Putting place value in its place

ATM article putting place value in its place by Ian Thompson. Also the first I’ve really read on developing students’ understanding of place value in the classroom. Thompson uses a research study to argue that:

…children are able to add two-digit numbers successfully using partitioning without needing to have an understanding of place value.

Which is interesting for me, because up until now I hadn’t appreciated that there might be four properties of place value: positional, base-ten, multiplicative and additive.

Thompson’s central idea: moving digits to the left requires very sophisticated understanding of place value – it may be better for teachers to accept and teach the ‘rule’ ‘add a nought’ whilst later recognising (and discussing) the misconceptions this idea creates.


Boaler: What is maths? And why do we all need it?

Another recommended reading from Teach First. Selected quotes below:

Mathematics is a performance, a living act, a way of interpreting the world. Imagine music
lessons in which students worked through hundreds of hours of sheet music, adjusting the notes
on the page, receiving ticks and crosses from the teachers, but never playing the music. Students
would not continue with the subject because they would never experience what music was. Yet
this is the situation that continues in mathematics classes, seemingly unabated.
Those who use mathematics engage in mathematical performances, they use language in all its
forms, in the subtle and precise ways that have been described, in order to do something with
mathematics. Students should not just be memorizing past methods; they need to engage, do, act,
perform, problem solve, for if they don’t use mathematics as they learn it they will find it very
difficult to do so in other situations, including examinations.

We cannot keep pursuing an educational model that leaves the best and the only real taste
of the subject to the end, for the rare few who make it through the grueling eleven years that
precede it. If students were able to work in the ways mathematicians do, for at least some of the
time – posing problems, making guesses and conjectures, exploring with and refining ideas, and
discussing ideas with others, then they would not only be given a sense of true mathematical
work, which is an important goal in its own right, they would also be given the opportunities to
enjoy mathematics and learn it in the most productive way.

Boaler’s vision is an inspiring but ambitious one. Like Swan, Boaler discusses the end goal of a maths classroom without always making explicit the precursors necessary to allow students to work like mathematicians. Working like a mathematician is hard and, amongst other things, requires grit, persistence and a willingness to be wrong before being right. None of these things will come naturally to a mathematics student.

On the other hand, each of these things can be encouraged. The challenge is that it will take time, persistence and agility on my (the teacher’s) part to encourage students to work like mathematicians. It isn’t easy, but if Boaler is to be believed it’s worth the effort.

Boaler: What is Maths? And why do we all need it?

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

Embedded Formative Assessment Ch 7: Activating students as owners of their own learning

  • Self-assessment depends on both meta-cognition and motivation
  • The most important thing for self-regulated learning is to encourage students to pursue personal growth not well-being. This can be encouraged by:
    • Sharing learning goals
    • Promoting a belief that ability is incremental
    • Make comparisons between students more difficult
    • Make feedback a recipe for future action
    • Use every opportunity to give learners control of their learning.
  • Practical techniques:
    • Traffic lights used to focus revision efforts.
    • Green/red disks on desks to indicate when a teacher is going too fast.
    • Or use red to ask a question of another student.
    • Choose three of a selection of reflection statements to respond to at the end of a lesson.

Embedded Formative Assessment Ch 6: Activating students as instructional resources for one another

  • Peer tutoring can be as effective as teacher tutoring because of the different power relationships.
  • Effective cooperative learning requires group goals and individual accountability.
    • Some schools achieve this through a ‘secret student’ approach whereby a single student’s behaviour is reviewed and if good the class gets a reward.
  • Practical techniques:
    • 3B4ME
    • Peer evaluation of homework
    • Two stars and a wish
    • Getting students to write down end-of-topic questions in groups
    • Classifying errors and sharing with a peer with strengths in that area.
    • Student reporter summarises the lesson in plenary and answers remaining questions.
    • Get a buddy to sign off on a checklist e.g. labelling a graph, with accountability for their peer’s work in this area.
    • Assign group reporter at random so everyone has to be prepared to do it.
    • Get students to review and present on a particular aspect of work.

Embedded Formative Assessment Ch 5: Providing feedback that moves learning forward

  • Giving scores can completely wipe out the positive effects of giving constructive comments.
  • What matters for learning is the mindfulness with which students engage with feedback.
    • This can be achieved by scaffolding feedback.
  • “The best learners consistently attribute both success and failure to internal, unstable causes” (p.117)
    • The feedback we give should support this view, by being used by the learner to improve performance.
    • To achieve this, feedback must provide a recipe for future action, broken down into a series of small steps.
  • Never grade students while they are still learning.
  • An idea for grading is on p.126
  • Practical techniques
    • Mark work with a -, = or + to mean not as good as, equal to or better than your last piece of work.
    • Don’t provide feedback without class time to act on it e.g. write 3 questions next to specific aspects of a piece of work, and give 10-15 minutes in the lesson to respond.
    • Write feedback on strips of paper and give to groups to decide which feedback goes with which piece of work.
    • Relate the feedback to the learning objectives.
    • “Five questions are wrong; find and fix them.”

Embedded Formative Assessment Ch 4: Eliciting evidence of learners’ achievement

  • Generating questions that genuinely assess students’ thinking is surprisingly difficult (but crucial).
  • Whether to go over something again is a professional judgement, informed by how crucial that topic is to progression.
  • Practical techniques:
    • Asking questions that cause new learning through analysis, inferring or generalising.
    • Picking students at random (pose-pause-bounce-bounce) rather than accepting hands up.
      • Use a random name generator for this.
      • But not always popular.
      • If a student tries to opt-out by saying ‘I don’t know’, come back to them. Its important to establish that classroom participation is not optional.
    • Increasing wait time after an answer as well as before.
    • Make statements that can be debated rather than closed questions.
    • Listening interpretively not evaluatively – listen to understand what pupils think, not for the right answer.
    • Use question shells such as ‘Why is…an example of…?’ or ‘Why is … x and … isn’t?’
    • Hot seat questioning using a series of q and a’s to one student that another then summarises (chosen randomly).
    • Eliciting all student responses to cognitive questions, for example by using ABCD cards (require pre-planned questions) or mini WBs (for more spontaneous questions) or exit passes.
      • Exit passes can be used to group people for the next lesson.
    • There is an important distinction between discussion questions and diagnostic questions
      • In the latter they are designed so that it is very unlikely that the student will get the correct answer for the wrong reason.
      • The idea is that a lesson will be designed with at least one ‘hinge point’ where the teacher uses a diagnostic question to check whether the class is ready to move on.
        • The questions should take no longer than 2 (and preferably 1) minute to answer.
        • Responses should be analysable in 30 seconds or less.
        • It should be very difficult for students to get the right answer for the wrong reason.
        • Incorrect answers should be interpretable.
      • Multiple choice hinge point questions reduce possible answers so make them more analysable in real time.