Making sense of a trial maths intervention program for students with disability in Australia: Interim report

forms of teaching and learning advocated in the program increased students’ engage and performance. The adoption of reflective teacher portfolios demonstrated their usefulness for engaging teachers in appreciative inquiry and action research to monitor the implementation and impact of the program in their schools and classrooms.

against the OECD average of 1.6. Forty-five percent of Australians with disabilities live in, or near, the poverty line [9]; more than double the OECD average of 22 percent [6,8]. Their preparation in literacy and numeracy through formal education for post-school life in activities such as employment, vocational training or higher education is critical to moving towards an independent adult life and breaking the cycle of such disadvantage.
Tied to this complexity is research evidence reporting that teachers, including special education teachers, lack sufficient mathematics content knowledge and pedagogical content knowledge [10,11]. Currently in Australia, students with intellectual impairment and with additional disabilities struggle to learn essential concepts and skills at primary and secondary levels of schooling. Whilst there is a strong commitment from teachers to support students with learning mathematics, their preparation and capacity to teach it is of current concern [12]. Research indicates that many have a poor understanding of teaching and learning for specific content areas, resulting in an overemphasis on procedural and low level skills and limited use of multi-modal and multi-sensory instruction that more fully engages students' development of conceptual understandings of use of mathematics ideas, equipment and materials [12,13]. This concern provides the context for the mathematics intervention program used in the study. Whilst there are interventions widely endorsed for reading there is much less evidence for mathematics although interest is gaining in mathematics [14].

The Intervention Program
The intervention program was designed for teachers (Preparatory to Year 12) who teach students underperforming in mathematics in special education schools in Queensland. Titled Yumi Deadly Maths, it was originally developed by a team of researchers at the Queensland University of Technology (QUT) [15][16][17][18] and focused on schools with high Aboriginal and Torres Strait Islander students who were identified by their classroom teachers as underperforming in mathematics. More recently the program has been used in classrooms in special schools in Queensland and Victoria. Funding for the program that is the basis of this report was derived from twelve participating schools through the State Schools Queensland, Great Results Guarantee, a four-year funding initiative to improve student outcomes [19].
This program passes on to the schools, funding provided by the Australian Government's Students First initiative [20]. The program is underpinned by Payne and Rathmell's [21] theory of mathematics learning and Bruner's [22] three modes of representation (enactive, iconic and symbolic) both are represented through a four phased instructional cycle, reality, abstraction, mathematics and reflection (RAMR).
The RAMR instructional cycle has four phases of learning. Each phase builds on from The phases are interconnected and not viewed as discrete and isolated throughout instruction. Teachers' and students' explicit connections from one phase to another are essential for learning concepts and skills. Without this awareness students, are likely to feel as though they are memorising isolated procedures that have little connection to what they are learning.
There are several benefits for teachers who use the instructional cycle for teaching mathematics to students with disability. First, the RAMR instructional cycle provides multimodal forms of learning and opportunities for students to see their realities of mathematics in everyday life, orienting themselves to those ideas and the context from which they arise. These forms of learning include seeing, hearing, touching and muscle movement-visual, auditory, kinaesthetic, and tactile learning aids memory and retrieval skills [13,23]. Second, students with disability and those who struggle because of other factors have multiple characteristics that affect their ability to learn mathematics. These characteristics include impulsivity, language deficits, hyperactivity and lack of prior knowledge, memory difficulties and motivation problems. They create the need for connecting the importance of content to everyday life to increase motivation [12]. Third, body movement and manipulation of materials in the reality and abstraction phases allows students to represent their reality using their hands, body and mind, materials, symbols and language in a range of ways to create meaning [21]. These phases allow students to recognise new experiences as having the similarities of an already formed experience [24]. Fourth, through this process, the construction of knowledge and meaning making becomes a necessary condition for mathematics learning [25]. Finally, the setting of problems back in reality enables students to validate, justify and generalise their own knowledge so that they can extend on ideas.

