Background noise and awkard silences

Have you ever experienced a moment in a lesson where, without warning, the whole class suddenly goes quiet for a while?

I had one of them today. As ever, it was an unplanned moment. The students had been working on task for a while, with the usual level of classroom noise, then suddenly, as though there was some telepathic signal between them, silence descended.

This happens in my lessons every so often and it usually feels a bit wierd. For a moment it's calm, then it feels a bit awkward, then I feel a sense of pressure descend on the class. I always worry that one of the students might want to ask a question, but they don't want everyone else to hear, so they don't ask and end up feeling a bit trapped.

Inevitably, it never lasts more than a few seconds before one of the more vocal students says something inane.  "Why has it gone quiet?" is a popular one, asked in indignant tones, as though it's the responsibility of the rest of the group to keep up an incessant stream of background noise. Or they say something like, "It's really quiet. I don't like it!" as though the brief silence is something to be endured rather than a moment of calm.

To be honest, I'm usually as relieved as the students when the spell is broken. A few nervous giggles and we're back to normality.

Recently however, I've been wondering if silence might actually be a useful pedagogical tool. It's not something that comes naturally to my style of teaching, but I think it's worth pondering.

So what are the benefits? Well, working in silence would give students a chance to become more engrossed in the task at hand rather than getting involved in off topic conversations around them. Teachers should never forget that school is a highly social experience for students and they can often feel pressurised to manage relationships with their peers at the same time as learning anything academic. This sort of social scenario in the classroom is the antithesis of the psychological concept of "flow" when you become completely absorbed in what you are doing and you don't notice anything else that's going on. I think I've experienced "flow" when playing the piano, writing certain essays and even when solving some particularly engaging maths problems. It's a great feeling and a very productive state of being. A normal classroom environment, full of distractions, doesn't seem conducive to inducing a state of "flow" in anyone.

Working in silence might also give students a chance to be more reflective about what they are doing. Rather than racing through their work, they would have more time to concentrate on the task at hand (because they would spend less time chatting to each other). This may particularly be the case for problem solving questions, where students have to approach something from several different angles and refer to various different facts or techniques in a single quesiton.

Hmmm. I'm definitely not suggesting that I will try to insist on silence for a whole lesson, or even the majority of a lesson. As a general rule, I always encourage students to talk to each other throughout lessons and I have no intention of dramatically changing this modus operandi. My classroom is arranged in groups, not rows, specifically to facilitate discussion, even though I know it means I have to work a bit harder to get them to focus on me when I'm teaching from the front. And I'm not at all averse to background noise. Whenever I work at home I tend to have music on, particularly when I'm writing. Currently, iTunes shuffle is soothing my ear drums with "All the young dudes" by Mott the Hoople, which I consider to be pretty good writing music.  Oooooo. It's just changed to "White Winter Hymnal" by the Fleet Foxes. Even better writing music.

But I think it might be worth trying out shorter periods of silence, maybe 5 minutes at a time, after students have had a chance to talk to each other about what they are doing. I like the idea of giving a class an extended question, asking them to discuss a strategy for answering it, then telling them that it's time to put their plan into practice, but they have to do so in silence so that they can concentrate better. I don't know how well it would work. It might just feel incredibly awkward. They might completely resist it, or they might embrace it. Whichever class I choose to try it with though, I know that I'd have to explain my reasoning first. I wouldn't want them to associate the period of silence with punishment or boredom. I'd have to explain that we were going to use silence as a way to help them concentrate and produce their best work. It would be a planned silence, a friendly silence, and most of all, a productive silence.

Hurry up Gove! Post GCSE Maths Qualifications

A y11 student asked me today if it was a good idea for her to study A level maths. "I like maths" she said, "I don't want to stop learning it after year 11".


I wish I could respond with unhesitating encouragement. I'd love to advise all my students to continue with maths post-16 and I know that many of them would like to. I reckon that about half of my y11s have talked about taking A level maths, or have said that they will miss maths next year and wish they could do it in the sixth form.

So why didn't I just say go for it?  Why don't I encourage all my students to take A level maths?

Well first things first, I firmly believe that everyone can achieve highly in maths, its just that some people need more time to understand things, or need to learn in different, often more visual ways. I also can't stand putting limits on myself or anyone else. I think anyone can do anything if they put their mind to it.

