This first problem blog is an outline of one of our messy maths problems. Over the next term, I will collect the ideas and suggestions from the MathsConf5 and MathsConf6 workshops to produce more examples. Please feel free to share your thoughts and ideas.
This was our story from then to now.
Over the last two years there has been lots of discussion about two key issues in our workroom:
- How do we teach the new GCSE topics?
- How do we improve our pupils’ problem solving?
Well the second issue also included discussions about how we improve pupils’ marks in problem based questions, but let’s hope that good mark harvesting is a consequence of improved problem solving skills. So if we solve that thorny issue the rest will follow, field of dreams style.
At MathsConf5 I gathered over 200 teachers’ views on which were the hard to teach topics for the new GCSE and collated the top 10 topics which students find difficult to grasp, and at MathsConf6 I ran a workshop about how we had tackled these topics through a problem-based approach.
As a department we talked long and hard about what we meant by ‘problem solving’ so we could then discuss how to get better at it.
We agreed on a few core principles:
- It is not a problem if students can immediately solve it.
- A problem occurs when students are confronted with a task where there is no prescribed way of solving the problem.
- A problem can be solved in different ways.
And while we were at it we also wanted problems that:
- begin where the students are
- are presented in an interesting way / have hooks
- have an engaging mathematical aspect or a motivating context or both
- do not have an obvious answer but require justification and explanations for answers and methods.
And to be better problem solvers we expected that our pupils would need to be better at:
- independent thinking
- avoiding distractions; being absorbed
- perseverance – to try, try again and be resilient
- collaborative and creative working.
Which is all lovely in principle, but what does that mean in practice?
Well for us it meant looking at the problem and the pupils’ learning habits, not the subject content. Looking at the various routes through the problem and considering the multiple pathways you could take to solve it. Also looking at the skills pupils need to develop to become better problem solvers and developing these skills in our pupils. Then using these skill judgments to track and assess our pupils.
Our planning sets each problem up with a starter, a main course and a desert, and usually each meal is finished by the end of the lesson, but more than rarely it will run on as the pupils go and do some extra, independent and unprompted research!
Holding this all together was a BIG QUESTION for each lesson.
In our planning we considered the various routes and methods that pupils might take. We thought this problem would be a good one to develop ideas about mean, deviation, sampling and being ‘fair’ mathematicians. We also hoped it would be a good launch pad to ferment disagreement. The starter activities were an ideal opportunity to revisit previous topics, especially on working out the circumference of the earth, and manipulating large numbers such as global populations etc.
The starter in this lesson was some ‘facts’ about Lego that may or may not be true, and the pupils work out if they are reasonable.
Everyone on earth would have 50 pieces of Lego if they were shared out.
The entire mini figures in the world lined up would go round the earth 4 times.
Then the main activity was based on an ‘it depends’ question – in this case the question is ‘how much is a piece of Lego?’ We have loads of old Lego boxes donated by students, with prices and numbers of pieces on. They choose a method, then justify and present their answer to the big question.
And then the real substance of the lesson begins. This is where pupils start to say things like…
‘Yeah, but…some pieces are bigger / move/ have movie tie ins / were bought 3 years ago / have tags…’
‘…but that‘s not fair because this set is much larger so it should count for more…’
‘Isn’t every set at least a 3 in 1? How many models can you make from 3 pieces?’
It is in this discussion and debate and justification where we can start to develop those learning habits that make for good problem solvers.