On being FUN!

My tutor thinks my lessons are not fun enough; I think I am not fun enough. I also think Maths is not fun enough. If I really want my students to have fun I should try teaching them something other than Maths. My tutor’s lessons, however, are fun; I cannot compete with the fun that is had in his lessons because I do not know how to make Maths fun. Why do they not already find Maths fun? Why should I have to dress Maths up in funny clothes before it becomes palatable? I am not even sure that I find Maths fun; it is interesting for sure, and amazing, but it is not fun unless you are doing something specifically designed to be fun, and that is usually just for fun. I loved Topology, Metric Spaces and Game Theory because I thought they were fascinating, not because I thought they were fun. While I was learning these things I used to turn to Carrol’s puzzles or BMO questions to have fun, but I was not learning new and exciting Mathematics from doing these. Marcus Du Sautoy makes Maths fun but he always chooses things like “the 4^th dimension”, you do not get popular Maths books about using a protractor.

Clinging to idealism

I do not suppose I am alone in saying this, but I came into teaching with some rather high-minded ideals, indeed my motivation to become a teacher lay partly in the reading of The Elephant in the Classroom [10]; I wanted to change the world. In fact I was not quite so naïve: in May 2013 I wrote that “Hopefully, as a teacher, I’ll be able to give the impression that maths is at least marginally interesting to at least a small majority of my pupils.” so my expectations of myself did have a degree of realism about them.

I can identify myself with Alf Coles at the start of his career: I feel that I am articulate about teaching around my peers and my tutors but there is “a gap between these ideas and the reality of my classroom” [27 p.7]. I did not expect to have implemented ground-breaking reforms in my placement school but I am disappointed with the formulaic sameness of my lessons: textbooks, exposition, PowerPoint… And I am somehow unable, whilst not unwilling, to take risks. Upon entry into the classroom I become absolutely mechanical, which I did not expect to happen; I have found that my main difficulty is not in subject knowledge or dealing with unruly children, as usually anticipated by starting teachers [28], but in transforming the good insight I have – concerning what went wrong with the planning – into action plans for improvement.

The issue lies, I think, in my expectation that the students will guide me in working towards good planning; in fact, I am supposed to guide them towards good learning. It is not the case that I can expect the students to be creative or independent: if they have finished the questions set they will not go on and do more, or question ‘why’ the method works, as I think I expected. If I want creative students I have to facilitate their creativity and if I want independence, I have to establish that culture.

To focus on Creativity, I found that all of the students in my class approached Mathematics methodically, indeed they “long for rules and procedures” [29], and Gardiner’s article leads me to believe that it is not just my class. Gillian Hatch [30] describes my experience perfectly when she reflects that her secondary pupils, unlike at primary, are not prepared to be “autonomous learners”:

Sadly, I remained in conflict with the class as to what they should be able to do for the whole term. I found it impossible to accept that I could not tap back into the behaviour they would have shown only a year before. All their energy for learning Mathematics had been sapped. (p. 129)

Although I know that it is I who must make the change, not the students, I have not yet truly accepted the fact. I have a sense that they ought to be doing it, not that I ought to have told them to do it; and I have a feeling that I will not have Boaler’s perfect classroom until I have remedied this.

Playing the fantastic game

In the afternoon session on Thursday, Laurinda wondered out loud why it is that some people (present company included) found doing Mathematics exciting at school. Why is it that although most children loath it, some of us love it? Why is it that some students can only be inspired to take an interest if Mathematical topics are introduced wrapped in a context, presented as a useful tool to solve a pre-existing problem; whilst others (like myself) can find enjoyment in doing Mathematics purely for its own sake?

I recently finished reading Keith Devlin’s 2011 book Mathematics Education for a New Era [21], which predictably advocates the use of hitherto unimagined video games in Mathematics education. Devlin’s main point is that most people are good at Maths in day-to-day situations but if you present them with school-style Maths problems they suddenly freeze. Unable to access the part of the brain which ably handles Maths in “real life”, they attempt to remember half-learnt facts from the classroom with almost no success. To make this point clearer Devlin makes frequent reference to two studies: the Recife “street kids” [22], and the Adult Mathematics Project (AMP) [23]. In the former study, the researchers observed Brazilian street children in Recife whilst they sold goods from market stalls. Over the course of the study the children got 98% of their arithmetic correct when computing prices and change. One vendor, for example, was able to correctly, and quickly, give change of Cr$168 from Cr$243 when selling an item for Cr$75. The same child could not have found the correct answer to “243-75=” presented as symbolic maths. In fact, the researchers gave the children a school-style maths test featuring many of the same problems they had been observed to get right on the street. The test average was only 37%.