But we have to be realistic. It's not fair, in fact it is irresponsible, to encourage students to do something that they are unlikely to be successful in. Maths is a linear subject where you need to understand basic principals before you can move on. There is a substatial gap between achieving a C or a B in GCSE maths and tackling the subject matter in the A level course.

So when students who are on track to achieve a C or a B at GCSE ask about doing A level maths, I never tell them that it's out of the question, but I am honest with them. I tell them that it is hard work, much harder than GCSE and I tell them that they will need to do a lot of work in the summer between y11 and sixth form, to bridge the gap. I try to strike a balance between encouraging their ambition and being realistic about what can be achieved.

Even as I write this I feel awkward. I didn't become a teacher to put limits on students or to tell them that something is "probably too hard for you". But I have found some cause for hope from an unlikely source: the department for education. The turrential downpour of new initiatives, reforms and policy changes from the DfE has been pretty difficult to keep up with over the past couple of years and I have to admit that I hadn't seen this http://www.acme-uk.org/media/10520/20121217acme_post_16_strategy.pdf which is a report from the  ACME (the advisory committe on mathematics education) about options for post-16 mathematics, published in December last year.

I was very encouraged to find this paragraph:

A new [mathematics] qualification should be developed and introduced as

part of wider A level reforms.

This qualification should:

•  Be distinct from A level Mathematics, with an emphasis on solvingrealistic problems, using a variety of mathematical approaches, and should be for students not currently doing AS or A level Mathematics

•  Give students the confidence to consolidate their understanding ofmathematics by using and applying mathematics already learned in GCSE and new mathematics beyond GCSE developed during the course.

•  Have a smaller volume than AS level and be designed to be studied over two years

It sounds good to me. I like the phrase "give students the confidence to consolidate their understanding of mathematics by using and applying mathematics already learned in GCSE". I'm feeling cautiously optimisitc about the idea of a sixth form course that focusses on using and applying GCSE mathematics plus some extra content. I certainly agree that we need a course with "a smaller volume than AS level" for students who aren't ready for the onslaught of AS and A level maths. 

So hurry up Gove, get a move on with this one. After all, its not like you to wait around!

I just have one plea: make sure you introduce it as an optional course, with engaging real-life content, suitable for those C/B grade students who want to keep going with maths. If you try to make it compulsory it will be "one-size-fits-all" and it won't work. Use it as an opportunity to give teachers and students more choice, not less.

 

Practical Volume Activities

Here's a quick post this evening about a sequence of lessons that I've done with y10 on volume.

I often find that y10 and y11 lessons fall foul of time pressure. There is so much content to cover, in such a limited amount of time, that there doesn't seem to be much room for creativity. In y7, y8 and y9 I'm always keen to make lessons as creative as possible but with the older year groups the prospect of exams always seems to loom large and we don't get to have as much fun.


I usually plan y10 lessons with a colleague of mine who teaches a similar, middle ability group and I always find it really useful to talk to her. Two brains are definitely better than one! When we saw volume on the scheme of work, we both agreed that we'd like to get some practical activities involved, and this is what we came up with.

In the first lesson we gave the students some nets and asked them to work out the surface area. We took the opportunity to recap how to work out the area of simple shapes (rectangles, triangles, circles) and some of them took on the challenge of working out more complex shapes (trapeziums, pentagons, hexagons).

In the next lesson they cut out and made the nets into 3D shapes. We then talked about classifying the 3D shapes and they got into the idea of prisms. Not content with the shapes they had made, the girls wanted to look for prisms in the rest of the room. Surprise surprise, there were quite a few dotted around the place (it's not like I was collecting examples or anything) including a lovely octagonal quality street tin that just happened to be sitting at the front.

It was a really worthwhile lesson, although I'm pretty sure the pace would have been considered too slow by Ofsted. It took a while to cut and stick all the shapes, but once they had made them, the girls clearly understood how all the faces fitted together and they had no problems appreicating that the volume of a prism was cross-sectional area x length.

