M4 - Conducting mathematics lessons

Introduction

This module develops practical strategies to support effective teaching. It is not a comprehensive guide, but rather is meant to start you thinking and reflecting about your practice. It covers presenting information and acting on stage, delivery of mathematical lessons, introductions to establish a positive relationship, timing strategies, and effective use of technologies.

See Murphy (1998) for an excellent, easy-to-use resource on lecture presentation. Although it is not mathematics-specific, it contains more information about many of the points covered here. Krantz (2000) gives mathematics-focused advice.

Learning Outcomes

By the end of this module you will be able to:

encourage enquiring minds as opposed to delivering a suite of mathematical facts
effectively communicate with both large and small classes
utilise a range of teaching strategies to help students achieve learning outcomes
implement a range of technological options for unit and session facilitation, and describe how each of them can contribute to effective teaching.

Module Structure

This module outlines the following aspects of conducting a mathematics lesson:

Establish positive relationships
Follow good practice
Interact with the class
Engage students through technologies
Follow good practice in lecture recording.

Establish positive relationships

As a teacher you are not just delivering content, you are on stage and expected to perform just like an actor. This means that you will create an impression on your students not only through your body language, but also by the way you talk to them. The main aim should be to come across as a human being, not a robot delivering the material, and you should aim for a dynamic environment. A textbook can provide content; the advantage of a lecture lies in what you can achieve through your presence, and it is well-known that this can make a real difference to student learning outcomes (e.g. Ware and Williams, 1975),

Read this reflective article (Jungic, Kent & Menz, 2006) in which three lecturers describe the two-way, respectful relationship that must be established when teaching a large mathematics class (if your main interest is tutorials, see the alternative resource suggested in Task 4.1.)

Task 4.1

Read the following sections of the article by Jungic, Kent & Menz (2006) Reflections on large class teaching: pp. 1-6; also read Lectures on p. 8 and Notes on p. 9.

If your main responsibility is for tutorials, read this advice on Problem-solving classes given by the University of Edinburgh (1995). Note down points from your reading that you think are particular to mathematics lessons, i.e. that would not be found in a general article, or in an article on lessons in another discipline.

We will now explore some of the aspects of good practice in more detail. While we will primarily discuss lectures, many of these ideas apply or can be adapted for tutorials also.

Follow good practices

The first impression is very important. When you walk into a room full of students, don't look down at your feet, but look around and make eye contact with your students - smile into the class. Say hello to nearby students.

Introductions

Task 4.2

It is Week 1 of the semester and you are facing your students for the first time. You have been assigned to teach an introductory calculus (or statistics or discrete maths) unit to first year students. You are about to start your very first lecture. Write down a list of dot points that will cover your introductory remarks for this unit.

Here is a checklist that may help you get started:

How do you introduce yourself? Write your name on the board, indicating how you would like to be addressed.Are there administrative issues you should address immediately? For example, tell them the unit code (and check you are all in the right room and at the right time), how to contact you and others teaching in the unit, how to pass the unit, how to seek help, how to access the unit on the Learning Management System (LMS) and website (if there is one).
How should you present this information? Give as much information in writing as possible, so students who did not attend class can access it via the website, but also to reinforce the information for those present - information that is spoken but not written down is easily misunderstood especially by international students, and students will often fail to note it down. You should also advise the class when you want certain points to be written down.

Example: Good morning. This is the first class of MATH1000. My name is Joe Bloggs and I'm your lecturer for this semester. My consultation hours are from 2pm to 4pm on Wednesdays, and I am usually available to answer questions immediately after lectures.

Are there any techniques you could apply to get your students onside from the start?
Do you want or need to establish any ground rules for mutual respect?
Should you use an icebreaker in your class?

