Tag Archives: primary classrooms

Ability grouping and mathematics: Who benefits?

During the primary years students either experience mathematics lessons within mixed ability classrooms or are ability grouped across or between grade levels. In mixed ability classes and to a lesser degree in ability grouped classrooms teachers often have to cater for a range of students whose previous attainment varies considerably. The gap between low and high achievers is said to increase as students progress through school, with as much difference in performance within year levels as there is between Years 5 and 9 students overall (State of Victoria Department of Education and Training, 2004). This concurs with findings by The Middle Years Numeracy Research Project (Siemon, Virgona, & Corneille, 2001) that cites teachers in the middle years can and should expect a range of up to seven school years in numeracy-related performance. Australian TIMSS and PISA reports show our middle years students, when compared to students in other countries, are not keeping up with some aspects of mathematics (Lokan, Greenwood, & Cresswell, 2001). Among the strategies teachers implement in order to cater to the diverse needs of the classroom is the use of ability grouping and the incorporation of differentiated tasks.

Although there are arguments both in favour and against the use of ability grouping, it is not uncommon for primary and secondary schools in Australia and internationally to use such grouping in mathematics classrooms. The following is an exploration of both arguments. Clarke (2003) maintains teachers need to think carefully about reasons for choosing to place students into groups according to perceived ability. Teachers need to consider the potential impact upon students’ self-esteem, and the ‘self-fulfilling prophecy’ (Brophy 1963, as cited in Clarke, 2003), in which students perform to the level expected of them by their teacher. He concludes that ability grouping is used in mathematics for teacher convenience more than for student benefit. This is a sentiment echoed by Boaler, Wiliam and Brown (2000), who also believe ability grouping is one of the main sources of disaffection.

Ability grouping, such as that which is common in Australian secondary schools, became the focus of a study conducted in the United Kingdom (Boaler, et al., 2000). The grouping of students into ability ‘sets’ emerged as a significant factor that influenced students’ ideas, their responses to mathematics, and their eventual achievement. The study found that students in the school that used ability grouping were significantly disadvantaged by their placement and this disadvantage was not restricted to students in the lower ability groups. Approximately one-third of students in the highest ability groups felt disadvantaged because of high expectations, fast-paced lessons and pressure to succeed. Students from a range of groups were ‘severely disaffected’ by the limits placed upon their attainment. Students reported that they gave up on mathematics once they discovered their teachers had been preparing them for examinations that gave access to only the lowest grades. Large numbers of students, in the study by Boaler et al., experienced difficulties working at the pace of their particular class. For some the pace was too slow, resulting in disengagement, although for others it was too fast, resulting in anxiety. Both responses led to lower levels of achievement.

In addition to the findings above, there is research that claims ability grouping causes behavioural problems for some within the mathematics classroom. Teachers in a study conducted by Ventakatakrishnan and Wiliam (2003) found behavioural problems more common in mixed ability groups than in their fast-track, higher ability group. These behavioural problems were compounded by the weak literacy skills of some individuals in these groups in addition to peer self-management skills. Interestingly, placing students in ‘tracked’ groups had an effect on students’ perceptions of themselves as learners of mathematics. Those who were fast-tracked perceived themselves as ‘doing well’ while those in mixed ability groups perceived themselves as ‘low’ in mathematical ability. The teachers involved in the study also noted they had problems motivating the higher attaining students within the mixed-ability groups – students who had ‘just’ missed out on being placed in the fast tracked group. Ventakatakrishnan and Wiliam also note that mixed-ability grouping decreases the opportunities for higher-achieving students to interact constructively with peers although ability groups have the same effect on lower achieving students. The study found that advantages of grouping by ability are limited and restricted to higher achieving students while causing disadvantage to those who are the lowest attainers.

Although there are advantages and disadvantages in grouping students according to ability, it is also reasonable to expect that a result of such grouping is differences in teacher expectations and instructional techniques. There is a tendency for instruction in lower ability groups to be of a different quality to that of higher ability groups (Ireson & Hallam, 1999). There appears to be a concern that instruction in low ability groups is conceptually simplified with a higher degree of structured, written work, as in the traditional method of teaching mathematics. Higher ability classes appear to include more analytic, critical thinking tasks with pupils allowed greater independence and choice along with opportunities for discussion, reflecting a more contemporary style of mathematics lesson. Based on stereotypes and past experiences, some teachers hold low expectations for low ability groups, further decreasing students’ own expectations and self-esteem, and leading to decreased levels of engagement with mathematics (Ireson & Hallam, 1999).

