The language of mathematics, maths, math.....

As someone who makes a reasonable use of mathematics on a daily basis, I thought it might be useful to anyone coming to the subject either for the first time, or because you need to solve a problem, and they think maths might hold the answer! I shall explore not only the value of maths (or math as it is called in the USA), but also one of the barriers that makes maths so unpopular amongst the majority of the population, starting with school students: the language of maths. First, let us look at the origin of the word itself. It is thought to have its roots in the Greek word mathema, meaning knowledge, and matos, the Greek word for thinking or learning. From these ancient roots, it would suggest that mathematics underpins all knowledge, and in Science we tend to believe it does. Just think of the famous equations of Physics, such as E=mc², an equation that links energy (E), mass (m), the velocity of light c, and provides us with a way of understanding the Universe and the energy trapped in atoms. However, this doesn't explain why this equation is correct (or should I say a reasonable fit to the observations), it is perhaps better described as a short-hand for words.

Before we get too advanced let us begin with another word that is often used interchangeable with maths, arithmetic, which comes from another Greek word: arithmos, meaning number. From the beginning, these terms cause anxiety and confusion. What is the difference between arithmetic and maths? Well arithmetic comes before maths. Arithmetic is the manipulation of numbers which largely grew out of the needs of merchants and farmers: addition, subtraction, multiplication and division. As an employer of a store, you need to add up your day's takings, you need to subtract the wages you need to pay your staff and you need to multiply the amount of wages per month to work out how much profit you need to make to offer someone a job for a year, and then divide that number by 12 to work out the monthly wage (or 52 if you pay them weekly). In the language of maths, this is simplified by using the symbols +, -, x or ÷. There is no doubt that these four symbols cut through a lot of words; and moreover they are recognised by anyone, no matter their native language. 

Mathematics is a little more difficult to define: even the mighty wikipedia tells me that "mathematics has no generally accepted definition". Let's start by thinking of mathematics as a word like "sport". How do you define sport? Well you could say it is an activity that some humans engage in that requires a level of effort to get from A to B more quickly than by walking at normal speed. This would be one definition of a race. You would probably want to add that sometimes there would be more than one person involved, although solo sports are not uncommon. Some sports are competitive (think of the Olympics) and some are not (a jogger keeping him/herself fit). There are team sports: football, rowing, polo (which requires the participation of horses, or a suitable swimming pool!). I hope you can see, defining sport is complex, and so too is defining mathematics. It is easier to consider each sport or each branch of mathematics separately. If I want to describe the 100m sprint, it would go as follows. Typically eight runners (either all male or all female), line up and, when the starter fires a loud pistol, the competitors run as fast as they can until they reach a tape at chest height, exactly 100m from the start. The winner is the first to touch the tape. I am not going to define every sport here, but you can imagine a definition of cricket or rugby, would require a much more detailed explanation. The same is true in maths. Some areas  of maths, such as geometry (which deals with points, lines, shapes and solids) and trigonometry (which is the maths of angles and the lengths of the sides of triangles) require an explanation of the objectives and the rules, which are more sophisticated than the rules of arithmetic. It is trigonometry and geometry that provide the mathematical foundations for architects, builders and sailors.

In order to construct great monuments (mainly to keep the Gods happy), it became popular to factor in predictions about, for example, the forces that might act on the foundations of the "temple", or whether the land would be able to support the "Pyramids". Or how to design a roof of a particular span, without the need to incorporate supporting pillars that simply might "get in the way"! In fact as time went on, the desire to build bigger and more elaborate structures fueled the desire to understand the forces associated with geometry and trigonometry, thus began the marriage of mathematics and materials science, which we now call engineering. It is for this reason that engineers place such an importance on trainees having a sound appreciation of mathematics. Just think for a moment about your own home. It will typically be constructed from wood and bricks (or something similarly strong and rigid). It will have water flowing through pipes and electricity through cables. Some walls will be cosmetic (ie not structural) and some will have windows in the middle. An architect and an engineer, will have to know the relationship between the load bearing capabilities of wood, brick, metal etc and will have to "integrate"this knowledge with the forces that will act on the floors, since a typical household has items like beds, cupboards, washing machines, as well as people, to accommodate. If they get it wrong, the house will simply fall down!