Devlin says that the reason the Recife kids dropped from 98% to 37%, and the reason so many people struggle in the classroom even though they can do the Maths outside the classroom, is that Maths inside the classroom lacks meaning; it lacks context. Video games, it is argued, can provide that context; if problems requiring to be solved with Mathematics don’t pop up often enough in the real world, maybe we can engineer a virtual world where these kinds of problems abound. So how come some people are able to do classroom-variety Maths, and enjoy doing it, even without the context? In the final chapter, reflecting on his clarion call for Mathematical video games, Devlin observes:

Of course, there is one group of students for whom learning advanced Mathematics with the aid of a video game will not offer much advantage, if any. Indeed, those students will likely shun Maths video games altogether, just as they shun Maths textbooks. Those are the future Mathematicians of the world. …. We discover very early on in our lives that there is a virtual world far more intricate and exciting than any video game will ever be. That world is Mathematics itself. We play in the most fantastic video game there is. (p. 194)

The idea that Mathematics is a virtual world to which Mathematicians have access is by no means new; Plato predates Euclid by about 50 years so the concept is practically as old as Mathematics itself (Mathematics as we know it today, courtesy of Thales and Euclid). It is an idea to which I strongly relate. The key to enjoying doing Maths, and to playing the video games that is Mathematics, is to make as much contact as possible with Plato’s world. This is probably quite difficult in some cases but everyone is capable of some contact: it is evident that someone has made contact with Plato’s world when they say “ah, I see!”  The person who makes this remark has not made a leap into the dark; they have uncovered something solid and permanent: a necessary Mathematical truth. Roger Penrose describes the experience nicely by comparing the feeling of being on the verge of mathematical discovery to the feeling of having ‘drawn a blank’ trying to remember someone’s name: “In each case, the sought-for concept is in a sense already present in the mind” ([24], p. 555).

Is there a maximum age limit for playing the video game of Mathematics? I cannot recall a time when I wasn’t a player in the game but Keith Devlin says that his own revelation came during adolescence. Laurinda too remembers a time when x was scary and algebra didn’t make sense. I suspect that there is no limit and no child is too old to yet make contact with Plato’s world and enter Devlin’s “fantastic game”. G. H. Hardy [2] famously said that he had no passion for maths until his Professor Love advised that he read Jordan’s Cours d’analyse, but he was already a Mathematical prodigy at a very young age. There is a difference between the passion for Maths – which implies an appreciation of the “fantastic game” – and the ability for Maths – which can be due either to playing the “game” or to simply following orders (Hardy’s ability clearly attributable to the former).

On Being a Hypocrite

All of the student teachers I have spoken to have lamented the necessity of being a hypocrite. Obviously we cannot have students running around, talking over their teachers or exhibiting any bad behaviour which could be disruptive to the learning of others. In a better society children should not feel the need to behave badly and, if they did, they would be given a way to express themselves which would not impinge upon their peers’ enjoyment of life. Unfortunately we do not live in such a place, we live in place where everyone needs a ‘C’ in GCSE Maths and English and where specialist support is given only sparingly to people with ‘real’ problems. We also live in a place where teachers are put under pressure by the Government, the Governors, the Head teachers and the Inspectors to deliver the full curriculum to each member of a class of up to 30 in a short allotted time. All thirty of these students have five different subjects every day, and they have one, two or even six weeks off at regular intervals throughout the year, during which time they struggle to retain a little knowledge of each subject but cannot possibly recall details of every topic in all of their subjects.

Clearly, under such conditions a teacher will have to reprimand disruptive behaviour, even if that teacher behaved exactly the same way during their own childhood. I was never a disruptive force in my classes and I always resented those who were, I therefore would not be a hypocrite if I shouted at a student for their behaviour, although I still would not like to have to do so. However, whist being a keen, well behaved learner, I was a badly organised one, or at least I had the outward appearance of being badly organised. Every year I was given a planner at secondary school. In year 7 I wrote down all my homework in the planner for the first two weeks and then I left it in a draw at home. In years 8, 9, 10 and 11 I put the planners straight into this draw without so much as having written my name on the front. I never had the intention to build up a collection but I know that I did because I found all five planners in that draw when I was clearing out my room to move. Since then I’ve done five A-levels, had a full time job, and completed an MA degree in Pure Mathematics; in all those six years, as in the previous five, I’ve never once used a diary to plan my week or to remind me of deadlines. Yet I have never missed a deadline or forgotten to go to a meeting or an exam, I have managed to remember to do everything I have needed to. I have tried to organise myself, occasionally I receive a diary for Christmas, and I have even been given a Filofax: each time I write everything down for about a fortnight and then I stop. I’m saying all this to illustrate how possible it is to get through life using only a brain as a reminder; using a diary and being organised is not a life skill, not by any means. It is something that some people have to do to keep on top of things but there are others who are quite capable of storing information in their heads. I know the birthdays and ages of everyone close to me and, because I don’t have a mobile phone, I can recall at least ten people’s phone numbers. But I am not special, I am no memory champion and I do not mean to brag (I’m sure that at least 30% of people have better memories than mine) I simply mean to point out the fact that not writing down your homework is not a cause for concern.