An icebreaker is an activity designed to ease tension or relieve formality; it briefly takes attention away from the unit material and attempts to help everyone to feel relaxed and get to know each other. An icebreaker can help set a positive tone for the unit. Note that this is something mathematicians are often bad at, and mathematics students are also likely to be uncomfortable with active participation. Do not push too far past your comfort zone or your students’; it may be sufficient to introduce yourself, say a few words about your research or other interests, then move on to an overview of the unit. On the other hand, if you want to use interactive and collaborative exercises in your lessons, or to regularly call for responses from your class, introducing interaction from the start establishes this as an expectation in the subject. Examples include a mild joke (warning: attempt at own risk!) or getting students to say something about themselves, such as what they are looking to get out of their university studies. In a large lecture, this would be to their nearest neighbours. Can you think of two other possible icebreakers?

Example: Possible icebreaker activities include putting up a few background problems to solve in small groups, or asking the class to solve a mathematical puzzle, or holding a short quiz with a small prize. These ideas have an added benefit of revising needed background material, and allowing you to evaluate where the class is starting from, but your icebreaker does not have to be mathematical at all.

Module 11 includes a more detailed discussion of icebreakers and collaborative learning.

So you’ve introduced the unit, and given your first class. Something else to consider:

Are there benefits to having an introduction to every class?

Task 4.3

Once again, you have been assigned to teach an introductory unit to first year students. It is the first minute of any teaching session. How should you start the session? Write down a list of common dot points that will cover your introductory remarks (see Murray & Murray 1992, p. 111). Points that may help you to get started are:

Include an overview of what you are going to cover in the session, and put this in context of the overarching topic under discussion, and the whole unit.
Include a brief motivation for why today's topic is important. You shouldn't see this as a waste of time, it is important for students to make the link between classes and topics and to know what will happen next. Show important information on slides/whiteboard as well as this helps visual learners.
Include a brief review of what was covered in the last class, e.g. by giving a summary, or by posing questions that students answer.

Example: To give an overview of the current lecture, and put it in context, you could say something like: Today we will finish the topic of differential equations by looking at an important application, known as a damped oscillator. Although we will be considering a simple system consisting of a just a weight on a spring, damped oscillators occur in many real world systems, for example car suspension. We will model this system with a homogeneous linear second-order ODE with constant coefficients, which we learned to solve last lecture. By the end of the lecture, you should understand how the different types of solutions to this equation correspond to different types of behaviour in the physical system.

Although the ideas above may appear to be commonsense, there are sound pedagogical reasons for providing introductions, making connections and facilitating student interactions in your classes: re-read the Seven principles of learning from Module 1.

Timing

Simply getting through a set amount of material should not be the major aim of your class - the aim should be that your students understand the material and learn to solve mathematical problems on their own. Covering all the material might make you feel good as a lecturer, but your students may not take it all in.

The timing of your class presentation is very important as it can have a deep impact on your students' learning and your own stress levels. While you may perform well under stress, your students may feel rushed and won't appreciate it if you explain material faster than they can write it. You can gauge your students' response to your delivery by watching them and asking questions as you go along. An increase in noise level may indicate that you are losing their attention. If you try to cover too much content, you will inevitably run out of time and start to rush through the material in the final minutes of your lecture, which would be a waste of effort for both you and the students.

For example, you might be taking an introductory calculus lecture where you have been explaining that every differentiable function is continuous. You've given the first of two very good examples you have prepared, and you find yourself at the end of the lecture time. While you had planned to give the second example, even if you think it to be extremely important, the reality is that the students will be thinking of the next class and simply not be absorbing what you are saying. If the example is really that important, it can always be posted on the LMS.

Disruptions can occur, such as fire alarms, technical equipment failure or power loss. You should be aware of the time constraints and be prepared to make a clear decision about material that you don't have time to cover properly. This is true even if you were at fault through poor planning, or if it simply took longer than expected to properly explain something, or if you had lots of questions during the class. It is also courteous towards the lecturers and students who follow you in the same room to finish your class on schedule and give them sufficient time to set up for their class.

When you reflect on a class in which you had timing problems, you should consider the reason/s it happened. Even a bad day can eventually have a positive outcome if you learn from it. One strategy is to annotate your notes immediately for next time you present the class, suggesting where you think you could shorten the presentation. If your second example really is important, should you plan to keep a better eye on the time and abbreviate the first example, or some other part of the lecture, if necessary? If you are regularly running short of time, then you should reconsider the amount of material or number of examples you prepare for your future classes in the current semester.