We had another lesson on calculating the volume of more complex prisms including problems where they were given the volume and hade to work out a missing length or area. Then in today's lesson we looked at the cuboid challenge where you give students a single piece of paper and ask them to make an open-topped box with the greatest possible volume. I think this task is usually done as a nice introduction to calculus, or at least plotting the graph of a cubic equation (nrich, as always, have explanations of how it works if you're not sure http://nrich.maths.org/6399/solution ). But, much as I feel a bit pained to say it, I wasn't bothered about the algebra this time. I wanted students to get a feel for dimensions and to have a go at a practical version of trial and improvement. I also wanted to make a big deal of the second part of the challenge, where I gave them 4 multi-link cubes each and asked them to work out how many whole cubes they could fit in their boxes.

The picture above shows the efforts of the winning team of 4 students, and the photo on the right shows a section of working out in another student's book. I asked the girls to work out the volume of every box made by their team and only once they had finished that were they allowed some cubes so they could work out how many cubes would fit inside. At first they were unimpressed with the quota of 4 cubes each, but they quickly figured out what to do and even the weakest students in the class seemed pretty motivated.

There is much more you could explore with volume, but I feel pleased that even in a limited time we managed to cover a lot of content in a way that didn't feel pressured and allowed the students time to become really familiar with 3D shapes. I'm sure that making the shapes, physically measuring them and turning them round in their hands helped the girls to understand what they were doing far more than simply working from a textbook would have done.

Reader, I married him: Going with the flow on ratios

I overheard some year 11s today talking about one of their earlier lessons and getting quite agitated about it. “It’s just structure, structure, structure in those lessons” said one of them. “I think she plans for every second” said another.

At first I wanted to laugh. I have no idea who they were talking about (and have no desire to know either), but whoever it was sounded like an excellent teacher and I thought the students just didn’t know what was good for them. “Every second planned” they’d said. Crikey. It sounds like the work of a very organised person. Hats off to them.

But then I wondered; why were the girls so agitated about it? They are highly motivated students, the sort that have been turning up for after school revision every week since they were in year 9. They’re not afraid of working hard.

The answer seemed to come in the final comments I heard:

“She never lets us just go with the flow” said a different student. “Yeah,” said the first, “she should just let us get on with it”.

This little conversation links in to an issue that I come back to time and again in my teaching practice. How much independence should we allow/encourage? How much is too much? Do students do better in lessons where “every second is planned” or in lessons when teachers “let us get on with it” a bit more.  An interesting seam, but I think I’ll save mining that one for another day.

Following another thread, it made me think about the lesson I’d just taught. It was year 9 and we were doing ratios. I knew that I’d put together quite a boring lesson. Originally I’d wanted to do something really creative involving the ratio of the string lengths of musical notes, but I hadn’t sorted things out in time and I was stuck with a boring stack of textbook questions. So I started the lesson feeling a bit uninspired, and explained how to simplify ratios into the form 1:n.

The girls were not impressed. What was the point? You can’t get 0.64 of a person. Why not just simplify them in the normal way?

So I said the first thing that came into my head and it went a bit like this:

Well, when I was at university there was a bit of an issue about the ratio of boys compared to girls. My university was split into colleges and some colleges had very different ratios. One college might have had a ratio of 11:10, but another might have been more like 7:3.

They looked interested, so I ploughed on….

So in some colleges, the number of boys and girls was fairly equal, in others there were lots more boys compared to the girls. If you were choosing a college, you might want to know the ratio of boys compared to girls and pick somewhere based on that. It might be difficult to compare the ratios, because you can't tell straight away if 3:7 is better than 16: 21, but if you had them in the ratio 1:n you would always be comparing like with like. Ideally the ratio would be 1:1 or pretty close (I was rambling at this point) although to be honest….. my college wasn’t that equal (really should have stopped to think at this stage)…but I suppose it did have its advantages … I mean I’m marrying one of them so it worked out quite well for me

Cue raucous laughter and bright red face from me.

This definitely wasn’t a case of every second planned, but it certainly livened things up. It took a while for them to calm down and stop pointing out how red I’d gone, but they were clearly interested in the idea. So I ditched the textbook and made up some ratios on the board saying that they were the male/female ratios of different colleges. The students had to simplify them into the form 1: n, then explain which one they would like to go to.

Nearly all of them said the one with the ratio closest to 1:1, so we then ranked the answers in order from most fair to least fair ratio and had a useful discussion about ordering and comparing decimal numbers (is 1:1.6 equally as unfair as 1:0.6? Some of them thought it was at first).