In all the schools I have seen, form tutors prowl around their classrooms, usually once a week, and tut-tut at anyone who has not been using their planner “appropriately”, and they are required to sign the planners to show that this check has been made and that this tut-tutting has been done. It cannot be long until the day when I will be squeezed by the expectation of conformity to perform this dreadful ritual. I know I will come across students who, like me, never write down their homework and have even decided to discard their planners completely. Why should I have to tut at these students? Why, if they manage to get their work done and keep their parents informed, verbally, of upcoming events, the dates of which they might be quite capable of remembering?

I’m not saying that all schools should be free (in the true sense of the word), I know that we need to teach to the GCSE standard; but for God’s sake, who cares if they do not write down the due dates of homework? Who cares if they take up the habit of smoking, so long as it is outside the school grounds? In fact, I care. I hate smoking and think it is a horrible addictive habit, and a dangerous activity at that. I think that we would be better off if tobacco were illegal completely. But no, I would not reprimand a student for smoking. I would explain the dangers scientifically but before I uttered a word of judgement or declared a punishment I would have to demand that all members of staff, and anyone else associated with the school professionally, who smokes, be fired. It is one thing to be a retrospective hypocrite, to tell a student off for something you once did yourself; it is quite another thing to be a true hypocrite, to tolerate staff behaving badly and breaking school rules but not to show the same tolerance to students.

First Lesson

Year 7 high achieving class at an all-girls’ school – group of 8 students

I thought I might recap on the idea of perimeter but since the class unanimously agreed that they knew what perimeter was I decided to introduce the first activity directly. I asked the class to draw a square in their work books, using a ruler, of side length 10cm. This was completed quickly and with little fuss. I then asked them to draw, with a pencil and ruler, a quadrilateral inside the square with as large a perimeter as possible. They should measure and write down the perimeter and then possibly try again if they think they can do better. This task prompted a range of reactions: some girls drew a skew square touching the sides of the 10cm square, possibly because of a memory of another mathematical task, or possibly because of a self-imposed restriction on simply retracing the large outline square. Other girls just played around drawing small squares of rhombuses; and two girls drew a square of side 9cm neatly inside the big square, exactly 0.5cm from each side, again because of a belief that “the big square doesn’t count”. I challenged these girls by asking what their perimeter was: “36cm? But I can see a square with a perimeter of 40cm; if you were going to do a square you may just as well have done the 40cm one.” To this one of them moaned: “Oh… I didn’t think we could touch the big square because you said it had to be inside the big square.” I didn’t mention it but I suspect this girl would have considered herself to be inside a house irrespective of whether she was touching the wall. She clearly had some appreciation of the thinness of the ‘walls’ in this case, which I was not expecting. For an older class I would have clarified that by “inside” I meant to include the boundary but it can never hurt to be clear in your explanations.
Once one girl hit upon the idea of drawing a concave quadrilateral and succeeded in getting a perimeter greater than 40cm, I held her work up to the class and mentioned it as a suggestion. The girl who first did this must have had some conviction in her idea because she tried again even though her first and second attempts at a concave shape had a perimeter smaller than 40cm. She had started to draw three sides of her shape, saw that the fourth side would lead towards a concavity and somehow realised that this was the way to proceed. From observing her at work I think that she recognised a fundamental difference between the convex shapes and her new shapes. She probably also realised that the 40cm perimeter of the square was the largest possible with a convex shape and had worked out that there must be a larger perimeter available otherwise I would have already moved on with the lesson. This girl was not the first to realise that concavity was the best way to proceed with the task- another had said that if she made the sides of her 9cm square into zigzags, keeping them within the boundary, she could increase the perimeter. I was about to intervene with a question when she realised by herself that this would not be a quadrilateral: “oh, but then it would be 10-sided shape, or maybe more”- I think she was being quite conservative with her use of zigzags in her imagination, or she wasn’t really sure what the thing she had imagined would look on paper and couldn’t estimate the number of sides. She might also have been thinking of 10 sides on each side of the square.

