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


Teaching Mathematics Ch 8: Reflections on mathematical investigations

Maths investigations clearly have huge benefits; the example in the book was of a Happy Numbers investigation. The open-ended approach not only allows for differentiated approaches, but also encourages mathematical thinking and the development of process skills.

In planning for an investigative task in the classroom, I clearly need to plan carefully the scope of the task, and also the scaffolding I will provide to learners who are stuck, or are struggling for ideas. For example, I might include a list of approaches that students could take to get started on the task, as well as a ‘hints’ table if they are stuck.

Assessing performance on a task like this clearly requires a more considered approach than simply ticking correct answers and giving feedback. I need to consider what approaches a pupil has taken (which may not be explicit – thus I probably need to award marks for clarity of presentation as well as outcomes) and how they might improve their approach. However, this seems to accord closely with good practice in assessment anyway, so I don’t think it should put me off!

The Lazy Teacher’s Handbook: Ch 4 Great ideas for lazy lessons

As I read through this book I realise the ‘lazy’ moniker is simply a marketing ploy – this is a book on how to teach much like any other. There are some good ideas in here all the same, based on the principle that some of the best lessons occur when you are forced to think on your feet, rather than having planned in depth. The key idea in this chapter is to apply that principle, but in a systematic way i.e. planned spontaneity. Ideas that can allow you to achieve this include:

  • Quick ideas:
    • Arrest me! Tell a pupil to imagine they have been arrested for being an outstanding mathematician. What evidence would be used? What evidence might be used in 4 weeks’ time? [To me this is a slightly forced way to achieve meta-cognition goals – I feel as though there might be more effective ways to achieve the same outcome by being more straight with pupils]
    • Choose a letter/number and come up with the specified number of mathematical words starting with that letter.
    • Just a minute [Love this idea!]
    • Thunks (get students to create their own)
    • Killer questions on a topic being studied – with someone in the hot spot.
    • Get students to create questions that keep them awake at night [I could have got these already from the first lesson questionnaire]
    • What if…ask odd questions and see what creativity it sparks in pupils
  • Chunkier ideas:
    • 5-3-1: Think up five ideas – pick the top three – justify one
    • Link two items together in five steps
    • Create a presentation of exactly 60 seconds to prove they know something
    • Pick the odd one out
    • Create your own report (then post it to them a month later) [I really like this]
    • Plot quirky variables on a graph e.g. happiness of a water molecule throughout the water cycle [another great idea!]
  • Huge ideas:
    • Circle time to discuss emotive topics such as dreams or ambitions.
    • Get students to think up the most unbelievable way to teach a topic, and then come up with more reasons why it could happen rather than couldn’t, covering all areas of project management.
    • Set up a trial for a controversial historical (or present) mathematical figure [I adapted this one a bit]
    • Create a crime scene in your classroom and get pupils to solve it.
    • Rewrite history – imagine what might have happened if oil had never been discovered, or the computer hadn’t been invented?
    • Predict future events, and the likelihood of these
    • ‘Resign’ and get pupils to plan the next five lessons (great for revision!)
    • Get pupils to present a topic they have learnt in the style of a TV show.
    • Evidence hunt in the textbook.
    • How could they improve the textbook?
    • Tell students they have the lesson to do something that would help them be a better learner, mathematician, or person.


I really like these ideas, but I think an important criterion for each is what the objective and outcome will be, and how this is related either to improving mathematical learning, or assisting pupils in building agency, self-esteem and an internal locus of control.


The Lazy Teacher’s Handbook: Ch 3 Structuring the lazy lesson

  • Prepare – do – review “learning loops” allow flexibility in lesson planning rather than sticking to a ‘three part lesson’ or similar.
    • By reviewing what you are doing throughout the lesson, this allows students to develop their abilities as autonomous learners.
  • In the Prepare phase, you need a good starter, clear outcomes and task introduction. All of these can be student-led.
    • At this stage it is worth having a discussion about the learning you will do in the lesson; if you are asking students to learn differently, you need to prime them to do so by e.g. asking them what they want to learn, where they have learnt something similar etc.
  • In the Do phase, let the students work! Don’t interrupt or try to lead.
    • Use strategies such as 3B4ME: use three strategies (brain, book, buddy, boss) before putting your hand up and asking the teacher.
    • FAQs whiteboard – students write questions, the teacher tells only them the answer and others must ask them.
    • Set up a ‘help desk’ with useful info/an email contact.
    • Use meta-questions: what are you currently thinking about?; what are you stuck on?; what do you need to answer the question?; what connections have you made?; what progress have you made?; have you seen anything like this before? Etc.
    • Again in the Do phase you can have a ‘meta-discussion’ about students’ learning.
  • In the Review phase (not necessarily the end of the lesson but the end of a learning loop):
    • Use a plenary to review the learning or reflect on progress so far
    • Again have a ‘meta-discussion’: how are you going to remember this learning?; what is the key point you will remember?; what has this lesson reminded you of/what connections have you made?; what did you learn that you didn’t know before?; where can you use this learning?
  • At the end of the lesson, hook them into the next lesson by introducing a teaser, and also dismiss them with praise and encouragement.