Task 4.4

What do you do when you've run out of time? Watch the screencast below and observe the strategy that is being used to deal with a shortage of time. Then write a list of possible strategies that you could follow.

Screencast: Out of time

Some possible examples of strategies that we have identified are:

Skip examples and set them as exercises to be done at home. You could then distribute solutions later, for instance through your unit website.
Cover some of the lecture material in tutorials. Give your tutors clear instructions, e.g. hand them the material you were going to cover and walk them through it.
Or, if your tutorials are based on students working from problem sheets, forewarn the tutors as to which parts of the sheets are likely to elicit the most questions.
Simply postpone material to the next class, and make adjustments to the material for the next class to accommodate the change. (Don't forget to tell the tutors!)
If all else fails, remove a segment of material from the unit, making it clear that it is not covered and that it will not be assessed.

Dealing with a situation where you run out of time in class will become easier with practice. Watch the following videos, where two very experienced mathematics professors talk about their approaches.

Video 1: Professor Walter Bloom - Running out of time

Video 2: Professor Nalini Joshi - Losing time

Interact with the class

Every group of students is different, within the same semester if you are teaching several classes, but also from one unit offering to the next. This means that your approaches to teaching a topic may need to be adjusted depending on your students' existing skills. For example, in one year you may have a strong class that picks up the Chain Rule quickly without further explanation; in another year or to a different class, you may need to give several examples and explain in different ways if you detect blank stares from your students.

This is an important point: If a student does not understand a particular mathematical concept, use a different approach to explaining it.

Don't repeat the same explanation several times as this may lead to frustration in you and your students. Try to find out why they don't understand, e.g. by asking which steps they did not understand, or by asking them to explain the steps to you. In this way, you will be able to identify gaps in their knowledge, and you can target those directly. If you find that many students have the same difficulty, then you will need to adjust your teaching. Podcasts or videos can save repetition of the explanation.

Keep this in mind: Adjust your teaching to your students' understanding.

Quite often, you will have a class with a diverse range of skills, particularly in first year units. You will need to find a balance between explaining in too little or too much detail. You should be offering additional help to those students who are struggling, asking them to attend lecturer consultation hours, or referring them to mathematical learning support at your university. You could also offer extension material to the brightest or most eager students.

Student difficulties may be identified by observing attention in class and student responses as well as their written work. It is important to respond quickly and positively as soon as you become aware of issues. Refer to Module 7 Collecting evidence about your teaching for further discussion of this practice.

Task 4.5

Write down a list of strategies to help you assess your students' understanding, and approaches you can take to lead the students through these difficulties. To help you in this task, you could ask a more experienced colleague with a reputation for good student feedback about their strategies. To get you started, we have given you two examples of strategies that you could try.

Example: Ask a revision question at the beginning of each class. Reveal the solution, and ask for a show of hands to gauge how many of your students still remember the concept.

Example: Where you have explained a new concept or while explaining it, seek constant feedback from your students by asking if they understand, and encourage questions if they don't. Look at them to gauge if you have made sense to them. And establish an environment where students don't feel stupid for asking a question. Clickers could be used here.

See Module 7 for more detailed strategies of collecting evidence from students and colleagues; and see Module 6 for more detailed strategies on how to create assessment tasks to monitor students' learning.

Engaging students

Among the Seven principles of learning (Module 1), recognising different learning styles was mentioned. Traditional lectures appeal to auditory learners. For some students who are visual or kinaesthetic learners, diagrams, physical objects and demonstrations will help them better understand mathematical concepts, and these cannot necessarily be found by reading the textbook. Social and collaborative learning is also important to many students. Variety in your teaching strategies adds interest for all students, and does not need to be elaborate or take a lot of time or special equipment.

