Chapter 2

Curriculum Design in MathematicsTeaching

2.1 The Importance of Good Curriculum Design

H Exercises

1. Remember the following letters in any order:

E G T A S C N I M H

2. Remember the following letters in any order

TEACHING MATHEMATICS

Given sufficient time each of these is easy - most people will eventually learn the letters in either exercise.But the more restricted the time the easier it will be to remember the letters in the form of Exercise 2. Ifonly one second is allowed in each case almost all people will find Exercise 1 impossible, while almosteveryone will still be able to do the Exercise 2. Even though this is a very easy learning task it illustratesthe following important points.

Underpinning our expectation that the learner will find the second version easier to learn there isan implicit theory of learning (more correctly theories of learning). That is we base our teachingstrategy on some ideas or principles about how we think the learner actually learns most effectively.

Designing the learning activity to suit a given objective can make dramatic differences to ease oflearning.

Redundancy/repetition has been introduced in Exercise 2 to assist the leaning process.

Some people may do Exercise 1 faster than others - they may have the sort of mind or aptitudethat quickly devises mnemonics for themselves - e.g. GET HIM CANS; but few will do this in one

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second. They may even enjoy the process, but some may just be irritated by it and resent the wasteof time, much preferring the teacher who uses version 2.

Version 2 is not spoon feeding - even the brightest student may appreciate the second method if itsaves them time so they can get on and learn more important or interesting things quickly. Andin general, efficiency and effectiveness in the design of teaching is equally important at all levels.Even the UKs best students must compete with those in other countries, who may have been taughtmore efficiently and effectively.

So good design of the curriculum and its delivery, based on sound theories of learning, is essential andcan pay useful dividends. It is also a skilled activity for which training is valuable and cost effective. So,this chapter will firstly review ideas in theories of the learning of mathematics. Then we will use this inthe practical issues involved in designing the curriculum and the teaching of mathematics.

2.2 MATHEMATICS for Curriculum Design

Curriculum design may range from the design and planning of a single lecture or tutorial to the produc-tion of a completely new undergraduate programme. In all cases however there are some basic issuesthat we need to consider when sitting down to plan any aspect of the curriculum. Teaching is a compli-cated activity involving intellectual, organisational and administrative challenges, all bound up with thecomplexities of human interactions. In Chapters 3-5 we will be looking at the main teaching activities forthe average mathematics lecturer - giving a lecture, running a tutorial class and assessing students. Eachof these has many component activities, and different practical features, yet they share similar principlesof learning and all require a general overview of the entire teaching activity. So, whether you are prepar-ing a whole degree programme, a single course, or an individual lecture or tutorial class, there are largenumber of things to think about, most of which can be summarised in the apt acronym MATHEMAT-ICS(Section 1.7).

Apart from the all important issue of resources(from materials to (wo)manpower to time) available toyou, the main things we need to consider when planning and preparing teaching can be summarizedunder the following headings ([24]):

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 components fit togetherStudents.

The acronym (not one you are likely to forget!) is too good for us to be worried about the order of theimportance or treatment of these issues, but in fact we normally need to consider them in parallel anyway.There will inevitably be some overlap and repetition between the different items, but it is better to bereminded more than once than to forget altogether. Throughout the book we will use MATHEMATICSfor different teaching and learning activities, from curriculum deign to assessment. A quick run throughthe rough checks it provides can be a useful reminder what we have to do.

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So far as curriculum design is concerned, for a single lecture or a whole degree programme, the math-ematical content and the aims and objectives form the starting point. We need to design teaching andlearning activities that will achieve these, including any help and support that we need to give the stu-dents. As with all good design, evaluation and any administrative aspects have to be built in from thestart. Subject to resources available we will have to prepare the teaching materials we will need, fromour lecture notes to any student handouts we will be issuing. Although assessment normally comes atthe end of a course, we have to think about it from the start, because it will influence how we teach thetopics, and such things as course-work have to be prepared early on anyway. And course-work remindsus of the need to think about how we will use the time available, and schedule the different activities.This will be influenced by the initial position of the students knowledge, their background relative tothe aims and objectives of the lecture or course. When planning how we will cover the mathematical con-tent, we will need to think about the coherence of the curriculum, how different lectures relate to eachother, how the course as a whole relates to other courses in the programme, and so on. And to supportall these considerations we need to have a good knowledge and understanding of the students, theirabilities, motivations and priorities.

2.3 Theory in the Learning of Mathematics

So, the MATHEMATICS acronym is a good reminder of the practicalities of teaching - what to do when,and so on. But there is more to teaching than practical issues. Teaching is really an applied theory oflearning. You may protest that there are no theories of learning, and you simply teach in the way thatseems sensible to you. That just means you are applying your own personal theories about how studentslearn. As professionals we would naturally want to see what others think and so in this section we lookat current theories of learning mathematics.

In fact, there are too many theories of learning for the average academic to assimilate, compare and usefor their day to day teaching. Choosing to invest time and effort in such theories is similar to choosingwhether to spend time learning a new mathematical topic that we are not sure will help with our research.For example, if your work is in non-linear differential equations you may have heard about Lies grouptheory methods [31]. To understand and use these requires considerable investment in time and energyand you will need to have some confidence that this will pay off. So it is with theories of learning,they require investment of time but there may be a great deal of practical usefulness to be learnt fromtheoretical insights produced by deep thinking of teachers, psychologists, neuroscientists, and others.Here we can only give a hint at the sorts of things that might be useful, and hopefully persuade you thatthere is value in delving into such material and using it to inform your own practice.

2.3.1 What is a Theory of Learning?

Given that there are more than one theory of learning, as there are more than one Grand Unified Theoryin Physics, we have to look at what qualifies for such a theory. A workable definition of a theory oflearning might look like:

a system of ideas formulated (by reasoning from known facts) to explain the acquisition of a form ofknowledge or ability through the use of experience

Now it has to be admitted that many widely accepted theories of learning, even some used in educa-tional practice, do not in fact satisfy this definition, if only because of the clause in brackets. The commonsense advice of an experienced lecturer often relies more on gut feeling than known facts, yet is stillinvaluable. So we are going to be a little cavalier in what we take as a theory of learning.

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It would be arrogant to believe that mathematical learning is other than a subset of learning generallyand so much of what we discuss concerns learning in the generic sense. However, there are at least fourreasons why mathematicians are in a privileged position so far as the study of learning is concerned:

mathematics is one of the core curriculum topics in the broadest sense - it is the only truly interna-tional language, and its importance is timeless

partly because of its precise and unequivocal nature mathematics is often the test-bed of choice forlearning theorists

increasingly, the language used in many learning theories is mathematical (Indeed Piaget based hisideas on a group theoretical notion of learning)

mathematical education is a long established discipline in its own right with a large output thatcould be put to more use in practical teaching.

Thus, while many of the examples we give and work we describe is in the area of mathematics, there ismuch we can learn from other subjects.

2.3.2 Types of Learning Theories

There are any number of learning theories that one may broadly classify as looking at the exterior be-haviour of the learner, or looking into and hypothesizing about the internal thinking of the learner. Ofthe former, major examples are Piagets work, Pavlov, Skinner, etc, while introspection such as that ofDescartes and Polya and also neuro-biological work are examples of the latter. One of the difficulties ofgetting to grips with any learning theory lies in spanning the wide range of subjects required to studythe different approaches to learning. Thus, theories may originate from sources such as generic educa-tion theory, psychology, sociology, neuro-biology, computer science, mathematical education, cognitivescience, philosophy, teaching practice at all levels, measurement theory, and combinations of all these asdescribed for example in [17] and [29].

Here, we will only choose what appears to be useful from the major influential theories from these differ-ent disciplines, particularly so far as mathematics teaching in HE is concerned. It is important to realisethat in fact many theories say similar things, and are only really divided by differences in language andterminology. Another point to remember is that virtually everything done on learning theory relates toschoolchildren. This means that developmental theories such as Piaget, that track learning behaviour byage, have limited application in HE teaching, although they have been extended by people like Skemp[66].

