Dr Beryl Exley is a lecturer in language and literacy education, Queensland University of Technology. Keiran Abel is a recent graduand from Queensland University of Technology.

There is widespread acceptance in the teaching community that each subject has its own content focus and ways of representing that content. Wyatt-Smith and Cumming (2003) refer to the specialised forms of representation as literacies. Importantly, the emphasis is on the plural form: literacies. They advocate that there is no single set of literacy knowledge or skill that covers the demands of all school subjects. While it is fair to say that all school subjects share some commonality of literacy knowledge and skill, each subject has its own peculiarities and makes idiosyncratic literacy demands of its students.

Mathematics is no exception. Each state and territory mathematics syllabus recognises the important link between mathematics content and literacy or literacies. Each affirms the need for all students to become citizens who are mathematically literate. Yet no state or territory syllabus explicates what the particular literacies of mathematics might be or what teaching strategies provide the necessary scaffolding for students. In other words, these documents lack a metalanguage, both for teachers to consider the literacies of mathematical representations, and also as a point of study for students.

Recent research undertaken at the Queensland University of Technology in Brisbane explored the literacies of mathematics problems in the early years. Dr Beryl Exley, lecturer in language and literacy education, and Keiran Abel, a recent QUT graduand, used Halliday’s (1990) functional grammar to analyse the literacies of mathematics problems in the early years. Their study was undertaken in two parts: firstly, an examination of six worded mathematical problems; and, secondly, an examination of the reading alternatives of the accompanying pictures. The worded problems and visuals were taken from the Primary Mathematics series from RIC Publications in Western Australia (Way 2004, p 87). The figure, below, shows the first three problems that students working at Level C (Year 2 in Queensland) had to ‘read, choose an operation to use, set out a number sentence and answer it’ (Way 2004, p 87).

 Figure 1: ‘Number Stories. Which Sign?’ (Way 2004, p 87)

Our examination of the worded mathematics problems revealed seven complex grammatical devices. We identified these devices as ‘complex’ because when we examined the English syllabus for this age group (Queensland Studies Authority 2005), the devices were not considered to be part of the learning outcomes for this student group (refer to Abel & Exley 2008 for a detailed discussion).

1. For each of the six worded maths problems, students had to bridge their understanding from a statement to a question. Statements and questions differ in structure and function. In this case, the statement provides information about a seemingly everyday event whereas the question makes a ‘How many…?’ request for information. This sort of questioning requires a more complex response than one that requires a ‘yes’ or ‘no’ answer. According to the QSA English syllabus, students are not typically proficient at these moves until the end of Year 3.
2. Generally speaking, students of this age are more familiar with the action, thinking or saying processes found in narratives. Yet, some of these problems required students to make meaning of a relational process. A relational process relates one concept or entity to another. Problem (a) uses relational processes: Jess has 12 toy trains. Jess is set up in a relationship with his 12 toy trains. Relational processes are complex because their ‘action’ is not easy to identify.
3. Both the ‘addition’ and ‘multiplication’ operations use the key word ‘altogether’. This has the potential to confuse students and encourage them to add the two numbers in the statement. 
4. The (b) word problem uses an existential process (eg ‘There are’), a process concerned with ‘existence’ rather than ‘action’. Again, this makes it difficult for students to identify the ‘action’. This more complex process is not a part of the QSA English syllabus for students of this age.
5. The (a) word problem uses complex references, where two individual people, Jess and James are later referred to as ‘they’. Moreover, students have to carry their meaning-making from the first and second clause to the third clause, which is in another sentence. Such devices may be very confusing for early years students who are not explicitly taught to deal with these more complex references.
6. The (a) word problem also leaves out the main part of a noun group. In problem (a), what the ‘15’ is referring to is left out. The problem says, ‘James has 15’, instead of ‘James has 15 toy trains’. Students of this age have usually spent more time working with the narrative genre where they are used to more information in noun groups.
7. The (c) word problem uses a dependent clause with a complex conjunction of cause and effect (eg ‘until’). Students in Year Two will typically be working with conjunctions of time and sequence rather than conjunctions of cause and effect.

