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Submission to the National Numeracy Review

The Mathematics Education Research Group of Australasia has a membership of about 550 researchers from across Australia and New Zealand, as well as international and institutional members. MERGA aims:

to promote, share, disseminate, and co-operate on quality research on mathematics education for all levels particularly in Australasia;

to provide permanent means for sharing of research results and concerns among all members through regular publications and conferences;

to seek means of implementing research findings at all decision levels to the teaching of mathematics and to the preparation of teachers of mathematics; and

to maintain liaison with other organisations with similar interests in mathematics education or educational research.

MERGA is well placed to provide the evidence base sought by this review because the primary role of its members is carrying out and supporting research into the teaching and learning of mathematics as well as mathematics teacher education and development. To this end, this submission draws mainly on research completed by MERGA members.We are pleased to have the opportunity to make a submission to the National Numeracy Review and appreciate the work that has gone into the background and discussion papers, and the expertise of colleagues in the review committee. However, we note the absence of a tertiary mathematics educator and request that this is attended to in the appointment of the writing team. Given the evolving nature of current tertiary mathematics studies, changing needs in the professions, changes in technology and its use, and increasingly broader university entry cohorts, the time is right for secondary-tertiary collaboration on the way forward for mathematics education.We are also concerned that there have been many national reviews — the most recent being The National Strategic Review of Mathematical Sciences Research in Australia held in Canberra in December 2006 — with little action resulting from these reviews. We draw the writing team’s attention to a recent report by the Australian Council of Deans of Science about The Preparation of Mathematics Teachers in Australia (Harris & Jensz, 2007); the major DEST publication Numeracy, a Priority for All: Challenges for Australian Schools (Commonwealth of Australia 2000), and the DEST-funded Primary Numeracy: A mapping, review and analysis of Australian research in numeracy learning at the primary school level (Groves, Mousley & Forgasz, 2004, 2005). The DEST-funded Science, ICT and Mathematics Education in Rural and Regional Australia: Report from the SiMERR National Survey (Lyons, Cooksey, Panizzon, Parnell, & Pegg, 2006) surveyed teachers and parents throughout Australia on key issues contributing to the poor performance of rural and regional students, and offered 23 recommendations. The Review of Teaching and Teacher Education (Kwong Lee Dow et al, 2003) resulted in a published, “action plan”. There are also similar reviews that have been held recently in other countries, such as New Zealand’s Effective pedagogy in Mathematics/Pangarau: Best evidence synthesis iteration (Anthony & Walshaw, 2007), and a lot that Australia can learn from elsewhere.MERGA members have made significant contributions to all of the documents listed above as well as similar productions and discipline reviews in the past. Each of the resulting documents summarises relevant research-based evidence, includes recommendations that could be attended to, and provides answers to some of the questions asked in this current review’s Discussion Paper.

MERGA recommends that the writing team include recommendations made in previous national numeracy reviews, and reports progress made so far against these.

Summary of the MERGA submission1. Numeracy and Mathematics in AustraliaThere is little point arguing the difference between numeracy and mathematics. “Numeracy” does not suggest higher-level thinking and “teachers of numeracy and mathematics” implies teaching numeracy without teaching mathematics! Focus on the development of students’ mathematical knowledge, not on “parallel” or “diverging” streams of mathematics and numeracy.While Australia generally performs well in problem solving and mathematical literacy, many MERGA members believe that there is not enough emphasis put on the development of higher-level thinking skills in mathematics classrooms. The shortage of well-qualified mathematics teachers in secondary schools is a serious, on-going problem that is currently no being attended to well enough at state or national levels.

2. School organisation and support structuresStructure is not as important as leadership, planning processes, high teacher expectations and curriculum. MERGA members do not support ability grouping, but flexible and purposeful grouping is supported. There is agreement with differentiating curriculum in upper secondary area, but concern that many students at this level are not challenged. At primary and lower secondary levels, it is possible to cater for a range of students with the whole class working on the same topic and level using strategies such as open tasks and developing communities of inquiry. Use of simple, scientific, and graphic calculator use has implications for curriculum development and curriculum processes, and teacher development. Likewise, effective computer use can clearly enhanced children’s learning of mathematics. However, students in different states experience very different opportunities to learn how to use graphic calculators and computer algebra.Algebra must be recognised nationally as a strand in the curriculum framework from K-12. At the middle school level, number work could better prepare students for formal algebra.

