Teaching without Teaching (an experiment)

I’ve been looking forward to writing this post, having had an exciting/terrifying idea two weeks ago, but had to do a test with the class before I knew if it worked (or not). 

Inspired by a link from @mrpcollins’ blog, inspired in turn by daratodifferentiate’s article on think tac toe boards, I was looking for a way to try something similar in my lessons. 

During the first week back, after a successful lesson on surface area of cuboids and relatively ‘easy’ prisms (e.g. trapezium-based), Y9 ran into a stumbling block in lesson 2 (surface area of cylinders) when it turned out there was a VERY wide range in their knowledge and understanding of circles. I asked them to comment in their books, as part of their end-of-lesson self-assessment, on how much practice on circles they wanted in the next lesson. Predictably, there was a 3-way split amongst

• “I don’t get circles at all, can we spend a lesson on the basics?”
• “The basics are fine, but I’d like to do harder ones” and
• “Everything in circles is easy, and so is surface area of cylinders. I want to do something harder.”

Reading this at 18.30, when the next lesson is in the morning, and you’re rubbish at differentiation…

Assuming that there was no way I could successfully run a ‘normal’ lesson with such a range of knowledge, skill and confidence, it seemed like as good a time as any to try think-tac-toe. I made a ‘step ladder’ (not *quite* a level ladder) showing the different sub-topics in order of difficulty so pupils could decide how much they wanted to challenge themselves (and select the right resources for their objectives), and a 3×4 grid. They had to complete at least one activity from the centre of the grid, and had three lessons to complete 3 activities from the grid. 

Here are the stepladder and grid: 

Think Tac Toe (circles)

[Full disclosure: I should add, at this point, that the class I did this with are very able (although this is still very spread), very motivated and - with rare exceptions - utterly delightful.]

On Tuesday morning I explained how the TTT would work (they chose their objectives and their activities, and could work alone, in pairs or in groups), that there would be ‘red time’ when they weren’t allowed to ask me any questions (as I was teaching a very small number of pupils who were really struggling), green time when they could ask, and ‘amber time’ when I was inspecting pupils’ work at random. I also explained that we would have a test the next Thursday (lesson 6), that on Monday (lesson 4) we would revise as a class/clear up misconceptions and that on Tuesday (lesson 5) they would work independently on revision websites (e.g. Mangahigh/MyMaths/Bitesize). Finally, I set two pieces of homework on mymaths: a compulsory one on surface area and a choice of pieces on circles/volume/cones, etc, of which they should choose one.

They launched into it (totally bizarrely, at least 1/3 – all boys – went for exam questions as their activity of choice! Weirdos), and were quite quick to choose their activities and get going. However, I was disappointed to see that at least 1/3 weren’t really challenging themselves in terms of content/objectives (e.g. making posters on how to find circumference, or what radius/diameter means), and one boy completed little work. That said, at least 10 produced some really fantastic stuff (board games with increasingly difficult questions on mensuration, or independently researching the volume of cones and spheres). 

On Thursday I reiterated that I wouldn’t be teaching them any content directly, and that it was their responsibility to make sure that they were prepared for the test (pointing out that everything on the ‘ladder’ would be on the test). I was quite scathing about some of the activities chosen (saying that anyone who needed to make a poster about what the diameter is shouldn’t be in the top set, or even Y9) and – on the advice of a colleague – warned them that the class had to achieve an average of a C in the test if we were to move on to the next topic. I’ll admit I made the ‘not moving on’ scenario sound really grim (“you’ll have to copy examples from the board, then practise them silently until I’m convinced everyone can do it”…even I was dreading that scenario).

The transformation was amazing: by the end of Friday (lesson 3) at least 1/3 of the class had taught themselves how to find the volume of cones and spheres, and we had a host of ‘products’ from a “Pi in the Sky” boardgame to a “Circles for Dummies” workbook with examples, tips and questions to ‘try at home’, to a rap. The class was asking much more sophisticated questions of me, and were working in productive little groups of their own making (e.g. one group of 5 were doing exam questions together and marking each other, one group of 4 were making ‘instruction books’ and reviewing each others’ work and three pairs were developing board games). 

On Monday (lesson 4), about half the class was absent due to a trip, so the revision lesson was very focused but did miss out half of the pupils. On Tuesday (lesson 5), they worked on Mangahigh challenges and asked questions (6 spent about 20 minutes with a senior maths teacher, who was ‘interviewing’ them about the work they’d produced). I felt quite nervous about them taking the test today as, barring the first two lessons on surface area (the second of which was quite rubbish due to the mixed knowledge), I’d done little direct teaching. There was lots of evidence of progress and work, but many worked almost exclusively on laptops, or in groups, and I had no individual work to look at how they’d been doing. 

We took the test today, with questions starting at ‘this is the radius, find the circumference’ and finishing with ‘find the volume of the frustum’, with some really nasty ‘functional’ questions in the middle. I was so nervous while they were taking it, and had to wait until the evening before there was time to mark them. 