Methodology
The project adopted a collaborative approach which has horizontal and vertical dimensions. The vertical dimension involved exchanges of views at different levels, e.g., between twelve schools and principals and forty-eight teachers. The project facilitated the engagement of two teachers to actively champion and promote the project across the twelve schools. Through regular communicative tools such as email, telephone conferences and intensive face-to-face PL meetings, the project team and participants, regularly collaborated across all aspects of the project. These strategies contributed to the "population of values" [26], influencing participants and other teachers horizontally within schools. Given the substantial significance of the issue that this project aimed to address, it promoted ways for bringing people together to participate in organisational learning and change, knowledge sharing and making sense of impact. It advocated considerable dialogue about whether the proposed change was sustainable, who benefited and, would other supporters of the project like it-all of these elements personified views about priority values.
The project adopted an appreciative inquiry (AI) approach to monitoring and evaluating impact and interconnects with the change process for schools, principals and teachers [3,27].
AI has been identified as a reconfiguration of action research within organisational settings such as schools. It is described as a strategic planning model, participatory and a systemwide approach that seeks to discover what works based on solutions that exist currently within organisations such as the schools.

Participating schools, principals and teachers
The twelve participating schools were from regional and metropolitan areas of Queensland. The participants in the project included twelve Principals and forty-eight teachers. Principals were required to attend the first day of workshops in 2014 and 2015; this was to ensure that they were fully aware of the program, its intentions and purpose. As they were responsible for distributing the funding the program, building their capacity about it and the change processes involved, including the demands of teachers, were important to its successful implementation. The Principals, in conjunction with individual schools' leadership teams selected four highly motivated teachers from a range of school year levels to participate and lead the program in their schools. Two project champions were nominated by the Principals' leadership team to guide the overarching implementation of the project.

The professional learning program and data collection strategies
There were several elements to the professional learning (PL) program, including six days over two years of PL workshop attendance, school visits, resource provision and action research support. The timeline and strand focus (Table 1)   The questionnaire focused on demographic information, for example, formal qualifications, students' mathematics areas of difficulty. The survey focused on practices used in classrooms.
Schools were provided with 1 x 1-day visit per year (2 days across 2 years) to support teachers through lesson modelling, observations and critique of practice to develop knowledge and understanding of the learning needs of students with disabilities. Participating schools were provided with supporting documents about the approach to teaching mathematics and resources on how to implement this approach.
Participants were trained in action research and inquiry to monitor their progress with the implementation of the program. To do this, participants were workshopped on how to monitor their activities in their schools and gather data by way of a reflective portfolio that contained: (a) teaching plans (RAMR cycle), (b) student pre-post test results, and (c) analysis of teaching plans and student results.

Analysis: Making sense of the intervening program's impact
Building the capacity of the schools and teachers in relation to teaching mathematics to their students was central to the project. Because of the scope of the project, the analysis will draw on elements of the questionnaire and examples from the portfolios.

The Questionnaire
A questionnaire was administered to participants at the Round 1 workshops (N=60) with 93 percent response rate (n=56) so as to build knowledge about the participants. A range of questions focused on demography and identifying background variables including diversity of school community, teacher qualifications, number of years at previous/present school and views about teaching and student learning.
Students attending the participating special schools included students with intellectual impairment and with additional disabilities (Autistic Spectrum Disorder, Physical Impairment, Hearing Impairment and/ or vision Impairment). Their needs were identified by the teachers as complex and therefore were not always met through procedural approaches to teaching mathematics. Coupled with this complexity was the range of cultural and social groupings. Of significance was that nearly 72 percent of the student population of the 12 schools were low socioeconomic, English as a Second Language and refugee. The challenges that the schools faced in responding to such diversity included trying to support students who experienced multiple and cumulative disadvantages because of their disability and belonging to a number of disadvantaged groups.
The results from the teacher questionnaire indicated that the strand of Number was strongly identified as an area of student difficulty. Of significance was the large percentage attributed to Number (n=68%) (identified by participants as before and after, more than/less than, trust the count, value of numbers, number formation, teen numbers, place value, renaming/regrouping). If we include "other" with this portion, nearly 97 percent of difficulties are associated with this topic (and including language, multistep problems, generalisation, abstract ideas, conservation and comparing). This information is particularly significant for the project and teachers and highlights a range of issues for such a diverse group of learners. It creates a challenging situation for their teachers and their expertise with how to address the issues. Of the participants who completed the questionnaire (n=56), figure   1 documents the highest university qualifications gained.
Participants were asked about the their highest university qualification, specialisations, year of completion, perceptions about mathematics, approach to teaching mathematics, role in school for implementing PL program and students mathematical areas of difficulty. Of significance is that of the participants who completed the questionnaire 98% have a university qualification ranging from Bachelor of Education to a Graduate Diploma and or Certificate. Of interest 35% identified as having a qualification in the special education field, for example, Autism studies.

