What your future lecturer wishes you knew
This is a post that I wrote in 2014 after attending a university readiness panel discussion in Cape Town. I have adapted it slightly since then. It is worth a read for anyone wanting to study Mathematics in the future, or is aiming to write NBTs in Matric.
The panel consisted of Maths lecturers from the Cape Peninsula University of Technology, the University of Cape Town, the University of Stellenbosch and the University of the Western Cape. They discussed what they wished students knew when they started university. I am going to share a few of their points of discussion with you. I think that they are relevant, not only for those of you planning to study a degree that involves pure Mathematics, but for anyone studying further after school. Maths is used by virtually everyone: Business Science, Commerce, Engineering, Science, Law, Medicine, Psychology, etc.
Lecturers expect you to be familiar with the basics of school Maths, such as algebra, trigonometry and coordinate geometry, and they do not revise these again. You should have a “toolkit” of Maths skills that you can do almost without thinking. So pay attention in class, and ask, ask , ask until you understand! If you have a weak background, you may find that you get bogged down with details and are not able to get to grips with important new concepts.
Arguably the biggest change in moving from school to university Maths is the shift from asking “How?” to asking “Why?”. At university, there is more emphasis on understanding rather than just doing. This means that completing homework exercises or tutorials is not enough. Reading lecture notes and worked examples is not enough. Following algorithms, the strategy of relying on a given template to do problems of a similar kind, is not enough. These approaches to learning, if done in isolation, will no longer give the same rewards as may be the case in school. You need to engage in aspects of Maths that are sadly not emphasised enough at school: Mathematical reasoning, problem solving and reflection.
Teachers can help you to prepare for this change in emphasis by giving you problems to do on your own that go a bit beyond the type of questions that you can expect in your Matric exam (or your grade-relevant exam). But do not wait for your teacher to offer you these questions. Ask. Or go find them yourself. A good source of such questions is the Varsity Test, or Varsity Readiness Test, that regularly appears in the quarterly magazine, Mathematical Digest. You can also buy a National Benchmark Test (NBT) book at most bookstores. These tests include questions that are designed to highlight mathematical skills that are of particular relevance for students planning to study Maths at university level. These features include: Arithmetical ability (without needing a calculator); algebraic skills; thorough knowledge of trigonometry formulas and how to derive and use them (without a formula sheet); geometrical and three-dimensional perception; and understanding of logic (proofs, counter-examples and contradiction).
There is far more emphasis on proofs in university Maths courses than there is in the current school curriculum. This is linked to the emphasis on understanding and logical reasoning in university courses. So, when you learn or do a proof in trigonometry or geometry at school, don’t just learn it off by heart. Think through the steps and try to understand the logic and reasoning. This will help to prevent a “culture shock” when you go to university.
Learning to read Maths on your own is very useful. You can know all the theory in the world, but if you do not understand the question you won’t know where to start or what theory to apply. When you read a question, pick out the instruction words. Are you being asked to calculate, solve, explain, simplify, prove, etc? Also look for any keywords. Do not just brush over the little “blurb” at the beginning of a question. It usually contains a roadmap for navigating the correct solution.
The mathematical environment at university is very different from that at high school. Lecturers give their lectures and walk out of the lecture hall. Classes may be very large (in the hundreds). Your lecturer may not know your name or notice when you are in a lecture. Nobody checks to see that you do your work and keep up to date; this will be your responsibility. So, take ownership of your learning from the start. Revise daily and form study groups. Most universities have tutors ready to help you. Go see them. Attend tutorials. Use online resources like Khan Academy and Wolfram Alpha when you do not understand a concept. You should even take the bold step of introducing yourself to your lecturer and asking him or her for help when you are struggling. But be sure to take a worked attempt at a problem as a starting point for your discussion.
Some practical matters: For many Maths courses no formula sheet will be provided. In addition, you will probably not be allowed to use a calculator (so you had better learn those trig special angles). Exam papers may also contain multiple choice questions.
This last point, I believe, is one of the most important. LEARN TO STRUGGLE! Do not give up if a problem seems difficult. Approach it from different angles, work on it collaboratively with other students, verbalise your thinking and strategies. You will learn so much more if you struggle through a problem than if you just read through the solution or have someone else explain it. I encourage my students to have three different coloured pens in front of them when they are working through a question: A blue or black pen, a green pen, a pink or purple pen, and a red pen. Write down your first attempt at the question in blue or black. Especially when you are preparing for exams, do the question in the allocated time. When your time runs out, if you are still busy, complete your solution in green. If there are aspects of the question that you are still struggling with, take out your notebook, textbook or study guide as a resource and answer the rest of the question in pink or purple. Then mark your solution in red. Do full corrections. If necessary, go back again later and do that question again.
I am sure that there are many more tips for coping at university. If you have any that you would like to share, or if you have any questions, please comment on this post on my website. I would love to hear from you.
Yours in Maths,