Mathematical Problems

Mathematical Problems

Arithmetic is often not problematic. Children and adults can mostly manipulate numbers arithmetically. (There is a condition called innumeracy which is recognised).

But arithmetic is only one branch of Mathematics. To be good at arithmetic is one thing; to transfer that ability to the study of Mathematics, is another thing – a different thing.

If it were possible to make an analogy, then I might say this:

To be able to work successfully in Arithmetic is not automatically a precursor to being able to work Mathematically. It would be the equivalent of saying that because one can remember some melodies, one is capable of composing musical works. The one thing simply does not follow the other. (Or another way, that because one might be able to speak, that one can write creatively. Again, this does not follow).

Sadly, teachers do not understand this, and neither do Mathematicians.

I can honestly say that with one exception, it was never my lot to meet a Mathematician who could (or was prepared to try unstintingly) explain mathematical principles without tiring, or becoming increasingly annoyed with the idea that pupils cannot grasp mathematical facts with ease.

It is true that some pupils can. But it is also true that many – and probably most – cannot.

Mathematical explanation as received in school, is generally poor. Those who understand the processes find them difficult to put into words, so that those who have difficulty, might also learn. Consequently, learning is patchy.

Mathematical people often deride those who cannot understand, and quite often laugh at, or shout at those who find the mathematical world bewildering – even those who are good at arithmetic.

LET US QUALIFY

When I was 11, I could process arithmetic. Firstly, I could count. Secondly, I could put things in number or numerical order. I could also ADD. I could also SUBTRACT. I could also use MUTIPLICATION. I could also DIVIDE using both simple and long DIVISION. I was also able to ADD, SUBTRACT, MULTIPLY and DIVIDE fractions (of any denomination); and I could also ADD, SUBTRACT, MUTIPLY and DIVIDE decimals. All this was done without a calculator of any description – which back then, would be either a pre-reckoned and printed calculator, known as a “Ready Reckoner” or a mechanical one, which nobody had, unless they lived in a shop.

I could also work out simple percentages. I could also solve what were called “PROBLEMS” which were questions presented as prose exercises, where the child had to deduce and divine which numbers might be relevant and also how they might be processed.

I could MEASURE, both WEIGHT and CAPACITY, and AREA. I could calculate square feet and square yards.

In addition to this above, I could work in any base at all, having been taught the principles of that by a visionary or forward-looking headteacher, who, when we were aged 7, made us work in different bases on different days of the week. I might add, that to survive in the day to day world at that time, it was necessary to have a daily and easy working knowledge of bases 10, 12, 14, 16, 7, 20, 112, 56, and 28. We also needed to know how to use several groups of numbers for particular functions, which were again, bases, and these included 3, 36, 8, 22, as well as the functions of 1760, and 5280, and indeed, the limitations imposed by 60, 30, and 15.

Let me state that at NO time whatsoever, was any of this material skill recorded for anybody to peruse in any report of my ability.

In my transfer to the next school, at age 11, there was no interest taken at all in any of these skills, and there was no transfer of paperwork from my First school, to my Second. Indeed, the whole transfer was fraught with such massive uninterest, that those of us who were in any way sensitive to the disinterested behaviour of our “Maths” teacher, knew at once, that there were going to be difficulties on the horizon. The norms were as follows:- to laugh, to deride, to scold, to hit, to send out, to mock, to shout.

Nor was the Secondary curriculum in any way matched to anything that we had worked at Primary School.

I now know that we had become part of a peculiar experiment (entirely unannounced) in which we were told nothing at all, and given Pure Mathematics lessons from the age of 11 plus. It was a terrible error, and many of us did not recover from the experience.

The teacher was unhelpful. He told us that we had “wasted” our time with our earlier work. We had not been “taught properly”. And on one occasion, that we ought to “forget all that baby stuff”.

Yet I do not ever remember any visit, official or otherwise, from the Secondary schools to the Primary schools, or a transfer of teacher, to see what we were doing. I recall no letters from the Secondary school to the Primary School, to ask “that the following things might be taught in advance”. In other words, there was no preparation at all – and that was not the fault of the Primary School teachers.

