Chapter 4 Learning Mathematics from Lectures and Notes

Figure 4.1: Baby holding mathematical textbook.

This section is predominantly devoted to live, in-person lectures and how to get the most from them. However, since the Covid-19 pandemic, some lecturers/institutes use an online delivery method when it comes to core lectures. You will be made aware of the delivery method used for your course/modules at the start of the semester/term and full instructions will be available on how to attend lectures, whether in-person or online.

We discuss the differences between the online and in-person methods in the ‘Online learning’ section below.

Note taking and note using

It is a common misconception that every detail of a lecture should be understood at the time when it is given – this is very rarely possible in mathematics. If possible you should try to follow the steps and understand what the lecturer is doing.

The most important information about any lecture course will be the set of notes which you have obtained by attending it; you may also be given a digital copy but the exact form of ‘lecture notes’ will differ from course to course and lecturer to lecturer. In addition, you may have to make notes yourself from reading relevant textbooks. It is essential that these notes should be as clear as possible and arranged so that you can find information in them with ease.

Acquiring the skill of making good lecture notes is essentially a matter of common-sense and practice; you must also make the effort yourself and be conscientious. Some of the suggestions given below may seem so obvious that they hardly need to be stated but it is amazing how easily they can be ignored or neglected. You should always:

• Arrive at the lecture in good time and with all of the necessary equipment. Some may like to make notes with pens, pencil and paper, while others may like to write on a tablet or annotate digital notes. However you like to take notes, it is useful to highlight important points and theorems. Sit in a position that allows you to see and hear well.

• At the outset you should be given a provisional course plan together with a list of recommended texts by your lecturer. If not then ask for them. During the course this plan will give you warning of the introduction of a new topic and it will help you to structure your revision.

• Number the pages of your notes; mark the place where each lecture starts with the date.

• Leave plenty of space so that you can annotate your notes easily when you are studying them afterwards.

• Use headings carefully to distinguish major from minor results, proofs from examples, theory from techniques.

• Make sure that your notes are accurate. You can waste a lot of time subsequently if, for example, you have written down an inequality the wrong way round or written \(x\) instead of \(x^2\), say. So take care and check with a friend if you are unsure.

• Your lecturer may use abbreviations or notations not found in textbooks so make sure you understand the meaning of these and what equivalents are more common.

• Your lecturer may make an error – everyone does from time to time. If you suspect this do something positive and active to correct it. Ask your friends what they think; approach the lecturer after the lecture; approach your tutor; or simply ask a question during the lecture. More often than not you will be helping everyone. Even if no mistake has been made the question will help you (and others) understand.

In the lecture

A constant problem in lectures is that you are required to listen to the lecturer, look at diagrams/graphs, read explanations and make your own notes, all at the same time! To do this successfully requires concentration and skill. What the lecturer is saying is heard only once; you do not have much time to decide what part of it to write down. A balance must be achieved between taking no notes of the spoken word and trying to make a word-for-word transcript, and the art of doing this must be mastered as quickly as possible.

When you don’t understand the lecture, don’t panic; keep taking notes and seek help as soon as possible from the lecturer, your tutor and other students. Don’t be afraid to ask questions during or at the end of the lecture; usually other students don’t understand either and will admire your courage. Lecturers, like other human beings, make mistakes – sometimes ‘slips of the pen’ and sometimes mathematical errors – so do point these out as soon as you spot them! And remember that difficult ideas become easier to understand if you work at them through reading and doing problems.

You should identify the style of lecturing. Here are some common types:

• Continuous written presentation without discussion
  This is a very formal style of lecturing which is sometimes still encountered. You should copy everything down and ensure your record is accurate. Don’t hesitate to ask the lecturer to slow down, to write more legibly or to re-display slides if you are having difficulty in keeping up. It is essential to re-read the material as soon as possible to consolidate it, since there will be no chance during the lecture.

• Written presentation punctuated by spoken discussion
  It is essential to be able to distinguish between the formal notes and the discussion – ask for clarification if necessary. Ensure accuracy etc. as for the case above. Summarise the discussion by making comments at appropriate places in the margin.

• Two stage lecture (Class discussion followed by formal lecture or vice-versa)
  Try to participate in the discussion even if you feel self-conscious or nervous; others certainly share these feelings. It is often not necessary to take notes during the discussion stage – listen and participate.