A further instruction at the top of this worksheet told students, ‘You may wish to use the pictures or counters to help you’ (Way 2004, p 87). The second part of the research examined the possibilities for making meaning from these accompanying pictures. We again drew on Halliday’s (1990) functional grammar and the practical work of Hart (1999) to consider the field (the actions, setting and participants of an art work), tenor (the form and construction of an art work, or put another way, the elements and principles of design) and mode (how the visual journey unfolds for the viewer) of each. We noted four ambiguities that left the meaning of the supporting pictures unclear.

1. Our analysis revealed that none of the drawings were representations of still life, which would have concretised the situation to the students and helped them to relate it to something in their own experience. Rather they were decontextualised representations. For example, picture (a) was a representation of two boys who each own some trains. The boys are represented by their heads (no bodies) and their trains appear to be suspended in mid-air. Students might expect to see a child’s train collection in a box or on the floor.  
2. We found each picture employed inconsistent design features. For example, dominance is the design feature that dominates the viewing. It is a way of emphasising what is most important. Dominance was represented in picture (a) by the use of dark colour on the boy’s hair, yet this participant was not integral to the mathematical operation. This design feature may serve to distract or confuse the student.
3. We also found that the rules of entry young students are taught for traditional print-based text do not hold up for these mathematical pictures. Students are traditionally taught to read text left to right, top to bottom. Yet pictures (a), (b) and (c) all presented a different visual journey. For example, the dark colour in the top left of picture (a) would probably take the viewer’s eye from that position, and the repetition of lined-up shapes could take the viewer’s eye left to right, top to bottom. A viewer would also start to read picture (b) in the top left corner, but could opt to move horizontally, vertically or on a diagonal as there is no particular journey created through dominance. The block of colour for the black cat creates dominance in the bottom right of picture (c), and from there the repetition of line takes the viewer’s eye around the birdbath. Importantly, it seems the entry and viewing routes for a picture are not always the same. Students have to be taught to bring a different reading route to these pictures.
4. In each of the three pictures, the subjects are not engaging directly with the viewer, save for the black cat in picture (c). This means that the subjects that are not engaging directly with the viewer are not making any demands of the viewer. In contrast, the black cat is, but the cat is peripheral to the operational work of the number story. The problem here is that students don’t have reliable clues for reading the mathematical pictures. 

This examination of the written and picture representations of mathematical operations brings into focus the task faced by young students. Students need to learn to decipher some complex written and picture codes. Our findings raise two questions: Where is it that the complex written and pictorial representations of maths are carefully scaffolded for these young students? Is it happening in maths instruction? In conclusion, we strongly suggest it is important that those who teach maths also explicitly prepare students with the essential skills necessary for carefully and appropriately dealing with the written and pictorial demands of mathematical problems.

References:

Abel, K & Exley, B 2008 (forthcoming October), ‘Using Halliday’s functional grammar to examine early years worded mathematics texts’, Australian Journal of Language and Literacy.  

Halliday, MAK 1990, An Introduction to Functional Grammar, Hodder & Stoughton, Melbourne.

Hart, P 1999, ‘The practised eye: Ways of seeing in visual arts’ in J Callow (Ed.) Image Matters: Visual texts in the classroom, Primary English Teaching Association, Newtown, New South Wales, pp 75–84.

Queensland Studies Authority 2005, Years 1 to 10 English Support Materials: Elaborations for Levels 1 to 6 outcomes. Retrieved February 12, 2007 from http://www.qsa.qld.edu.au/yrs1to10/kla/english/support.html

Way, C 2004, Primary Mathematics. Book C, RIC Publications, Western Australia.

Wyatt-Smith, C & Cummings, J 2003, ‘Curriculum literacies: Expanding domains of assessment’, Assessment in Education,10 (1), 47–49.