After all the girls had been able to appreciate roughly what the optimal solution to the perimeter problem would look like, I moved on to the MATs. First I explained what MATs are and then I said that we would be measuring perimeter in the natural units, rather than centimetres. I held up a triangle and a hexagon: “so, this is perimeter 3, what is the perimeter of this?” They all said “six” so I held up a square and a triangle attached an a side: “five” they all said; so I handed them out and explained clearly what they had to do concerning the questions on the sheet. The only part that caused confusion, and it did cause quite a bit, was how they should record their answers.
I’ve spoken before about the need that (mainly) girls have to write down a nice neat answer. Many girls consider a maths lesson a waste of time if they have nothing to show for it and believe that how much you have learnt correlates directly with how much written work you have done [20],this despite the fact that maths is something that takes place inside your head, not on paper [21]. The maths teachers I have spoken to at my current school, all of whom have also experienced co-educational schools, agree that this does indeed apply to girls a lot more than it does to boys. The MATs activity was an active exploration with no numerical answers to write down; I did, however, want the girls to have some record of what they had done in the form of a small diagram to represent how the MATs fitted together in their solutions. This resulted in a lot of questioning about the expected quality of the diagrams and although I repeatedly explained that it would certainly be sufficient to produce a cluster of blobs with labels ‘T’ for triangle, etc., there were still four of the eight girls who drew very neat and accurate regular polygons (one girl took this to a painstakingly careful degree). This attention to detail meant that these girls had much less time actually ‘doing’ maths than did their peers.
Having found solutions to all of the set questions the girls moved on to the next activity; making their own shapes with which to challenge each other. This was quite pleasant and took up the rest of the time until the end of the lesson, I did give my plenary activity for consideration after most of them had made at least one of their own shapes but I didn’t do any work on it as intended. Most of the girls saw the pattern and managed to determine the 10th shape’s perimeter, but there was too little time to begin talking about the nth term.
During the make-your-own exercise all of the girls decided to make the most fiendishly complicated shape they could think of. They fitted together all of the available polygons in whichever way they could to achieve a massive, sprawling pattern. They then counted how many side lengths fitted into the perimeter and challenged another pair of girls to replicate their shape using the same MATs and the perimeter information. Obviously none actually replicated the shape exactly but it is usually fairly easy to find some shape or other with a given perimeter. There are often several solutions. That the shape wasn’t the same prompted many to say: “oh, no that’s wrong, ours wasn’t like that.” I had to remind them that the task is to achieve a given perimeter using given MATs – the result could be any shape. They all wanted to have made the most difficult puzzle: “oh you’ll never solve ours!” But they had misinterpreted the task. I asked right at the end what the properties of a difficult puzzle would be and explained that it would be better if there were only a few possible solutions, therefore making these harder to find. I think a few of them took this on board.

Learning and teaching

It occurred to me that I do not care for teaching; I only really care about learning. I am not sure how normal it is to feel this way – possibly a majority of young student teachers feel the same. Certainly I know a few more experienced teachers who do not share the sentiment, and neither does the one other student teacher with whom I have the most regular contact. John Holt writes that he went a long time observing teachers at work before he realised that he should instead be observing the learners. Only then did he notice that the children were not in school to learn, they were in school to survive – they were in school simply to attract or shun attention depending on disposition, and to avoid getting any answers wrong. Only once his attention shifted towards the learners did he notice their survival strategies at work; he noticed how their survival could be ensured without any learning actually taking place. Intense effort was expended in teasing answers out of the teachers’ intonations and in the avoidance of the teachers’ questioning.

I find it more interesting to observe the learners but I know what Holt meant when he described being draw towards the teacher. When the teacher is speaking at the front of the classroom I naturally focus on her, it is difficult to refocus that attention onto what the children are doing in reaction to the teaching. The assumption is often that they are doing the same thing as I am: concentrating on the teacher. This is rarely the case. The best teachers are those who are able to practice this redirection of attention from teacher to class even in their own classroom; they concentrate on what the children are doing. By this I do not mean that they have their eye on the class’s behaviour but that they actually know what learning is happening. That they know when a child is only appearing to listen or that the child is desperately trying to disguise the fact that she does not know the answer. That they know the difference between a child deep in thought and a child deep in distress – these two states are often indistinguishable.