Those of you who are already interested in maths will be aware that I have been mainly discussing "Applied Maths", that is mathematics that underpins practical applications, such as building houses, or cars, or aircraft, or ships....But there is the area referred to as Pure Mathematics, which usually has to wait before an application comes along that can benefit from its incorporation. When Sir Isaac Newton and Gottfried Leibniz independently developed the mathematical field of calculus, which many students find particularly challenging and esoteric. However, much of the motivation to develop calculus came from the need to solve astronomical and navigational problems: two geniuses creating a system of maths, without which, many of the subsequent breakthroughs in science and engineering would simply not have been possible. However, you may have read the following headline in the Daily Mail

Scientist discovers there are 177,000 ways to knot a tie after being inspired by Matrix villain

Now, I can hear you say: how is the world served by working out how many ways you can tie a knot in a necktie? Knots have a practical importance, but some mathematicians spend their lives trying to understand the theoretical basis of why only a certain set of knots are ever possible. This is an area called topology and although it might sound a little odd, it may shed light on how we can pack a 1 meter fibre of DNA into every cell in our body. Don't forget each cell is about one thousand times shorter than a meter! It think we should let the bright you maths geniuses follow their noses, since you never know when an equation or a concept might come in handy! And anyway what else would they do?

Let's look at the language in more detail. First of all, we have introduced the symbols for addition etc, but we now have to appreciate the use of letters, from both the Greek alphabet and our own. Next up is another word that strikes fear into the hearts of many: algebra, from the Arabic al-jabr, to restore from its broken parts. This is where letters and symbols are used to replace numbers in equations. [It is important to recognise the contribution of the Egyptians and Babylonians to maths: the introduction of a zero, for example and the movement from Roman Numerals to the forms derived from India and the Arab world to the familiar characters we use today]. Back to algebra; see Einstein's equation above as an example, but more typically, the letters a, b and c, or x y and z, are used. As the definition goes, letters are used to "simplify" a calculation and then the solution is "restored" by putting the real values in. As an example, 

y=mx + c

Some of you may recognise this as the equation for a straight line. Try replacing the letters m, x and c with numbers and see what happens to y. For example, let us make x increase by 1 and keep c at 1 (a constant) with m held at 5. The value y when x is 1 is simply given by (5 x 1) add 1, or 6. When x is 5, for example, y becomes (5 x 5) add 1, or 26. With x at 7, y is 36 and so on. The relationship therefore defines how y varies with respect to x, in this case. There is some explanation needed here for novices.

First, mx means that m and x are multiplied together. The symbol x (times, or multiply) is not used in algebra, we say it is implied. (I suppose it is because you might mix up the multiplication sign for x!) Mathematicians also use the phrase, "the product of", when they multiply two or more numbers together. For example, 10 is the product of 2 x 5. The second thing to say is that if m and c are held constant, then m influences the slope of the plot of y versus x, while c determines where the graph line crosses (or intercepts) the y axis (otherwise known as the ordinate: the horizontal line is called the abscissa). Now we have moved from algebra to graphs. The two are like opposite sides of the same coin.
 
Now, why not try the following. With m set at 2 and c at 4, calculate y for x = 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. Plot a graph of y versus x (y is the ordinate and x is the abscissa). See left as an example plot. What do you notice?

Time to take stock of the language, since I do not intend to cover the maths curriculum in full! There is a great deal of discussion about STEM subjects, and I am in full agreement with this. However, the difficulties that people have with maths, for example, often come from a language problem, rather than a STEM problem! Reading the above post back to myself, whilst trying to imagine once again, that I am a newcomer to the subject, I can see the language is a major hurdle to overcome. Even when I think I am using "plain" English to explain a mathematical manipulation, the added problem of the specialised words (or the vocabulary) of STEM subjects make matters worse! 

My suggestion here is pretty simple: delivering early maths lessons should be shared by Maths and English teachers? All English teachers will be comfortable with arithmetic, and most will be proficient at maths up to Key Stage 3 (pre GCSE, age 14). But importantly, they will be less familiar with the language of mathematics and will therefore be more likely to appreciate the student perspective. And since early school maths is something all high school teachers will be comfortable with, this approach might open doors for the more challenging STEM subjects.

I would welcome comments and suggestions as always, particularly from Maths and English teachers. As a simple way of combining literacy with numeracy, while capturing the ethos of STEAMco, maybe the following would work?
 

ABC + 123 = STEAM