Figure 1. Highest university qualifications of participants
Current data from labour market research on school teachers [29, p. [16][17] reports that 48% of teachers of special needs in primary school had less than one year of tertiary study in this area. School employers identified that it was difficult to attract teachers with relevant experience in special schools but were willing to compromise to fill their positions with many hiring learning support teachers instead of special education teachers. Reasons applicants were unsuitable included that they did not have specific special education qualifications and lacked experience teaching students with special needs or a particular learning disorder such as autism.

The Survey
A Likert scale survey was administered in 2014 (n=36). It asked several questions related to teachers' practice. A value (1 strongly disagree to 5 strongly agree) were assigned to each response allowing for reporting a single average for each response. Table 3 shows the results of the survey. The results captured a range of item difficulties including critical reflection in mathematics lessons and connecting realistic experiences to abstract language symbolic language. There are reasons for why this might be the case including that student characteristics such as impulsivity, language difficulties, hyperactivity, lack of prior knowledge, memory difficulties and motivation problems may impact on their capacity to reflect on their learning.
Most of the items in the survey were identified as easier to endorse. For example, there was a degree of agreement with statements 2, 3 and 8. Statement 2 (4.1) which focused on the link to reality suggests that the teachers recognised the importance of developing students' awareness of maths ideas in the local environment. Statement 3 (4.4) highlights the importance of providing verbal and non-verbal feedback to students. Feedback is critical to the teaching and learning process [30]. It leads to students recognising their next steps and how to take them. It is underpinned by confidence that every student can improve and it involves every teacher and student reviewing and reflecting on the teaching and learning.
These characteristics contrast with assessment that simply tests procedures.

Teachers as researchers -reflective portfolios
Of the total number of teacher participants in 2014 48 forty-eight portfolios The project strongly believes that the successful implementation of the program in the participating schools required participants, as "teacher researchers", to actively engage in conducting collaborative research to learn about their practice, the teaching of mathematics and how students learn from that teaching. As the teachers' trialled ideas with other teachers and their students, their efficacy was shown to increase. Efficacy was a critical factor that emerged from the portfolio data as playing a key role in the process of implementing, trialling and changing their practice.
Efficacy is described in various ways including; the motivation that teachers expended on effort to implement the program and a willingness to set challenging goals and the persistence to see them through. Such ways were evident in the portfolios and influenced their determination and adoption of new approaches to teaching by increasing their willingness to take risks and persist with difficulties and setbacks that came with the implementation and change process [31]. Reflection was a critical element as it was through this process that major themes in the portfolio were identified.
Overarching themes The adoption of reflective portfolios as a research strategy in the project aimed to engage teachers in their own learning and reflective practice as well as that of students.
Through this process, teachers could trial new ways of teaching as well as create new professional learning collaborations and believe that they could perform instructionally related tasks that were likely to bring about increased student learning. There were ten overarching themes and relationships identified in the portfolios (Table 4). Identified steps to implementing the program The RAMR instructional cycle was not designed as a guarantee that schools with students with disability would achieve accuracy in all areas of the mathematics curriculum.
Nor was it a guarantee that once teachers were professionally developed in the program they would be successful in their mathematics instruction. It did however provide teachers with an instructional cycle for incorporating the use of multimodal and multi-sensory forms of teaching and learning. The following seven steps were identified in the portfolios, providing an example of how the program was implemented.
1. identify the mathematics concept to be taught and learned;

identify what comes before, what comes after and what connects;
3. identify what local knowledge and previous experiences students have had with the concept and draw students' attention to it using visual, auditory, kinaesthetic and tactile activities; 4. match the concept from students' knowledge and experiences with appropriate multi-sensory experiences, materials and language so that representations can be created using the body hands and mind; 5. match the conceptual representations with formal symbols, signs and language; 6. practice with students and make connections to other maths concepts and student experiences; 7. assist students with applying knowledge to other areas and validating their knowledge.
The steps were found to contribute to how the teachers worked through a sequence to ensure they were addressing key components of the program and adopting multi-modal and multi-sensory forms of teaching and learning which provided flexibility and transferability across the mathematics curriculum. This outcome supports the findings from international research which evidenced improvements in mathematics achievement of students with disability through these forms of teaching and learning [32][33][34]. The portfolios showed that the teachers could design instructional sequences of mathematics lessons that engaged their students in ways they had never experienced before. An overview of one teacher's, Talina, RAMR planning is shown in figure 2. To identify what place value knowledge the students had she tested the students using a Cognitive Diagnostic Assessment [17]. She reported that overall the students did not achieve well and demonstrated substantial weaknesses in their place value understandings as can be seen in the examples in Table 5 below. Table 5. Examples of students' responses The intention of these questions was to specifically identify if students could rename 2digit numbers in both directions (number -tens and ones; tens and ones -number). Talina reported that they could not. She then designed a teaching sequence that focus on renaming as part of place value learning. This sequence is presented graphically in figure 3.

