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DEPARTMENT OF MATHEMATICS Private Bag X1314, ALICE, 5700 Tel. 040-6022369 Fax. 040-6022369 Email: bmakamba@ufh.ac.za HOW TO STUDY MATHS AND PASS AT TERTIARY LEVEL Most learners find Mathematics hard and frustrating. They then conclude that Maths is meant only for a small group of intellectuals. They develop a negative attitude towards Maths and may even hate teachers of Mathematics. Some Maths teachers, unfortunately, deliberately make Maths appear as being reserved only for the small elect. Of course a lazy person is not likely to excel at Maths, not because he does not have the intelligence to cope, but because he is not diligent. At this point let me offer a few suggestions to those who want to excel at Maths, irrespective of their natural intelligence. Make sure that you are motivated to excel at Maths, otherwise my suggestions may not help. 1. First realise that every subject or course has its own language that is unique to it. In Maths the language is further written in symbolic form. Learn the language and its symbols. When reading a mathematical statement whether in words or symbols, make sure you understand each and every word or symbol, otherwise you may learn wrong information. When reading a definition, do not simply memorise it, make sure you understand its meaning because a definition is a basis of argument in proofs and computations. 2. Having learnt a definition of a concept, look for easy exercises that test understanding of the definition. First you should start with examples in a book or as given by your teacher. Do not simply follow what is given in the examples, ask yourself the question ``why?’’ several times, and then attempt to answer the question yourself. 3. Understand various ways of proving: e.g. Induction; Direct proof; Contradiction; Contraposition. At a later stage you may also attempt to be innovative and defend your innovation. In proof by contradiction you will need to negate statements, this is usually a powerful tool. 4. Do NOT be quick to ask your lecturer or peers for assistance when you can’t solve a problem at first or second attempt. You will never develop selfconfidence if you are quick to ask; yes you may pass exams, but you will not excel at mathematics. I re-iterate, start with easy exercises that make you have a grip of basics such as definition and statement of a theorem. 5. Mathematics is very kind to those who are diligent. Diligence does not accommodate too much asking because that way someone else is doing the work for you. It is good to sleep and dream on a maths problem. This makes you try various techniques and ultimately to find the one that works. Thereafter you should go back to the other techniques that failed and ask why they failed. The answers should come from you as much as possible. 6. Do not be quick to give up; quitters usually fail. Do take a break after a struggle with a problem and then come back refreshed. Try another strategy or technique or approach. 7. Carelessness with use of words and symbols can be your downfall. Try to eliminate careless talking and writing. 8. Do involve yourself in a discussion group, but do not be passive, in fact do take a leading role. Others will question you and that way you learn to master the basics by explaining. 9. If you find that there are concepts you did not learn or master in lower classes, for example in school mathematics, make it your responsibility to find a relevant textbook and learn those concepts on your own. Most often students who lack the necessary background find it difficult to cope with Mathematics. Take this point seriously. 10. Do attend lectures regularly even if you do not follow what your lecturer is doing. When you read on your own you may relate what you read to some of the things that were said by your lecturer. It is suicidal to stay away from mathematics lectures no matter how busy or down-spirited you are. 11. Try to keep up to date with your work, even if you make a quick reading just before the next lecture so that you are not completely lost during a lecture. Most often mathematics builds on previous work, hence the need to keep up to date. It is sometimes not necessary to first understand everything before proceeding to the next topic. Thus you may have to accept a given fact (even if it is not convincing) for the sake of progress. 12. Try to be attentive in class and listen and try to follow the lecturer rather than copying notes you have no clue what they are about. It may be a good idea to work as a pair, with one of you taking notes while the other is following the lecturer. Then during your spare time you discuss what was done in class. If you prefer to work alone, have a textbook to refer to after the lecture. 13. Do not hate a bad or hard lecturer because by so doing you end up hating the maths course and thus you do not study it, leading to a fail. More frustration for you! 14. Do revise past test and exam papers but do not attempt to memorise and assume that your paper will be similar. Take a question as it is put, without thinking that it says what you have seen in some other test paper or exercise. Thus answer what you are asked, not what you think you are asked. 15. In problem-solving write down all that is given in words and then in symbols. Write down what you are required to find in words and in symbols. Then ask yourself ``how do I find what is required?’’ Answer your own question. If there is something else to find before solving the problem, find it and then solve the problem. 16. Finally check if the answer found is reasonable within the constraints given and within your general understanding. If not, trace the error. PREPARING FOR TESTS AND EXAMS 0. 1. 2. 3. Select a short topic or chapter. Start reading the theory given in class including definitions and theorems. Then practise ALL the class examples. Thereafter solve ALL problems given as exercise/ homework /tutorials. Do not read the solutions you have given because that exhausts your mind unnecessarily. Sleep early to give your mind rest, then go and write. 