Numeracy across the curriculum: demands and opportunities
Steve Thornton lectures in Mathematics Education at the University of Canberra. He is President of the Australian Association of Mathematics Teachers.
John Hogan runs his own consulting business, Redgum Consulting Pty Ltd., and is a former Learning Area Superintendent for Mathematics in the Education Department of Western Australia.
Numeracy and school mathematics
Numeracy has become very much an ‘in-vogue’ term in Australian educational circles. Systems across Australia conduct state-wide numeracy tests, linked to and reported against national numeracy benchmarks. Yet it is debatable to what extent these system-wide tests or school plans adequately reflect a view of numeracy as ‘having the competence and disposition to use mathematics to meet the general demands of life at home, in paid work, and for participation in community and civic life’ (Willis 1992).
Numeracy is portrayed in these tests and benchmarks as little more than school mathematics. This approach assumes that ‘mathematics can be learned in school, embedded within any learning structures, and then lifted out of school to be applied to any situation in the real world’ (Boaler 1993, p 12). In fact, evidence suggests that students do not automatically use their mathematical knowledge in other areas. Lave (1988) found that even experience in simulated shopping tasks in the classroom did not transfer to the supermarket. On the other hand, it appears that people use highly effective informal mathematics in specific situations (Carraher, Carraher & Schliemann 1985).
It would be easy to attribute this lack of transfer of mathematical skills to other contexts to a deficient mathematics curriculum and poor teaching, but the literature shows that even if mathematics were taught and learned very well people would not necessarily apply it to new situations (Griffin 1995). Researchers in the area of situated cognition argue that cognitive skills and knowledge are not independent of context, and that activities and situations are integral to cognition and learning (Brown, Collins & Duguid 1989; Resnick 1989).
Attempts to contextualise school mathematics using ‘real world’ settings relevant to the students (eg Cohen 2001) often appear contrived. Further, these attempts still have a primary purpose of teaching mathematics rather than developing numeracy. It would seem that if students are to learn to use mathematics outside the mathematics classroom then that is where they need to experience the use of mathematics.
The MYN Project
The Middle Years Numeracy Across the Curriculum Project commissioned by the ACT Department of Education, Youth and Family Services (DEFYS) attempted to address this issue. The project sought to help teachers identify the numeracy demands of their teaching area(s), support them in implementing numeracy, and facilitate productive professional discussion on numeracy within and across all curriculum areas in schools.
The project was conducted in over two years from 2002–2003, as a partnership between the Australian National Schools Network, Redgum Consulting and the University of Canberra. The principal researchers, John Hogan of Redgum Consulting and Steve Thornton of the University of Canberra, worked with officers from the ACT DEFYS and teachers from several ACT schools.
Teachers came together in two research circles to plan, evaluate and document their work. The research circles involved 11 and eight teachers respectively, and were drawn from small groups of schools covering primary, middle years and secondary students. The secondary school participants included maths teachers and teachers in a wide range of other subjects.
Each research circle met during four two-day workshops held over the course of the project. The participants were first asked to write down their perceptions of numeracy and what they hoped to obtain from their involvement in the project. Teachers were later introduced to the Numeracy Framework (Hogan 2000), which described being numerate as involving a blend of mathematical, contextual and strategic knowledge.
By the end of the first workshop each teacher developed a classroom-based action research task that would enable them to examine students’ numeracy in their classroom. These tasks focused on identifying and exploring ‘numeracy moments’, incidents in which students encountered mathematical ideas in other contexts. The teachers undertook to record in detail the circumstances in which students encountered mathematical ideas, the problems they had in understanding the mathematics and/or the context, the action taken by the teacher and what the student did next.
The second workshop commenced with a group discussion of teachers’ observations. It became apparent that the teacher-researchers had begun to see that a student’s numeracy problem might not be simply a matter of not knowing the mathematics, but might relate to the context, or their inability to continue work on the task once they confront something they can’t do. When it was seen to be an issue with the mathematics the teachers were now more sensitive to what the mathematical problem might be. Teachers were asked to plan a unit of work for the coming term, investigate possible and probable numeracy demands in the unit, and develop some strategies for dealing with them.
The third two-day workshop moved towards planning for a school-wide focus on numeracy across the curriculum. Some teachers planned staff professional development activities, some asked other staff to document numeracy moments in their classrooms, and some involved students and parents in discussing numeracy.
The fourth two-day workshop focussed on planning for school numeracy and asked teachers to take an active role in school numeracy planning, a task that had since become a requirement for every school in the ACT.
Between workshops each teacher was visited at their school by a researcher or teacher-facilitator to provide support and encouragement for their ongoing work. Teachers’ and school stories have since been collated and developed into a website to provide guidance for other schools in the system.
