Improving outcomes in Numeracy: 'Some things different'

— An edited version of Pam Sherrard's presentation to the second IESIP SRP Conference

The goal of this project was to improve the numeracy levels of 18 Years 3 and 4 Aboriginal students in a large district high school in a remote location. The targeted students in the group were all speakers of English as a second language or dialect (ESL/D) and described by their classroom teachers as being at 'educational risk'. Initial data, gathered through classroom teacher interviews, work samples and discussions with the students indicated the students' understanding of number concepts was below that usually expected of children in their year groups. One of the Year 4 students was working in Level 1 in Number as described in the Western Australian Student Outcome Statements; all the other students were working below this level.

The records of student achievement show that by the end of the project, less than a year later, seven of the ten Year 3 students and six of the eight Year 4 students were working in Level 2. Two students left the school during the year, but the progress which they had made indicated that they would have reached this level. Anecdotal data from classroom teachers suggests marked improvements in students' self confidence, which was transferred into other areas of the curriculum.

A glance through what follows may suggest that little occurred which was very different from current practice in many schools. Good practice has long incorporated such strategies as para-professionals working in classrooms, community involvement and a move away from rote learning. However, something was different for these students.

Maximising the conditions for learning

Three significant changes were made to the learning circumstances of the target group of students.

• The project provided a mathematics teacher to work 0.4 time to facilitate the support program for the targeted students. In addition, the school reallocated existing resources to enable Colleen Morris, an Aboriginal and Islander Education Worker (AIEW) to work full-time on the project. She liaised with the community and was a mentor for the students and teacher, making use of her distinctive perspectives and knowledge.
  
  
• The close working partnership between the teacher and AIEW over an extended period of time allowed the teacher to learn much about the children and their community, and helped to develop strategies which were more effective for the group and for other Aboriginal students in the school. In turn, Colleen was able to observe, practise, discuss and reflect on the strategies used in teaching the children mathematics. Her confidence and skills in assisting students' numeracy development grew substantially and thus some of these strategies were instituted in the school's homework program for Aboriginal students. She also instigated professional development in mathematics for the other AIEWs in the school, insisting that numeracy should be added to the current focus on literacy. This was different.
  
  
• Colleen had a liaison role in informing and reporting student progress to parents, as well as encouraging parent participation in the program. She organised parent visits to the school to allow them to watch their children's work on maths, and coordinated a workshop for parents followed by a celebratory lunch for all concerned. Students had the opportunity to share what they had learnt with their parents, and the parents were shown how to use a set of cards, developed specially for these students - the basis of practice at home. As a result, parents clearly felt more able to participate in their children's learning. They requested the continuation of the program the following year and its extension to more students. This was different too.
  
  
• The program was designed so that the advantages which students gain from peer modelling in a mainstream classroom were balanced with the value of learning through one-to-one or small group discussions with a skilled adult. The model included features to counter problems sometimes generated by withdrawal programs. These differences became key elements which were vital to the success of the support program.

The Year 3 support program was very flexible. The students were considered as members of the mainstream class, not the withdrawal group. One, two, three or up to ten students were withdrawn during any, but not all, maths lessons. Withdrawal depended completely on students' needs at the time. Student progress was closely, individually and constantly monitored. Strategies were developed to overcome misconceptions as they were diagnosed or to assist further development at the point of need. Sometimes the extra support was used to help students who had been absent catch up.

Given the students' language difficulties and the varying teaching styles of individual teachers, it could not be assumed that the students in the target group could transfer what they had learnt in their small group back into the mainstream classroom without this high level of support. Integration of the students in the program into the mainstream class was a priority.

Students who were not part of the target group were sometimes included in the withdrawal sessions. The support teacher planned and worked closely with the mainstream Year 3 teachers. Ongoing teacher collaboration ensured that strategies which were developed for the withdrawal students were used in teaching the mainstream class. The teacher of the withdrawal group and the AIEW often taught in the mainstream classroom, modelling strategies for other teachers and allowing members of the withdrawal group to see all children 'learning what they had learnt'. On these occasions the students in the target group were often the 'more able'.

The students in the Year 4 group were working at a level well below that hoped for from students of this age. For the first semester, they were withdrawn from all daily mathematics lessons before being integrated back into the mainstream classroom using a similar model to that described for the Year 3 program. Again, individual student progress was constantly monitored and strategies developed to deal with specific difficulties as they arose.

