work is that within a reasonable sized group there shouldbe someone who can make a start on a topic.

Make sure everyone is involved and engaged - for general ploys for encouraging participation see Section4.6. In the case of group work the main point is that the group should regulate itself in maintaining adisciplined, productive work ethos in which everyone contributes. However, in going round the groupsyou should be alert to such things and bring in the bashful members and keep a lid on the vocal ones.

Try to be mainly a supportive, positive and inspirational presence and not to intimidate, intrude unnec-essarily, embarrass or threaten anyone. Learn to stand back and let go when necessary. Really, groupsshould only interact with you to tell you about progress, or to ask for general advice. Sometimes youmay be asked to adjudicate on a disagreement between group members. Then you have to ask whetherthat is what you should be doing - isn’t the point of the group to resolve such issues themselves?

Get students to help each other - that is the whole point of a discussion session of course, but in the earlystages they might be a little inhibited about this. Emphasise that that is what they are supposed to bedoing. One important role you can play in the groups is as an impartial channel, helping the members ofthe group to talk to each other initially through you and then you can bow out and leave them to it.

Mason ([53], p. 86) gives examples of activities that might be useful for discussion. Exemplification - givea student a technical term, theorem title, or technique and get them to present examples of it to the group.They then have to guess what the student is exemplifying. This provides practice for the students indevising their own examples, modifying familiar ones, etc.

We include in discussion groups the sorts of collaborative exercises that might involve students in tacklingprojects, for example. This is of course highly valued by employers, and by mathematicians generallywhen working on large projects, or checking big theorems. Note the fine line between collaboration andcheating. The tutor has to incorporate mechanisms for ensuring that some students do not ride on thebacks of others and that true collaboration takes place.

Discussion classes might also be used to discuss the solutions to coursework problems recently submitted.They might discuss the marking scheme and try to understand where they went wrong (They of coursedo this already, informally!).

In running discussion classes the tutor needs to focus as much on the process of the discussion as onthe product of the discussion ([53], p. 74). To generate discussion you might get a number of studentsto present their ideas and solutions on the board and then guide and chair a mathematical discussionabout these. If silence occurs, break with simple questions such as ‘Can anyone provide an example?’.Ask one of the students to explain their view of the task to the others as the beginning of a debate amongthe students about what the question really is. Stop every five minutes or so during a session and pose aquestion to students. If you get several different responses get them to discuss in pairs and decide whichis correct.

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Finish the discussion session appropriately with a plenary session to allow groups to feed back on theirprogress, which you may then pull together to summarise the lessons that have been learned. If theexercise was structured properly then similar messages may emerge from the separate groups, and willbe all the more convincing for that. Give follow-up work, to consolidate and build on the outcomes, andpossibly to be handed in later. Describe briefly what you will do next session, and ask them to bringnecessary materials and do any preliminary work required. Make notes for yourself on how it went, howit could be done better, any promises to students (always do whatever you said you would do), etc?

Wankat and Oreovicz, ([72], p. 118) give a useful summary of the things to think about when runningdiscussion groups, some of which we have already touched on. They note for example that discussionsshould be structured to occur spontaneously. Also, the tutor needs to have an overview of the topic- a broad range of knowledge, not just be a little ahead of the students. The tutor needs a structuredagenda - don’t just say ‘let’s discuss’. Give the students a specific task. The purpose is not to find ‘thesolution’ but to introduce students to the process of looking for solutions. Break the problem down andeither allocate to different groups, or different times. Have a schedule covering different stages of thediscussion, but do not rush unduly. For example it may need two or three minutes silence before weget the first contribution, which is normally the hardest, after that others will chip in more freely. Ifnothing emerges get a student to enlarge on something they said earlier, or make a provocative statementyourself. Many students, particularly female, find it easier to ‘open up’ and talk in small groups (coffeehelps!) with the tutor not present. If we can get one or two groups started the noise level in the room willincrease and that will encourage the other groups to start. Once started the tutor can do the following tokeep things going:

• post ideas on the board as they arise, verify, correct obvious errors

• serve as gatekeeper, keeping students on track

• when discussion falters, request examples or illustrations

• encourage and recognise contributions, for example by writing down

• test the consensus - is class ready to move on to next part?

• summarise the discussion - for which you need to listen, another reason to talk less.

ExampleBaumslag ([7], p. 163) gives an example of a lecturer who spent the whole of a class findinga definition of the dual of a graph. One student thought this a waste of time, as a definitioncould have been given in minutes. But as Baumslag notes, this is to misunderstand the pur-pose of such a discussion. As given cold, the definition of the dual of a graph is uninspiring,mundane and almost meaningless, belying the great importance of the concept. If it takes adiscussion to uncover the depth and breadth of the definition, then that is time well spent. Butof course, in doing so we should ensure that we also visit other useful ideas in the process,so that the time is efficiently used. This is where the tutor can contribute by structuring andguiding the discussion to explore appropriate ideas.