For example, the Jungic, Kent and Menz (2006) article on teaching large mathematics classes suggests cutting up fruit in a lecture to demonstrate the disk and shell method (p. 5). Here are some other ideas: borrow a guitar to demonstrate properties of solutions to the wave equation; an egg carton has fascinating cross-sections and contours, and has saddle points as well as maxima and minima; use two long pointers to demonstrate skew lines in linear algebra. In pure mathematics, bringing in objects to demonstrate symmetries may help students with abstract concepts. In a statistics lecture, consider if you can collect (non-sensitive) data from your class to illustrate a point you are making.

Depending on the technologies you use (see the final sections of this module), video, web-based simulations or software output (e.g. from Maple or statistical software) can also be illuminating and provide variety in your presentation. Engaging students through technologies will be discussed next.

Engage students through technologies

While you might have been taught with a "chalk and talk" approach of writing mathematics on a whiteboard or blackboard, many universities now mandate the use of technologies for teaching. These usually include an online presence on a unit website, a LMS with forums, the use of lecture-recording software, and tablet technologies or interactive whiteboards. Some teaching rooms no longer provide standard whiteboards - they have been replaced by document cameras.

If you are overwhelmed by these technologies, take small steps to gain confidence. While technologies may aid your teaching, they have to be used in appropriate ways to be effective and should not drive your teaching approach or limit you in what you do. Also, always have back-up when you are new to technologies. For example, if you are planning to use a computer, bring a printed copy of your material just in case something goes wrong. If there is a whiteboard, even if you are not planning to use it, make sure that you have markers in case all else fails. Take a minute to think through your back-up plan to make sure that it is workable. Your university probably runs training sessions on using the technologies installed in teaching areas, and you should also find out how to call for technical support.

The following section is split into two part - technologies for face-to-face delivery, and technologies for online delivery. See Module 11 for more strategies for creating mathematics learning communities through technology.

Face-to face-delivery technologies

The most popular technologies for delivering lectures via computer (on Windows operating systems) are Microsoft PowerPoint presentations and LaTeX-based documents. Both have pros and cons for mathematics teaching. If they are installed in the lecture theatre, document cameras (also known as visualisers) can provide an appealing compromise between traditional and fully computerised methods, with all the advantages of using a blackboard, whiteboard or transparencies, but with fewer limitations. All of these technology options allow the presentation of prepared, typeset notes which can be reused or made available to students. In very large classes, the ability to project on to an appropriately sized screen is also a major advantage over a blackboard or whiteboard.

Task 4.6

Discuss the pros and cons of each of the following technologies:

Powerpoint presentation
LaTeX-based presentation
document camera
overhead projector
whiteboard
tablet PC.

Post some of your ideas to the discussion board; before you do, see what the other participants have already said, and see if you have any different perspectives or wish to respond.

With both PowerPoint and LaTeX-based presentations, and when using printed notes with a document camera, problems and all working can be shown to students in neatly typeset form. However, students learn mathematics better when they see the development of a solution, and when they can contribute to this development by taking an active approach rather than watching an already prepared and fixed path. For this reason, it is important that the mathematical solution is written down and developed during the lecture. In this way, the general skills of problem solving, communication and logical thinking - that are as important as the accuracy of the solution - are modelled for the students. This also allows the lecturer to respond to student questions or suggestions of alternative paths, and it gives students the opportunity to contribute to the solution and see their contributions written down for all to observe. It also means that the lecturer can react to student ability rather than being limited to what was prepared beforehand (Loch & Donovan, 2006). The following quote is very relevant.

"One reason we all use blackboards to write down mathematics is the symbols with which mathematics is communicated. Writing the symbols down gives the student a chance to read what has been spoken and thus access the content via several senses." (Townsley 2002, p. 2)

With some universities mandating the recording of lectures, electronic writing is gaining increased importance. For example, the Lectopia (or Echo360) lecture recording system allows recording of a document camera (if installed), or of a tablet PC screen. Camtasia Relay is another tool that allows recording of any screen movement. If using a tablet PC, note that at the time of writing this module, Classroom Presenter, PowerPoint and Windows Journal/MicroSoft Office OneNote supported the most effective writing with a stylus; and PDF Annotator was the easiest-to-use program to annotate PDF documents, including LaTeX documents converted to PDF format. If electronic writing is not possible, it will probably be necessary to complement a computer presentation with a traditional write-on board. Even if this is practical in the classroom set-up, it has the drawback that your writing will not be recorded (except possibly on video).