Figure 3. Example of renaming in place value
In this example Talina adopted a developmental approach to supporting students with learning about place value, in particular renaming. In this instance she drew on pasta (and small plastic bags) which was familiar to the students, captured their attention and provided opportunities for them to represent their developing knowledge and understanding about renaming. A significant outcome from using this approach was that students could represent their thinking as shown in figure 4.

Reality: Counting
As students could already count in 1s to 100 and back, this term we were going to focus on counting on and back in 2s, 5s, and 10s from any starting point.
Students participated in games such as 'leap frog' and 'kangaroo hop' that required them to jump over an object.

Abstraction
Students were introduced to the blank number grid. They practiced stepping or 'hopping' in every second grid space, ensuring that when they got to the end of the grid, they went back down to the other end and started along a new row.
Students then practiced stepping in every second grid space, dropping a counter in each space as they went along.

Mathematics
Appropriation: Students were asked to place the number symbols and language on the number grid. Practice: Just as students had practiced skipping along every second space on the blank number grid, they now practiced skipping along every second space on the grid, saying the number as they landed on it. Students were asked to relate the numeral to the position they were standing in along the number grid.

Reflection
Students practiced skip counting in twos across the duration of the term. They became very proficient in this skill. The fives and tens counting sequence was then introduced, using the same activities as outlined for the 2s sequence. The threes sequence was introduced for two students.

Teacher reflection on cycle
Students were able to reach the point of being able to count on and back in 2s from any starting point, some independently, some with assistance. Some students also had a go at counting in twos in odd numbers. Students also achieved well with the 10s counting sequence. Approximately half of students have progressed with the 5s sequence, however more practice is needed. Two students have progressed to learning the 3s sequence.
Katrina's overall reflection about the implementation of the program was documented in two levels: 1) engagement and 2) confidence.

Engagement levels
Students were found to be highly engaged and enthusiastic about maths activities. Attention spans and levels of disengagement improved. Some of our students with Autism found group work on the floor, away from their desks a little difficult so visual symbols of tasks, "first this, then this" cards, and reward systems were put in place.

Confidence levels
Students were so excited to see their progress, and to know that at the beginning of the term they could only do 'this', but now at the end of the term, they could do 'THIS!' Individual learning goal charts were created for each student. Each time a student demonstrated the knowledge that we were aiming to gain, they would get a stamp on their chart. A filled up stamp chart= a reward. engagement and confidence of her students in their learning. For example the phrases, "highly engaged and enthusiastic", "so excited to see their progress", "they could do THIS!" worked to evaluate learning as desirable. They also worked to demonstrate possibilities for learning-talking about mathematics with students. Further the statements reveal the impact of the program on Katrina's planning for teaching of number to the students. Of significance was the statement, "attention spans and levels of disengagement improved". Here, the issue might not be so much about engagement and disengagement but rather about supporting students with learning how to talk and communicate mathematically in multi-sensory and multi-modal ways that closely match the classroom community. Over time, students are more likely to produce skilled talk and gain validation from Katrina and their peers as they demonstrate their understanding.
The data from the portfolios indicate that the experience of moving the PL to the school, classroom and students allowed the teachers to reconnect with their professional expertise. Clemans [35] reinforces this aspect by stating that this process facilitates the teachers making connections between their teaching, student learning in the classroom and the staffroom. It encourages the teachers to build on their professional strengths and practices that they use to engage learners and share these with their colleagues. Excerpts from the portfolios express these connections.
Staff and Students found this topic fun and exciting. Also found that since students enjoyed it so much, they seemed to understand concepts quickly. As Karolina made the connections between her teaching, other staff and the students' learning, she will be more readily able to "see" in the future how with increasing efficacy she will be able to weigh her perceptions of her teaching and personal competence. Past successes will strongly influence her expectations about herself, her staff and students as they learning about mathematics. Most students were unable to complete the Schedule Early Numeracy Assessment (SENA).
QCIA assessment was more successful as it allowed students to be assessed by using observation of hands on activities (anecdotal records, photographs and video) rather than pen-paper assessment.
RAMR cycle was helpful as a planning tool. All students were able to access the 'reality' and 'abstraction' phases. Some students had difficulty moving into the 'maths' phase.
All students had difficulty moving from body-> hand-> mind activities. All students needed to continually participate in body activities to meet their sensory needs. Some students experienced anxiety when being asked to complete worksheets or written activities and these needed to be modified to include a body activity.
Students had difficulty retaining information learned from one day to the next. Point in time assessment was unable to be used, rather students needed to show that they could complete the task on several different occasions to show that they had retained what they had learnt.
All students had difficulty generalizing the skills learnt. All lessons needed to be repeated in different locations and with different staff (both familiar and unfamiliar).