H Exercise

Running a discussion group is just one more type of teaching activity to which we can applyMATHEMATICS (Section 1.7). Prepare for your next discussion class using this (or other)checklist, surveying the generic and mathematical literature on running discussion groups, anddiscussing with colleagues.

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4.12 Individual Tutoring and Consultation

By the time a student comes knocking on your door for your help they really do want to learn, and thatis the best time to teach them. So, when students come to see you it provides a golden opportunity. Notonly will you be able to help them, but by talking to them you will get an idea of possible problems thatother students might be having, and you can look at these in the next class. The same applies when astudent asks a question in class, or a group collar you at the end of a lecture - you can often do more goodat such times than at the scheduled meetings. All the previous points and ploys we have mentioned inthis chapter might be useful here of course, but usually with more focus as you may now be dealing witha particular individual in some privacy. In particular, your listening skills become even more importanthere as you can look for a greater range of clues in what the student says or does. You can be more alert tosubtleties in response, intonation, motivation, etc. By the tone of a student’s voice you may for exampledetect that it is not actually that they want you to show them how to do something, but that they can dosomething - that is he or she is looking for something to boost their confidence. Then you might not somuch tell them how to do it, but a story on the side about how you found this difficult at first and it is adifficult topic, etc. Convince them that their struggles are not unexpected. That might just give them theconfidence to give it another more determined try, and all they might then need is a small hint.

Wankat and Oreovicz ([72], p. 189) give good advice for eliciting contributions from students, and listen-ing to them. We first need to create a climate where students will talk freely. It helps if you are knownas someone who listens, and who is available and approachable. (An unavailable teacher might just bebusy, an unapproachable teacher is a bad teacher). Usually the easiest time to be available is just beforeor after class. We can use all the techniques discussed elsewhere for developing rapport. Be sensitive tothe inequalities between power and status of tutor and students, which can inhibit the latter and so wehave to work to overcome this. Reduce barriers like podium, desks, etc. Be relaxed, emit nonverbal cuesof receptiveness. Make it clear that you really care for the students and you want to help them. You wantto get good examination results while maintaining high standards, so you therefore need to help them.

Think about the student’s feelings and be alert to their sensitivities. Have a thick skin yourself. Whetheror not you like the students is irrelevant - be objective, polite and considerate. Be non-judgemental (thatdoesn’t mean anything goes or there are no standards, it means that actions and behaviours are evaluated,not the inherent worth of the student). As we have noted elsewhere, there are no silly questions, onlythose that show lack of understanding or that you don’t understand, both of which you can train yourselfto deal with. In any event, don’t be dismissive, defensive or aggressive. Acknowledge a student’s feelingsif necessary, but don’t let them influence your actions. Humour often works well, but not to the extent ofmaking fun of the students’ difficulties. Sometimes a student does need to be told bluntly that they needto get their finger out and get down to some work, but as information about the possible results of theiractions or inactions, and not as a character analysis.

Your focus should be on the student. Use eye contact, move or lean forward (but not too much!!), offeringnonverbal encouragement. Listen to what the student is saying and let them finish before you formulateyour response. Use the brain’s free time to think about what the student is trying to say. What is thereal underlying message? Paraphrase the question and see if the student agrees with your interpretation.This ensures you have understood the student’s question and that the whole class has heard it if it is ina classroom situation. Focus entirely on helping the student to learn and make it clear this is a sharedobjective, so use ‘we’ and ‘us’ instead of you and me. This is not being patronising, it is the real situation -you don’t know the best way to help the student anymore than they know how to resolve their question.Both tasks may be equally challenging. Aim for a balance between how much you do to resolve the issuefor the student and how much they do themselves. Also, think ahead and provide them with the meansto tackle similar problems in future.

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ExampleIf a student is having difficulty with an integral because they can’t remember, off the cuff, aparticular standard integral, don’t tell them the standard integral. Instead show them thatthis is the source of their difficulty, get them to go off and learn all the standard integralsthoroughly and then get back to you if they are still having problems. Emphasise that theirunderlying problem here is not that they can’t do this particularly tricky integral, but theyhaven’t consolidated the basic material sufficiently. In this way they will learn to re-examinetheir basic knowledge when they come to have difficulty with other problems. They will cometo ask themselves if their difficulties arise simply from lack of background knowledge ratherthan from lack of ingenuity (Principle 8).

