#MBToS Challenge 3: Collaboration Nation

This weeks challenge on the Math(s) Twitter Blog ‘o’ Sphere is to check into one of the collaboration projects set up by Maths educators around the world (mostly in the US but at least one of which comes out of the UK), to explore a bit and reflect on what we find.

I’ve been lurking around the edges of Maths education blogs for a long time, so I remember a lot of these projects starting up.  I’ve dropped in but for whatever reason I haven’t contributed much – I thought I’d use this as an opportunity to drop back in on one of them, namely 101questions which is a project of Dan Meyer.

Dan has done a lot of work thinking through 3 Act lessons.  When I describe that work to others I talk about trying to get a picture or video that poses one key and interesting question that could be used as the launching point for a lesson or series of lessons.  I talk about seeing lessons as stories with a beginning, middle and end, with possible sequels.  I talk about using the digital projectors in our rooms to share digitally rich materials.  I talk about having something beyond the answer key or my decision being the measure of correctness or success.

101questions is an online community set to pose videos and pictures to see what questions come up – to see if other people intuitively jump to the question you have in mind.  The people posting the pictures or videos and the people suggesting questions are mostly Maths teachers but given that level of homogeneity, there is a great diversity in the types of answers that are given.  You still get different questions in the classroom but this gives a good starting point to make sure you aren’t completely blindsided.

I have taken some of these and used them in the classroom.  We’ve watched the video or examined the picture, come up with our questions, asked what information we might need to begin to solve that question and get stuck in doing some mathematics.  It takes work to find an Act 2 (the information) or Act 3 for some of these problems which is where I have greatly appreciated the work of others doing this (I’ve benefited a lot from Dan and Andrew Stadel in particular).  The site now has functionality to search for those resources that have lessons associated with them.  I’m pretty sure this is new and will make it much more likely for me to use these in my lessons more often.

I find  that these problems are very engaging but not often fully mathematically rich.   They do present problems that can be solved in a variety of ways but as we are aiming toward one key question most students are working to try find out the same thing.  Maybe the sequels is where the juicy richness comes in – we could encourage pattern spotting and generalisations as students develop their own problems in our context.

Do you think 3 Act lessons are mathematically rich?  Am I way off base here?  Are there ways to make them more rich?

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Thinking about indices …

Okay, maybe I’m a bit slow but I said and thought something today that I’ve never thought about before.

We’re looking at indices & surds in my Year 11 class.  Today we were looking at the rules of indices: multiplying and dividing indices was “insultingly easy”.  Negative indices seemed to push it over the edge … some very confused faces.  We struggled with it and I think most people managed to get an instrumental understanding but not a relational one.  We’re working on that!

I was trying to explain that 6^{-3} was equal to \frac{1}{6^3}.

Now, in explaining this I said it was 1 divided by 6 three times.  Seems fine but where does the one come from?  We talked about reciprocals, we talked about the pattern that follows as we look at 6^3 6^2 6^1 6^0 6^{-1} … and we talked about a few more bits and pieces.

I then thought about 6^3 could be thought of as 1 x 6 x 6 x 6.  This adds a bit of consistency to the system and made me think of an inaccuracy that I sometimes hear myself making or confirming.

“What does 6^3 mean?”

“6 times by itself 3 times”

But does it? 6 times by itself three times could be 6 x (itself three times) or 6 x (6 x 6 x 6).  I suddenly thought surely this could be 1 times by 6 three times.  This helps with consistency in two ways.

Firstly, with our negative powers – we can say that it is 1 divided by 6 three times.  This has internal consistency and I like it.

Secondly, raising something to the power of 0 makes sense – 1 times by 6 zero times is obviously 1.

I’ve never seen or heard this anywhere before – am I wrong?  Does this break down somewhere that I’m not aware of?  Do you think this could be a helpful way to see it?

Also, is there a richer way to introduce this apart from pattern spotting?  Any ideas gladly received.

Level 6 Maths in Year 6 – what’s going on?

I recently gave some input at a primary Maths training event in my local area.  As more and more schools need to move students on to level 6 at the end of primary to hit their league table targets, there is a training need to enable our teachers to be able to teach the content effectively.

I think primary teachers are great – I’m pretty sure I could not deal with planning for subjects that are not my specialism.  I love Maths – I get a buzz from planning for Maths lessons – I have taught Maths at P-Level and A-Level – I can plan engaging lessons with background and context and extension and support really well in Maths.  I’m confident with the vocabulary, the progression, the pitch and can address misconceptions well.  All that being said, ask me to teach history and I’d flounder.  Last year, I had to teach some science – it wasn’t difficult stuff – but I really struggled to know enough to make the lessons meaningful and useful.

Primary practitioners are expected to do this for every area of the curriculum every day!  No way I could do that!  Now, with the expectation that Level 6 is achieved by some students, there is a danger of slipping into old ways of working – teaching verbatim from a textbook, not knowing where this goes, not fully understanding where a concept has come from.  I have observed some amazing Maths lessons in primary – using concrete resources and visualisations, pair talk and problem solving – all of which were used excellently to help their learners learn and make progress.  But, if the content is something unfamiliar, if you’re worried about being ‘good’ at Maths or that your students might be ‘better’ than you, if you are being pressured to get the magical level 6 – then it would be easy to forget about all the excellent practice you do and slip into something from a box.

