Logical Sequences

Everything has to be broken down into a logical progression from start to finish. I have seen lessons of experienced teachers where this happens; they know what the aims of that lesson are and they know how to take each student on a little journey from what they know to what they should know by the end of the lesson. My lessons conform to two types, neither of which coincide with the above description of a good lesson. My lesson planning often turns out a plethora of activities which, together, would give a student enough knowledge and practice to have mastered the topic, if only they had the mental equipment to piece together this jigsaw of Mathematical activities. Basing a lesson around this planning outcome defines Lesson Type A: a lesson characterised by quite rightly splitting up a topic but then reassembling that topic in the wrong order. Lesson Type B is characterised by a lack of splitting; in a B-type lesson the students have 15 minutes of instruction and a load of questions. The questions explore the whole topic and problems are dealt with as they arise; my broad intention is that the students will end up discovering something for themselves, which would be lovely.

Finding 10%, check they can do it. Finding 5%, check they can do it. Finding 15%, check they can do it. Deducting 10% from a price, check they can do it. Deducting 5% from a price, check they can do it. Finding 1%, check they can do it. Finding 27%, check they can do it. Deducting 27% from a price, check they can do it. Finding 27% of a weight or distance, check they can do it. Finding 27% of a weight or distance, check they can do it. Increasing a weight or distance by 27%, check they can do it. This probably needs a few more steps adding in. Of course, each individual “Finding 10% of a price” is a learning objective in itself and will be broken down further into achievable little chunks.

I am not against this style of teaching. I want to be able to do it! In fact I can do it. When I am tutoring one-to-one I always break things down like this, in reaction to the student’s current needs. What I do not seem able to do is plan this sort of thing on a whole-class scale. What I need is Devlin’s Game. I want them to wonder around WoW learning bits of Maths as they go, everything being broken down into achievable little chunks. They can go at different paces and I will be there to deal with problems as they arise. They will be guided on a journey but they will also be independent learners, occasionally discovering things for themselves. I thought I understood these concepts before I started my journey but now it has become so obvious how useful such a thing would be. I do not so much want to use Devlin’s Game as to help to design it; that would be my ideal job.

 

On The Values of New Teachers

In regards to my previous post Caring, I recently attended an interview for a teaching post at Clifton College in Bristol; whilst there I mentioned my misgivings about not being able to properly teach Mathematics to the Deputy Head. I had thought that being quite honest on such a matter might have been inadvisable but, on the contrary, I found that he and I were on common ground; he personally expressed great regret that he had been unable to teach everything to everyone. Doing so would be an exceptional achievement indeed, and conceiving that one might be able to do so in one’s early years of teaching could be regarded as being tremendously naive; however, such a goal is – I have learned – not at all uncommon amongst new teachers. My interviewer told me that in the early days of his own career he had been mindful that his students did not fully understand the current topic despite having done quite some work in that direction. He came under pressure to move the group on to the next topic regardless so that they would at least know a little piece of each chapter without knowing the full contents of any one: the received wisdom had been that: “according to their ability this class will never be able to understand more than the surface details. It is therefore advisable that they learn a lot of surface details without the complication of also trying – and failing – to gain some deeper understanding.”

Because I cannot conceive that there is any value in knowing only the surface details, it seems a waste of time to teach a class at all if their highest aspirations do not penetrate significantly into the depths of the subject. This may not be true of mental calculation, or of other aspects of school lessons which will certainly be of considerable importance to students throughout their life, but it is true of the higher-level Mathematics, and indeed of Physics, Chemistry or English Literature. What good can it be for a student to have learnt that f=ma for the purposes of passing an exam if there is going to be no appreciation of how forces work, or even what forces are. If a student has been told that when he opens his exam paper he must write that Lady Macbeth is evil, but yet he has never come to appreiciate Shakespeare’s verse or to understand the moral of the play, then what has been the point in his education? The government may be happy for students to get a C, but teachers want their students to learn, even if they get a D. They can fail the test but if they have wept over Grapes of Wrath, if they have spent two hours solving a problem in Mathematics, if they been able to grasp, even for a second, the scale of the universe and immense complexity of the human brain, then their education has been worth it.

