Swan: Collaborative Learning in Mathematics

From Swan’s paper (available below):

Teaching is more effective when it …

  • builds on the knowledge students already have;

  • exposes and discusses common misconceptions

  • uses higher-order questions

  • uses cooperative small group work

  • encourages reasoning rather than ‘answer getting’

  • uses rich, collaborative tasks

  • creates connections between topics

  • uses technology in appropriate ways.

Types of teaching activities that can achieve these principles:

  • Classifying mathematical objects
  • Interpreting multiple representations
  • Evaluating mathematical statements
  • Creating problems
  • Analysing reasoning and solutions

But what are the precursors to this? Swan talks about the challenge of ensuring all group members participate, and some ways to encourage this, but reflecting on what I’ve heard from other teachers in the past few weeks, I think we need to take another step back first.

The challenge that I’ve seen and heard from several teachers is to create the conditions within their classroom that allow productive collaborative work, rather than students spending group work time chatting about something other than maths.

Those classes that have used group work time productively have all had some common features:

  • The level of challenge is just right – students want to do the work because they want to find a solution to the problem the teacher poses.
  • There is a balance between letting students spend quality time on a difficult problem, but not letting activities run for so long that attention wanders.
  • Most importantly, the teacher has a good, professional, mutually respectful relationship with the class.

These points are hard as a teacher to achieve, but worth working on when I get to school in September. The last one, particularly, is what I keep coming back to – mutual respect and a professional relationship with the class are absolutely central to good teaching.

Another reflection is that the stuff Swan is suggesting is tough for students too. That’s part of the point of his activities. But if my classes are to have success with strategies like this, I need to scaffold them through Friday skills lessons or similar. Asking students to undertake activities like this when they’re not used to them will take time and effort on both our parts.

Swan: Collaborative Learning

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Sunlight on a gremlin

Before starting the Teach First Summer Institute I’ve read quite a bit about maths teaching. One thing that I’ve really struggled to make sense of is how to map progression in key mathematical topics. In particular, being able to track back from what I see as the key learning outcome of a particular topic to the simplest building blocks of that topic is a real struggle for me.

This post, then, is a bit of a celebration. If progression in maths feels like a bit of a gremlin, at the end of the first week of the Summer Institute I feel as though I’ve found the string that’ll let me pull back the blind and expose the gremlin to the sunlight (excuse the awful analogy!)

Taking advantage of some excellent subject studies sessions this week, and using the example of linear equations, I am starting to grasp what progression means in practice, and most importantly how to plan for progression.

It is surprisingly simple in practice really. Taking examples of linear equations, I worked with fellow trainee maths teachers to write all the linear equations we could think of, and then number them according to our perception of their increasing complexity.

Focusing particularly on finding the simplest linear equation we could, this process allowed us to look at equations at several National Curriculum levels. Most valuably, it means that when we come to teach solving linear equations and try to judge the level of the class, we will know where to go if pupils find the content either too easy or too difficult.

Just as important, going through this process is helping me to formulate a strategy for planning for progression in other topics too. Brainstorming all aspects of the topic, and the assumptions that underpin each of those aspects, before ranking these in order of increasing complexity will help me to know where to go when students find a topic too easy of difficult. The result is that I will be better able to help all students learn in every lesson, which is the point of it all after all!

After a busy first week of the Summer Institute, this nugget of progress stands out as the most exciting for me.

Instrumental and relational understanding in maths

Fascinating exchange with @Kris_Boulton about his post on simplifying area. This post is in part a response to that post, in part me trying to make sense of the relational and instrumental understanding debate.

Kris pointed me towards Skemp’s original article on relational and instrumental understanding, as well as Willingham’s article on the value of what Willingham calls inflexible knowledge.

Starting with the caveat that I don’t yet have any experience of teaching instrumental or relational understanding in the maths classroom, the exchange has stimulated lots of thoughts. Pulling them all together, I’m left with a few conclusions:

1. Relational understanding, whilst the ideal, is rarer and more difficult than Skemp implies.

…even relational mathematicians often use instrumental thinking.

This quote from Skemp sums it up for me – however valuable relational understanding is (and I do think it is very valuable), it just isn’t realistic for mathematicians in many day-to-day applications, be that as simple as working out the area of a trapezium or as complex as factorising quadratics that don’t have integer solutions. The instrumental approaches offered by the respective formulae are vital for mathematicians operating at all levels.

2. BUT in the case of area, a drive for simplicity actually leads me towards starting with relational understanding not instrumental understanding

In the simplifying area post Kris highlights an interesting approach to reduce the complexity of the formulae for area of four basic 2D shapes to three words: ‘length times height’. Keeping in mind the caveat above, it seems to me the approach works really well, and if anything promotes relational understanding more effectively than the four formulae.

But the point I want to focus on is whether it would be even simpler to start with relational understanding, and then teach the instrumental approach.* Perhaps its the naivete of the beginning teacher, but my instinct is to take an approach where students (who are confident in how to calculate the area of a rectangle) mark length and height on card rectangles, then cut them up to make the other three basic 2D shapes.

This approach seems engaging, manageable and most importantly gets at the ‘why’ of both the formulae for area and Kris’ simpler ‘length times height’. Subsequently teaching ‘length times height’ offers a handy shorthand for the relational approach, and is hopefully more meaningful as a result of previously exploring the relationship between each shape and the rectangle. Of course, revisiting and revising both the relational approach and the instrumental approach will aid both recall and understanding.

3. Inflexible knowledge is in a different category to instrumental understanding

Willingham makes a great case for inflexible knowledge – it seems clear that inflexible knowledge is a necessary step on the road to relational understanding. But it seems to me that inflexible knowledge is quite distinct from the knowledge that underpins instrumental understanding. Inflexible knowledge is narrow but meaningful. Instrumental understanding can be useful, may be quite broad, but isn’t meaningful in the way I understand the term. An example of the former might be knowing why the area of a rectangle is ‘length times height’, but not being able to use that to investigate the area of a triangle because I can’t visualise the relationship between the two shapes. An example of the latter might be using the ‘length times height’ rule to work out the area of a whole range of shapes, without understanding why it works.

As a result Willingham’s article isn’t, for me, a case for promoting instrumental understanding. It’s a case for promoting a particular, narrow form of relational knowledge.

Putting all this together my current thinking is that relational understanding is often tricky to promote. In some specific cases a mathematics teacher is justified in retreating from this ideal and instead fostering a narrower form of knowledge, at least initially. However, wherever possible this narrower form ought to be inflexible knowledge rather than instrumental knowledge.

I’m just beginning my teaching career and feel as though I’m stretching the limits of my own knowledge with this post – any comments on anything I’ve written would be really valuable for my own learning!

*As an aside, I am aware that this is almost exactly the approach that Kris describes as ‘doomed to failure’. Eek!