3. Effective Teaching PracticesQuality experience with mathematics at all levels includes a focus on student learning, a challenging curriculum, high expectations of all students, mathematical investigations, problem-solving.Areas of current concern include many Indigenous students, vulnerable students, students with disabilities, and rural students. There is also concern amongst MERGA members about broad-based state and national testing that is proving counterproductive. We report how improvements in teachers’ knowledge have resulted from major research projects on assessment.

4. Teacher Education and Professional DevelopmentMERGA recommends that the AAMT Standards be made a basis nationally for the initial accreditation and higher-level reaccreditation of teachers of mathematics. We summarise recent research related to effective professional development and maintain that teachers’ need to be able to take ownership over their own professional development. One concern shared amongst pre-service teacher educators is the difficulty of translating theory into their teaching practice. The practicum is an issue for remote, rural and lower SES areas in particular.

Signatory and contact personJudy MousleyMERGA PresidentFaculty of EducationDeakin UniversityGeelong, Vic. 3217

Phone: (03) 52272661Fax: (03) 52272014Email: judith.mousley@deakin.edu.au

MERGA recommends: that the writing team include recommendations made in previous national numeracy

reviews, and reports progress against these. that COAG abandons the term ‘numeracy’ and emphasises the breadth of mathematical

knowledge and skills required for personal, professional, civic and academic activity. that curriculum developers are directed to put emphasis on learning to think and work

mathematically at all levels of schooling. that national and state departments stop “passing the buck” and take positive steps to

addressing the need for well-qualified teachers of mathematics. the use of flexible, purposeful grouping of students, as appropriate for the task and types of

interaction sought, in mathematics classes. that curriculum be differentiated at upper secondary levels but not in primary of junior

secondary classrooms, except through the use of strategies that allow all students to engage fully with class tasks.

that the value of appropriate and well guided technology use for the teaching and learning of mathematics be recognised nationally and encouraged throughout the development of K-12 curriculum and assessment as well as targeted professional development.

that a Year 12 mathematics subject should be an entry requirement for all teacher education courses and that improving students’ own mathematics knowledge should be a key responsibility of teacher educators in universities.

that algebraic thinking be recognised nationally as a strand in the curriculum framework from K to 12 and a major focus of mathematics teacher education.

that the features outlined on page 11 and the back page of this submission be used to underpin every national development in school mathematics and teacher preparation/support in the future.

that DEST to sponsor some young researchers to find and draw this literature together so that the general findings may be used as a basis for a national drive to update teacher knowledge and qualifications as well as a comprehensive set of resources to support this work.

that coherent and substantial programs be developed to attend to these areas of need, with nationally focused re-training of teachers of mathematics, provision of supporting personnel and resources especially through the early years and in rural and remote areas, and the provision of intervention-style assistance programs.

broad-based sampling to assess standards and identify issues, banning comparisons between children, teachers and schools, and emphasis in teacher education ways of developing students’ appreciation, knowledge and skills in mental mathematics and investigations.

that these be made a basis nationally for the initial accreditation and higher-level reaccreditation of teachers of mathematics.

that this review, its findings, and its recommendations be taken seriously enough at national and state levels to result in a significant nation-wide program to improve teachers’ knowledge of mathematics and their understanding of evidence about how it is best taught and learned.