At the primary level classroom, research suggests students in mixed ability classrooms display more positive attitudes towards school in general (Ashton, 2008; Ofsted, 2008). However, the more common practice of ability grouping in the secondary mathematics classroom is a complex issue. Although it appears to perpetuate the inequities associated with mathematics and the notion of mathematics being an elitist subject, it is often an attempt to address the needs of all students as well as the needs of mathematics teachers.

So what is the answer? My advice is to group flexibly. There are circumstances that require children to be grouped by ability, and other circumstances where it is more appropriate and beneficial for students to work in mixed groupings, where students can learn from each other. Another reason to have flexibility is that often children may excel in one area of mathematics, but may be challenged in other areas. Those who appear to struggle, may not necessarily struggle with all concepts. The key is knowing your students and their needs across all aspects of the mathematics curriculum, and ensuring that assessment of students’ ability informs teaching and the way students are grouped. And finally, have high expectations of all students!

References:

Ashton, R. (2008). Improving the transfer to secondary school: How every child’s voice can matter. Support for Learning, 23(4), 176-182.

Boaler, J, Wiliam, D, & Brown, M. (2000). Students’ experiences of ability grouping: Disaffection, polarisation and the construction of failure. British Educational Research Journal, 26(5), 631-649.

Clarke, D. (2003, 4-5 December). Challenging and engaging students in worthwhile mathematics in the middle years. Paper presented at the Mathematics Association of Victoria Annual Conference: Making Mathematicians, Melbourne.

Ireson, J., & Hallam, S. (1999). Raising standards: Is ability grouping the answer? Oxford Review of Education, 25(3), 343-358.

Lokan, J., Greenwood, L., & Cresswell, J. (2001). 15-up and counting, reading, writing, reasoning. How literate are Australia’s students? The PISA 2000 survey of students’ reading, mathematical and scientific literacy skills. Melbourne: ACER.

Ofsted. (2008). Mathematics: Understanding the score. London: Ofsted.

Siemon, D., Virgona, J., & Corneille, K. (2001). Final report of the middle years numeracy research project. Melbourne, Victoria: RMIT University and the Department of Education and Training (Victoria).

State of Victoria Department of Education and Training. (2004). Middle years of schooling overview of Victorian Research 1998-2004 Retrieved July 7, 2005, from http://www.sofweb.vic.edu.au/mys/docs/research/

Venkatakrishnan, H., & Wiliam, D. (2003). Tracking and mixed-ability grouping in secondary school mathematics classrooms: A case study. British Educational Research Journal, 29(2), 189-204.

Programming & planning dilemmas in primary mathematics

Often when I work with teachers I am asked for advice regarding the design of a scope and sequence for mathematics. The programming and planning of mathematics seems to cause much concern, and often the reason is that there is no ‘magic fix’ or one-size-fits-all solution.

Traditionally, schools have planned their mathematics teaching using a topic-by-topic or strand-by-strand approach. Sometimes there is a formula for teaching the Number and Algebra strand for a certain number of days per week, with the other days dedicated to the remaining syllabus strands. Often, the strands are split into single, stand-alone topics. Unfortunately, there are issues with this approach. Teaching individual topics in mathematics hinders students in gaining a deep understanding of mathematics and the connections that exist between and amongst the strands. Teaching in this way can promote a traditional, rote learning approach where the opportunities for mathematical thinking are limited. Our curriculum places the proficiencies (Working Mathematically in New South Wales) at the forefront of teaching and learning mathematics – teaching topics in isolation does not promote the proficiencies.

So what’s the solution? Consider planning and programming using a ‘big idea’ approach. What’s a big idea? Big ideas are hard to define and different people have differing ideas on what the big ideas in mathematics actually are. However, all the definitions in literature have one thing in common – they all refer to big ideas as the key to making connections between mathematical content and mathematical actions, and they all link mathematical concepts. Take, for example, the big idea of equivalence. This relates to number and numeration, measurement, number theory and fractions, and algebraic expressions and equations. Connections can be made across the strands and these links should be made explicit to students.

