Summer 2004

Talking Maths

Catching Them Young

What sort of classroom practice helps young children to enjoy and become engaged with mathematics? DOUG CLARKE AND BARBARA CLARKE describe a classroom research study that identified particularly effective teachers, then watched how they worked with their students.

ANNE JOYCE’S year 2 class at Ararat North Primary School loved problem solving. During their involvement in the Early Numeracy Research Project (ENRP), the project team occasionally sent them problems to explore. At one stage, we sent this problem:

A man goes into a shop and says to the shopkeeper, ‘Give me as much money as I have in my pocket, and I will spend $10’. It is done, and then the man does the same thing in a second and third shop, after which he has no money left. How much did he have at the start?

The reader is encouraged to work on this problem for a little while. This is a very difficult problem for this level, but the children seized on the challenge, and worked on this problem for three weeks. All contributed— making money, acting out the story, suggesting strategies. Although this is a bit of a ‘silly’ problem, it has a lot going for it mathematically.

The year 2s sent a poster to the project team outlining their solution process. They concluded: ‘It took a lot of thinking and working out.’ These children loved problem solving, and they had a teacher who demonstrated clearly one of the characteristics of effective mathematics teachers. Anne Joyce ‘promoted and valued effort, persistence and concentration’.

As part of the ENRP in Victoria, 354 prep to year 2 teachers in 35 schools participated in a three-year research and professional development project, exploring the most effective approaches to the teaching of mathematics in the first three years of school. There were four key components of this project:

1. A research-based framework of ‘growth points’ in young children's mathematical learning (in Number, Measurement and Space)
2. A 40-minute, one-on-one interview, used by all teachers with all children at the beginning and end of the school year
3. Extensive professional development at central, regional and school levels, for all teachers, coordinators, and principals
4. Case studies of particularly effective teachers.

Student interview data from the first two years was used to identify particularly effective teachers, the ones whose children showed the greatest growth over two years, for intensive study in the third year. Six case study teachers were studied intensively using lesson observations, detailed observer field notes, photographs of lessons and collection of artefacts (worksheets, student worksamples, lesson plans), and teacher interviews following the lessons.

The description of effective teachers, as revealed in this study, is shown below.

Snapshots from the classrooms of effective P–2 teachers

Sometimes, a list such as the one on the previous page can seem removed from the life and colour of the classroom. The following classroom examples illustrate the themes evident in the table.

Effective teachers of mathematics structure purposeful tasks that enable different possibilities, strategies and products to emerge

The one-on-one student interviews made very clear the considerable range of knowledge and understanding within any classroom. As a result, teachers made extensive use of open tasks, encouraging children to share their solution strategies. Examples of such tasks included:

• Two children measured the basketball court with rulers. Huong said it was 20 rulers. Yasmine said it was 19. Why might that be?
• I drew a shape with four sides. What might my shape look like?
• I bought some things at the supermarket and got 35 cents change. What did I buy and how much did each item cost?
• What can you find that is lighter than an orange but bigger than it?

Children eagerly accepted these kinds of challenges, responding at their own level of understanding.

Effective teachers of mathematics focus on important mathematical ideas and make the mathematical focus clear to the children

Teachers noticed during the interview that although many children could read and write twoand three-digit numbers, quite a few had difficulty ordering onedigit numbers.

In school teams, they developed a range of games and activities that focussed on this important mathematical idea.

One teacher asked children to cut up magazines and catalogues, taking out any numbers they could find. They then sorted these out from smallest to largest. The two examples below give a sense of the range of children’s responses.

Another teacher developed a card game, where the picture cards were removed from a standard pack, and children had half the pack each. At the same time, each child turns over a card, and the person with the larger of the two numbers takes both. Once again, there was a clear focus on ordering numbers from smaller to larger.

Effective teachers of mathematics encourage children to explain their mathematical thinking/ideas and build on children’s mathematical ideas and strategies

ENRP teachers noted that they found themselves using many more questions than previously that probed children’s thinking. Examples included:

• How did you work that out?
• How are these two objects the same, and how are they different?
• What could you do next?
• Can you see a pattern in what you’ve found?
• Can you make up a new task using the same materials?

It is interesting to consider the extent to which the teacher behaviours (for effective early numeracy teachers) and characteristics applies to other grade bands and learning areas.

It was a wonderful privilege to be in the classrooms of dedicated mathematics education professionals. We have described classrooms where the enthusiasm, curiosity and strategies of young children are valued and built upon, with lasting effects upon their understanding, their attitudes, their love of mathematics and their confident views of themselves as learners of mathematics.

References

Clarke, D M & Clarke, B A (2003). ‘Encouraging perseverance in K–2 mathematics: The tale of two problems’, Teaching Children Mathematics, 10 (4), pp 204–17.

Clarke, D M & Clarke, B A (2004). ‘Mathematics teaching in Grades K–2: Painting a picture of challenging, supportive and effective classrooms’, in Rubenstein, R N & Bright, G W (eds), Perspectives On the Teaching of Mathematics, Reston, VA: NCTM.

Doug Clarke is a professor of Mathematics Education at Australian Catholic University.

Barbara Clarke is a senior lecturer in Mathematics Education at Monash University, and is associate dean (Staffing) in the Faculty of Education.

The author owns the copyright in this article. For information related to the reuse of this work in any form please contact the publisher denise.quinn@curriculum.edu.au