Caring

A girl was having some conceptual trouble in my lesson yesterday. We had some linear equations, such as y=3x-2, and were going to try to draw the lines by joining up some points. Her first trouble was that she was struggling to recollect that 3x-2 meant “three times x, then take away two”, but she got there after a while. I came to understand that she was having trouble thinking of x and y as variables rather than unknowns. It seemed strange to her that she was supposed to pick some random values for x and then work out what y would be in each case. This is what I love about teaching: students having problems; students trying hard to understand something and requiring a few Socratic nudges in the right direction. I love it. Unfortunately there were several others in the classroom, not struggling conceptually, but struggling to sit still and resist the urge to start throwing paper aeroplanes around; struggling and, for the most part, failing miserably. Who could expect me to care? I do of course care about the futures of these children. If they end up throwing not just paper aeroplanes at teachers but throwing nail bombs and bricks at unfortunate members of a racial minority group then their teachers will have failed them and failed society as a whole. It is a huge amount of pressure and something to which teachers must eventually become immune; how many children leave school without having learned basic manners and respect? Without having learned basic social skills? It must be hundreds every year, teachers cannot constantly feel guilty, feel that they have contributed to the downfall of society, for every single child they let slip through the net. Of course it troubles me that these kids were showing a total lack of any social niceness towards me, but who could expect me to care then? Right then when I was helping this girl to actually do some Mathematics? I was doing exactly what I love doing and the rest of the world could just wait a while.

Telling kids about the real-life applications of Maths

My tutor, the engineer, keeps asking me to think of real life applications for the things I am teaching. Once again I just cannot quite say why it is that I find the idea, or the implication imposed, so offensive to my sensibilities. It is a good thing to introduce children to the real life applications of what they are learning. It gives a purpose to Maths at last. If they thought English was their favourite subject then hey! Maths is actually useful, trump that! To be sure, Geography, IT, Science and MFL have more usefulness in their little fingers than a lesson on protractors ever possibly could, but at least we beat English (unless, of course, the student in question cannot even read and write at level 5, then being able to do so is of primary importance). I would be stupid indeed if I deliberately tried to hide the real-life applications of any Maths I taught; my fault is that I simply do not trouble myself to think what applications my Maths might have. Ruler and compass constructions? It’s just fun, just enjoy it, do not worry about how you might have to use it one day. If I went around telling my students that they will fail in life without a basic knowledge of Maths I would probably provoke anxiety. I ought rather to tell them that they will be more likely to fail in life, based upon the latest statistics. Now there’s a nice application of Maths.

 

Lesson 13 – Common Factors

For this lesson I had planned in great depth and detail; I was trying hard to hit the ‘structured lessons’ teaching standard. Unfortunately I misjudged the access point and it all went downhill after the starter. If only they had behaved long enough for me to reach the well-differentiated activities and the excellent explanations which would have helped them all to make progress and reach the Los. If only.

Actions targets: Clear guidelines and Quick consequences.

On being fun, part 2

I have been advised that 11-year-olds like cartoon characters and video games; true, and in fact using cartoon characters and video games ties in with Keith Devlin’s philosophy which I have long supported. Why, when I go around advocating Devlin’s ideals do I find the prospect of using Homer Simpson in my lessons so vulgar? I think it is a problem of intent. My tutor says that “they should come to lessons to have fun, and if they learn some Maths along the way then that’s fantastic.” Shouldn’t we encourage children to come to lessons to learn Maths, and if they end up having fun then that’s even better?  After all, if they truly came to lessons in order to have fun then they should logically not come to the lessons at all and instead do something guaranteed to be more fun than even the most cartoon character infested lesson I could devise. This is the Summerhill philosophy, I am sure that I do not know why it is not more widely accepted.

If children come to lessons to have fun then they will be consistently disappointed. You may think your lessons are fun but they are not as fun as football or skateboarding or wonton vandalism or whatever else kids do these days. Given the choice, a child whose primary motive was fun would stop coming to your lessons, I guarantee. However, children are naturally inclined towards learning (as are, I believe, adults) and they will eventually tire of having fun all the time. Eventually they might venture back to the classroom and this time their primary motive will not be to have fun, it will be to learn. This, I think, is a far better reason to come to school; it, after all, validates the whole purpose of school.