I wish I’d thought of this idea before the lesson. I’ve since been online and looked at the male/female ratios on different courses at Oxford and the results are sadly unsurprising, but they are interesting and hugely relevant to a group of bright young things at an all girls school. The male/female ratio in history is 1:1.02 and in English it’s 1:1.7, but in maths it’s 1:0.44, in engineering it’s 1:0.3 and in physics its 1:0.2.

So definitely an idea worth using again, and despite the momentary embarrassment – I think it shows that it doesn't hurt to “go with the flow" occaisionally.

Egg Box Update!

Note - it is best to have this link playing in the background while reading this post. Open it in another tab and enjoy. http://www.youtube.com/watch?v=2wdWGtTQ7hA


The first set of egg boxes are complete and I have photos!

I'm going to reflect on the project as a whole in another post, but for the time being I thought I'd assume the role of artistic director of the gallery and give you a guided tour. I've put in a selection of designs from top and middle sets. I haven't got a picture of any work from the lower set groups, but I have seen a couple of really nicely decorated cubes and cuboids from them.

(N.B. My tongue is now firmly in my cheek)




1. Untitled
On the left here we have what is considered to be the pre-eminent piece in the collection from two of our gifted and talented artists. The talented duo who created it declined to name the work, but at the gallery we have chosen to refer to it as "Possibly the head of a panda princess?"  A great deal of skill went into the production of this piece and the artists should be commended for their ingenuity in creating the spherical shape. They were also highly commended for their excellent research skills which enabled them to look up the formulas for the surface area of each section.

 

2. Bright Yellow Rabbit with Oversized Teeth and Cavernous Eyes

This is the work of another one of our most distinguished artistic duos. The work encourages us to reflect on the contrast between the jaunty yellow colour that dominates the head of the rabbit and the deep black colour used for the creature's eyes. Again, this pair of artists were highly successful in calculating the surface area of each section of the work, not forgetting the ears.

3. Chateau de l'Oeuf (unfinished) 

This ambitious work was sadly unfinished as the artists had to run to their next lesson, but one can get a sense for the scope of their design by this partially completed exhibit. This exemplifies a trend in some of the middle set groups towards creating "buildings" for their eggs. Judges were particularly impressed by their use of the formula 3/4 x pi x radius squared, for the cone shaped turrets, which were made using 3/4 of a circle.

4. The Eggcitement of Easter
This egg-box design, showing a rabbit in open-mouthed delight as it is about to consume a chocolate egg, encapsulates the joy of the Easter season. Although they had never been taught how to calculate the correct dimensions when designing the net of the cylinder, this group should be particularly appluaded for their "can-do" attitude and problem solving approach. When discussing how to work out the length of the rectangular part of the net, the group realised that it had to be the same as the circumference of the circle, but they had not been taught a formula to work this out. Instead, they took a piece of string and used that to measure the circumference of the circle. Ingenious!

5. The Nodding Rabbit

 This design demonstrates that sometimes simpler is better. The group used 2 cubes, attached with a piece of card at the back. Viewers in the gallery are encouraged to interact with this exhibit by tapping the head of the rabbit, which bobs up and down. Although not particularly ambitious from a mathematical point of view, this is a much valued contribution to the gallery and will be used as a model to inspire future generations of artists, especially those who are less confident.

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6. Sheep
As artistic director, this is my personal favourite in the gallery. Again, the acutal net is not particularly mathematically adventurous (it is a cuboid) but who could fail to love an egg box covered in cotton wool and made to look like a sheep? 



7. Egg-Barn

Our guided tour concludes with a very accomplished piece of work from two artists who have made fantastic progress this year. Not content with a simple cuboid shape, the pair decided to construct a triangular prism for the roof of their Egg Barn. They realised that the length of the triangle had to be the same as the length of the corresponding side of the rectangles in the net. With some assistance from their artistic director, they were guided to experiement with using compasses to construct inter-secting arcs to find the third vertex of each triangle.Once they had mastered the technique, the artists became experts in their field and showed another group how to do it.

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Musings on the pedagogical uses of laughter....

One of the most important things in teaching is building a relationship with your students. I always think things are going well with a class if they can laugh at/with me in a good natured way AND I can laugh at them (just a little bit of course).