Task 4.7

Imagine that you are planning to deliver a session in a unit you teach. Decide on a platform to use, and your teaching strategies, and prepare all appropriate material for the lesson, including handouts. (Don't forget to start by writing down your goals and objectives.) Do this exercise for the most appropriate scenario for your teaching context- for instance a small tutorial and limited technology, or lecturing a large class in a lecture theatre with a full range of technological tools. You should reflect on how best to achieve the goals and objectives with the available tools. (This task relates directly to Assessment Task Two.)

The use of “remote personal response systems” or clickers (Educause Learning Initiative, 2005) has both advantages and disadvantages; consider whether you can capture some of the benefits with simple polling of hands or “thumbs up/thumbs down”. If you don't have access to clickers hardware, you might consider benefiting from students’ own portable devices that connect wirelessly to the internet, e.g. mobile phones, iPads or tablets. These can be used to record responses from students using software available at low cost.

Online delivery technologies

Online delivery is vital for distance education, but has also become very popular to complement traditional face-to-face teaching in a "blended" approach. Use of a standard website is perfectly adequate for the distribution of material including sound and video, but more options and greater engagement are facilitated by a LMS (or unit management system) such as Blackboard (proprietary), or Moodle (open source). The LMS you use is the one provided by your university.

Recordings are an important means of online delivery. You may choose to make a video of you giving a lecture (so that the students can actually see you talking), or you may use screen-capture technology (in which case the students hear you and see your working on the computer in real time, but cannot see you); unfortunately audio alone is of little use for mathematics. Most lecture theatres these days have recording technology installed, and screen capture is also available outside the lecture theatre using a tablet PC with software such as Camtasia Studio. Screencasts can be useful as a supplement to full lecture recordings. These are short recordings, usually of difficult concepts, which can be made available on demand for students who are having trouble understanding a particular idea.Use of a LMS and lecture recordings are discussed further below.

Other technological tools, although less widely used in teaching, can allow a higher degree of interaction, more like a lecture or tutorial; this is particularly useful with distance students. Synchronous chat support or tutorial sessions can be run using a standard client such as Windows Live Messenger (Loch and McDonald, 2007), where an "electronic ink" tool allows instant sharing of handwritten text or pictures. This is useful because mathematics generally requires special symbols and diagrams, and so ordinary text typed on the keyboard is too restrictive.

Another option is the use of web conferencing software. This - along with other approaches, particularly techniques using a tablet PC - is discussed by Galligan, Loch, McDonald and Taylor (2010). The software tested in their study provided several tools. Based on student feedback, the most important for mathematics teaching were a "shared whiteboard", audio chat, graphing calculator, and typed chat for short messages. In addition, screen sharing was identified as most important for working through study material, assessment tasks and MATLAB code, and was also identified as the most valuable tool in those statistics tutorials that were included in the trial.

Follow good practice in lecture recording

Task 4.8

Loch (2009) identifies several attributes of recorded lectures may help you explore whether or not lectures should be recorded:

saves time, (as opposed to recording a lecture in your office afterwards)
gives students a real classroom experience without entering the physical classroom, but
takes longer to watch than recordings created in an office (because live lectures usually take longer than when you record with no audience in your office)
is therefore also of larger file size,
often contains plenty of ums and OKs and sooos,
needs to be managed well to ensure there are no silent gaps when a student is talking without a microphone.

Communication face-to-face is ideal. If this is not possible for some students, the next best is to record the material. If recording using audio and screen capture, be aware that students will not see your face and you can no longer rely on body gestures or facial expressions to put emphasis on what you say. This now needs to be communicated through the use of your voice alone. That said, most of the important points to remember when making a recording are also worth remembering in a live lecture (or tutorial).