There are two types of motivation: intrinsic and extrinsic. Both can be positive factors when it comes to learning: extrinsic motivation, such as the view the perform well in exams, is very motivational but should be kept in check because there is a danger that it might transform into a fear of not doing well in the exams, it might even become anxiety. The intrinsic motivation of actually enjoying Mathematics is obviously very positive (although it could lead to a child reading widely around the subject and not focussing on exam content). So what of fun? If a child is motivated to come to lessons because their teacher cracks jokes and the worksheets are chock full of cartoons, what then? This motivation seems more inclined towards the extrinsic than the intrinsic but, unlike exam motivation, it doesn’t particularly encourage the doing of Mathematics, just a presence in the classroom. There is, of course, nothing wrong with teachers being fun; I had a few really fun, and funny, teachers at school and I really liked them and I did well in those subjects. My point is that fun should not be the primary motive since it is not necessarily a productive form of motivation.

Lesson 12: Moving towards fing prime factors

After yesterday afternoon’s lesson I was trying to be really intolerant of noise and misbehaviour. I finished 10 minutes early and practised the packing-up routine (yesterday they had been particularly noisy at the end). I did the same thing to my year 9’s just after. One boy told me that he had to go because he had a break time detention, to think that he would have to explain that he was late to a detention (presumably administered because of naughtiness) because I had kept him in for being naughty! The fact that these children even bother coming to school is quite amusing, and a great testament to the power of herd mentality.

Action targets: Plan the lesson by starting with “what do I want them to be able to do at the end?”

Lesson 11: Factors

Last period again, they were a riot. Not only is it last period, it is also right after PE so it takes at least 5 minutes before everyone has arrived; this makes it difficult for me to adhere to my beginning routine. At one point I spent quite a long time waiting for silence and three boys (who were not talking) looked visibly bored. I felt really sorry for them, upset with myself that I was unable to continue with the lesson because of my lack of control. I mentioned this to my tutor who, later on, took my words out of context when talking (again) about how my activities are not fun enough. He exclaimed “you said yourself that those three boys were visibly bored in that lesson; your activities are just boring”. Well, my activities may be boring but at that moment the class were not doing an activity, boring or not; they were waiting for me to give them an activity, which was not going to happen until they were quiet. Those bored boys were bored because they had nothing to do, not because they had something boring to do.

The high point of the lesson was I went to see what W had been doing (W is always reluctant to do any work at all). It turned out that W had been quite productive: in his book he had written “1×1=1 2×2=4 3×3=9 …” and so on up into the mid-teens. What I had asked him to do was factorise the numbers up to 20, as an example I had written, on the board, “1=1×1 2=1×2 … 4=1×4 or 2×2 …”. I told W that I was really pleased with how much he had done, but it was not what I had asked; in response he motioned to the first thing I had written on the board: “1=1×1”, “well I thought we were doing that”. I, in turn, replied “well, what did I write underneath 1×1? I did not write 2×2 did I?” It turned out that he had not listened to my explanation, he had not listened to my instruction, he had not seen me write up my examples; all he had done was glance at the very first thing written on the board. The very first thing; he had not even bothered to look at the second example. He had made a guess at what the activity was supposed to be merely on the basis of the very first thing on the board. I am quite at a loss to think what the word is.

Lesson 10: factors

New topic, Laurinda Brown observing the lesson. I had a good start and stuck to my routine (almost). I had prepared a variety of worksheets at different levels but not got on to the “extra hot” which would have talked them through prime factorisation. Laurinda’s main problem with the lesson was that I spent too long explaining what to do after I had already explained what to do. It seems that sometimes some students may not quite be paying absolute attention to my instructions; I then have to go around answering questions during the 5 minutes I set aside for the activity. By the time everyone knows what to do, the time is up. Laurinda also noticed that my differentiation is not up to scratch; it is something I am really struggling with when I am having so many other problems. It also seems quite odd, to me, that the students do not have their own text books. It might not be fantastically good form but it is useful to be able to say “go on to the next question”, it is especially useful when textbooks have “challenge” questions or investigations at the end of each chapter. Here I have to create and print every resource, if they finish what I have planned, there is nothing left! Even when I try to give an investigation, as I did with my extra-hot sheet, they are just something I have printed out and lack the lustre of a well laid out text book.

Action Targets: Stick to beginning routine, start end routine, differentiate. I could also ask “does anyone have any questions?” before each activity.

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.