You have already met a lot of general strategies for reading and understanding scientific texts in Chapter 2. These techniques – scanning, active reading and making notes – work just as well when you are studying maths. To see how, suppose Sushma has decided to study the ‘Approximations and uncertainties’ section of Maths Help on pages 346 to 352. To get an overall idea of the main points, and as a guide to how much time and effort she will have to spend on it, Sushma uses about five minutes of her lunch hour to skim through the section. Here are her comments, with the main points that I would like you to note about Sushma’s notes written in ‘clouds’.
At her next study session, Sushma plans to study the section in detail. Following the advice in Chapter 2, she works through the examples, makes notes and tries some exercises on her own. To keep her learning active, she asks questions of her own and makes notes of her answers.
X How does this fit in with what I already know?
You use the same ‘5 or more’ rule for all the different ways of rounding, but significant figures allow you to deal with very big or very small numbers more easily.
It’s about rounding numbers off. I know how to round numbers to the nearest 10, 100 etc. and to decimal places, so I should be able to get through that first section fairly quickly. Allow 30 mins.
The section on errors is new. I’ll have to spend quite a lot of time on that. And I’ve not heard of
significant figures before. I need to find out the difference between the sig. figs and the decimal places. Why do we need these different ways of rounding?
Main idea of section
Clear objectives for studying: errors, significant figures and the differences between them
Assess your knowledge and the time it will take This will help you to plan your study time
Questions to think about and search out the answers
X Have I seen or used anything similar to this before?
Yes, I know how to round to the nearest 10, 100 and so on – this just extends those ideas.
X Can I relate it to some everyday experience or practical example? Yes, it’s a bit like estimating how much my shopping bill will be, before I have to pay.
X Where could these ideas be used?
A lot of the numbers in the media are rounded off. You could use these ideas whenever you need a rough idea of the size of a number and aren’t too fussy about its actual value (for example, in estimating answers to calculations).
These kinds of questions work well for most topics. Try using them! They challenge you to take a broader view of the topic and deepen your overall understanding. Another important question to ask is ‘What would happen if …?’ Try changing some of the initial assumptions or values. You need to explore where the ideas apply and think about their limitations. For example, Sushma might ask ‘What would happen if you rounded down when the next digit is 5, instead of rounding up?’ or ‘What would happen if people
sometimes rounded down and sometimes rounded up?’
You can see that the general skills you are developing for tackling scientific texts will help you to make sense of a mathematical topic as well. Experiment with different approaches until you find those that work well for you.
But if you compare a piece of mathematical writing with a scientific article, you will notice that the language used and the style of writing are slightly different. Although both kinds of writing are written in sentences and can be read aloud, maths language tends to be more condensed, with more notation and abbreviations. Before you can make sense of the mathematical ideas, you have to understand this language. Imagine a story written in a foreign
language using a alphabet different from your own. First, you would have to master the alphabet, then the vocabulary and grammar, and finally, you could try to understand the story. You would expect all this to take a lot of time and effort. You have to spend time learning a language before you can understand ideas expressed in that language, and still longer before you are sufficiently fluent to be able to think in the language.
2.1 Learning the language
Here are some ideas to help you to get to grips with maths language. Making an MST dictionaryYou should be introduced to new notation as you need it, and you should be given plenty of opportunity to practise using it. To help you to remember the language and check meanings quickly, try using a small notebook to make
yourself a maths (or MST) dictionary (Figure 4.2). You can use it to store new words and abbreviations alphabetically. Preparing definitions in your own words or diagrams will help you to remember them. Remember, an example may illustrate an idea more clearly than words ever can. For example, how would you explain the square root of a number? Recording symbols in your dictionary is a little more tricky. Try including a section at the back of your dictionary divided into separate topics: Numbers, Geometry, Statistics and so on. Then store each new symbol in the appropriate section. Remember to write down how to say the symbol as well as any specific meanings. You’ll find your dictionary useful when you read pages 121 to 130 of Chapter 5 which explain the meaning and pronunciation of some of the symbols and abbreviations frequently used in maths and science. A separate section for formulas is also a good idea.
words symbols formulas
Maths
Dictionary
α
µ
π
θ
ω
ε
×
≠
Figure 4.2Practising using new language
Make an effort to include new vocabulary when you are discussing maths. Practise new notation as you work, making sure that you can read it properly (either in your head or aloud) as you write it down. For example, if you have written 32, say to yourself ‘three squared’, not ‘three with a little two at the top’. Note that some mathematical expressions do not quite follow the left to right, top to bottom conventions used in reading English. For example: 2
3 + 4 5 is read ‘two thirds plus four fifths’ (see Maths Help on page 317).
Noting differences in how words are used
Maths, like other subjects, has a specialized vocabulary. You will meet a lot of new words that describe particular properties or processes – for example, ‘radius’ or ‘circumference’ (see Maths Help on page 386).
Maths has borrowed quite a lot of English words too, but often these words have a very specific meaning in maths – for example, ‘power’, ‘odd’ or ‘root’. Borrowed words are sometimes combined to describe particular mathematical ideas as well (for example, ‘the highest common factor of two numbers’). These words and expressions can be baffling if you use the English
interpretation instead of reading them mathematically. So, it is important to spend some time making sure you understand these words in a mathematical context. And remember that the same word may have more than one
interpretation in mathematical language. For example, ‘solution’ can mean ‘answer’, as in ‘the solution is 4’, or it can mean a complete piece of work, explaining how you worked out the result and including the final answer. In this chapter, the latter meaning is used.
2.2 Learning to read mathematical texts
When people are writing maths, they tend to use a lot of notation. Thisincludes abbreviations (such as d. p. or sig. fig.), special symbols (such as > or ⯝), and letters from different alphabets (such as π, α or β). Providing you understand the notation, it can help you to express ideas in a clear and concise way, making it easier to visualize the problem and understand and work with the ideas. As an example, imagine what life would have been like before the decimal number system was adopted. (For instance, think about working out your shopping bill using only Roman numerals.) So, good notation can save time and effort, and that’s one reason mathematicians use it a lot.
However, since mathematical writing is concise, you do need to read each sentence slowly and carefully, checking that you understand every word and symbol. Then think about the whole meaning. You might have to look up work you’ve done earlier, or check the meaning of some special notation. In mathematics, the argument follows on, line after line, in a logical way, leading you down a very straight and narrow path to the conclusion. Of course, when you tackle your own mathematics, there isn’t a well-marked path: you are free to go in whatever direction you like, including down complete dead-ends and into huge pits of unfathomable calculations. So, to help you keep on the right lines in your own work, you need to understand both how the individual steps work in a piece of maths and why they were taken.
To illustrate key ideas, you often see worked examples in course materials. It’s important that you understand these examples, as you can use them to check on your understanding as well as models for your own solutions. The ideas overleaf will help you to follow worked examples.
2.3 Understanding how the steps work
To understand what is happening, you need to relate the steps in the example to what you already know. You can do this by writing notes and explanations on the text (providing it’s your own!) and by drawing diagrams. For example, consider this newspaper extract.The extract made me wonder whether I was overweight, so I worked out my body mass index. Here is my calculation with comments written on it to help you understand it.
When you are tackling any mathematics, it is important that you build up internal ‘pictures’ of the problem in your mind; diagrams, physical models or practical examples can help you to do this. For example, I find it difficult to visualize three-dimensional problems, so I almost always resort to building a structure out of pencils, boxes, straws or anything else I can find. Sometimes,