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.
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?
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