Responses to students can be non-verbal or verbal - but they must be congruent and give the same mes-sage. Non-verbal messages are in fact stronger and verbal messages will be ignored if incongruent withnon-verbal messages. Interpretations of non-verbal can be wrong and lead to wrong impressions. Also,such things are culture dependent. Minimal verbal messages can be used to indicate attention, and en-courage the student to keep talking. Probes can be used to get the student to enlarge on what (s)he hassaid. It is best if these probes are open-ended, so that the student can have control over how they respondand what to. Use silence to encourage not punish students. A period of silence gives the student timeto assemble thoughts after a question. It is best if we do something else like cleaning the board whilewaiting for the response. Also, silence is useful when a student is trying to manipulate us into, say, anextension on coursework.

4.13 Supervising Undergraduate Projects

The use of projects is now becoming commonplace at the undergraduate level [41]. The issue of super-vising projects is quite sophisticated, but obviously it is analogous to a one-to-one tutoring situation, soall the skills discussed in the previous section may be relevant to project supervision. The supervision ofprojects is an area where you cannot talk to enough people. There will be departmental protocols, pro-cedures and assessment criteria which you should of course familiarize yourself with. Talk to colleaguesin your own department and others, and if possible in other institutions. Maybe your staff developmentdepartment runs courses on project supervision. Have a look at some past projects and if possible talkto people who supervised and marked them. Consult the literature (E.g. [56]). Here we will give somegeneral advice by running through MATHEMATICS for project supervision.

The Mathematical content of the project assignment will be less governed by a syllabus for a course ormodule and more by what is needed to tackle the project, and this might not be clear from the outset,part of the exercise being for the student to determine what is required and learn for themselves any newmaterial needed. But part of the supervision is to judge what is appropriate content for the student toexplore and advise them on how best to study and use it, depending on their prior knowledge. The Aimsand objectives of a project will in general be broader, less content-based, and more focused on transferableskills such as communication, enquiry, presentation and so on. It is particularly important, because of theopen-ended nature of a project that the aims and objectives are clearly stated and that the student isaware of how they will be assessed - there are no ‘past exam papers’, although if possible they may beable to see previous projects in a similar area. Also, in most projects we have to allow for the possibility ofmodifying the objectives as we go along, maybe we were over ambitious, or something turned out easierthan we expected and we can push the work a bit further. The Teaching and learning strategy is of coursecompletely different to a conventional module, relying on the student to effectively teach themselves,with your guidance. The problems they face will be more prolonged and require far more independentstudy. This can lead to frustrations and fears for the student and so the supervisor’s strategy will involve

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more emotional and moral support than is usual in teaching. There will need to be a balance betweengiving the student their head and possibly suffering disappointment when things go wrong, and leadingthem by the hand through difficult areas. This brings us to the Help and support for the student doing aproject. This is an issue where even experienced staff have difficulties, and you should seek as much helpand guidance for yourself as possible. The problem is that by giving different degrees of help you canproduce a project mark of almost any value. Some supervisors do little more than point out references andtheir students struggle and possibly underachieve. Others almost write the project for the student, whothen goes on to get more credit than they deserve. A large part of departmental guidelines on projectsis devoted to providing consistency in such things across the department and it is where new staff needa lot of support. Key areas in which students are likely to need help are in the literature survey, settingand modifying objectives, balancing the depth and breadth of the investigation, formulating specificresearch questions, disengaging from the work when appropriate, expressing arguments and conclusionsconcisely and convincingly, writing the report, presenting the viva.

Evaluation of project supervision tends to be very much outcomes-based, with such things as secondmarking and external examiner oversight providing the main quality assurance mechanisms. However,for new staff there should certainly be close mentoring and ongoing advice as the project develops, withregular consultation and guidance from an experienced staff member. On the plus side, the issue ofpreparing Materials for project supervision is less onerous, since basically the student has to producethese themselves! Essentially, we just need the project specification, student and staff guidelines onproject work and a reading list. However, if for example the project involves computational work thenwe need to ensure that the student has access to appropriate hardware and software. Other resourcesmay also be required, particularly in a modelling or statistical project.

The Assessment of projects could fill a chapter itself. This is where the new lecturer will need mostguidance. Fortunately there will almost certainly be some departmental pro-forma for recording theassessment, and this will provide some idea as to what is expected. For example there may be marksawarded for an interim project report designed to check that the student is progressing satisfactorily.There will be marks for the final report both in terms of scholarly presentation and work accomplished.There may be a viva at which the student can be encouraged to bring out their achievements and explainthings from the report. The thing is not to be too hard on the student, or yourself, especially if this is anew experience for both of you.