Table of Contents

1. Numeracy and Mathematics in Australia....................................................................................1What is numeracy?............................................................................................................1Mathematical thinking.......................................................................................................2

Teacher qualifications....................................................................................3

2. School organisation and support structures................................................................................4Student grouping...............................................................................................................4

Group work.....................................................................................................4Differentiated curricula.....................................................................................................5

Open-ended tasks............................................................................................5Communities of inquiry..................................................................................6

The use of technology.......................................................................................................6Calculators......................................................................................................6Computers.......................................................................................................7

Primary teachers’ knowledge............................................................................................9Algebra...........................................................................................................9

3. Effective Teaching Practices........................................................................................................10Features of effective mathematics pedagogy..................................................................10Areas of specific concern................................................................................................13

Indigenous students......................................................................................13Vulnerable students......................................................................................14Early Intervention Programs and Assistance Programs...............................15Students with disabilities..............................................................................16Rural Students..............................................................................................17Student attitudes and motivation..................................................................17

Classroom assessment.....................................................................................................18Significant developments in early years’ assessment...................................19

4. Teacher Education and Professional Development...................................................................21The mathematics teaching standards...............................................................................21Characteristics of effective teachers and teacher development.......................................21

Lesson study.................................................................................................23Other teacher research..................................................................................24The practicum...............................................................................................26

References.........................................................................................................................................29

1. Numeracy and Mathematics in Australia

What is numeracy?There is little point arguing the difference between numeracy and mathematics. Motherhood statements such as “every student should be numerate” serve little purpose, and too often “numeracy” and “numeracy education” are thought of as lowest common denominator and basic life skills. The development of “numeracy” does not suggest exposure to higher-level thinking and problems that every student of mathematics — at any age — should have. Thus the terms are not interchangeable, but as the AAMT asks in its submission to this review, “Does any of this really matter?”MERGA members recognises the many and varied definitions of numeracy attempted by respected authors in Australia and overseas, such as the Australian Association of Mathematics Teachers Inc. (1997), Brown (2001), Hogan and Kemp (1999), Primary Mathematics Association (1997), Scott (1999), Siemon (2000), Skalicky (2007), and Willis (1990, 1998) — as well as Neill (2001) who examined at least 40 definitions! We are also familiar with variations that have more situated meanings, such as “critical numeracy”, “quantitative literacy”, “financial literacy”, “democratic numeracy”, “functional mathematics”, “cross-curriculum numeracy”, and “adults’ numeracy levels”, and see the relevance of the terms within appropriate contexts. We are also vitally interested in the development of different kinds of “academic numeracy” that are an issue for students studying nursing, business, engineering, etc. We share concerns of the Australian Council for Adult Literacy who noted that

Ten years ago Australia was at the forefront of adult literacy and numeracy teaching, learning and research. Now, we do not even have a national policy and we are going backwards.

(Australian Council for Adult Literacy, 2001, cited in Coben, in press).

We also recognise the strategic convenience of the “literacy and numeracy” combination often used in political contexts, but in fact when awards, grants and developments in this combined area are made, numeracy invariably suffers through lack of equal emphasis and funding. (The same happens with Science and Mathematics awards and grants.) The goals in all of all of these contexts, though, are mathematical. Research into numeracy such as that undertaken by Zevenbergen (2004) and by Taylor and Galligan (in press) generally just recognises and highlights the complexity of workplaces where workers require a strong general mathematics education. Further, the expression “teachers of numeracy and mathematics” (Discussion Paper) implies the ridiculous proposition that one can teach or learn numeracy without teaching or learning mathematics! Proposing definitions of “numeracy” is a futile exercise because of the variety of meanings used within and across countries and in the context of rapidly changing societies. Energy is better put into considering who needs what mathematics and how best to give them access to it. Separating everyday and workplace skills from broad higher-level skills will only create a longer tail of underachievement and limited opportunity. The key to achievement is high expectations for all students. The point is that students need to learn to think and work mathematically, they do this in mathematics classrooms, and they come to understand how mathematics can be used and how to use it productively. The aim of mathematics education is develop these understanding, capabilities and attitudes. Curriculum documents, teacher education, and talk between classroom teachers should focus on the development of students’ mathematical knowledge, not on “parallel” or “diverging” streams of mathematics and numeracy. Playing with language and wasting effort distinguishing between numeracy and mathematics has served little purpose in this endeavour over the past two decades.