• Lecture supported by set books and/or duplicated notes
  Read the relevant section before going to the lecture. Make notes of any points causing trouble so that you can raise these difficulties during the lecture. Supplement notes with written comments from the lecturer. Above all, take your notes or books to the lecture for reference.

Aim to make out a set of neatly laid out, coherent notes during the lecture. Some people prefer to make rough notes and re-write them afterwards but very few can sustain the effort required for this. In any case, the time after the lecture can be spent more profitably.

After the lecture

As soon as possible, work through your notes tidying up and expanding where necessary and checking the accuracy wherever there is any doubt (where, for instance, you may have copied something down in a rush). Try to learn any new definitions straightaway, make sure that you understand the details of any worked examples, pick out the important features of any new concepts and proofs. You may find it helpful to extract the key points, such as definitions, theorems, important examples or counter examples etc. and write them on your own ‘cheat sheet’ or summary page; this will prove invaluable when it comes to revision time. An alternative method is to highlight the key points in your notes. However you choose to do it, you should make sure that you make a good index and a brief summary for each set of notes.

So, with pen and paper at hand, read your notes actively. This is different to the way you read a novel, for example, and involves you analysing and clarifying your notes - do you understand the important points, the steps of a proof (can you fill in the details?), can you replicate worked examples and apply your notes to tutorial/homework problems?

Read your lecture notes, together with problem sheets, trying to sort material into order and into different categories; definitions, examples of definitions, techniques, minor theorems, major theorems, applications of theorems and techniques and finally, proofs of theorems. Look for examples both in your notes and in other sources (books, problem sheets, exam questions) and add these to your notes in the appropriate place. You could work together with other students in doing this.

If you have been told to read part of a textbook, or handout, do so as soon as you can while the subject is still reasonably fresh in your mind. Make notes on what you read – doing this is rather like working over your lecture notes and picking up the ‘key points’. It is advisable to cross reference the material so that annotations in your lecture notes point to relevant parts of notes from books etc. and vice versa.

If you are having difficulty with the lecture content or the reading which your own efforts and discussions with other students cannot resolve arrange to go to see your lecturer or tutor. Do this as soon as possible, otherwise you will fall behind and find yourself with several ill-understood lectures to sort out. It will be easier for your lecturer to help if you can be precise about your difficulty. Take your notes with you and be sure to have relevant places marked.

If all of the above advice seems rather daunting and hopelessly time-consuming, don’t despair! Do what you can. You will find that you get quicker and better as time goes by - you will become a more efficient student. By following these ideas, you will acquire a record of each lecture course which can be referred to easily at any stage, is a clear account of the topic, and is an invaluable aid at revision time.

Online learning

We all hope the days of forced lockdown are over and we will not be returning to a fully online learning experience again! But some aspects of online learning have their benefits and have persisted.

While most lecturers have moved back to lecturing in-person, in the styles described above, many have developed an hybrid method.

A couple of different methods adopted are as follows:

• Flipped Classroom
  In this method the lecture material is presented asynchronously, in advance of the class meeting. This is usually via video recordings or other online materials. You are expected to engage with the online content; listen/watch/read, take notes and formulate questions on any topics which you did not understand. There may be examples which you are expected to try. In the live class the discussion is then led by the students (guided by the lecturer); it can involve individuals or small groups presenting examples/proofs/problems in front of the class or working in small groups.

• Online Live
  This method is very much like a traditional lecture (presented synchronously), but delivered online. The lecturer may share slides, or write live on the screen for students to follow. This may be in the ‘continuous presentation’ style or it could be punctuated by class discussion. The interactive portion of such lessons can be achieved in a variety of ways; a non-exhaustive selection includes:

  • Polling,
  • Breakout rooms,
  • Use of the chat function,
  • Or simply opportunity to ask questions.

Summary

It will take time to hone your note-taking skills, but you will get better and the more effort you put in at the start, the easier it will be as you progress.

• Be prepared with whatever materials you need for taking notes in your lectures (pen/pencil/paper/tablet/laptop).
• Try to keep your notes as neat as possible, and organise by dating the notes and numbering the pages.
• Consolidate your notes by going over them afterwards, clearing up any misunderstandings or working through relevant problems.
• Speak to your lecturer, tutor or friends to correct mistakes or clarify tricky sections.
• Finally, summarise your notes to allow for easier revision and access to key points/theorems/examples.