Task 4.9

Watch the following screencast on proof by induction. See if you can identify examples of the following in this recording, or if you can you suggest where the presenter could have approached the problem differently to achieve these ends.

Also consider why the following issues are important:

confidence
preparation
being relaxed
enthusiasm
careful timing
combining written and verbal cues to the mathematics.

Screencast: proof by induction

Task 4.10

Have you ever listened to a recording of your own voice, and thought it was absolutely terrible? The best way to check the quality of your recording is to listen to it yourself, as you are your own best or worst critic. Record yourself teaching the same Proof by induction (or part of one of your lectures). Listen to your recording. Did it exhibit the above characteristics? Ask yourself the following questions:

Did I get my message across?
Was I speaking too fast?
Did that go too long?
Did I fall asleep?

If you can answer the first question with YES, you are on the right track. The other three aspects may require a bit of practice, and discipline unless you are a natural (Slow down! Keep it short! Show enthusiasm!).

You don't need to listen to all of your recordings - just listen in from time to time. Also, you could ask a colleague to listen and to comment on your recording.

Many universities offer presentation skills workshops or voice training. Try them out, they can be fun.

Review and conclusion

After implementing the above strategies you will have established an effective learning environment, and one in which students will be willing to contact you, thereby creating a feedback loop. Constructive interactions with you will allow students to understand their difficulties and how to best improve. Observing their difficulties should in turn help you identify problems in your lessons, and provide suggestions about future steps. This will be covered in more detail in Module 7, where you will plan and implement activities to obtain feedback from students. But first the next module discusses the important area of service teaching.

References

Arthur, D. (1995). Problem-solving classes. In F. Forster, D. Hounsell, & S. Thompson (Eds.), Tutoring and Demonstrating: a Handbook (pp. 25-38). United Kingdom: Centre for Teaching, Learning and Assessment, University of Edinburgh. Retrieved February 17, 2014 from http://www.docs.hss.ed.ac.uk/iad/Learning_Teaching/Tutors/Handbook/Tutors-Chapter4.pdf
Educause Learning Initiative (2005), 7 things you should know about clickers. Retrieved July 13, 2011, from http://net.educause.edu/ir/library/pdf/ELI7002.pdf
Galligan, L., Loch, B., McDonald, C., & Taylor, J. (2010). The use of Tablet and related technologies in mathematics teaching. Australian Senior Mathematics Journal, 24(1).
Jungic, V., Kent, D., & Menz, P. (2006) Teaching large math classes: Three instructors, one experience. International Electronic Journal of Mathematics Education, 1(1). Retrieved from: http://www.lon-capa.org/papers/JungicKentMetz.pdf
Krantz, S. (2000). How to Teach Mathematics (2nd ed.) Providence, Rhode Island: American Mathematical Society.
Loch, B., & Donovan, D. (2006). Progressive Teaching of Mathematics with Tablet Technology. e-Journal of Instructional Science and Technology, 9(2). Retrieved 16 February, 2011, from http://eprints.usq.edu.au/archive/00001716/
Loch, B. (2009). Web 2.0 for all. Podcasting and screencasting professional development material. University of Southern Queensland. Retrieved 16 February, 2011, from http://www.sci.usq.edu.au/staff/lochb/relay/study_modules/Relay%20Good%20Practice.htm
Murphy, E. (1998). Lecturing at University. Bentley: Curtin University.
Murray, J. P., & Murray, J. I. (1992). How do I lecture thee?. College teaching, 40(3), pp. 109-113.
Townsley, L. (2002). Multimedia Classes: Can there ever be too much technology?, in Proceedings of the Vienna International Symposium on Integrating Technology into Mathematics Education, Vienna Austria. Retrieved from: http://rfdz.ph-noe.ac.at/fileadmin/Mathematik_Uploads/ACDCA/VISITME2002/contribs/Townsley/Townsley.pdf
Ware Jr, J. E., & Williams, R. G. (1975). The Dr. Fox effect: a study of lecturer effectiveness and ratings of instruction. Academic Medicine, 50(2), pp. 149-156.