The overall results were: 

1 D+ 

2 C-

1 C

1 C+

7 B-

4 B

5 B+

A- 3 

A 1

A+ 2

A* 0

(3 were absent)

Given that their levels at the end of year 8 were 5a-6a, this represents unbelievable progress for all of them (even the pupil who got a D+ made almost a grade of progress from his end of Y8 level, although it was the least progress). They’ve been making good progress with ‘normal’ lessons, but this was definitely much better. 

So I guess the trick is to stop trying to teach them:)

(jks – this format was well suited to this topic and this group – it wouldn’t work for many parts of the curriculum). 

Next challenge: trial it with a less amenable class… I’ll report on how that goes, assuming I/we survive (and I’m not still cleaning up my classroom).

New Year’s Resolution: more words

It’s been such a long time since I last posted; the last term felt pretty hectic, so I couldn’t justify spending time on a blog. However, I think it would be good to start recording things again properly (especially in the (happy) absence of a reflective journal).

Prompted by a TF colleague, I’ve been thinking about how to help my pupils – particularly those outside of the higher sets – remember important vocabulary and know how to ‘react’ to it (e.g. understanding the different implications of ‘simplify’ and ‘solve’). Having marked a mock exam for one KS group, it’s become really clear that literacy/limited mathematical vocabulary is preventing them from demonstrating what maths they do know.

As a result, I’ve gone back to including keywords explicitly in my lesson planning. This is something we learned to do during ITT but I didn’t really know how to embed it, and hadn’t’ understood the depth of the problem or how weak pupils’ retention would be if keywords weren’t pushed explicitly and reinforced.

Tomorrow’s ambitions are that Y7 will understand the difference between expression, equation and term, and that Y10 will understand and use the words vertical, horizontal, origin, x-axis, y-axis and quadrant. I feel a bit sorry for Y10 – that’s a lot of vocab and I’m struggling keep it from being dry…which means I feel sorry for myself too (in anticipation!).

Ideas so far (I’m still refinining) include:

- connecting words with pictures/examples and then connecting those to written definitions

- evaluating if statements using the keywords are sometimes, always or never true (e.g. (3, 0.5) is a coordinate; 3+4 is an expression, etc)

- unscrambling at the start (standard…not the best, but will hopefully help with reinforcing correct spelling!)

- using coordinate ‘code’ to rewrite the key words (tying the two parts together)

- using ‘cold call’ to test understanding and improve pupils’ definitions/explanations (search ‘cold call’ in the Teach Like a Champion channel on youtube for more on this).

I’ll see how those go and then try to improve it for next week.

For the rest of the week we have acceleration day (I’ll be with 10 pupils from Y7/8/9 respectively for an all-day maths session), so planning for a ‘strong start’ for the term has rather gone out the window – oh well!

Unsolved mysteries of our time: pupil talk

Many of my lessons that aim to incorporate more group and partner work, and generally increased levels of ‘talking about maths’,  leave me wanting to bellow – for that is the only way to be heard in such lessons -

“How can you be so good at chatting yet so bad at talking?!”

 

Given that pupils in my school arrive with far-below-average skills in writing, a focus on speaking seems the obvious route towards building a foundation for writing elegantly about maths and using maths to write persuasively. However, this revealed (what used to be) the unexpected reality that many simply don’t know how to talk about maths, articulate their ideas, or work in pairs or groups to solve a problem that requires resilience.

Perseverance has begun to pay off with some classes, particularly higher sets – perhaps due to the unhappy, if imperfect, correlation between social skills and academic ability (or, to be precise, attainment) – and I’m beginning to understand how to run effective group work.

Last week, in collaboration with a very experienced and talented colleague (credit where it is due:), I prepared a lesson for my Y7s that required them to work in groups to solve a series of problems and try to get the most points (by writing their conclusions in a mathematically convincing way). Predictably, they were motivated by the competition aspect – or at least by the shiny scoreboard – but I think/hope they found the maths inherently interesting*.

Most groups had, by now, learned how to work together, express their ideas and, most importantly, get themselves unstuck when needed. One table – you can probably see which from the board above – struggled to get going and, when approached, complained that no one was listening to them, they couldn’t decide what to do to solve it, etc. It was clear that they weren’t actually talking to each other and that some people were simply opting out, either due to being fed up, not being bothered to put in the effort to listen and to be understood, or lacking the resilience to stick with a problem that seemed difficult to solve.

Discussion on Saturday at the ‘Learning Together Mathematics Group’ (really trips off the tongue, eh?) at SHU brought up a really good resource that I think I’ll use with many classes to get them to better focus on group work, speaking and listening.

I like that it compels all members of the group to participate (and listen) and is a non-threatening starting-point for getting pupils to practise talking in a logical and reasoned way without the additional barrier of symbols and numbers. I plan to use it with Y7 and Y8 although, sadly, I think the reading burden may be too much for one group of Y8 pupils I teach I’m trying to think of how it can be adapted to be accessible to pupils with severe learning difficulties or who have extremely limited English – suggestions very welcome.

Anyway, I’m looking forward to using this resource and, if it meets with success this half-term, I’ll probably use it early next term with all classes as part of an as-yet-vague campaign to have more groupwork and rich tasks in my lessons. A girl can dream!