Another way to divide learning theories is into the soft end (psychology, etc) and the hard scientific end(neuro-biology, computer science, etc). Two books that nicely summarise each of these two perspectivesare Bigge and Shermis [9] for the former and Newell [58] for the latter. In their fields, both of these areuseful references. Bigge and Shermis is particularly useful because it considers the teaching implicationsof the various theories. Both books are easy to read and give coherent well structured overviews ofthe main theories in their respective domains. Both are generic in nature, but often use mathematics asexamples. Schoenfeld [64] is a good reference for areas of interest to the mathematics teacher, bringingtogether contributions from a wide range of disciplines.

Bigge and Shermis [9] list and describe ten psychological theories of learning that are influential inschools, and divides them roughly into those that originated before the twentieth century and later the-ories. The split is appropriate because prior to the twentieth century few theories relied on experimentand scientific observation, being mainly predicated on such humanistic prejudices as religion or personal

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introspection. Before embarking on the study of the individual theories we need some way to categorisethem - some audit of concepts that usefully characterise theories of learning.

Bigge and Shermis also consider how we view students when they present themselves for teaching, andwhat governs our basic attitudes to them as people that we wish to educate. Of course we are interestedin what they know already, but many teachers will also be interested in what might be broadly termed thestudents moral and actional nature. These dimensions bear on how the student will behave if left entirelyto themselves. We all know that this is highly relevant to their learning behaviour and it influences ourview of how we should try to teach them. Bigge and Shermis classify the ten main theories they studyaccording to the moral/actional conception of Humankinds nature. Despite containing a great deal ofinterest to studies of human learning none of these theories are really relevant to modern learning. Forexample none of the theories allows for good interactive learners, or bad passive learners! Really, themoral/actional stance taken by modern theories can be summed up by saying that no assumption ismade about a students moral nature, and only the issue of whether they are passive or interactive isconsidered.

Newell [58] offers a candidate theory of general cognition (SOAR). There is no need to subscribe to this, orany other theory, to get a great deal out of this book, it being just a good readable summary of the artificialintelligence end of learning theory with a wealth of experimental material. It has the limitation of coursethat it relies on believing that the learning brain operates in a way similar to how we believe machinesthat purport to emulate the brain are supposed to operate. However, [58] also draws in neurophysiology,and so far as the hard scientific approach goes, this has to be the ultimate objective of theories of learning.For a readable account of relevant developments in this see Koch [48]. While this is devoted to searchingfor the neural correlates of consciousness, and understandably advances Kochs own ideas, one again isnot obliged to go along with these to benefit from the excellent summary material in the book. The keypoint is that it demonstrates how things are advancing and should finally lay to rest any thought that onecannot erect respectable theories of how we think and learn.

2.3.3 Generic Learning Theories

In the previous subsection we have tried to offer a range of references that should satisfy the most ardentand technical interest in theories of learning. The intention is to at least convince you that teaching, whichis really applied theory of learning, is not merely a craft in which anyones opinion is as good as the next.It is a discipline to be studied and learnt in depth. But, to be pragmatic, not here. We will have to becontent with accessible distillations of current thought that will guide us in our practical job of teachingand helping students to learn. First there are the sorts of generic overviews of the processes one goesthrough when engaging in learning activity. The classic example of this is the Kolb Learning Cycle [44],which is often the standard model used in generic teaching and learning courses. It is an experientialcyclical process comprising concrete experience, observation and reflection, abstract conceptualization,and testing of concepts in new situations. It is not very sophisticated, being analogous to the way weusually approach modelling in mathematics - model the situation, abstract the mathematical model, solvethe resultant mathematical problem, compare the results with experience, etc. Since most mathematiciansknow what a circle looks like little of this is really challenging, and does not really help us to understandhow the student struggles with ideas that we find so clear, nor helps us in helping them. For this reasonthe generic stuff is usually of little help to us in mathematics. Fortunately, mathematics education hasa long history of research and development both by specialist mathematics educators, and practicingteachers and psychologists.

As a good summary of what generic learning theories tell us we suggest you read Section 7.1 of Baumslag[7, p. 81]. This, in the space of three pages, gives as good an overview of generic theories of learning inrelation to mathematics as you are likely to find or need. It is practical and refreshingly frank and honest.

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It highlights what, to the practitioner, are shortcomings of many theories, for example the neglect oflearning by watching and imitating, the neglect of affective aspects such as competition, co-operation,fear of failure, pleasure in success, or in learning for its own sake. However, I believe Baumslag is undulypessimistic about the prospects of making a science out of the study of how we learn.

Baumslag [7] goes on in Section 7.2 to distil the following ten fundamental rules of teaching:

teach at the right level make the class as uniform as possible remember to deal with the obvious ensure your students are active make demands encourage and give students opportunities to ask questions motivate make lectures interesting treat students as people lecturers, keep on learning.

Many authors seek to encapsulate the skills of teaching in a few principles in this way (See for anotherexample, Chapter 1 of Krantz [49]), and we will give our own version of this below. But you can thinkabout it yourself - see exercise below.

We might also mention that there is much to learn about learning from other subject areas, particularlythe sciences and engineering. Useful recent references here are [44] and [62].

H Exercise

Summarise what you think are key principles that underpin effective teaching.

2.3.4 Theory in the Learning of Mathematics

Thinking about learning mathematics dates back to the origin of mathematics itself. Much of this relatesto the teaching of mathematics in schools, to schoolchildren, and so relates to elementary mathematicsteaching. This may of course have relevance to the more advanced teaching at university, but specificattention has been focused on the latter only in recent times, and is still not a large part of mathematicaleducation. We might conveniently summarise the history of this development as follows.

In 1913 Poincare gave a now famous lecture in which he tried to analyse the thought processes involvedin mathematical invention [60] - which was essentially about how we come to develop new (for thelearner) ideas in mathematics. He based this on introspection about how he himself learned and devel-oped mathematics, and also on his historical knowledge of other mathematicians. Others were thinkingabout similar things at the time, but not in a coordinated way, and it was not altogether clear how thisrelated to the general theory of learning mathematics, as applied to student learning at university forexample. It was after all focused on how researchers came to find new results in mathematics. In an

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almost equally classic extended essay Hadamard [39] built on Poincares work, summarised others con-tributions, in other subjects, as well as mathematics, and linked this with student learning of mathematicsgenerally. Hadamard also sought to establish the results and ideas in a coherent semi-theoretical context,linking to contemporary views from the psychology of learning, and one can see in his essay the seedsof many of the ideas and structures prevalent in modern mathematical education at the advanced level.Perhaps the key lesson to learn here is that Poincare, Hadamard and others, the leading research mathe-maticians of their day, thought education was important enough to study in depth and worth devotingthe time and scholarship to do it.

It was many years before anyone really took this early work further, but in 1971 Skemp, coming essen-tially via school mathematics and psychology, consolidated the ideas and established a more theoreticalframework. Work since then, in what has now become a specialised area of mathematical educationis summarised in David Talls excellent book Advanced Mathematical Thinking [69]. This book can beconveniently taken as the starting point for modern ideas in this area, since it collates and organisesthe previous work and contains the recent research by the leading activists in the field. It is an essen-tial read for anyone serious about their teaching. It is also an easy read, and makes every effort to linkwith classroom practice. A very accessible introduction to the ideas of Mathematics Education, designedfor the practical needs of the busy mathematics lecturer while maintaining the scholarly, evidence-basedunderpinning of sound research, has recently appeared from Alcock and Simpson [3]. This, in an easyevenings read, describes examples of work from Mathematical Education and gives examples of applica-tions of these outcomes in the classroom. With authoritative and comprehensive references this providesan excellent warm-up to David Talls book and can be highly recommended to the lecturer serious abouttheir teaching. For further detailed surveys of current state of the art see [43] and [65], although the latteris probably beyond what most of us need. So, in practical terms what are the outcomes of all this work,and what can we learn from it?