Maths in the new school was the equivalent of being given lessons in Chinese, or Ancient Egyptian, without being first taught to use characters or hieroglyphs.

It was very wrong, it was totally educationally unsound, and it was cruel, and it was the reason why so many of my contemporaries, myself included took no interest in “Maths” from the very first lesson. (and don’t go hiding behind the usual teacher excuse that it was “our own fault”.)

We didn’t understand it; it isn’t something that one might pick up by “the need to survive” or perhaps pick up by “immersion”. It was hopeless, uncharitable, ill-considered and abusive, centring only on the satisfaction of the adult, usually male, and almost always not articulate enough, or having anything like the amount of patience or empathy required to teach children. And if anybody reading this taught Maths, and sees a pattern emerging, then if the cap fits, as they say, WEAR IT.

The failure in all these schools of the child to achieve comprehension of the subject of Mathematics, requires a level of care that is not met. People who teach mathematics often think and believe that those who are being taught, merely SHOULD understand. But it is not so. One does not understand on the basis of turning up to school. Showing people things is not teaching. Repetition of exercises is not teaching. Child minding and overseeing maths (“turn to page 47 – quietly”) is not teaching.

To teach is to lead out and forward, having discovered the starting point, and preferably one that can be shared and built upon. It is a difficult and problematic thing to do. One example is not sufficient.

So, for those who teach mathematics, here are some problems for you:

a) If I have “2” and I multiply by “1”; why is my answer “2” – ? (That was never even considered worthy of response). Yet I don’t think that it can be explained in lay terms.

b) If I have “2” and I multiply it by “0”; why is my answer NOT “2” – ? I had “2”. Say I had “2 apples”; then explain in lay terms why “multiplication by zero” negates the apples with which I started. (Please note, that I did not start out with “zero apples”, in which case I knew, that if I had “nothing”, if I multiplied my “nothing” by any number, I would still have “nothing”).

c) In the case of both the above questions, is it not the case, that the use of the word “multiply” is wrong? That the function of “multiplication by 1” and “multiplication by 0” is not multiplication at all, but something else entirely? If this is the case, then the layout of the “times tables” contains at least two lines that do not belong there.

d) Explain how division of fractions actually works using the word “reciprocal” and do it in language suitable for children aged 11.

e) Explain carefully and slowly how “2 to the power of 3” works.

f) What are Tangent, Sine and Cosine? Why are they important? Give at least two tangible applications of each one, in daily life.

g) What is “2 to the ‘minus 11’ ” and how or why might this be important? Where would we meet this? What are the implications of it? How do we use it, or how are we affected by it in our daily lives?

h) What is “an equation”? Can you show me in simple terms what it is? Also how can you recommend the simplest way to go from addition and multiplication and division, or subtraction, to equations?

i) What is a “simultaneous equation” and where would I see this happening? Does it have an application?

j) How can I turn an idea into an equation, and work it out?

k) At what point does Mathematics begin? For example, we were told that “arithmetic isn’t maths” but, we were not told what “maths” actually was. So what actually is “maths” – ?

l) What is a “formula” and where does it come from? Why does a Formula exist?

m) What is a “set” and why do we need to know this? What is the application? and if there is no application, what is useful at a theoretical level about a “set” – ?

I probably have more question, but for now, those are enough.

At the present time, our Prime Minister would like us all to know Maths better in the future. That is quite a noble aspiration. But I think he would do better, to understand first the starting point. And I don’t think that he knows or realises where that might lie.

The fact that many people do not understand maths, and we might all “do better”, is somewhat like blaming the dog for crapping on the house floor, when really what happened was, you tied it to the table, left no instruction or message and somebody else in the house couldn’t be bothered (or didn’t want to be bothered) to find the key to open the door to let it out.

n) If you are clever enough, maybe that last paragraph might make a simultaneous equation. Would that be possible?

Frankly, I have no idea. But I’ve never given up hope, and never lost my interest, I just wasn’t taught. (And Richard Skemp’s book “Understanding Mathematics” was no help, and was a waste of paper and card).

Glove Down.