In my input, I wanted to encourage primary teachers to keep doing what they are doing.  I’ve embedded the Prezi I used below.  Saying all that, there is a training need that is there.  How can these teachers plan effectively if they haven’t made these connections themselves in a number of years, if ever?  With the demise of the National Strategies, I think it is the responsibility of local schools to help each other.  Effective secondary Maths departments that have the capacity to do so, should be helping to develop and equip our primary colleagues to know the content well, to make the connections more easily so that when our students arrive with level 6, we believe it.  Surely, a secure Level 6 rather than one coached for a test is a benefit to the secondary school who receives that student and is then judged upon their progress?

What do you think?  Is this the responsibility of local cluster groups to source funding and time to deal with?  If not them, then who?

Answers on a postcard please.

#MTBoS Mission 2: Coping with the Steam

I think teaching Maths is the best job ever – I love doing it and I love improving at it.  I actively seek out ways to do that and the #MTBOS has given me some great ideas over the years I’ve been lurking.  I’ve a full twitter feed and a bulging blog-reader.  I also have a full-time job, a family, a church and (though I haven’t been there in a while) an allotment.  Getting the balance is something I find tough.

There are times of the year when I am on top of my blog and twitter feed – I think I sometimes see it as a competition and if I can get everything down to zero then that’ll be great.   But, this is unsustainable – there is so much great stuff out there that I don’t want to be left behind.  When I see something I like, I add that blog to my blog feed – on average that clocks up 100+ posts a day. I need some survival techniques and I’ve come up with two.

For twitter, it’s using notifications.  I found out about this over the summer and I’ve loved it since.  If you use an iPhone or similar device, you can select particular users to get notifications for.  Then, when they tweet, it pops up on your screen.  I use this to make sure I don’t miss tweets from friends and people I’m finding interesting.  If I find someone tweets too much for me, I can switch them off.  If there is something I want to respond to, it is there straight away.  Occasionally, I turn them all off if I want a break.  This has helped me not open twitter like a nervous tick which I like a lot.

For blogs, it’s quality over quantity.  I’m still struggling with how to do this well – but I want to read blogs until I find a post I find interesting, stop and engage with it.  Too often, I find myself skimming and racing to get to the end, to get to the next post, to get to the end, to get to the next post, …..  That’s probably a sign I shouldn’t be reading that post, right?  Or do you persevere with the hope there might be a great idea coming up?  I want to allow what I read to affect what I do.  Otherwise, what is the point?  I love reading but skimming blogs sometimes leaves me feeling a bit morose – either overwhelmed because there is no way I can implement all those great ideas or numb because there are so many ideas I haven’t engaged with any of them.

Do you feel my pain here?  Do you have any award winning strategies to help make the deluge of information manageable and useful?  How do you make sure that your engagement with the #MBToS changes your practice?

Aside

#MTBoS Mission 1: Rich Task – Painted Cube

I really wanted to be involved in the MTBOS (Matho Twitto Blogo Sphere) challenges this year – I was a bit lame last year and never got off my bum to do anything about it.  I’ve been a bit late on last week’s Exploring the MTBOS challenge but I thought I could catch up today.

I wanted to talk about a problem that my Year 8s have been working on this past week – The Painted Cube.  Now, this used to be one option for GCSE coursework and I’ve used it for that in the past.  I think it is a pretty good example of a ‘Low Threshold, High Ceiling’ task – the problem is easy to pose, everyone can do something and the students can run with it, extending it to the limit of the their Mathematical ability.

Here’s the problem:

3x3x3Imagine a large cube made up from 27 small black cubes.

Imagine dipping the large cube into a pot of yellow paint so the whole outer surface is covered, and then breaking the cube up into its small cubes.

How many of the small cubes will have yellow paint on their faces?
Will they all look the same?

nRich have a nice animation to help with the launch here.

There were a number of skills that I wanted to be used during the few hours that we were looking at this task.  I wanted students to collect results systematically, I wanted them to look for and spot patterns, to come up with general rules, draw isometrically, consider the volume of cubes and cuboids and to relate their rules to the physical reality of the problem.  I knew I could encourage some to look at graphs to summarise and present their results, to begin thinking about algebraic proof, expansion of quadratic and cubic brackets and, potentially, propose results in 4D (sadly, none of my 12 year olds got that far this time round).

I had students working in pairs or individually – I gave them access to some cubes but encouraged them to move towards visualising using diagrams because we didn’t have enough cubes for a 10x10x10 example. Lots of discussion and lots of excellent thinking ensured.  The quality of the questions was superb and the student support for each other was excellent.  Some students weren’t clear on drawing isometrically, so a student expert demonstrated with the visualiser while another narrated and answered questions.  There was some tussling over the multilink cubes but that calmed down after a while.

We spent two hours of lesson time looking at this problem.  We looked at finding nth terms earlier in the term and are about the begin a section on volume, so I thought this was a good place to drop this problem in.

I’ve asked every student to spend another hour on their own for homework to take the problem further, so I’m looking forward to see what they bring back after the weekend.

I love that from a simple problem we can get so much Maths that connects to everything else – when they come back on Monday we’re going to mindmap all of the Maths we used in this one simple problem and marvel at the wonder of Mathematics.