Firstly, so far as mathematics is concerned, we can view the main current thinking on learning as theconvergence of two main strands. One is that emanating from the Poincare mathematical inventionroute [39], the other from the experimental psychology route from the work of Piaget [66], originallyaimed at the study of learning in children, but adaptable in large part to student learning as well. Perhapsunderstandably they overlap considerably and so we can weave their stories together. But of course wemust provide a health warning. Much in each is controversial or even wrong, and many may argue forcompeting theories. In the interests of brevity we are also going to paraphrase and simplify many of theideas involved. That said, they provide a start and the idea here is to establish the principle that one canunderpin ones teaching by reasonable theoretical ideas, and if you want to take this further, there areplenty of sources in the references.

We begin with Poincare. He viewed mathematical discovery or learning as a four stage process:

Preparation - incubation - illumination - verifying and precising

These stages of Poincares encapsulate how many of us work in mathematics, whether simply learning anew topic, or conducting original research. We first do a lot of preliminary groundwork, make a numberof attempts at understanding (preparation), but if it is anything at all difficult we then need to leave italone for a while and give time for our unconscious to work on the problem (incubation). Hopefully, butnot invariably, this is followed by the Eureka (illumination) stage. But for most of us this stage is usuallyreached via intuition or rather loose thinking - and for that reason may be short-lived! We have finally toverify our ideas and precise it, with formal mathematical/logical argument. And you know how manygood ideas evaporate at this stage!

The above Poincare stages are, like the Kolb cycle, somewhat vague and bland. They dont really tell usmuch about how or what we do at each stage, and might not appear to help us very much in our teaching.But in fact they contain an extremely important message for our teaching, which is fundamental to how

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we approach it. When we teach a topic, because we feel we are conveying a well defined body of knowl-edge we sometimes tend to present in a polished and packaged form - the standard example of which isof course the definition - theorem - proof structure beloved of many pure mathematics books. But thisis really presenting just the preparation and precising stages of Poincares framework. It is certainly notthe full story of how we do mathematics, and gives the student a false impression of what mathematicsis all about. The real challenge of teaching is to present the whole picture, expose the student to all of thestages, but in a way that will benefit the students learning processes. Later we will be suggesting waysthis can be done, but for now it is sufficient to realise the problem - we sometimes teach the product ofmathematical thinking alone, without revealing the processes of mathematical thinking.

Now lets think about adding more meat to the Poincare stages. How does one go about each stage -more generally, how does one actually learn new mathematics? Whether it be a new definition, new tech-nique, proof of a theorem, etc we are essentially learning new concepts and fitting these into serviceablestructures or schema for subsequent use. The concepts are the atoms of a subject, and the schema arechunks, collections of concepts (correctly or incorrectly fitted together), perhaps like molecules or morecomplex structures, even quantities of thinking material. Skemp [66], building on critical evaluation ofPiaget, gives perhaps the most accessible introduction to this for mathematics. Tall and others furtherdevelop such themes in detail, looking at the technical details in terms of learning mathematics. Here wewill take a superficial but hopefully serviceable overview to illustrate the use of such theories in practice.

First look at how new concepts are developed and learned. Essentially one builds on already existingideas and knowledge. One may do this by generalising or abstracting for example, or modifying existingknowledge. Either way the learning is only likely to be permanent if there is some reconstruction ofalready existing concepts or schemas. This is often a difficult job, and requires time and effort. It involvesgoing through the full Poincare cycle, maybe a number of times. It is not sufficient for the teacher to telland the learner to listen. The teacher can intervene in such a way that better facilitates the process for thelearner, and can provide exercises, time and support in the course of the processes, but it is the learnerthat actually has to do the cycling! Techniques by which the teacher than provide this assistance, whichwe will continually come back to, are, for example:

helping students in chunking information providing hooks, mnenomics, etc step-laddering in arguments presented or problems set regular overviews of progress highlighting the key ideas identifying the essence of ideas encouraging students intuition and instilling guess/check techniques alerting students to patterns of thinking and links between ideas providing appropriate examples and counter examples providing signposts, road maps, etc identifying and highlighting the difficult bits and providing extra support through these giving a number of alternative perspectives, which might suit different students ways of thinking

about the concept.

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A key feature in the development of concepts and schemas is the idea of a relay result or formula. Essen-tially this is an intermediate result used as one of the steps from initiating the collection of ideas to formthe concept or schema to the actual concept or schema itself. Such intermediate results may be very sim-ple ideas, or very sophisticated ones, but to be useful and effective they must be thoroughly understood,and usable (Hadamard says utilizable), otherwise the development of the full concept/schema will beimpeded. We all know this effect, and it simply reflects the obvious notion that in order to develop somemethod, concept, etc, the results on which they depend must be accurately known and at the learnersfingertips. For example, in developing the idea of the integrating factor in differential equations, an es-sential relay result is the product rule. A student who has an incomplete grasp of the product rule hasno hope of getting the idea of the integrating factor method. A more sophisticated example of a relayresult is the idea of a coset in the development of the quotient group. A thorough grasp of the former isessential to develop the concept of the latter. The lesson for the teacher in all this is that when teachinga particular concept or schema one must analyse its structure and identify the key relay results, ensuringthat these are sufficiently understood by the student. This is the common sense notion that students haveto have a strong grasp of the intermediate results needed to develop a concept, method, schema, etc. Thusfor example we often go straight into the integrating factor method in first year mathematics assumingthat our students have done the product rule. They probably have, at A-level, but most of them will havenothing like the strength of grasp and the facility to use it in the integrating factor context. So it has to beconsolidated before embarking on the integrating factor method. So, the idea of a relay result is simplyto remind us that we need to think about what the current topic relies on and the extent to which it needsconsolidating until it is utilizable.

Of course concept, relay-result and schema formation are rarely likely to be perfect, and authors likeTall [69] have made much of the difference between a concept definition and a students concept image- that is, often a student can have a false impression of a concept. This can both get in the way of newconceptual development, and also lead to further errors in concept development. Such things are difficultto deal with. You will soon meet examples of this problem after teaching for a while.

ExampleA good proportion of students enter university believing that eA+B = eA + eB and it is verydifficult to change this - some even graduate with the same misconception! Such erroneousconcept images do not usually respond to normal teaching and tutoring. To shift them it isusually necessary to engineer some cognitive conflict that brings the student face to face withthe error. For example we might get them to work out e = e1+0 = e1 + e0 etc. deducing 1 = 0.

It is not always easy to identify precisely what a particular students concept image is - entire researchpapers have been devoted to just this problem [69]. There are ways in which the teacher can help (orhinder) the student to develop concept images. For example, one step sometimes used in concept forma-tion is generalisation. Many students find this difficult and the teacher can help them by giving hints,providing appropriate examples, testing specifically the generalization they are supposed to have made,etc. Again abstraction is a very difficult process in which many students need help. For example, thestudy of abstract equivalence relations gives a lot of trouble, especially the appreciation of the transitiveproperty. The teacher can ease the latter by giving the example of the transitive nature of the relation be-tween cups, saucers, and drinkers ([66], p. 147) Another example is the abstraction of division requiredin the definition of the quotient group.

Hopefully, the points and examples above illustrate that thinking about theoretical structures in learningcan lead us to improve the effectiveness of our teaching. Note that the nature of the evidence for theoriesexpounded by Poincare or Hadamard or Skemp is important. Some of it is survey or anecdotal evidenceand does not have the same rigour and validity as that we might be used to in mathematics and science(Small numbers, subjective, introspective, no control groups, etc). But, if enough qualified people are

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saying the same thing then there is nothing wrong with this - it is more reliable than for example talkingwith ones colleagues in the department.

2.3.5 An Example of Schema Formation - the Group

Consider the mathematical idea of a group. To the well trained mathematician the single word Groupconjures up an entire collection of ideas, experiences, results, facts, etc that can be used. A group is morethan a collection of a few axioms. It is essentially the simplest mathematical structure in which we cansolve equations for unknowns. It is the structure that reflects all kinds of symmetries. As such, it formsa ready made tool for use in further advanced mathematics. As soon as we recognize a group structurein our problem we can bring to bear everything that is known about groups. Indeed Piaget based hislearning theories on the notion of a group. However, his was a non-dynamical group, not allowing forthe evolution of learning through teaching. Tall, on the other hand envisages a dynamical analogy towhich one can apply catastrophe theory [69, p. 8].

