4.3 Australia
Introduction
In terms of international mathematics assessments, Australia’s achievement on the PISA 2009 assessment is slightly lower than Ontario (i.e. Australia - 514; Canada - 527, Ontario – 526) (EQAO, 2010; OECD, 2010). They also score somewhat lower on the TIMSS 2011 Grade 8 assessment (Australia - 505, Ontario - 512) though their achievement on the TIMSS 2011 Grade 4 assessment is very close to that of Ontario (i.e. Australia - 516; Ontario – 518) (Mullis et. al. 2012). Overall,Australia’sperformance on international assessments can be described as comparable to Ontario, which is one factor that suggests Australia is a suitable comparison jurisdiction. However, the main reason we chose to include Australia as a comparison jurisdiction is that the states of states of New South Wales (NSW) and Victoria have both developed a type of learning progression or developmental continuum for mathematics in recent years.
New South Wales has a mathematics continuum for grades K through 10 thattakes the form of a poster summarizing the key ideas in mathematics for “Number”, “Patterns and Algebra”, “Data”, “Measurement”, “Space and Geometry” as well as a category called “Working Mathematically” (NSW, 2010). The continuum is presented as a table detailing the mathematics expectations within each strand or sub-strand across the grades. The contents of the table are aligned with the NSW curriculum and the poster is intended to be displayed in classrooms and staffrooms so that teachers, students and parents can see “the big picture of mathematics learning” across the elementary and secondary grades (NSW, 2010). The NSW continuum is similar to the mathematics learning progression for Quebec in that it is essentially a scope and sequence document. For this report, we have chosen to focus on the “Mathematics Developmental Continuum P-10” for Victoria rather than the NSW continuum because Victoria’s continuumincludes elements that go beyondscope and sequence.
Description of Victoria’s Mathematics Developmental Continuum
The Mathematics Developmental Continuum P-10 for Victoria is presented as a website of inter-related links that address five dimensions: “Number”; “Space”; “Measurement, Chance and Data”; “Structure”; and“Working Mathematically” (Victoria Department of Education and Early Childhood Development [VDEECD], n.d.)check ref. The continuum spans grades from “P” the “Preparation” year (age x years) through Year 10 (age y years). For each dimension, the continuum is presented as a two-column table. The first column lists the “Mathematics Standards and Progression Points”, which are aligned with the mathematics curriculum for Victoria. The second column provides “Indicators of Progress” which are points on the learning continuum that highlight the critical understandings students must have in order to progress in their mathematical learning. Indicators of Progress are intended to highlight common misconceptions of students and to support teachers in deepening their understanding of student learning in mathematics by providing“research-based descriptors of achievement”[need to find quote & ref]. The Indicators of Progress give teachers a sense of the progress students should be making and the types of learning and teaching experiences that are appropriate for further progress(cite users guide here).Table x shows the number of Indicators of Progress and the span of grade levels for each dimension in the continuum. Reviewing the table provides a sense of the grain size maybe mention grain size in introof this developmental continuum. For instance, there are 19 Indicators of Progress associated with “Space” across the span of mathematics instruction from P to Grade 10.
Dimension / # Indicators of Progress / *Levels CoveredNumber / 41 / Level 1 (K & Grade 1) to Level 6 (Grades 9 & 10)
Space / 19 / Level 1 (K & Grade 1) to Level 6 (Grades 9 & 10)
Measurement, Chance and Data / 25 / Level 1 (K & Grade 1) to Level 6 (Grades 9 & 10)
Structure / 16 / Level 3 (Grade 3 &4) to Level 6 (Grades 9 & 10)
Working Mathematically / 14 / Level 1 (K & Grade 1) to Level 6 (Grades 9 & 10)
* each level in the curriculum (VELS) and the continuum includes two grades
For each Indicator of Progress links are provided to “Illustrations”, “Developmental Overviews” and “Teaching Strategies and Activities”. “Illustrations” are suggested focused observations and diagnostic tasks intended to assist the teacher in determining where students are in their mathematics thinking and to help them uncover misconceptions. Developmental Overviews map student progress in key concepts through all levels of learning. There are ten developmental overviews which cover various “big ideas” in mathematics including: “Proportional Reasoning and Multiplicative Thinking”, “Numbers and Operations” and “Methods of Calculation” addref here. Teaching Strategies and Activities are suggested tasks teachers can use to support students in developing a conceptual understanding of the progression point. We have provided a few selected screen shots of the Indicators of Progress, Illustrations, Developmental Overviews, and Teaching Strategies and Activities in Appendix x) .
Another interesting feature of Victoria’s developmental continuum is the Mathematics Online Interview ( a secure part of the website that can be accessed only by school staff. The Mathematics Online Interview is an assessment tool that can be used by teachers to determine students’ existing mathematical knowledge in relation to progression points. Analysis of the responses provided during a one-to-one interviewprocess are intended to provide teachers with information to use when planning to meet student’s learning needs. While we could not access the interview process itself, the descriptions of the tool suggest that this may be a valuable resource for teachers.
Relationship between Curriculum and Learning Trajectories
Australia has a new national mathematics curriculum as well as a detailed scope and sequence document for mathematics which covers the foundation year to year 10 [ages for these years] (ACARA, 2013). Australia also has a “General Capabilities in the Australian Curriculum” document (2013) which is a document that details seven general capabilities similar to the 21st century learning skills documents that are appearing in many educational jurisdictions. One of the general capabilities in this document is “Numeracy” and there is a learning continua for numeracy included in the document. This learning continua is entirely separate from Victoria’s developmental continuum but it suggests that presenting information using a continua approach is viewed as worthwhile in Australia.