The teaching and learning of the mathematics

For the students to come to view themselves as successful mathematicians they needed to feel in control and have ownership of their mathematics. At the beginning of the program the students were happy to complete many repetitive 'sums' rather than be challenged or think about new ideas. As long as the 'sums' were done, whether answers were copied or even wrong, they believed the maths was done. They became agitated if asked to solve a problem or to generalise an idea. They saw the sums as an end in themselves and viewed maths as a series of rote learnt facts.

While rote learning may provide immediate success, that success is usually only temporary, setting students up for failure when basic conceptual understanding is required. It also encourages the behaviours of students who already have the tendency to be ritual learners. The aim of the teaching program was to show students that maths is about ideas as much as it is about 'sums'. The mathematics was delivered in a way that allowed the students to work through understanding rather than by rote. It was important that the students were able to verbalise their thinking and to make generalisations about how the number system worked.

It was decided at the outset that an understanding of place value would allow students to access the ideas involved in calculations, operations and number patterning. Flexible use of place value and language development became the focal points of the teaching. If place value was to be the key to accessing other ideas of number then it was essential that students understood the concepts of counting, including skip counting, and partitioning numbers.

At the beginning of the program a particular student, typical of the students in the target group, could count forwards by ones to one hundred and by twos to twenty. She could not count backwards at all. This set the beginning point of the program for this student. The aim was for her to be able to generalise that 'If I can count forwards and backwards by twos I can already add two and subtract two. I also know numbers which are two more, two greater, two less or two smaller than other numbers'.

Although the student could count by twos to twenty, she could not answer such questions as: When you are counting by twos, what number do you say before twelve? What number is two more than eight?

Over a period of two months, this student received 145 minutes of individual help, in addition to coverage of the same ideas in the classroom. The extra time was spent considering terms such as 'before', 'after', 'more than', 'less than', and their relationship to counting. For example, a calculator was used to record the counting of objects by twos, and the connection between the next number on the display and adding two more to the group of objects was made explicit. The student was asked to commentate, using the words 'more than' and 'less than', on what was happening to the size of the collection of objects as the teacher removed two or added two to the group. After counting a collection of objects the student was asked to say how many were in the group as the teacher removed two or added two to the collection. (During that same two month period the student received a further 165 minutes of additional support to build her skills in and ideas about counting by fives and tens. This example illustrates the level of support required to make the difference for these children.)

The students in the target group were all ESL/D speakers. Each student needed much support to develop the language necessary to be able to make these generalisations about counting. But being able to make those generalisations was most empowering for them. Rather than believing counting by two was a process distinct from adding and subtracting two, the ability to see the relationships simplified the mathematics.

Often discussions about generalisations are reserved for more able students. In this program these discussions were used as a strategy to enable the students to become more able.

Students who are 'naturally mathematically able' make these connections for themselves and therefore view maths as making sense. 'Less mathematically able' students do not see the interrelationships and view maths as a plethora of isolated number facts. 'Less mathematically able' students can be led to see these relationships through explicit teaching. However, if students do not have the necessary language and cannot make the generalisations for themselves, they are denied access to these ideas and maths does not make sense. They resort to rote learning, construe incorrect rules for themselves to try to make sense of the numbers, or give up trying to understand.

The language of mathematics, in particular number, is sometimes thought of as a list of synonyms for the four operations. This language is essential, confirming that literacy is necessary for numeracy, but it is not enough to compensate for lack of understanding. Literacy alone will not ensure numeracy. The word 'subtract', for example, can be taught to be synonymous with 'take', 'less' and 'minus'. However, this is not very helpful when a student is asked a question such as 'If you had $57 and your friend had $34, how much more would you have?' Without adequate understanding of the subtraction operation the key word 'more' would probably trigger the student to add the two numbers.

We broadened the normal language of mathematics to include the language needed to discuss ideas. Conversations about how a student arrived at an answer, alternate methods of calculating an answer and their efficiency, and whether an answer was possible or made sense were used to help students to understand that maths is about ideas and not just about 'getting answers to sums'. The program incorporated opportunities for students to work individually or in small groups with the teacher or AIEW so that the students could be actively involved in these conversations.