This schema of a group, fitting together a number of already advanced concepts - set, binary operation,associativity, identity, inverse - can exist in a number of stages of development in the learner. For thestudent new to it the idea of a group can appear just as a list of pretty meaningless and only partiallyunderstood axioms that have to be continually referred back to when studying or applying groups. Theprecising and relay or utility stage has not yet been reached. Whereas the lecturer may have reached thisstage, the student needs lots of practice and support to get there, and that is a job for the lecturer. Theway the lecturer facilitates such schema formation can have a great impact on how the students learn. Itis a skill that can be developed and improved by training.

H Exercise

Choose a particular concept, technique, or schema from your teaching and analyse its logicalstructure, identifying the relay results required in its development. How can you ensure that thestudents understand these sufficiently and have the necessary facility?

2.4 Basic Principles of Teaching Mathematics

Even a brief overview of current ideas about how we learn mathematics is too much for most of us to di-gest and assimilate into our day to day teaching practice. As a compromise, we here suggest a number ofprinciples distilled from such theory, combined with other sources, that can be used to underpin practicein a fairly common sense way. These are by no means definitive or final, and the greatest service that thereader can do is to examine them critically. They are intended to provide useful guidance for teachingthat will help to support the students learning. If they do not, then they need to be changed. If they areincomplete they need to be supplemented. If they are inconsistent, they need to be rationalised. Modifythem as you wish to suit your understanding of teaching and learning, but at least have something thatprovides a robust principled basis for your teaching.The suggested principles below have been groupedunder the practicalities of setting up the learning environment, how we think students learn and the mainteachers tasks in helping them to do so.

Practicalities of providing the learning environment Even in a single lecture on a familiar topic there are manypractical issues to consider in providing a supportive and conducive learning environment. However, thespecifics of any situation will be underpinned by the following basic principles.

P1. All teaching and learning must take place within limited resources (the most important of which is

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time) available to you and your students

While obvious as a principle, this is often forgotten in practice. How much time do the studentshave for this coursework? Do I have the time to prepare the material for this mini-project? Whatare the hardware and software implications of introducing some computer aided assessment? DoI have a budget for these handouts, or do I charge the students? Such questions arise all the time,and so you need to have a good knowledge of your resources from the start.

P2. Teaching is a human activity that requires a professional management of the curriculum, the groupand the interpersonal interactions that implies

It is obvious that teaching and learning involves continual human interaction with all that brings,but the importance and difficulties of this aspect are not always realised. Work has shown [6] that inteaching and learning the three main factors exhibited by the teacher that best serve student learnersare:

congruence between the teacher on the inside and how they appear outside to others unconditional positive regard for the student empathy with the student

Most of this is clearly on the human, rather than technical side of teaching. Yet while much of teach-ing is about human interaction, it denies us many of the normal coping mechanisms that we employin everyday human interaction. We cannot get angry or impatient, or walk away or be dismissive.What we do can have an emotional and psychological impact on the students to which they mayrespond as any adolescents might. But the teacher has to have the professional detachment to riseabove this and respond in a professional manner. We have to do as good a job for those students wedont like as for those we do. We must judge students on their merits as people who want to learn,not on their mathematical ability.

P3. There must be clarity and precision about what is expected of the students and how that will beassessed

From statements about course content and objectives, through instructions for learning activities, tocriteria for student assessment, there needs to be clarity and precision about what is expected. Thisis the whole point of clear learning objectives. Also, in lectures and tutorials, in learning materials,or in assessment the teachers communication needs to be clear and precise.

P4. The teaching, learning and assessment strategies must be aligned with what is expected of the stu-dents

For given learning objectives, the teaching and learning strategies and activities must be designedto achieve them. The assessment strategy and tasks must be designed to measure achievement ofthose objectives. This linkage of the objectives, the teaching activities and the assessment is calledalignment. Again this is an obvious principle that does not always survive the transition to practice.For example a common learning objective in mathematics is the skill of unseen proof design, yet wesometimes just teach the students to imitate and regurgitate ready cooked proofs.

How students learn We have already emphasised the wealth of material on how students learn. For anygiven teaching topic or situation the literature is full of advice and information on theoretical ideas butfor practical application at the chalkface we suggest that four basic principles underpin how we can usewhat is known about how students learn.

P5. The workload, in terms of intellectual progression, must be appropriate to the level and standards ofthe course

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Even with the crudest tape recorder model of learning, it is obvious that any learner has a limitto what they can absorb in a given time. And teaching should progress at a rate that is well withinthat for most of the students in the class. Even if this were not obvious, there is abundant evidenceof the demotivating and counter effective results of intellectual overload in learning. To measurethe appropriate load we need an accurate assessment of where the students start from - and thisis where we as teachers sometimes fall down - we dont always have a good knowledge of thestudents backgrounds and so expect too much of them.

P6. Mathematics is best learnt in the way it is done, rather than in the way it is finally presented

It is widely realised that the polished Definitions, theorem, proof format in which some books andlecturers present mathematics is not how we actually do mathematics, in which the development ismore exploratory, less systematic and less inexorable. By first presenting mathematics to studentsin a more sensible rather than rigorous polished way we both make it easier for them to learn andalso induct them to the ethos and mathematical ways of thinking.

P7. Mathematics is most effectively learnt if the student reconstructs the ideas involved and fits theminto their current (corrected!) understanding

Again, the evidence for this (in any subject) is clear. Any learner absorbs the information they re-ceive only as the first step in internalising it. It becomes material in an internal dialogue throughwhich the learner reorganises the input and fits it into what they already know, modifying both toarrive at their own understanding. The teacher can help by more than just transmitting the infor-mation. They can help the learner as they develop the internal dialogue by providing appropriateexercises and problems, by intervening in external expressions of the internal dialogue, by provid-ing motivation, encouraging effective engagement, and explaining well.

P8. The meta-skills required for the previous learning points may need to be explicitly taught

By this we mean the skills that the student employs to monitor, adapt and apply their learning pro-cesses. That is we need to help them learn how to learn. This does not mean routine study methods,but encouraging students to engage in self-critical examination of their learning as it progresses. Forexample they need to recognize when they need to increase their rigour, dig deeper into an issue,or check what they have done so far. The well trained mathematician takes such things for granted,but the novice may need proactive help to develop such skills.

Teachers tasks Again, we cannot begin to itemise the mass of practical things we have to do in teaching,but we suggest these can all be founded on the basic tasks of explaining to students, engaging them andenthusing them in the subject.

P9. Good skills in explanation are required to assist students in learning efficiently and effectively(Explaining)

How well a teacher explains a point can have a dramatic effect on how easily the student learns it.There is no merit in presenting students with unnecessary challenges as a result of poor explanation,this just wastes time and discourages them. The art of good explanation is developed by practice.Some teachers are better than others and have a natural talent for it. However, the initial skills canbe learnt, and any teacher who takes the trouble to learn can improve their skills in this area.

P10. Students best learn mathematics if they are actively engaged in the process of doing mathematics(Engaging)

That mathematics is a doing subject is obvious to anyone who studies the subject, but consider-ation is not always given to how we get the students engaged, and with what. If we engage thenwith routine, repetitive, one-step exercises, then that is what they will learn to cope with, no matter

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how long they spend at it. If we want to develop higher order skills in mathematics and the solu-tion of major problems then that is what we have to get them working on, while at the same timeproviding them with the scaffolding to reach such levels.

P11. High levels of motivation are essential for effective learning (Enthusing)

No one learns anything well if their heart is not in it! And this is particularly so in mathematics.[29] argues that one of the major reasons that people find mathematics so difficult is simply becausethey dont want to do it, but often have to. In effect, many students who have to do mathematicsare press-ganged onto a voyage they dont relish. And even if students start off interested in math-ematics, it is very easy to turn them off by not paying enough attention to keeping them motivated.Generating enthusiasm and motivation is therefore another skill for the teacher to master.