Nobody taught me Mathematics at school; I learnt Mathematics. People taught me History and Geography hence I was not interested in these subjects. I suppose it is possible for teaching and learning to happen in a classroom. The danger with teaching, however, is that the class will become passive; the teacher is now the active agent, not the children. I’ve never met a child who does not prefer learning to being taught, there is no gratification is having been taught something. Have learnt something, however, one is quite entitled to feeling of pride and satisfaction.

I enjoy teaching because of what I can learn from my experiences, rather than enjoying teaching in and because of itself. I like it when learning happens and I strive to provide the opportunity. I would much rather provide children with the opportunity to learn than to teach that child, to paraphrase A. S. Neill: every time some old humbug explains to a boy how an engine works he is depriving that boy of the opportunity to learn how an engine works, and to make that learning personal and meaningful. If we go about teaching children everything they will have nothing left to learn, and they will dislike all the subjects on account of them having been taught. 

The IMA scholars event

On Saturday I attended a welcome event for the recipients of the IMA teacher training bursary. I expected to see a lot of people like myself; those who absolutely love maths and have a clear passion. It was therefore a surprise to learn that over half the people in the room had not heard of either Mathsjam or even Martin Gardner. It was previously inconceivable to me that anybody with even a passing interest in Mathematics could have failed to have heard of the most famous populariser of maths. I happen to know that a subsection of the scholars are Mathsjamers and Ian Stewart readers and @MarcusduSautoy followers and all the rest of it, but overall I was very underwhelmed with the level of genuine interest in mathematics. There was a seating plan with tables grouped by geography and little opportunity to mingle throughout the day; besides the above mentioned statistics which came about via a hands-up question from the speaker, my judgement is mainly indicative of England’s south west. On my table there was placed a Hexa-Tetra-Flexagon, something I immediately recognised. The rest of my table had no idea at all, some of them decided to unfold it and exclaimed “oh, it’s just a square with a hole in!” and concluded that it must be  a precursor to some sort of geometry activity.

I’m not trying to imply that somebody won’t make an excellent maths teacher just because they don’t know what a Hexa-Tetra-Flexagon is, what non-transitive dice are or how to play Set, but I do think it’s a shame generally when people’s only knowledge of maths is that derived from their schooling. I think it’s an even greater shame when even the next generation of maths teachers, endorsed by the IMA, haven’t ever delighted in some recreational maths and believe that all there is to Mathematics is whatever happens to exist on the school syllabi and the University course option lists.

First impressions of Y6K’s mathematical ability

 

Today I had my first day of experience as an official part of my PGCE course: I observed Mr. King’s year 6 class for the day. I have to say that I actually found it more insightful than I had anticipated. The first lesson of the day was maths, followed by literacy. In the afternoon it was music and P.E. which is appropriate for a Friday afternoon I suppose.

 

To start off we had a mental maths test of the variety found in the SATs (preparation starts in September!) Mr. King joked to the class about the awfulness of tests (I used to love mental maths) and explained it away as a necessary measure for seeing where they were all at in terms of their ability. Before the CD was started Mr. King tried to encourage some good habits for the SATs, he got them to look at the paper and interpret the clues, one answer box, for example, had a picture of a pentagon and four tick boxes labelled ‘pentagon’, ‘hexagon’, ‘octagon’ and ‘circle’. It was fairly obvious what that question was going to be. Another contained the numbers 6 and 12, Mr. King wondered out loud “hmm… what are 6 and 12 added together? What are they multiplied? These are all things I might think when I see this paper”. I had spoken to him about this before the start of school, saying that it’s a good habit to acquire and that I would always try to anticipate the questions. He replied that although it seems an obvious thing to do, the majority of pupils tend not to look at the test paper for any longer then absolutely necessary. They often get overwhelmed and panicked, preferring that the panic should only last for the 10 seconds that it must before the next question is read out. Two of the answer boxes contained only the number 64. “The thing that stands out to me about this is that 64 is a special type of number. Does anybody know what kind of number 64 is? Yes, it is even. Yes, the two digits add to ten. Yes, right, spot on, 64 is a square number. It is something times by itself. 3 times 3 is 9 so 9 is a square number. 4 times 4 is 16, 16 is a square number. 64 isn’t 4 times 4 but it is something like that.” I noticed many interested faces and hushed whispers of “ah, eight” going around. When the test started and the question was read “what is half of sixty-four” I started to question the judgement of stressing the squareness of 64 so much beforehand, lest they all get confused. Then the second such question was read “what is a quarter of sixty-four”. Nothing about square roots at all. I was pleased when I marked the tests to observe that only four out of 29 had put 8 for either of these questions, all the rest had actually listened to the questions and not simply dwelt on Mr. King’s speech about square numbers.

