Doing maths in your head takes time

How do children learn to understand mathematics? And what is the best way for teachers to support them? These questions are being asked directly in the classroom where a LiU researcher is working together with teachers. Children need to be given time to think about their mathematics. 

Emme in the first gradeis deep in thought. Do the pictures in the mathematics book show addition or subtraction?

Karin Bengtsson, mathematics teacher in Vikingstad elementary school is crouched close to her, waiting calmly. And then Emme is ready to express her thoughts in a quiet murmur of mathematics.

Make use of silence

“A lot can happen in a period of silence. As an educator you can make use of silence, or rather silences, in an incredible number of different ways,” says Bengtsson shortly afterwards.

Silence gives us time to think, to gather our thoughts, attract attention, to point out something important, to bring down the tempo. Just to mention a few examples.

“By being silent I can send the message that it’s not only OK to reflect, but also that it’s necessary. Especially when we want to understand mathematics.”

Last autumn, Bengtsson was one of four in the first group of teachers that took part in a new mathematics education research project at LiU, financed by Linköping and Norrköping municipalities with SEK 10 million (ca EURO 1.2 million) over five years.

Produced results in the classroom

“It’s been fantastic, simply having time put aside to talk with other teachers and researchers about teaching. The best thing is that it has produced results in the classroom – more mindful teaching,” she says.

A cross between development and research projects. This is how Lisa Björklund Boistrup describes the action research about assessment and feedback that she is leading. She has been employed as a researcher at the LiU Department of Behavioural Sciences and Learning since August, specialising in mathematics projects.

“We head straight into the classroom. A group of four teachers per term and the municipality participate. The research is based on what they do, their teaching forms the basis of the analyses and together we discuss our way towards the answers to the questions the group has asked.

How the groups choose to deal with the questions will vary.

“This all means we will gradually build up a ‘map’ of experience and useful knowledge,” Björklund Boistrup says.

Start out from a narrow idea

From the two groups in the autumn the Linköping group chose to start out from a narrow idea, such as silence, in order to broaden the perspective to a more general approach. The Norrköping group began from a broader concept, that of feedback, later on focusing on something more specific: how the teachers’ approach influenced the pupils’ desire to continue to talk about mathematics even when the teacher has moved on from them.

“We were keen to know if they immediately switched to talking about what they were having for lunch,” says Annika Knutson, mathematics teacher at Vibergsskolan in Norrköping, with a laugh.

But no, they didn’t.

The pupils’ capacity to maintain their concentration on mathematics is highly dependent on the teacher’s approach, for example when they need help. More “feedforward” than feedback, as Knutson puts it.

The approach is one area Björklund Boistrup herself has researched. Generally speaking, the approaches can be divided into four categories, as she showed in her doctoral thesis in mathematics education.

“‘Quick and direct’ might be a description of one approach. ‘Anything goes’ is another very common approach: Everything the child does is valid; making demands is almost seen as a hindrance to learning.”

“In the third category the teacher focuses on the mathematics and asks open questions. They do this in the fourth category too, but here there is another angle on learning: reasoning must be given time.

The approaches in categories three and four invite the pupils to maintain their interest in the mathematics and move on.

Mindful feedback

“We can see how there is more and more mathematics as the teacher provides mindful feedback. We can even observe how pupils who are nearby are positively influenced,” explains Björklund Boistrup in relation to her own research results.

“Our group became curious about to what extent her conclusions were right. How active we are as teachers and how we approach our pupils is absolutely decisive,” says Knutson, whose fourth graders took part in the research.

A series of lessons were filmed and formed the basis of individual analyses and group discussions.

“It was no problem getting the children or their parents to go along with this. On the contrary, they were proud to be part of an important research project,” Knutson continues.

Saw behaviour patterns

In Vibergsskolan in Norrköping classes were filmed during previous development projects, otherwise it can be a sensitive issue for many people.

“As a teacher you got used to it rather quickly. Soon we stopped seeing ourselves and instead we saw behaviour patterns. For me it was incredibly beneficial to be able to watch my own teaching afterwards, to be able to pause and reflect. And to get to see what happened with the pupils, which is something you don’t really get to see normally,” says Bengtsson.

A lot of hands in the air, restless scooting around on chairs and chatter are situations that can make any teacher quickly intervene with a “do this, do that”, and hurry on to the next child.

“Today I can see that it is better for me to find time in one lesson to help four pupils properly than to run around between them all,” she says.

The participating teachers got to keep a logbook of their teaching during the weeks when the research took place and those who wanted to write articles received support for it.

“We are currently putting one together that we want to put in the online journal Venue,” says Knutson.

For the sake of continuity, the group of teachers from one term also gets to be in an action research reference group for the following term.

Professional development

On the part of the municipalities there is, of course, the hope that the money invested will result in goals being better met – that is, better mathematics knowledge for the pupils and also preferably quick and measurable improvement in the results in the national tests. International surveys carried out at regular intervals have for a number of years shown that Swedish school children have become increasingly weak in mathematics when compared with other countries.

“It’s not easy to bear the weight of such heavy expectations. I try to think that there are a lot of us doing this together. That’s the point of action research, of course. But I really believe in this,” says Björklund Boistrup, who is now running the project together with Joakim Samuelsson, senior lecturer at the Department of Behavioural Sciences and Learning in collaboration with both municipalities.

“Of course the expectations can feel a little burdensome. On the other hand, participation in the research project is fantastic professional development. This is something that all teachers should take part in,” Knutson says.

Footnote: Both Bengtsson and Knutson graduated from LiU as teachers, in 2001 and 2006 respectively.

Published in LiU magasin 1-2013.