The above principles will underpin everything we do in this book. To emphasize the point we will occa-sionally highlight where they buttress teaching practice. Combined with the practical summary MATH-EMATICS (2.2) these (or your own version) should help you to design and deliver your teaching basedon a sound theoretical foundation. In the rest of this chapter we will now look at the issues around cur-riculum design in mathematics teaching, considering first the content (Section 2.5), then defining aimsand objectives (Section 2.6) and devising appropriate teaching and learning strategies (Section 2.7)andfinally looking at the preparation of teaching materials (Sections 2.8, 2.9). We are going to treat all theseissues in some depth and detail because they underpin much of the practice in teaching described in laterchapters.

2.5 Mathematical Content of the Curriculum

In deciding on the mathematical content of the curriculum we are rarely given a clean slate. At oneextreme you may be given an already long established module, for example a first year methods course,in which the content is pretty much prescribed. At the other extreme you may simply be asked to puton a new final year module, possibly based on your research area, in which you have a reasonably freehand. Reasonably, because you will still need to ensure coherence and consistency with the rest of theprogramme to which the module contributes. You will need to ensure that the content is fully supportedby pre-requisite material for whatever pathway by which students arrive at your module. In any event,in the initial planning stage of module you will have before you a fair idea of the content that needs to beincluded, say a rough list of topics to be covered (syllabus). You will have your own interpretation ofwhat the syllabus means, or what you would like it to mean, but you must be objective about this andlook at it from the point of view of the department and the students.

ExampleIf you are asked to teach calculus to first year engineers, and you indulge yourself by doingthe proofs that interest you, using limit definitions, etc, far beyond the capabilities and pri-orities of the engineers, then you will turn them off and annoy the engineering departments.This may lead to them taking the teaching off your department and doing it themselves, los-ing your department student numbers. Conversely, if you are teaching calculus to first yearmathematics students then the rigorous proofs may be exactly what they need. If you thenteach a purely techniques course with few proofs, then again your own colleagues in the de-partment may be annoyed because the students wont develop the in-depth understandingthey need. So, interpreting content even in the form of a detailed list of topics, is a difficulttask that must be done with others in mind.

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You can get ideas and guidance about content from books and your colleagues. If you studied the topicat university, you may also have your old notes you can consult. This is the way many of us prepare ourfirst module - but always remember to modify and update the material where necessary, tailoring it toyour own students.

For the first, but not for the last time we will here remind you that you were NOT the average student.As a lecturer in HE you will almost certainly come from the top few percent of the achievement range,and you will be far more interested and committed to mathematics than the average student. Rather thanmaking teaching an easier job for you, it can actually make it harder. You cannot work on the assumptionthat the students are just like you - far from it. You have to learn about their interests and capabilities.This is difficult. Almost invariably, new lecturers that cannot step outside themselves in this way makethe classic mistake of expecting too much of the students and putting in too much content. You mustdetermine what the average student in your class can cope with, be realistic in your expectations and beready to be flexible if you get it wrong first time. Talk widely with experienced colleagues, and listen towhat they say, even if it seems to conflict with your own views. If anything err on the safe side and aimfor a lower content treated in more depth, rather than risk it being too high.

Above we have, for simplicity sake, identified mathematical content with a list of topics. Of course, thereis much more to it than this, and it is important to be aware of this right from the beginning of planning.In most cases you want the students to not only know the material, but know it well, really understand itand be able to use it in new situations. We will later formalise this in terms of the MATHKIT (See Section2.6), but for now simply remember that we not only need to specify what we want the students to know,but how we want them to know it.

ExampleBelow is part of a syllabus for a first year course in elementary algebra:

Multiplication of linear factors (x+ a)(x+ b)Definition of a quadraticFactorisation of a quadraticSolving a quadratic equationCompleting the squareFormula for the solution of a quadratic equation

These are such elementary and basic topics that you may feel there is no problem in seeingwhat is involved here. Each topic can be taught in a straightforward routine fashion as simplya set of techniques that the students remember, practice and can demonstrate on particularexamples. You may have an idea about how long you would need to teach these topics. Butwhat about the proofs and derivations, which are you going to do and to what depth? Pre-sumably we want the students to be able to really understand this material, and be able to useit in novel contexts. Developing these aspects, in addition to simply memorising the detailsof the techniques and proofs is just as much content as the topics themselves. It constitutesspecific learning objectives that take time, resources, teaching materials, etc, and have to beassessed. This is one of the main reasons for the current trend of expressing content in terms oflearning objectives - it spells out exactly what mathematical skills we want to develop. Eventhough the students will do the bulk of this work for themselves, you still need to supply thematerials for them to do this, and you can guide and help the process - all this is content.

When considering the content of a particular teaching activity, such as a lecture or tutorial, there arenumber of things to keep in mind:

the content needs to be clear to the students it should fit naturally into the overall structure of the course

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the level should be appropriate to that stage of the course and should take account of students backgrounds.

This is because we usually learn most easily when material is well packaged, is clearly set in overallcontext, is well sign-posted and is at a (not too) challenging level. Designing content to meet these needsrequires good subject knowledge, a keen sense of what are the really important issues in the subject, goodpedagogical skills, and good understanding of the background mathematical capability and motivationof the students. In the rest of this section we will examine these requirements in detail.

Content should be made clear to the students

Most of us feel more comfortable about undertaking a difficult activity if we have a good idea of roughlywhere we are going, why, and how we will know when we have arrived. For this we have to use languageand ideas that the students are already familiar with. Analogy sometimes helps - expressing the idea,purpose, content in terms of some other ideas that the students have already met. Another approach is togive a specific example of a problem that the activity will address. Some topics actually undo somethingthe students have covered at an earlier stage (Integration undoes differentiation, etc). Sometimes we aresimply proving some already stated result, in which case an outline of the steps of the proof may beuseful. In all such cases the students need to be made aware of this overview.

Examples

1. If the topic of a lecture is the chain rule in partial differentiation, then refer back to therule in ordinary differentiation with respect to a single variable, and explain that we areabout to extend it to functions of more than one variable. We can give the single variablerule and alongside it the partial differentiation form, and explain that the objective of thelecture is to derive the latter. Here we are generalising an already familiar result.

2. When about to teach integration by parts one can start with the product rule and explainthat the content is devoted to reversing this in order to integrate certain types of product.One could do an explicit example of such a reversal, before going on to generalise this.Here we are reversing an already familiar process.

H Exercise

Think of a topic you are soon to cover and write a few lines that will tell the students what they areabout to do.

Content should be coherently structured within the context of the overall curriculum

It is rare that a topic is totally isolated from the rest of the course. It will usually be set in some particularcontext, related to other parts of the course. This allows the students to orientate themselves and mentallyassemble the background they will need to understand the topic, and to understand why we are coveringit at this point. In a textbook this might for example be illustrated by appropriate sectioning. For example,linkages and relations between topics can be illustrated in an overview in a course guide. And when youcome to cover a particular topic you can refer back to this, linking it with other topics as much as possible.There should be a natural progression from one topic to another. If there is a complete change in material(as there might be in a methods course for example) try to explain why, and give the students time to

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adjust to the new area. The idea is to aim for coherence between and within topics. We can often usehistorical references to provide coherence - they may help to explain why we do things in a particularway. The following give some examples of how we might set the context of a particular topic.

Examples

1. When covering Lagranges Theorem in Group Theory it can be emphasised to the stu-dents that this is one of the key elementary theorems of subgroups. The proof is asimportant for its structural properties (leading to the important idea of the coset andhence the quotient group, which plays such a vital role in subgroup analysis) as for theresult it produces. Thus the rounded mathematician would not view it just as an inter-esting isolated result, but along with its proof, as a pivotal stage in the study of subgrouptheory.

2. Completing the square is not just an algebraic trick related to quadratics but is an impor-tant method for making the maximum and minimum of a quadratic manifest, which isuseful because near to a turning point many continuous functions can be approximatedby a quadratic. Here, the quadratic and its properties are set within the context of moregeneral functions. Also, it is by completing the square that we provide a link between thesolution of a quadratic equation and the simpler problem of solving a linear equation. Soagain the trick of completing the square has a broader context, and this is something wecan point out to the students.