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MERGA recommends that COAG abandons the term ‘numeracy’ and emphasises the breadth of mathematical knowledge and skills required for personal, professional, civic and academic activity.

Mathematical thinkingSpecific aspects of mathematics such as number, measurement, geometry, probability, etc. and processes that allow estimation and calculation of these, are used in everyday life. Most importantly, however, mathematics involves ways of thinking and working that are useful throughout life. People need to develop abilities to tackle, interpret and solve problems; to observe, describe, interpret and predict numerical and other sequences; to think abstractly and manipulate ideas; to comprehend differences between general, specific, and unknown; to understand how to use representations effectively; to analyse and reason about situations logically; to monitor progress and apply reality checks; to summarise, communicate and generalise findings — and to use many other skills involved in working mathematically, Such skills and experiences underpin effective everyday citizenship as well as Australia’s scientific, economic, security and everyday well-being and development. They also form the basis for higher education and the preparation of mathematicians, statisticians, operations researchers, etc. In short, learning to think and work mathematically involves using higher-level processes, and one does not learn this through focusing on basic numeracy. While Australia generally performs well in problem solving and mathematical literacy, many MERGA members believe that there is not enough emphasis put on the development of higher-level thinking skills in mathematics classrooms. Learning activities are often routine, few are challenging for capable students, and there is a lot of repetition in curriculum documents, textbook examples, and learning activities. A focus on specific, repetitive and closed examples and types of activity may be a safe environment, but many students express their sense of boredom and few experience the usefulness of mathematical thinking. It seems from recent reports such as findings in the Maths? Why not? project (McPhan, Morony & Pegg, 2007) that this situation has not improved in recent times.There is little emphasis in teacher education on teaching mathematics across the curriculum, and little recognition in schools of the mathematical concepts that underpin the study of most subjects at secondary level, or professional development for mathematics and other teachers that attends to this issue. There is ample evidence that higher-level mathematical thinking can be taught from the very first years of primary school — and indeed in pre-school contexts. MERGA members could direct the Review’s attention to the work of Lyn English, for example, who has investigated children’s reasoning ability and strategies used in solving deductive problems. English (1993, 1996, 1998a, 1998b, 1999a, 1999b) found that the deductive reasoning of even those children who had been classified as low achievers in mathematics had potential: they readily saw relationships and connections, used these to find appropriate solutions, exhibited self-monitoring processes, and independently developed sophisticated and effective procedures. It is not unusual for non-routine problems to prove very popular with such students.Another approach is the notion of mathematics classrooms functioning as “communities of inquiry”, based on the work of the Philosophy for Children movement. Groves, Doig, and Splitter (2000) found that while there is much support for this notion among principals, teachers, and mathematics educators, it is rarely a feature of current teaching practice. They proposed that the common perception of the curriculum being fragmented and outcomes-focused does not encourage the development of such tenor in mathematics classrooms. Principals and mathematics educators rated the cognitive demands of typical lessons as low to very low, and noted that they are not challenging for most children; while teachers saw the cognitive demand as being determined by the tasks (Doig, Groves & Splitter, 2001; Groves, Doig & Splitter, 2000). The researchers later compared Australian and Japanese teaching in terms of cognitive demands of instructional tasks, and suggested that more attention in Australian classrooms needs to be given to conceptually