 

There was one question that I thought looked particularly difficult: “nine and a half is half of a number. What is that number?” Even though an earlier and very similar question: “two and a half is half of a number. What is that number?” attracted almost unanimously correct responses, so they knew that doubling was required in this type of question, there were still many confused answers. Nineteen got is right, one pupil just doubled only the 9 part and got 18. Two added a half and got 10. Five tried to halve the number and got either 4.5 or 5. Two others had no idea; one put 10.5 and one didn’t even guess. Still, despite the difficulty of that question it only had the second most wrong answers. There was a tricky question about time and one about area but the top wrong-answer question was “what is a quarter of sixty-four”. Considering they’d already done “what is half of sixty-four” with much success I was very surprised by this. The answers, apart from the twelve who got it right and the seven who put 14, 15 or 17, weren’t even close! Three pupils put 8, possibly thinking of the square numbers, and the other answers were 1, 10, 10, 12, 22, 54 and ‘left blank’. I thought it was brilliant that somebody, clearly not having a clue, had written down the number 1. Mr. King had advised “if you don’t know, just put down something, it might be right!” clearly this had been taken a bit too literally. ‘1’ obviously could not have been right but this pupil wanted so much to do what the teacher had said. After I’d done the marking I asked Mr. King if he’d have expected this, after all, they’d already found half of 64 in a previous question with almost total success. Most of them, I was told, simply don’t see the connection between a quarter being ‘half of a half’ (a fact they all know) and that talking a quarter of something is the same as taking half of a half of that something. It is connection like these which permeate mathematics and make it such a lively and fascinating subject, but seeing the connections takes a lot of practice for some. That is why, in the next part of the lesson, on place value, the connection between, say, ‘0.05’ and ‘5/100’ was repeated and emphasised to such a degree.

 

For the high scorers, those with 18, 19 or 20, the only issues were with vocabulary: they either didn’t know how many meters in a kilometre, or they didn’t know what ‘product’ meant, most just did the sum instead.

 

After the ordeal of the test the lesson moved on to place values. Mr. King put _ _ _ _ . _ _ _ on the board and filled in numbers. For each one the class had to tell him, via mini whiteboards what certain digits represented so 5487.756 was ‘0.05’ or ‘5 / 100’ or ‘five hundredths’. To practise saying these strange words we then played a game. In pairs they put _ _ _ _ . _ _ _ on their board and took turns to roll one die and write the outcome of that roll in one of the positions on their board. The object is to make a larger number than your opponent, obviously if you got a six you should write it in the thousands column, small numbers should go in the thousandths or hundredths columns. First I played against Tom, who played the game very well and seemed to understand the objective and how to achieve it. After that first game I swapped partners and played against Ellen, the most noticeable difference was that Ellen didn’t make the effort to announce out loud “I got a 4, I’m going to put that in the tens column”; Ellen just wrote her numbers down very neatly and very quickly then passed me the dice. To remedy this I said that I would write her numbers and she would write mine, then she would have to tell me where to write it. Still, it took a couple of throws for the message to get across; at first she just pointed. In another situation I might say that this was down to shyness, I was a new adult after all, but Ellen was happily chatting to me whilst playing and was otherwise very outgoing. What Mr. King confirmed to me was that girls are often very concerned with writing down their answers, making them very neat, and completing the task a quickly as possible. In literacy or science this kind of behaviour is warmly rewarded, as it is in maths classrooms which are more concerned with answering questions than actually thinking. It is maybe one reason why girls might become disaffected with maths: their neatly produced answer booklets are not given due credit, and nobody tells them that the way to get credit is to think mathematically, or if they do then they don’t tell them how to do it.

 

Given Ellen’s write-it-down-without-thinking strategy (she did put the numbers in appropriate places, she just wasn’t fully engaging with the main intended outcome – actually understanding what the place values mean, and to understand the associated language), I was very pleased when I did get Ellen to think, and she displayed very good powers of thought indeed. We got to a point where I had ‘_ 5 5 _ . 2 _ 1’ and she had ‘6 6 _ 2 . _ 1 _’. I asked her if it was necessary to continue, and I led her on a little bit “can we already tell who will win?” After about 30 seconds of concentration she announced “yes, I will win no matter what.” To reinforce her own thinking I filled in the blanks, mine with all sixes and hers with all ones, showing that even in this scenario she would be the winner.