Time and scheduling considerations are crucial in a project, because for administrative reasons there willbe tight deadlines, while at the same time the student, and possibly yourself, are likely to underestimatehow long such things as writing up take. This is definitely a situation in which leaving it for ‘last minuterevision’ is a big mistake. Writing up a project is nothing like last minute mugging up for an exam, andyou will need to keep well on top of the student’s progress. And of course it is likely that the submissiondate for the project report will be in the same time frame as for the rest of the students’ exams and thiswill be a period of great stress. Also, all sorts of unforeseen delays can get in the way - data may takelonger to collect than expected, university IT and administrative systems may get overloaded and causedelays, the student may go up one blind alley too many. Make sure you are aware of the deadlines, knowthe student sufficiently well to be able to judge how they will cope with these, be able to judge when todraw the work to a conclusion and start writing up. And of course you have to be clear about your owntime commitments to the project. Find out roughly how much time you should spend supporting thestudent and manage this sensibly.

The issue of Initial position of the student now relates to the general background they have in the areasrelevant to the project, and your first early meetings with the student should be devoted to discussing thiswith them so that you can assess what reading they need to do. And bear in mind that they need to knowthis background material well enough to use it in new contexts. For example if they need to use somemathematical software such as MATLAB the facility they need with this may far exceed that developed

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in a module taken the previous year, so they will need time to develop this level of facility. Also ofcourse students (and particularly mathematics students) may be unused to writing reports, or presentingmathematics in vivas, so they may really be starting from scratch in this respect. And indeed one of themost difficult things for a student is presenting the outcomes in a Coherent and organised way. Oftenthey write the report almost in ‘diary mode’ with objectives, methodology and conclusions jumbled upwith no overall structure. Your department will probably provide detailed guidelines for students on this,describing the structure of the report, length, bibliographic conventions and so on, but you may have tointerpret these for them. And in general you will really need to care for the Students you supervise. Theproject will probably be a new and daunting experience for them, they will have no benchmarks on whichto judge their progress or likely outcome, and their performance might have a significant impact on theirdegree results. It will be a stressful time for most of them, so as with assessment generally, be considerateand understanding and give them plenty of support. You will also need to get to know them very wellas early as possible in the process - their background knowledge, motivation, independence, enthusiasm,stamina, imagination, etc. The easiest students to supervise are the self-propelled, keen types, perhapslike yourself, enthusiastic and capable. Often you can leave them to themselves, perhaps only having torein them in when it is time to write up. Much more difficult is the average student who treats the projectjust like any other part of the course, a hurdle to be overcome. Their enthusiasm and interest may bemore restrained. It will be harder for them and for you. But your job is not to make up for any lack ofapplication - they have to demonstrate independence, stamina and dedication to the task.

4.14 Working with Teaching Assistants

It may be that you have postgraduates or postdocs to help you with your tutorials, and possibly withsome marking. Remembering that tutorials are front line teaching and learning activities, it is clear thatsupervision of such assistance is very important. In recognition of this the MSOR Network runs trainingsessions for postgraduates who teach mathematics. The materials for these sessions can be found onthe Network website (http://mathstore.ac.uk/) and may give you some ideas of the sorts of problemsthey experience (You might have already been there!). So far as you are concerned you just need to liasewith them regularly, ensure that they can do all the problems in a number of ways, and that they toounderstand the need to engage, enthuse and explain in dealing with students. If they mark work setin the tutorials then provide them with good marking schemes, and regularly check their marking andfeedback to students.

H Exercise

The previous three sections cover topics that are rather specialized and rely very heavily ondepartmental policy and procedures. Survey those that you think are appropriate to your duties andgather the necessary documentation, discussing with colleagues what actually happens on theground.

4.15 Evaluating Learning and Teaching in Tutorials

Morss and Murphy ([56], p. 71) quotes Jacques on evaluating tutorials. They give the following self-evaluation checklist.

• Did I ask questions which stimulated lively discussion?

• Did I manage the time well?

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• Did the students all participate in discussion and tasks?

• Were there any difficulties?

• What would I change if I did this again?

• What notes do I want to make for the next time round?

Student evaluation questions might include the following.

• Did you enjoy that activity/task?

• Which aspects of the group activity worked well?

• Which aspects did not work well?

• What skills do you think you were developing as a result of the group work?

• What is the most important thing you learned today?

But how can we be sure, at the end of the class, that the students have learned what we had intended?The honest answer is that you can never be sure what they have learnt! You can try though. One way isto give a short quiz at the end of the session - specifically, something like ‘What are the three key points ofthis session?’. During the session monitor their behaviour and any contributions (or lack of contribution)they make. This is of course much easier in a small tutorial. You have to think carefully about the sortof questions you ask, and your ability to get responses from across the class. For example, if you havejust been discussing the various uses of integration by parts, give the students a couple of minutes at theend to simply write down the methods they would use to integrate each of x exp(x), x exp(x2), x2 exp(x),x2 exp(x2). If you then collect their responses you will soon see if they have assimilated the key points.By the way, don’t regard this as a waste of precious class time - it will also help the students to learn.

H Exercise

Design a checklist for evaluating how a tutorial went, including questions for the students andyourself.