The decision to help students progress in their understanding through conversation was confirmed during a lesson with the Year 4 group early in the program. The students were very angry when they came to their session. It became apparent that they had been teased. They believed that they were blessed with different, even inferior, brains to their non-Aboriginal peers. It seemed to them that non-Aboriginal children were born with answers to all questions already present in their minds, waiting to be called upon when the right question was asked. This fitted well with their disposition to be ritual learners.

To challenge the students to see themselves differently a 'brain check' was carried out on each child. Each child was asked a place value question, for example, 'How many tens and ones are there in thirty-seven?' Every student passed the first section of the brain check. Each child was then asked a basic addition or subtraction number fact. Every child passed the second section of the brain check. With both questions answered correctly it was jointly concluded that there was nothing wrong with their brains! They were working quite adequately.

The teacher then described thinking as putting little ideas together to make big ideas. She explained that to think about adding 42 and 23 in her head she just used the two little ideas which the students already knew and then put the ideas together. Firstly, she thought about the tens and ones in each number and then she did 'little sums'. She thought about the four tens and two tens which together gave six tens and then she thought about the two ones and three ones which added to give five ones. Six tens and five ones was the same as sixty-five. The teacher modelled the steps in a cartoon of her brain as she described the process.

This lesson marked the turning point between the students looking at mathematics as a mass of isolated facts and seeing it as a network of interconnecting ideas. The focus shifted from getting the answer to the thinking involved in reaching an answer. The students took turns to 'draw their brains' and explain to the group what was happening in their brains as they mentally processed a problem. The strategy was used to encourage students to verbalise their thinking, drawing on the relevant language. A deliberate emphasis was initially placed on mental calculation as it forced the students to use strategies which made sense to them and over which the students had control. After reaching the point where the students could orally describe their strategies they were asked to write down their ideas on paper. Written work, as informal algorithms, was therefore an account of their own thoughts not algorithms imposed by the teacher. The students came to view maths as making sense and being under their control. They gradually came to see themselves as successful mathematicians. Conversations in small groups with a skilled adult, in which everybody could actively participate, were vital for students to make this step.

A final example will demonstrate further that seemingly small differences from common practice in classrooms contributed significantly to the success of the program.

A decision was made not to use the Multibase Arithmetic Blocks (MAB) commonly used to model place value. For younger students these materials are used to demonstrate the equivalence of ten ones and one ten. Students who understand place value have the flexibility to move between both representations as the situation demands. For example, to add 46 and 23, the forty can be considered as four tens and the twenty as two tens. Each ten is thought of as one unit. However, to add 7 to 37 the situation can be managed by thinking about three tens and fourteen ones. To compute the answer, ten of the fourteen ones must be thought of as one ten. While MAB are used to model these ideas, the concrete aid itself embodies ten ones as one ten. For students to manipulate a ten as ten ones teachers introduce a system of trading or exchanging. This process involves much language to explain and understand. For students who have difficulty with numbers or do not have the necessary language, the concept of place value is actually made more confusing by the use of this aid.

We used plastic straws held together in bundles of ten by rubber bands. This concrete representation allowed flexible, easy and obvious conversions between one ten and ten ones. While the replacement of MAB by plastic straws may seem a trivial point, its significance cannot be underestimated if it removes a barrier to understanding for children who have difficulties with number concepts which are compounded or caused by a lack of Standard Australian English.

This short description of some of the teaching strategies used is an attempt to capture the approach taken to the teaching of maths in the program. But any one of these ideas in isolation is unlikely to bring about improved mathematics outcomes for students.

The aim of the program was to support students to make sense of mathematics and to have ownership of their thinking. The strategies used clearly demonstrate that empowerment to access real understanding of mathematics was dependent upon language, conversation and explicit teaching.

The project provides evidence to show that it is possible to improve numeracy outcomes of Indigenous students within the relatively short time span of less than one year. Although the project was small and took place at one isolated site, the findings could be considered by many schools across the country. But there are several matters to consider.

The main one is that a very high level of support was required. Any level of support below this would have been unlikely to have made such a significant and measurable difference. The success of the program was dependent on the combined effect and intensity of a series of factors used to support students in their maths learning. Each on its own may be commonplace; but they must be carefully integrated and used in combination for maximum effect. Finally, the success of the program was very much dependent on the approach taken to the teaching of the mathematics. It could be glibly suggested that it is possible for teachers and schools to reflect on their view of teaching mathematics and adopt changes to their teaching practice. However, it would be unreasonable to expect that significant change would or could occur without professional support for teachers.