 

After maths was literacy, the class were in the middle of writing their own Martin Luther King style speech about their dreams for the new school year. I went around helping with spelling and suggesting how they might use rhetorical devices to good effect. One interesting common problem I noticed was the incorrect placement of the apostrophe in the word “would’nt”. The reason I was surprised was that about half the class had done it and they had all correctly spelt “don’t” at least once. I may be a mathematician but I don’t think it is a difficult connection to make. If in one word ending in “nt” the apostrophe is after the “n” then why is that not extended to other words ending in “nt”? Don’t schools teach lessons on apostrophe usage, establishing that connection? I got my answer when I discussed the issue with Kate, a KS1 classroom assistant. Kate told me that “don’t” is a keyword in the curriculum so many pupils will have taken the effort to learn it by wrote, rather than thinking about why this particular word has an apostrophe, and what that apostrophe represents. Had this effort been made then there would surely be a larger proportion of the class knowing that, when a word or pair of words is abbreviated, the apostrophe goes in the place where a letter (such as ‘o’) has been missed out.

 

 

 

Maths and ‘other Subjects’ (part II)

The other day I had my revelation. My wife’s friends were talking about the new style GCSE organisation that many schools are adopting. In this new system students spend half their time in year 9 doing GCSEs, sitting those at the end of year 10 (a year early). For the other half of their time they do ‘filler subjects’. In year 10 they take up the other half of their GCSEs and sit these in year 11 as normal. I don’t see any reason for this change, half of year 11 is then just free time (which I suppose bright kids could use to do even more GCSEs, the rest do filler subjects and lose all interest in school) and if you do, say, German over years 9 and 10, you have a year 11 of no German so that when you start A-levels in year 12 you have been without practice and fallen behind. This is exactly what my brother has done, not to mention that he got a much worse grade than he could have done had he sat the exam in year 11, like everyone use to, with that extra year of practice, and this came up in the conversation. I made the point that for languages it is particularly important to practise for as many years as possible. Speaking a language for an extra whole year can make an enormous difference (similarly, I was thinking, to maths and unlike, I imagined, essay writing where you either can do it or you can’t). In agreement to what I had said (out loud) Manda gave the following response:

“well say for maths, if you have a maths brain anyway then I don’t suppose it makes a difference if you sit it a year or two early. But for subjects where you have to think like languages, or history and English…” At this point in the conversation I found that I could no longer believe my ears and became lost in thought instead.

What used to frustrate me was people saying that I simply hadn’t the brain for it: “I’ve got an essay brain. It’s okay though, you’ve got a maths brain instead.” As if you could either have one or the other.

“No!” I would explode, “I haven’t got a maths brain; there’s no such thing.  I’m good at maths because I learn all the things there are to learn, I pay attention in class, unlike you. If you tried in maths you could do it too, but I do try in English and I still can’t do it!” It was frustration to the point of madness, and anger that a subject existed in which there was no right answer. A subject where the only skill required was ‘being good at writing essays’. Everything hinged on anticipating what the mark scheme, distributed to examiners and teachers by AQA, would comprise. I still don’t know how this feat is achieved; the damn scheme was different for every essay anyway. Where’s the logical rigour? Where is the mathematical consistency? In maths it genuinely didn’t matter if you got the correct answer by a unique and creative method not taught in any school or included in any mark scheme; so long as the correct answer was there on the paper (even if no working had been shown at all) you got the marks, hooray!

So I felt like essay writing was a skill I couldn’t possibly acquire, indeed, I literally couldn’t. At the same time I strongly believed, as I still do, that maths is a language that can be learned by anyone. I am a native speaker and before I really thought about it and started to read books on the subject of educating, I maybe didn’t appreiciate the difficultly that others can have in learning. Just as we native English speakers don’t give a thought to why it’s ‘bought’ instead of ‘buyed’ or why it’s pronounced ‘live’ and not ‘live’ (said ‘liv’ and ‘lyve’). This does not imply that I have a ‘maths brain’, just that I learnt maths in infancy. The Englishman’s and the Chinaman’s brains are not constructed differently; a Chinese baby raised from birth in an English-speaking environment will not struggle to learn English any more than an ‘ethnically English’ baby.