H Exercises

1. Provide an overview of your research area, or methods used within it, that could be appreciatedby the average first year undergraduate.

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

Content should progress at a reasonable pace in timescale and intellectual demands

By this we mean that the pace should be reasonable - not rushed or too slow - and ideas and intellectualchallenges presented should be fair to the majority of the students. The bulk of the students shouldbe able to assimilate the main steps within the time scale of the course or lecture, although of courseconsolidation may rely on further work by the students. The reason for this is that there is plenty ofevidence that many people learn most effectively this way - in small bite-sized chunks which are thenput together coherently later. The traditional lemma, theorem, corollary format is one manifestation ofthis.

ExampleWhen working through elementary methods of integration, it is easy to simply run throughthe methods, derivations, examples, in a seamless flow from use of standard integral tablesto integration by parts. To the students new to the subject this has a tendency to blur the dis-tinctions between the methods, derivations, examples, and gives them no milestones or highpoints to use in order to organise their new knowledge. Slowing the pace, by for exampledevoting separate lectures to each of the separate methods, breaks down the material into

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more easily handled chunks, each lecture representing a compact package on some particularintegration method. All this can be put together later by devoting a lecture entirely to theskills of choosing the best method for integrating a particular function. In this way the paceis more reasonable and the lectures themselves provide a more easily organised structure forlearning the subject.

Content should take account of the mathematical background of the students

While on the face of it this would appear to go without saying, it is not always easy to ensure this inpractice. Indeed the transition problem, particularly from school to university, is one of the major prob-lems of modern HE mathematics teaching, and has spawned a vast literature. (For relevant resourcesand examples of good practice in this area see http://mathstore.ac.uk/). It is very difficult for an expertmathematician to judge levels of difficulty at an elementary level. This is certainly one of the areas wherethe new lecturer can find difficulty. It involves being able to put yourself in the students shoes. Makingjudgements about level is something that only really comes with experience and detailed knowledge ofthe students. It is not always useful thinking back to your own student days, because you were almostcertainly one of the best students in your cohort. And of course, you may be a foreign lecturer, unfamiliarwith the UK school curriculum. Finding the right level to pitch at in the transition from school to univer-sity is a major headache, because you do not have direct access to either the material or the teachers thathave preceded you. A useful exercise is to work through a few appropriate school examination papers,such as GCSE A-level papers. As you do so try to think yourself into the students position and imaginehow they would have tackled the questions.

In your first years of teaching you will probably be assigned a mentor, or at least a trusted experiencedcolleague, who can help you in pitching your material at the right level and be able to give you accurateadvice on what the students are likely to know. And be sure you understand what know means for yourstudents. Just because a topic is on the syllabus of the pre-requisite courses for your module, that does notmean that the students necessarily have the required fluency and grasp of the topic - their knowledgemight be quite superficial. Even when you have a good idea of what they do actually know, then youhave to continually seek feedback from the students to check that they are following you. It is probablybest to get the level too low initially rather than too high. Ideally your first teaching assignments shouldrecognise these problems and there should be plenty of guidance for each lecture.

Examples

1. In the UK, skills in curve sketching have seriously declined in modern school syllabusesand many first year students are hard put to sketch even the simplest linear or quadraticcurves. So, for example, if you have to teach maxima and minima or definite integrationto such students, that is anything that requires rough sketches of curves, then choose onlythe very simplest functions to start with. So the level of the material that you can dealwith is determined by the sorts of curves the students can sketch easily - you are goingto treat quite simple maxima and minima or integrations, simply because the studentscannot easily recognise more complicated graphs.

2. When teaching separation of variables for partial differential equations the level of suchequations that you can present to the students may be determined by their prior knowl-edge of ordinary differential equations. Therefore you may have to compromise on theco-ordinate systems you use and treat the simplest PDEs (while still illustrating the sepa-ration of variables technique) because the students do not have the background to handlemore sophisticated ordinary differential equations. Again the level you can cover is de-termined by the students background.

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3. Most first year students will claim that they have done the product rule for differentia-tion at school, and most will have. However, their level of facility may be very low - forexample some will have to substitute u = , v =, etc and use the product formula. Otherswill confuse it with the chain rule. So, for example when they come to do something likesolving linear differential equations by the integrating factor method, they will perhapsnot follow easily the subtle reversal of the product rule that is needed to express the lin-ear combination involving the integrating factor as the derivative of a product. It maytherefore be necessary to build some extra explanation and examples into this step, andprovide some consolidation in the basics.

In fact, most first year classes present special problems in this area of matching to students prior knowl-edge. Students will have come from a range of backgrounds, particularly in a service class. Even studentswith the same formal qualification (For example, if all have Grade A A-level mathematics), can presentwide variations in their background knowledge. Up to a point, one can expect that they will mug upon any deficiencies they have themselves, but the lecturer still needs to have an idea of these in order toassess a fair starting point. If it is clear that the deficiencies add up to a substantial amount of revisionthen in fairness we need to allow for this and maybe modify the curriculum to match. A great deal ofwork has been done on this. For example it is possible, in the case of A-level to predict very roughly thebackground knowledge of students with a given grade in the core topics [22]. This enables a fair startingpoint to be designed. It is an open research problem to develop similar predictor capability in GCSE levelmathematics, and the various non-A-level qualifications such as BTEC, Access, and Foundation courses.

To familiarise yourself with the requirements at mathematics AS/A-level, the website www.qca.org.uk/14-19/6th-form-schools/downloads/mathematics.pdf gives an overview of the accredited specifications foradvanced subsidiary (AS level) and advanced (A level) GCE mathematics. This summary lists the aims,assessment objectives, content/options and scheme of assessment of each specification.

A warning is necessary here, however. Even for a students with a high mark in a particular qualificationthere is no guarantee that they have the facility you might hope for or expect in any particular topic theyhave covered. Sometimes the mode of assessment by which they have achieved the qualification doesnot encourage long term retention or facility in all topics. So it is always advisable to check directly withyour students what they can actually do.

H Exercise

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

2.6 Aims and Learning Objectives

So far the discussion of content has referred to what the educated mathematician might include at aparticular level in a particular type of mathematics course. To express this in terms that the student, orany other interested lay person can appreciate, we have to be more specific about what we mean by thecontent - we have to state precisely what it is that the student has to be able to do after completing thecourse. For example, our content may include the binomial theorem, but what do we expect in this - dothe students have to be able to prove it? Are they to be able to use it in a range of applications? Suchthings are expressed in terms of aims and learning objectives, which we look at in this section.

The broad purposes of a programme, module or even lecture are expressed in terms of aims. These

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aims may then be expressed in more measurable terms via objectives. The achievement of the objectivesconstitutes the achievement of the aims. As they express intentions, objectives can be those of the teacher,of the programme as a whole, or of the students. Those that refer to what it is intended the studentswill achieve are called learning objectives. The term learning outcomes is also widely used, but this issimply an optimistic version of learning objectives. Terms such as educational objectives are also used,but in this term it is not clear who is intending what to whom. We will stick with learning objective (LO)throughout.

This section summarises typical educational material on learning objectives. In practice, as with muchof learning theory, little of the vast material on learning objectives is used in the classroom (Although,for example Blooms taxonomy was used in the design of Physics A-levels some years ago). However,mathematicians of all people should be able to appreciate the need to be precise about ones intentions,and to classify those intentions in a way that facilitates implementing them and measuring the outcomes -that is what learning objectives seek to do in the context of teaching mathematics. The degree of precisionand classification is a difficult question however, and here we try to adopt a pragmatic midway approachthat doesnt involve too much effort.

2.6.1 Definition of a learning objective

A learning objective is a precise statement of something that the student is expected to be able to do afterthe teaching and learning has taken place. There are many definitions, but the most concise is probablythat of Mager [37]. He believed that a well formulated (behavioural) learning objective should includeeach of the following dimensions.

1. A statement of exactly what the learner should be able to do (a performance) at the end of a specifiedlearning experience.

