Math 111, Introduction to the Calculus, Writing Mathematics

Math 111, Introduction to the Calculus, Fall 2011 Writing Mathematics Although it goes against the stereotype, math courses have a strong focus on writing skills. Math is, in some ways, like learning a new language, and one of the things you have to learn is how to write it properly. (You also have to learn how to read, understand and speak it.) The main points to remember when writing mathematics are: • write in complete sentences; • write statements that are true. When I say complete sentences, I do not mean you have to use words. For example: • If x = 2, then x2 = 4. is a complete sentence. Many sentences you write will have an equals sign in: this is the verb in the sentence. For example, when we write (x + 2)2 = x2 + 4x + 4 we are saying that one object, namely (x + 2)2 is equal to another object, namely x2 + 4x + 4. You can see that expressions such as (x + 2)2 , or 13 , or sin(x), play the role of nouns. A common mistake is to write in a ‘stream of consciousness’ way. For example, if asked to solve the equation x2 + 2 = 6 a student might write √ x = 6 − 2 = 4 = 4 = ±2. √ √ This is obviously wrong: 4 is not equal to 4 (and, strictly speaking, when we write 4, we mean only the positive square-root, that is 2, whereas in this equation x could be either plus of minus 2). Even though you have the right answer and I know what you mean, you would lose points for writing something like this. In solving a problem like this it is better to write out equations on separate lines, making sure each one is true and is deduced from those before. For example: x2 = 6 − 2 x2 = 4 x = ±2 This example may seem obvious but when we come to harder equations involving the new ideas from calculus that you are learning, it is easy to forget the need to write clearly. Another mistake students make is to use a symbol such as “→”. Usually this means something like “and then I did...”. For example, in the above problem, a student might write √ 6 − 2 → 4 → 4 → ±2 to solve the problem above. The mistake here again is that there is no sentence, just a sequence of thoughts. You should instead make sure you use the equals sign when you mean equals (write 6 − 2 = 4 not 6 − 2 → 4) and when you really mean to start a new sentence (such as when you move from x2 = 4 to x = ±2), make sure you start a new line to indicate this. It may be hard to break out of old habits if you have been writing this way before, but you need to. Learning to write mathematics well is an important part of this course. The best piece of advice I have for you to improve your writing is to work out solutions to homework problems on scratch paper first. This gives you a chance to figure out what is going on for yourself, before trying to communicate what you have worked out to someone else (in this case, me or the grader). We’ll talk more about this as the semester goes on. If you have any questions about how to improve your writing, or do not understand why you have lost points when you got the right answer, please come and see me to talk about it.