The numeracy moments
Teachers observed and documented numeracy moments from different learning areas. One participant, for example, identified a discrepancy between students’ capacity to measure volume accurately in science and their ability to apply mathematical formulas for volume in mathematics. Despite measuring the same objects by displacement of water and using a ruler, students seemed unaware that obtaining divergent answers created a problem. They seemed unaware of issues such as appropriate levels of accuracy. In another example two participants undertook a joint action research project investigating students’ uses of formulas in mathematics and science. They found that, even though students had learnt techniques associated with changing the subject of a formula in the mathematics class, they rarely used that technique in appropriate situations in the science class. Yet most were able to successfully calculate the required quantity using other means.
These and other examples reinforce concerns that many students cannot readily apply the methods used in a mathematics classroom to other settings, and that students able to act numerately in the field are not always able to do so in the mathematics classroom. The context strongly influences the mathematics being used. Having dealt with a numeracy issue in one context, it seems to help to provide students with another context.
An important aspect of being numerate is knowing how accurate you need to be in a situation. Letting students make mistakes seems to be a key to developing their understanding in this area.
A close look at student work is necessary sometimes in order to diagnose their numeracy problem – is it a mathematical idea they do not understand? Or a mathematical skill they cannot do? Or is it they do not have the strategic skills to link the context and the mathematics? Is it that they are not fluent? Is it that they cannot do it so they give up? It seems that all students might need some practice at being critical of the mathematics they have used in given situations. Getting to find out what each individual student’s numeracy problem might be in any particular situation is difficult and may not be always possible.
There are many numeracy moments that potentially occur across the curriculum. Knowing when to explore them in depth and when to deal with them in a quick and pragmatic way is a difficult decision.
Sometimes the mathematics in a particular situation will be beyond what the students have done in mathematics itself. What a teacher should do in such situations is not always clear. Finding a way that stays true to the situation seems better than going too deeply into the mathematics.
Participants also reported their attempts to discuss whole-school approaches to numeracy at their schools.
They found that most teachers do not have numeracy as a priority in their teaching. Most teachers have a limited view of numeracy, usually as a subset of school mathematics skills (like number calculations) that were needed for low level social goals (like paying bills), and are surprised to see the extent of the mathematical demands in the work they ask students to do. Teachers tend to focus more on the possible mathematical demands of their curriculum rather than on the numeracy problems their students are actually having. Most teachers – primary and high schools – talked about the difficulty of working with other staff in their school who were negative or uninterested, or seemed uncomfortable talking about mathematics
The findings suggest that this is slow work. It is necessarily so because it is complex work. It involves working with the school culture, an understanding of school change, sophisticated skills in leading and managing change, and using a language about numeracy that is not well known or understood. So the confidence of teachers leading this work in a school is crucial.
Teacher’s experience of action research
The teachers enjoyed and appreciated the research circle methodology and action research model for giving them time to think about and issues, discuss them with teachers from other learning areas and different phases of schooling, and investigate their own classroom.
Issues arising and further research directions
A number of issues requiring further thought have arisen during discussions. They include:
Without an awareness of the underpinning role of mathematical ideas in problem solving, in communication and in public debate, it is debatable to what extent an individual can arrive at informed decisions or follow productive strategies. School mathematics alone is unlikely to develop this capacity in our students – it requires conscious effort by all teachers, and a willingness to engage in mathematical thinking in all learning areas. The action research approach used in the ACT Middle Years Numeracy Across the Curriculum Project provides a powerful model through which teachers can become aware of and plan for numeracy. Identifying and capitalising on numeracy moments not only develops students’ capacity to be numerate, it also enriches their learning in other areas of the curriculum.
Boaler, J. (1993), 'The role of contexts in the mathematics classroom: Do they make mathematics more "real"?', For the Learning of Mathematics, 13, 2, pp 12–17.
Brown, J.S., Collins, A. & Duguid, P. (1989), ‘Situated cognition and the culture of learning’, Educational Researcher, 18, 1, pp 32–42.
Carraher, T., Carraher, D. & Schliemann, A. (1985), ‘Mathematics in the streets and in schools’, British Journal of Developmental Psychology, 3, pp 21–29.
Cohen, P. (2001), ‘The Emergence of Numeracy’. In Steen, L (ed) Mathematics and Democracy: The Case for Quantitative Literacy, USA: National Council on Education and the Disciplines, pp 23–29.
Curriculum Corporation (2000), Numeracy Benchmarks Years 3, 5 & 7. Carlton, Vic: Curriculum Corporation.
Griffin, M. M. (1995), ‘You can't get there from here: Situated learning, transfer and map skills’, Contemporary Educational Psychology, 20, pp 65–87.
Lave, J. (1988), Cognition in practice: Mind, mathematics and culture in everyday life, Cambridge: Cambridge University Press.
Resnick, L. (1989), Introduction. In Resnick, L. (ed), Knowing, learning and instruction essays in honor of Robert Glase. Hillsdale, New Jersey: Lawrence Erlbaum Associates.
Willis, S. (1992), ‘Being numerate: Whose right? Who's left?’, Literacy and Numeracy Exchange, Autumn 1992.
Note: references refer to titles referenced in the abridged text.
Subject HeadingsProfessional development