What I never had realised was that many of those essay-brain people who speak of me having a maths-brain do not in fact accept that the other subjects require a specific brain. Essay writing to them is a totally normal thing requiring as little effort as it costs me to understand differentiation using the epsilon-delta definition. I still maintain that maths is a language, ‘other subjects’ (by which I mean essay-writing or any subjectively-assessed subjects) require certain skills and techniques; tactics for getting through. That people sometimes misinterpret maths and try to adopt certain tactics – missing the point of maths; that it just requires a bit of actual thought – is another issue. Of course it’s impossible for me to give any opinion on this which will be considered objective by any reader – I am a proud mathematician after all. It isn’t uncommon, however, to hear mathematics called a language – particularly by those who know what they’re talking about; I have never heard essay-writing referred to as such, or indeed as anything other than a specific ‘skill’ or a ‘technique’. It seems unfortunate to me that possession of this one technique counts for so great a proportion of the curriculum, along with that other useful skill: memorisation, whilst logical thinking and seeing connections, most useful of all in the ‘real world’ will only get you a mere handful of GCSEs: in maths, and maybe in a couple of modern languages.

Maths and ‘other Subjects’ (part I)

As a rationalist I like to be open to new ideas, there’s nothing that I believe in strongly enough to defend in the face of solid reasoning or greater evidence to the contrary (except, of course, the fact that my son is the most gorgeous baby in the world). Even so, I don’t expect to have to revise my opinions of assumptions very often. The other day I realised that maybe I’m not as ready to admit my errors in judgement as I like to think that I am; I had a small revelation but then realised that I had, in fact, known it all along; simply not being able to admit it to myself. The revelation concerned the way that other people view maths, school maths in particular.

I’m quite aware that a lot of children see maths differently (often unfavourably) to the ‘other subjects’ as a group; there is Boaler’s classic quotation of a school girl “in other subjects you have to think, in maths you just have to remember”. This seems a very odd thing in my opinion; I’ve always thought that maths is just about the epitome of ‘thinking’ in so far as any school subject requires thought. When compared with, say, science, I would say that in other subjects you have to remember, in maths you just have to think. And I always thought that maths was much the easier subject because of it; it was a nightmare trying to remember which cells contain which organelles or what the difference was between ionic bonds and whatever those other bonds were called. Science at GCSE is, very literally, a memory test. Maths on the other hand required no memorisation; a consultation with one’s common sense and a little knowledge of some basic principles would soon yield the answer to any problem. I found it easier to work things out from scratch than to memorise techniques, I never took pains to memorise the prescribed methods for long division or solving equations. By noticing the inherent order of things and the connections between them it all seemed fairly obvious and simple fell into place for me. Just like driving a car or riding a bike it seems impossible that I could ever forget maths: having understood it and been able to do it once it is now a reflex, and it has been that way since I can remember – well before secondary school.

Since science was actually a memory test it feels like the easy option to use in a ‘Thought vs. Memory’ comparison. The harder option would be languages. At GCSE I did Latin and French, the first of these, like science, required a lot of memorisation, a lot more than science in fact. I had to memorise Virgil’s aeneid word for word and how it all translated to English. In the exam a random passage would appear and I would write down the translation. French was different, in French I did have to memorise some things but a lot of it had moved past that stage where conscious effort was required. When engaging in very simple conversation I did not have to struggle to find the right words or translate my sentences from English in my head: they simply came out of my mouth fully formed in French. I was thinking in French, and I was never able to think in Latin. It could be said that French still required remembering but the remembering was done with minimal effort. If remembering Latin verbs was like catching flies with chopsticks then remembering French verbs was like coating myself in fly-paper and going about my normal business. I suppose that I had the same experience with maths, I was fluent in the language and it took no effort to learn the occasional new word. I suppose those students who struggle in maths feel like I did trying to memorise Virgil, no real understanding of why the sentences looked like they did, but with the ability to reel off whole passages by heart.

I always knew that there was such a thing as the “English student’s brain”, or the “Essay brain”. Those people (usually girls in my limited experience) who can, without knowing the text in question in any detail at all, and without even have read some of it, can write an essay simply by rewording the question and bluffing through thousands of words. When I asked them how they did it they would say things like: “oh, you just have to waffle on and sound like you know what you’re talking about” or “you just write what you know they want you to write”. There didn’t seem to be any technique that I could acquire. Unlike in maths I couldn’t just derive the answer from basic principles. There was no right answer and no algorithm for getting it, to me it seemed like you had to be born with the skill. Not even the teachers had a sure-fire way of getting an ‘A’ in an essay. I would spend hours analysing the book and write two thousand words of what I considered to be excellent prose only to get a ‘C’ or a ‘D’. Even when I took on board the teachers’ comments for next time I couldn’t improve the grades. English (and any essay writing at school) was a uniquely stressful experience for me; I could not see the point in doing something if there was no right answer. I couldn’t confidently claim that I’d got 90%, as I could in maths where I knew exactly what I either did or didn’t know. I had to wait patiently for the results and there was no way of knowing how well I’d done.