4. If you are preparing for exams, having followed what is suggested above, try to answer past exam papers. Again do not read or memorise your answers. Give your mind ample time to rest. Some students fail because when they write, their minds are exhausted, thus they are unable to think properly and they also make too many careless mistakes. 5. Do not panic before and during an exam. Start with easy questions so as to gain confidence. Thereafter proceed to hard problems. Now we look at an example. Suppose we are asked: Prove that square root 2 is not a rational number. This is often done by contradiction. So you have to assume that square root 2 is rational. But then you need to know the definition of rational number. Somewhere in the proof you need to know that if the square of p is even then p must also be an even number. Satisfy yourself by proving this fact (also by contradiction). Checking it for a few values of p is NOT a proof. This example brings me to DEDUCTIVE REASONING. This is something you learn by solving lots of problems. You will not succeed at Maths without deductive reasoning. You need to contradict something that you have assumed or something that is a known fact. In this example you will contradict your assumption that square root 2 = p/q with gcd(p , q) = 1. Previous knowledge in this problem is that the square of square root 2 is 2. Thus without previous knowledge we may not solve some problems. Knowing even integers and integers is also essential in this example. Deductive reasoning uses ``If … THEN’’ statements. Practise it. Consider a second example of an integral of a rational function whose numerator is a constant and whose denominator is a quadratic polynomial of incomplete square. Here a learner must know how to complete a square and must have knowledge of trigonometric identities. Manipulation of algebraic expressions may also be needed. This example illustrates the importance of previous knowledge in the understanding of mathematics. Most often learners fail to understand simply because they lack the necessary background skills. Go and acquire those skills concurrently with the course you are doing if you want to excel at Maths. In problem-solving strategies it is usually said: If you cannot solve a problem, there is an easier one (of the same type) you can solve. This basically says: Start simple until you understand, then proceed to complexity. Simpler versions of complex problems build confidence and ensure that definitions, theorems and techniques stick to your memory effortlessly. Some Problem-solving Strategies in Maths: Given a problem in Maths, how does one start solving it? Here we present some suggestions, NOT rules. We know that some people are innovative enough to find their own problem-solving techniques. 1. Be clear about what is wanted, underline it if necessary. If necessary, write it in symbolic form. 2. Write down, perhaps in symbolic form, what is given and maybe some consequences of what is given as well. 3. If what is wanted is clear, then look for a formula or a definition or a theorem or a known technique and use it to solve the problem. If there is no known technique that you have seen before, then maybe start asking yourself: if I were given certain information would I be able to solve the problem? Keep on asking this question until the answer is yes. Then try to find the information that is not given and start solving the problem stepwise. If you cannot understand what is wanted, seek help. If the question is clear but you cannot solve it, go back to the ``drawing board’’ and read your definitions, theorems and techniques. Study and practise the given examples. Consult other textbooks as well , then go back to the problem and have another go at it. If certain information from previous sections or previous year levels is required , go and search for such information and then use it in the problem as necessary. 4. In many cases it is highly recommended to DRAW A PICTURE of the situation that is related in the problem. This helps you to analyse the problem better and devise ways of solving it. If a graph in a plane or a 3-D space can be drawn easily, please draw it, it helps you and also the person marking your work. For instance a problem about a surface area of a cone or a volume of a solid of revolution certainly requires a picture. Area under a curve may require a graph in a plane. 5. In a word problem you sometimes need to set up an equation and then solve it. Thus you need to translate complex and abstract statements into plain statements in symbolic language if necessary. 6. Guess and prove method: If you are asked to say whether a statement is true or not, you need to take an ``educated guess’’ and then prove your guess. Your guess is not wild, it is usually based on inspection or substitution. For instance checking whether n! > 2n for every natural number n requires a guess and proof (by mathematical induction). 7. Remember this statement: If you cannot solve a problem, there is an easier problem you can solve. Thus you should start with a simpler version of the problem so as to gain confidence with some techniques and theory, and proceed carefully to the complex one. You modify the given problem by throwing away the difficult parts and then solve your simpler version. Then add one more of the complex parts that you replaced or threw away until you solve the entire problem. 8. In the case of proofs, if you do not know how to start, first write definitions of the given concepts and those that you want to prove. Sometimes you do not need a definition but a previous theorem (lemma). So write down what is given in terms of a previous lemma and what is wanted as well in terms of another theorem. Proceed from there to prove what is required. 9. Proof by contradiction is usually very useful. Start by assuming what is contrary to what you want to prove. Then negate a relevant definition or previous theorem, leading you to absurdity or contradiction of one of your assumptions. 10. Proving a contrapositive of a statement is also useful at times. For example: prove that if x is not greater than zero then ex < 1. It may be easier to prove the contrapositive of this statement viz. : If ex is greater than or equal to 1, then x > 0. 11. Learn to be innovative and thus use your own strategies. Many blessings. BB Makamba

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