Both of my current year 11 classes are pretty good at this.One class recently described me as "like a really funny Mum", which I took as a compliment, even though I'm only 10 years older than them and couldn't possibly be their mother. A student in the other class recently said "Miss, you should do that thing where teachers go on TV and teach difficult kids. I'd definitely watch you." I thought this was quite a big compliment, so I made the mistake of asking why. "Oh because you're really funny when you're cross".  Ah. That swiftly deflated my ego!*

Laughing at someone else is easy, but laughing at yourself takes real character. Even as adults, not everyone can do it. I have recently realised that some of the friends and family I admire most are people who can laugh at themselves and don't take themselves too seriously.


I'm pleased to say that most of my y11s seem to have this character trait. One of the most able students that I teach (and one of the most confident) stared at a distance time graph recently and shouted across the room "Miss this scale is all wrong. It makes no sense. It goes one thousand, one thousand and thirty, one thousand one hundred, one thousand one hundred and thirty. What is it doing?" Rather than give her the answer, I just burst out laughing. "You'll find it funny when you realised what you've done" I said. "Read the question again" (It was a distance time graph, so the scale actually read 10.00, 10.30, 11.00, 11.30). Similarly, in the other y11 class I asked the girls what the letter D stood for in 2D and 3D. "I know!" shouted one student "Dime... Dim..... Dementia?" To which I burst out laughing as well (along with the rest of the class). Again, in a great show of character, she took it on the chin and laughed along with the others.

We ask our students to cope with criticism every day. They are constantly being told how to improve their work, or being asked to correct their mistakes and think about which topics they need to revise. It can't be easy. In my first term doing a history degree at Oxford I felt completely out of my depth and I knew I was handing in rubbish work. I was so embarrassed that I couldn't bear to read the pages of comments my tutor wrote on every essay. If I had done, I would probably have improved much more quickly. As it was, it took about 3 months before a different tutor took me to task and verbally went through how I could make my writing better. She described the next week's essay as "a transformation".

By encouraging students to laugh at themselves when they have made a mistake, we can help them see that making mistakes is part of learning. But to have that sort of atmosphere in the classroom, we need to let them laugh at us too. Using humour in the classroom is a risky strategy, but I think that when it pays off, it's hugely worthwhile.

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* To re-assure readers that I'm not a complete idiot in the classroom, I am hardly ever genuinely "cross" with this class, because they are really nice. Instead, I tend to say things like "if you haven't brought your calculator I'm going to give you a dirty look" and I pull a face at them. 

Easter Egg Box Project

I don't remember a huge amount about maths lessons when I was at school. I remember enjoying the SMP system in year 7 where you worked through lots of little books and essentially taught yourself. I really liked that. I didn't like the GCSE coursework where you had to count words in sentences and then work out the standard deviation of the sentence lengths. As a lover of literature I despaired to see To Kill A Mockingbird reduced to such soul-destroying analysis.

However, some lessons did capture my imagination and I remember particularly enjoying a series of lessons where we had to design an easter egg box. Luckily for me, I was allowed to work with my hugely talented best friend who was both a mathematical high achiever and ridiculously creative. She came up with the idea of designing a net that looked like a hen's head.

It was pretty impressive. It consisted of an octagonal prism and a cone all in one net, which we decorated to look like a hen's head and beak. I've searched on the internet for something similar and this is the closest image I can get to it, but it doesn't really do it justice.......

So I wonder what our y7s and y8s will come up with when we give them the same project in a week or so's time.

Here are the resources on the TES: http://www.tes.co.uk/teaching-resource/Design-Your-Own-Easter-Egg-Box-Project-6323990/

If you are looking for a project to do before easter, I think it should be a good one. It wasn't difficult to put together and so far I think it's ticking all the boxes (ho, ho! I love a good pun)

Which boxes has it ticked? Well it......

*hasn't taken long to plan
*helps to deliver quite a lot of national curriculum content
*allows students to be creative
*has a cross-curricular element
*gives us the chance to assess objectives without testing students to death
*shows how maths applies to real life situations
*AND is open-ended enough for all students from level 3 to level 8.

The resources on the TES include a powerpoint  and a student assessment sheet. I haven't uploaded the nets that we are going to use because we've taken most of them from www.senteacher.org. We also drew some ourselves on sqaured paper, so that the level 2/3 students can count squares.

If we get any particularly impressive designs, I'll post them up on the blog too.