The state of Victoria has recently released their new curriculum document known as AusVELS (VCAA, 2013). AusVELS goes from Foundation to Year 10 and closely parallels the new national curriculum. For each year/grade level, the AusVELS curriculum presents content expectations for each mathematics strand and sub-strand along with achievement standards. Victoria also has a scope and sequence chart that is very similar to the national scope and sequence chart.AusVELSreplaced the previouscurriculum which was called VELS (Victorian Essential Learning Standards). AusVELSdoes not explicitly refer to the Mathematics Developmental Continuum P-10 perhaps because the continuum was developed when VELS was still in place(K. Stacey, personal communication, May 21, 2013). In fact, the “Mathematics Standards and Progression Points” in the developmental continuum are aligned with the VELS curriculum rather thanthe AusVELS document.Thus, the connection between the developmental continuum and AusVELSis a bit indirect. In any case, the developmental continuum is seen more as a teacher resource than as a curriculum document (K. Stacey, personal communication, May 21, 2013).
Research Base
As we reviewed each of the 115 Indicators of Progressinthe developmental continuum, we noticed that for some indicators references are provided either at the bottom of the web-page for the indicator or through a “See more about” link. However, following the links and examining the references for each Indicator of Progress revealed that many of the references cite general resources for teaching primary mathematics or identify the source of a specific activity recommended for teaching the progression point. In fact, very few of the references associated with each Indicator of Progress areto empirical, peer-reviewed research. Moreover, in the few instances where an Indicator of Progress is supported by peer-reviewed research, there is typically only one paper cited.
We contacted Kaye Stacey,a mathematics education researcher and professor at the University of Melbourne, who was been involved in the development of the developmental continuum to learn more about the research the continuum is based on. She confirmed that some research on Number was gathered in New South Wales and later drawn on by the state of Victoria for that dimension of the continuum. However, the other areas of the continuum reflect the mathematics educationresearchers’existing knowledge but the researchers involved did not conduct empirical studies of their own nor do they make explicit connections to studies done by others(K. Stacey, personal communication, May 21, 2013).
Challenges and Issues
One issueusing learning trajectory research to inform the curriculum development process for Australia is that the curriculum is written in one or two year stages whereas developmental trajectory research is often more fine-grained than this (Stacy, personal communication, May 21, 2013). This difference in grain size may make it more difficult to bring learning trajectories research into the curriculum realm in any jurisdiction. We also suspect that it was not feasible for the state of Victoria to provide the funding that mathematics education researchers would require to conduct empirical studies as the basis of each Indicator of Progress in the developmental continuum (maybe cite the email?).
Implications for Assessment
In general, the Indicators of Progress in the developmental continuum are intended to be part of the ongoing assessment practice of teachers (cite users guide).In addition, as noted above, the developmental continuum includes two specific tools that are intended to support teachers in their formative assessment activities. The “Illustrations” provided for each Indicator of Progress suggest observations and diagnostic tasks to assist teachers with determining where students are in their mathematics thinking and to help them identifystudents’ misconceptions. The Mathematics Online Interview provides another tool teachers may use to determine students’ existing mathematical knowledge in relation to an Indicator of Progress. Providing these formative assessment tools within the developmental continuummay help to ensure that assessment activities are integrated with teaching and learning activities (assessment as and for learning) rather than assessment being seen as a separate activity conducted after learning has taken place (assessment of learning).
Professional Resources for Use of the Developmental Continuum
The website where the developmental continuum is made available also includes a “User’s Guide” to provide teachers with information about the design of the continuum as well as suggestions for effective use of the continuum for planning purposes. The User’s Guide includes sections on navigating the continuum and on learning and teaching using the continuum. We did not see any evidence of professional development sessions focused on the developmental continuum though these may be available.
Comments
Victoria’s developmental continuum for mathematics goes a bit beyond a scope and sequence approach in that it provides Illustrations, Developmental Overviews, Teaching Strategies and an online interview tool. Both our review of online materials and literature and our personal communication with Kaye Stacey suggest that there is a limited research base for a few of the Indicators of Progress but no empirical studies for many of the Indicators of Progress. We acknowledge that the resources provided to teachers as part of the developmental curriculum are likely to be quite useful for teachers implementing the AusVELS curriculum despite their being based on the older VELS curriculum. We also see the emphasis on formative assessment as beneficial. However, we see a need for more empirical research for most of the Indicators of Progress before using this resource to inform the curriculum revision in other jurisdictions.
References to add:
EQAO reference was created for the Quebec text
OECD (2010)reference needed (used for PISA scores for some countries)
Victoria Curriculum and Assessment Authority (VCAA). (2013) The AusVELS curriculum. Retrieved from:
Victoria Curriculum and Assessment Authority (VCAA). (n.d.). The Australian curriculum in Victoria. Retrieved from:
Victoria Curriculum and Assessment Authority (VCAA). (2012) The Australian curriculum: Mathematics scope and sequence. Retrieved from
New South Wales (NSW) Department of Education and Training (2010) K-10 Mathematics continuum [Poster]. Retrieved at