*Email me if you want the resources and lesson plan from this lesson, as it’s a nice final activity for a L5c-6a group on fractions/percentages/decimals and could be adapted for other lessons.

Going around in Circles

With thanks to http://www.greatmathsteachingideas.com, I found the brilliant Circles are Awesome website.

The graphics and style are very user-friendly and accessible, with a pleasing simplicity matched by the clear explanations. The description and animation make a compelling argument for ‘any circle can be defined by three points’:

The interactivity here is really elegant – do give it a go for (nerdy) little thrill (and little dreams about how to get your students to be fascinated by circles).

Lastly, there is an interactive section that lets you play with circles AND colours – perfect. If you haven’t already gone to this website…go now!

Is it ‘proofy’ enough?

One thing I struggle with is getting my pupils to make the leap from creating examples of a proposition to giving a proof of it. I found the ‘Make 37‘ activity was great for getting them to start to explain ‘why it will never work’, instead of producing endless examples of it not working.

However, few of them were able to move from ‘an even number of odds can’t make an even’ to an algebraic proof (some version of 10(2n+1) = 37 -> 20n=27 -> no integer solution or “10(2n+1) divides by 2, so it is even, so it can’t be 37″ would have sufficed).

In addition to appreciating any advice or anecdotes on how to take the step to formal proofs, I would also like to know people’s opinions on ‘picture proof’ (e.g. proving that angles in a triangle must have a sum of 180 by taking the exterior angles and ‘shrinking’ them inwards to show they make a full turn – there is an animated example of this on mymaths). Mathsisfun do what is, to my mind, a more rigorous approach, but it still relies on ‘looking right on this occasion’.

Is it sufficient to stop with ‘word’ and ‘picture’ proofs, or is it inherently valuable to push on to algebraic versions?

Extension task:)  I puzzled over this with a colleague from another school – we could only show it with a picture. Is there a better way? Can it be done other than by pointing at an illustration? To our embarrassment, it’s from the Junior section of the UKMT mentoring resources. I’m avoiding the advanced section for now.

Answers to this questions would be appreciated, as the solution won’t be released for a few more weeks!

Mathematical Images: Symmetry

I love finding ‘maths pictures’, and thought it would be nice to start collecting and sharing them. Symmetry is the easiest to begin with, unsurprisingly. Images with simple line symmetry can be great for young (i.e. Y7) pupils to engage with, allowing for discussions about strategies to ‘test’ to how symmetrical an image is and how it could be improved to be ‘more’ symmetrical (this is a good ‘way in’ to using squared paper to complete reflections).

Rotational symmetry is fun to explore with an interactive whiteboard, and more practical activities abound (e.g. for homework pupils can draw their own Rangoli patterns, or make patterned wedges that can be repeated to build up a whole shape). Some of my most able Y7 pupils wrote excellent pieces discussing symmetry in buildings and gave their opinions on how symmetry affected the attractiveness of a building or shop display. A colleague in my department set up a lesson for his students to break up into small groups, each with a digital camera, and had to choose the ten best symmetrical images they could find around the school. If my memory serves me correctly, images were deemed to be ‘better’ if they were unusual or had more than one line of symmetry.

Some examples of line symmetry:

Some examples of rotational symmetry: 

Post links to any images of your own!

More Maths Games

I wrote earlier about how most online ‘maths’ games are very poor as games or even worse in terms of mathematical content. Happily, I’ve found another website, this time with a specific numeracy focus (making it a little less challenging and subject-specific than MangaHigh, and so a happy complement) – Sumdog.

The content is free and it is free to enlist your classes* and create competitions and challenges. It has a nice kids’-cartoon-meets-graffiti look and pupils can chart their progress with topics by whitewashing blocks on a spray-painted wall.

The main focuses (foci?) are addition and multiplication tables, although there are also challenges for negative numbers, subtraction, division and grouping (e.g. adding in lots of 25). Students can practise each through a range of games (e.g. practising their 7 times tables with an alien shooting game, a robot racing game, or a litter-picking game (!)), giving a bit of variety to otherwise monotonous drills.

I only just found it but was excited by the possibilities, either as a ‘last 5 minutes of class’ treat or as an additional homework task (or just for keen beans who want to do some numeracy work).

I’ll update soon with pictures and reports on how it’s been received by my discerning pupils.

[UPDATE: Y7 set 2 were extremely keen on it and really enjoyed that they could play against each other in challenges.]

- – - – -

*NB: I wrote to SumDog to enquire about the protection of pupils’ data, with the following response:

“Pupil information is securely stored on our server and can only be seen by us and other teachers at your school.  Normally, we would only record a student’s name and activity on the site.  Other users on the website will be able to see the student’s first name and last initial (we automatically abbreviate this), their avatar, and the name of their school.  We do not allow free chat or anything of that nature, and there is no way for another user outside of the school to find out any further information about this student.  The most they could do, essentially, is answer maths questions together!”

My own investigations confirm this to be case, so I am satisfied that it’s completely safe for pupils to use in school and at home.