2. A description of the conditions under which this behaviour can be demonstrated.

3. An indication of the minimum standard of performance which may be considered as acceptable.

So a written learning objective will contain:

an action verb describing a performance which is both observable and measurable

a statement of the conditions imposed on the performance

a statement of the standards to be reached.

Learning objectives are often expressed in some such wording as On successful completion of the coursethe student will be able to ....... (learning objective).

Example... will be able to apply Pythagoras theorem (performance), without any calculating aids orreference material (conditions), to solve any right angled triangle, giving the results correct tothree significant figures (standards)

This example is obviously not one that we are likely to set these days (Although perhaps we should ...),but it is intended to draw attention to how specific such learning objectives can be.

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2.6.2 Types of learning objective

Clearly there there will be as many types of learning objective as there are skills we wish the studentsto develop. There have been many attempts to classify such skills and the literature is vast (And largelyimpenetrable for the practitioner). Broadly, there are three domains of learning objectives (henceforthLO), as originally classified by Bloom et. al. [11], [4]:

cognitive: information and knowledge

affective: attitudes, emotions and values

psychomotor: muscular and motor skills

These broad types are split into further categories, and we will focus on the cognitive domain here. Thisis not to underestimate the importance (indeed some would say primacy) of the affective categories butwe address these in a more direct way later. The classification of the different types of LOs is called ataxonomy of LOs. There have been many attempts at designing taxonomies of learning objectives, butthese have had virtually no impact on the lecturer at the chalk-face. For example, in a recent survey ofassessment practices in the top-ten UK mathematics departments in terms of Subject Review outcomesrelating to student assessment [8] only two made any explicit use of learning objectives to frame assess-ment.

The seminal classification of educational objectives is that known as Blooms taxonomy. In this the maincognitive skills are categorised under knowledge, comprehension, application, analysis, synthesis, andevaluation [11, 72]. There have been many alternatives proposed, usually aimed at specific subjects orattempts to simplify the system for practitioners. The Bloom taxonomy itself has been revised in recentyears [4]. It is now two-dimensional, using a knowledge (noun) dimension comprising Factual, Con-ceptual; Procedural and Metacognitive knowledge, and a cognitive process (verb) dimension comprisingRemember ( = Blooms Knowledge), Understand ( = Blooms comprehend), Apply, Analyse, Evaluateand Create (= Blooms synthesis). The idea is that the knowledge/cognitive process cells can be usedto categorise most cognitive learning objectives. By similarly categorising teaching methods and assess-ment instruments one can then more easily investigate the alignment of learning objectives, teaching andassessment methods. But this is still far removed from most practitioners modus operandi. The simplefact is that most taxonomies are too complicated for practical use by the average lecturer.

In recent years simplified versions of the Bloom taxonomy have been developed specifically for mathe-matics. The MATH Taxonomy [68] - Mathematical Assessment Task Hierarchy - and the three-tier tax-onomy designed by Galbraith and Haines [34], again for the purposes of designing questions that testhigher order skills, both come pretty close to the intuitive approach used by most academics in describ-ing what they expect of students, and what they try to test in exams. However, they still do not seem tohave caught on with practitioners. Here we will use a simple three-tier categorisation, MATHKIT, whichis closer to how we actually work [23], yet gives a workable classification of the main cognitive skills. Itis one-dimensional, contracting the revised knowledge dimension back to that of Blooms original, andrepackages the cognitive process dimension into three categories rather than six. These are:

Knowledge/routine skills and techniques (Knowledge/remember)

Interpretation/Internalisation/Insight of these (Comprehension/understand, analysis, evaluation)

Transfer/Translation of these to new contexts and applications (Application, synthesis/create, eval-uation).

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As with the original and revised Bloom, and other such schemes (What we have called IT here are oftenreferred to as HOTS - higher order thinking skills - in some other schemes), these categories are notwatertight, and there will be inevitable overlaps. Also, the connections with Bloom, new Bloom, andMATH are easy to see, but the KIT acronym is hopefully easier for the busy academic to use. Also, beforethe days of Bloom many authors (E.g. [67]) used a similar three-tier classification.

Example - Pythagoras TheoremIn this case the item of Knowledge is the formula a2 + b2 = c2 for the sides of a right-angledtriangle with hypotenuse c. Under K we would, for example, expect a typical first year studentto know what a right-angled triangle is, to remember the formula and to be able to use it tofind the hypotenuse given the two short sides.

For I in this case we would expect the student to have instant recall of the theorem and thor-oughly understand it and its limitations. They should be able to rearrange it to suit the prob-lem. If they are maths students we would probably expect them to be able to supply a proofwith little trouble, whereas if they are engineers then maybe a heuristic proof/understanding- it should at least seem natural to them. They perhaps should be aware of the conditions andlimitations of the theorem - restricted to a plane, etc - or of limiting cases such as a straight-linesegment. Perhaps the acid test is that they should be able to explain it in their own words tosomeone new to it - I never really understood it until I had to teach it

For T we would expect the student to be able to apply and initiate the use of Pythagorastheorem in new and/or unfamiliar contexts. This does not necessarily mean, even in the caseof engineers, applications to real world triangles - diagonal of a field, etc. It may includeapplication and transfer to other areas of mathematics - for example the distance between twopoints, equation of a circle or length of a curve by integration. There is no doubt that forengineers one would expect rather more T in a course than I, depending on the discipline, butthe applications used do not have to be specifically in engineering.

In reading through this example you may disagree with some of the categorisation and some of therequirements of the students. This is not the point of the example however, it is whether the K, I, Tstructure provides a useful framework for discussing what it is that we want our students to learn - whatdo we want them to be able to do, the sorts of skills we wish them to develop. It formalises I wantthem to know it, really understand it and be able to use it (Many mathematics lecturers are happy totake this KUU as their taxonomy!). Also, in placing emphasis on the K, I, and T it enables us to discussin a systematic way how we develop these separate skills, what teaching and learning methods we use,and how to assess whether the students have developed them. It seems reasonable that at university weshould aim to develop all skills, K, I and T to higher levels. In Section 2.7 we will look at how we designteaching and learning strategies and activities to achieve these different types of objectives. In Section5.5 we will see how the MATHKIT classification can be used in writing examination and courseworkquestions for assessing students.

There is a practical issue about specifying learning objectives - a real world compromise that is essentialin day to day teaching. No document of reasonable length can spell out learning objectives to sufficientdetail to meet all eventualities. In fact, to be manageable they have to be quite brief (An A4 side at most).However as the teaching progresses we constantly add clarification and expansion on various objectives.These have to be in the materials and notes the students receive (Verbal information in the classroom isnot sufficient, because it doesnt reach absent students). For example we might add that the proof of aparticular theorem is not examinable. We might mention cases of a particular technique that might notbe included (E.g. repeated eigenvalues in the diagonalisation of matrices). All such things, spelled out inthe teaching as we go along, are effectively firming up on what we expect of the students. An additional

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means of achieving such clarification is by the use of past examination papers to illustrate the sorts ofthings expected.

H Exercise

Choose a topic you are soon to teach and write the learning objectives that express what you want thestudents to be able to do in regard to this topic, itemizing the different parts in terms of MATHKIT.

2.7 The Teaching and Learning Strategy

2.7.1 How do we Achieve the Learning Objectives?

Once the content of the curriculum and the learning objectives have been decided, we then have to devisea teaching and learning strategy and activities that will achieve them. In pure and applied mathematicswe are fortunate in that for the bulk of our teaching the strategy seems pretty straightforward. We teachby explaining to the students how to do things, showing them some examples, and then set them off ondoing it themselves through exercises and problem solving. Of course, in reality it is not quite as simpleas this. For it is important to remember that this is not just a matter of how you, the lecturer, will achievethem, but how you will, with their help, support the students in achieving them. This requires planning,since it is not only your actions that need to be taken into account, but also the students actions and youreffect on them. It is where much of the intellectual effort goes into teaching.

