#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?


#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.