Continue our committee fortnightly meetings.
To assist with Diagnostic testing as such we will be trialling for the rest of 2015 using the Numeracy Indicators. This will include a comments section that will include the level of support the student requires to complete the task is recorded. There will be a colour code key which will include baseline data and then different colours for when the student progresses.
The four of us who participated in the project will become support staff for different areas of the school. (Portfolio, teacher, Shakira) And finally, Shakira is confronted with the profound issues related to assessment for students with disability which fundamentally shaped the direction she would take with assessing student learning and planning for her teaching. She does appear to have confused the role of the SENA [see 36] as she implies that it is a pen and paper test, rather it is a diagnostic interview schedule. The importance of teachers like Shakira transitioning the knowledge and experience from the PL to her school, the staff and students was integral to the ways in which she would implement the program. She learnt about her role and the purpose of her work and developed effective approaches for collective transformations at her school.

Discussion and conclusion
This interim reports has only focused on "snapshots" from the project to demonstrate its position about the achievement of students with disability and what can be achieved.
Currently, there is limited research about students with disability, mathematics, multi-sensory and multimodal forms of learning and instructional pedagogy. Intervention studies of mathematics have focused on explicit instruction and concrete, abstract sequences of instruction [23,37], but the literature is largely silent on the prior and existing knowledge and experiences of students with disability, how teachers can build on from that knowledge and experience and why this process is crucial to students' development and teachers' instructional strategies.
The preliminary evidence provided in the portfolios and excerpts in this report show that schools are actively participating and engaging in the project's implementation. The research evidence strongly indicates the increasing efficacy of teachers to take risks, test their hunches and ideas and collaboratively examine their work as well as the work of their colleagues and students. As a consequence of the project there are several implications for consideration.
It is highly likely that teachers are using a range of assessment strategies to assess student learning, mainly in number; however, whilst this is positive, for students who require a highly individualised curriculum and for whom intellectual disability is significant and requires extensive adjustments that are comprehensive and ongoing, there exists no suggested means of assessing what students know and are able to do in the large area of mathematics.
This aspect is particularly evident in the portfolios. Furthermore, there currently exists no literature nor policy recommendation that advises schools as to the feasibility of administering diagnostic assessments in mathematics. What is known is that in current practice, schools take standard diagnostic assessments, intended for learners without intellectual disability, and individually modify these to ascertain student performance and to inform practice. Modifications are made on a teacher-by-teacher basis and often result in inconclusive results, leaving teachers to make assumptions about student performance. There is a considerable amount of work to be done in this area to ensure that teachers are better able to document student progress.
Efforts over the past decade to improve schools' mathematics performance have placed greater emphasis on students with disability to complete more challenging level. Given these expectations and the continued achievement difficulties students with disability experience, there is a need for special education teachers in the project to continue to build their repertoire of instructional strategies to assist students in their learning. The combination of wide-ranging deficits in foundational mathematics knowledge, experiences and skills and the pressure to increase student performance in the subject places students with disability at greater risk for failure unless specially designed instruction and resources are provided by their teachers.
In a new era where so much more is to be learned about how best to support special education teachers with teaching mathematics to students with disabilities, this interim report argues based on the PL, lesson modelling, questionnaire and portfolios submitted after Round 1 of the workshops in 2014, that a teacher who consistently exhibits a willingness to set challenges, persist with seeing them through and adopt new approaches to teaching, is highly likely to be effective with implementing the project more extensively and in doing so bring about successful for their students.