For any given learning objective there may be a range of teaching and learning strategies to achieve it.The point is more easily illustrated by taking an example from, say, Chemistry. One LO may relate tothe theory of chemical reaction rates - solving a system of differential equations. In this case one canteach much as we would in mathematics. We might give a lecture on the key points of the theory, witha number of examples. This might be followed by a tutorial in which students work through problemsthemselves, and then we may send them off with an exercise sheet to work through in their own time,perhaps later submitted as coursework. These activities may be an appropriate strategy for this objective.However, for a LO that related to practical aspects of chemical kinetics, through which students areexpected to develop experimental skills, then clearly one would have to run a laboratory class in whichstudents observe and then perform experiments, later writing them up in their own time. These days,with simulation displacing much experiment, another leaning outcome may relate to the use of computerpackages in modelling chemical reactions, in which case you would need to run computer practicals todevelop these skills. In this example taken from chemistry the link between LOs and appropriate teachingand learning activities is pretty obvious. Since much of mathematics is probably best taught in the sameway as the theoretical LO in Chemistry considered above, it may appear that in mathematics the linkbetween objectives and strategies is straightforward. However, there is actually more to it than that. Letsnow consider the different types of LOs in turn.

In Knowledge we are trying to develop students basic knowledge and skills - facts and formulae, orroutine procedures. There is no point plodding through these in a lecture. They are relatively easy tolearn by memorisation and drill. We can simply give the students a handout (or book reference), withplenty of exercises and instructions on how to learn the procedures in their own time. However, in thelecture you can briefly describe the items of K, set them in context, explain why they are important,motivate them, and perhaps give tips on the best way to learn them. And of course you must appreciatethat this has implications for the students in terms of time and effort - you need to estimate roughly howmuch of the module learning time it will take up. If it is not intended that they remember such K andwill be provided with a formulae sheet for the examination, for example, then they have to be told, and it

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wont use up much of their time. But if they have to have the results at their fingertips then that could takesome hours. Of course, not all students will put in the time required, but that is then their responsibility- you have at least discharged yours!

To develop I and T is much more difficult, and far more time-consuming. Sometimes there is a tendencyto include too much K, and then not have enough time for IT. So, if you consciously wish to embedthese higher order thinking skills (HOTS) then you must accept that you must reduce the content - cutdown on the K. This is not reducing standards - it is trading low level skills K for higher level skillsIT. This is particularly so in first year courses. In the last couple of decades, in efforts to widen theappeal of mathematics, there has been a move towards more concrete aspects of mathematics and awayfrom abstractness. Also, there has been a move from deduction to induction. As a consequence, thedegree to which schools now develop the higher order skills IT has reduced significantly and many firstyear students have not developed these to a great extent. This means that they have to be proactivelydeveloped in their first year courses, and this takes time and effort.

So, how can we develop, perhaps from scratch, these I and T skills? Of course, the first thing is to providethe students with the requisite K content - this is the raw material that we use to develop appropriateIT. This can be presented in overview in a lecture and then developed through exercise classes, tutorials,and independent learning. Starting from a given body of K we then start to develop IT. Contrary toclaims by some authors, a traditional formal lecture can go a long way to developing these. The lecturercan tell the student what these skills are, set them in context, describe means of developing them. (S)hecan explicitly illustrate these skills by describing their own thinking processes as they work througharguments on the board. Obviously, this is best done in an active lecture, with students working throughexamples themselves, with guidance, or an exercise or tutorial class. The sorts of examples and problemsone uses is important - lots of problems building up from very easy to hard, called step-laddering. Thisis certainly how most of us learnt our IT skills - there are few short cuts! And of course we must ensurethat students have the required problem sets and associated work to do in their own time of independentlearning.

ExampleWhen you demonstrate a traditional textbook proof you are often displaying a highly sani-tized cleaned-up version of the proof, and it is certainly not how you would wish the studentsto set about tackling their own unseen proofs. So, if the objective is to instruct them in formal-ized textbook proofs of standard results then you might adopt one approach of systematicallyproceeding through the proof, explaining each step and encouraging the students to rehearseit until they understand it. This effectively converts the proof into K, as the students are basi-cally remembering the proof and need little in-depth understanding.

On the other hand if the objective is to train the students in methods of constructing theirown proofs to unseen results then this requires high levels of IT. In that case you would nothave them imitate you, but would adopt a strategy of allowing them to do more exploratoryindependent work. You would want to give them opportunities to express the proof in theirown words. This is likely to take longer than mere reproduction of the proof and you haveto allow for this. So in mathematics, even though the teaching methods are fairly prescribedthere is still a need for careful thought about matching the teaching and learning methods tothe desired objectives.

Planning a teaching and learning strategy, for a whole course, or an individual lecture or tutorial, needsto be done early, to allow time for preparation of materials. In general we need to think about:

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the resources available the students abilities and background alignment with the learning objectives ways of engaging the students in active learning behaviour.

We expand on each of these in the rest of this section.

2.7.2 Resources

Resources tend to be tight these days, and it is probably best to aim for cheap and cheerful teachingactivities in the first instance. And remember that resources includes your time. You have to balanceyour time between teaching, research and administration, and it is doubtful whether you will feel youhave enough for either! Within the time you allocate to teaching duties you then have to decide the bestway to use it, which becomes a question of priorities. For example, is it better to spend time producingexcellent word processed slides on Powerpoint or to use that time thinking about how to get a particularlydifficult message across to the students using hand written material on the board? You may want to showthem some property of curves - do you use some computer package, implying set up time and maybespecial equipment, or do you simply demonstrate it with chalk and talk? It soon becomes clear that even asingle lecture can throw up such questions of objectives versus resources. Indeed, one of the reasons thatthe conventional chalk and talk mathematics lecture is so popular is because it cuts down time neededfor media and frees up time for the message.

ExampleFor larger classes one is limited in the teaching methods one can employ . For example unlessyou can bring in assistants it is difficult to set work for the students that requires your indi-vidual attention to all the students - your simply dont have time to get round to everybody,so there will be students who dont receive your attention. The use of handouts can becomeprohibitively expensive, and if you use a set textbook there need to be enough copies in thelibrary. If you decide to use some computer aided assessment you will need large enoughcomputer labs.

In practice it is likely that the resource questions will be out of your hands and you will have to workwith what you are given, but there will still be some room for manoeuvre within how you teach.

H Exercise

Itemise the resources necessary for a few of your lectures. Include your contact time, preparationtime (including thinking time!), costs of any materials, student time, accommodation facilities,equipment, etc. Now consider any efficiencies that can be made, without compromising the LOs. Inother words, do a cost benefit analysis on some of your lectures!

2.7.3 Students Abilities and Background

Any learning activities need to focus on and reinforce the main mathematical message one is trying toconvey, without relying on things the students might not know or be able to do. As a general rule, ifyou give an activity during a lecture for example then it needs to be quite simple, yet getting to the point

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quickly and effectively, using students intellectual and motivational energies efficiently and not wastingthem. On the other hand, in a tutorial more substantial tasks can be set and if the students background isa little sketchy they can be expected to consolidate this on the hoof. Whatever tasks are set instructionsneed to be clear, so that students can quickly judge what they will need to recall and use from theirprevious learning.

So the teaching and learning strategy may vary depending on the level of the group. Thus, the first yearsneed more supportive and interactive strategies where there are plenty of opportunities for interventionand feedback. Such things are needed less so in the second year, and you can start to expand theirindependent learning skills and expect them to rely more on their own devices to work though material.In the final year their independent learning skills should be fully developed and lectures can be moreformal, relying increasingly on students abilities to follow lengthy argument, sort out many of theirsticking points themselves and seeking help when they really need it.

Examples

1. Graphics programmes are sometimes welcomed by lecturers because of the way one cangenerate and display any sort of curve or surface - they provide a visual dimension to themathematics. However, to the student all you may be doing is adding more confusingcomplexity. It may be more useful to simply develop students elementary curve sketch-ing skills, getting them to construct their own examples. This of course is also cheaper interms of resources!

2. When teaching separation of variables in differential equations, it may be best to startwith examples where the integrations involved are relatively easy, so the students canconcentrate on the actual process of solving the equation. It is probably better to do twoexamples with very simple integrations, rather than one with