Charles (2005) presents a total of 21 big ideas across the mathematics curriculum, however he states that these are not fixed – they can be adapted. He also states that a big ideas approach has implications for curriculum and assessment and professional development – teachers need to develop their pedagogical content knowledge to ensure they have a deep understanding of the connections within the curriculum if they are to teach mathematics successfully.

Of course, there are challenges to teaching using a big ideas approach. Teachers often feel under pressure to address all curriculum outcomes, and often this is the reason that the topic-by-topic approach is adopted. Using a big ideas approach can feel messy – it is not linear and in some ways feels as though it is conflicting with the organisation of our curriculum. However, we must remember that although our curriculum is separated into strands and sub-strands, this is simply an organisational tool and does not mean that mathematics should be taught in this same way.

My advice would be to take our curriculum, pull it apart and try seeing it differently – what areas of the curriculum have obvious links? How can you link aspects of measurement to the number strand? Where does measurement and geometry link? And how can you use the statistic and probability strand to teach number concepts? Making connections will make your teaching easier in the long run, and more importantly, will result in deeper learning and deeper engagement with mathematics.

 

Randall, C. (2005). Big ideas and understandings as the foundation for elementary and middle school mathematics. NCSM Journal, 7(3), 9-24.

 

These are a few of my favourite things: Essential materials for every maths classroom

What concrete materials do you have in your mathematics cupboard and why bother investing in concrete resources? Concrete materials provide opportunities for children to construct rich understandings of mathematical concepts. In addition, providing opportunities for children to physically engage with materials is much more meaningful than working with drawn or even digital representations. For example, if you are teaching students concepts relating to 3-dimensional space, it makes sense that it is better for children to be able to manipulate objects in order to explore their properties and relate their learning to real-life. Concrete materials also promote the use of mathematical language, reasoning, and problem solving.

I often get asked about the essential resources required for primary mathematics classrooms. There are quite a few, but if you have a limited budget or space, there are a few resources that are what I would consider to be essential, regardless of the year level that you are teaching. My advice would be to invest in materials that are flexible and able to be used in a variety of ways, perhaps in conjunction with other materials. Also consider collecting things that are not necessarily intended as educational resources but may have some mathematical value, such as collections of things (keys, lids, plastic containers, etc.) for activities that require sorting and classifying. Here is a list of basics that can be purchased from educational resources suppliers (some of the items can also be sources at normal retail and/or discount stores):

  • Counters
  • Dice (as well as the standard six sided dice, you could purchase many other variations including blank dice)
  • Calculators (yes, these are great, even in the early years. Think about using them to investigate numbers rather than simply computational devices)
  • Base 10 material (be careful how you ‘name’ these – using terms like ones, tens, hundreds and thousands limits their use. It is best to use the terms minis, longs, flats and blocks so they can be used flexibly to teach a range of whole number and measurement concepts)
  • Pattern blocks (great for more than just exploring 2D shape – these can be used to teach fractions, place value, area, perimeter etc.)
  • Dominoes (one of my truly favourite things!)
  • Playing cards
  • Unifix blocks

Of course, any resource is only as good as the teacher using it and the way it is integrated into teaching and learning. Prior to using any concrete material, think about the purpose of the lesson and the mathematical concepts being taught. Also consider how you can make the most out of the resources – how will you differentiate the task, and how will you capture evidence of learning? This is where technology can play a useful role and allow teachers and students to capture evidence when working with concrete materials. Technology can also be used alongside concrete materials. For example, work with pattern blocks can be recorded using the Pattern Block App on an iPad. Or students could integrate their use of concrete materials with a verbal reflection or explanation using the Explain Everything app.

The best way to get the most out of concrete materials is to research. There are many high quality resource books and there are also many great websites such as NCTM Illuminations that provide excellent teaching ideas. Once you see the potential of high quality, flexible concrete materials such as those in the list above, you and your students will become much more cognitively, affectively and operatively engaged with mathematics.