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focused, robust tasks that demand and support higher levels of mathematical thinking (Groves & Doig, 2002).Teachers need to have strategies for engaging the full range of students —not only a range of capabilities but also a range of approaches and attitudes. Watson and Mulligan (1990) found that Year K to 2 children’s solutions to a variety of problems demonstrated operation at as many as six year levels. Lowrie (1998) concluded that individuals use visual or non-visual methods according to the nature and level of the task as well as their beliefs and values as well as the level of their mathematical understanding. Cooperative learning situations, which place visual and non-visual thinkers together, have been found to stimulate higher-order reasoning (Lowrie, 1998). It is clear that ways of working mathematically can be taught effectively by teachers who are familiar with ways of working mathematically. Bailey (2007) provides a detailed account of her work alongside student teachers who were learning about children’s mathematical investigations. Bentley (1996), and Cheeseman and Clarke (2007) researched the teaching of heuristic strategies and metacognitive practices. Clarke and Clarke, (2003), Taplin (1994), Williams (2003a, 2003b) and other MERGA researchers have studied how to develop perseverance and resilience. Diezmann (1998) used strategies, such as drawing a visual representation, finding that direct instruction improves children’s use of such tools. Hurst (2007) reported ways of helping middle-school students to develop “connecting” mathematical knowledge to a range of contexts. In a yearlong teaching experiment, Siemon (1993) taught students to use a metacognitive question and answer technique to reflect on problem-solving processes. English (1997, 1999b) focused on the development of understanding of relational properties rather than mere use of surface features, on use of analogy, and on hypothetical reasoning involved in deduction. Smith (2000) described other practices that promote mathematical thinking — and, in fact, the MERGA conference proceedings and journals (available on-line via www.merga.net.au) give many more examples of ways of teaching that promote mathematical thinking. The evidence base for such approaches is strong, but is largely ignored by curriculum developers in Australia, and so long as this is the case there will be little potential for development in Australia’s skill base.

MERGA recommends that curriculum developers are directed to put emphasis on learning to think and work mathematically at all levels of schooling.

Teacher qualifications The shortage of well-qualified mathematics teachers in secondary schools exacerbates the problem of under-qualified teachers. Teachers who have not studied mathematics to appropriate levels are bound to teach mathematics without fully understanding the content, likely problems students will experience in learning it, ways of explaining and illustrating it well, or its powerfulness in varied contexts and applications in other subject areas. The high number of teachers who are teaching out of field is an ongoing weakness in the Australian education system and this situation is worse in rural and regional areas (Lyons et al., 2006). It results in less-than-ideal teaching and test results as well as less-than-ideal numbers of students being well prepared for upper secondary and university mathematics. In contrast with other countries, there has been little progress in Australia in TIMSS results, and the number of students taking higher level mathematics and statistics subjects is falling. This must be addressed by making sure that secondary mathematics classes are taught by fully qualified mathematics teachers.

MERGA recommends that national and state departments stop “passing the buck” and take positive steps to addressing the need for well-qualified teachers of mathematics.

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2. School organisation and support structuresSchools vary in the way they are structured, within states and even within towns. The actual structure is not as important as leadership, planning processes that ensure a curriculum that builds on what has already been taught without gaps or undue revision, and ways of working in classrooms. The institutional practices that the discussion paper refers to are ability groupings, differentiated curricula, and the use of ICT.

Student groupingThere are good arguments for allowing students in the upper years of schooling to choose levels of study that are appropriate for their needs, interests and levels of attainment. It is important that students with a keen interest in mathematics be able to be taught in classes where challenging problems and activities are suited to their capabilities. It is also important for other students to study mathematics at appropriate levels and with foci that are attractive to and useful for students, including applications that will lead to meaningful use of mathematics and statistics in workplaces and professions.However, MERGA members do not support ability grouping, either for longer-term use within classrooms or whole-class streaming. In relation to primary and middle schooling, they ask questions such as:

What does it do for the remaining classes when the strongest 15% of students are removed? Why it is good for students to "complete a curriculum" early? Is it more important to attain higher levels of learning than a broader range and depth of

understandings at expected levels?There is ample research on the effects of so called “ability” grouping, with most projects world-wide finding evidence of negative self-concepts, of groups being formed on the basis of one aspect (e.g. Number) while ignoring different outcomes in other areas, of extended variance between strong and weak groups over time, of little ability to ‘climb’ groups, of parental and student concerns, of weak modelling and little challenge for lower groups, of evidence of social class difference between groups, and so on. Further, ability grouping and setting is not usually used in primary and lower secondary countries whose TIMSS and PISA results we would wish to emulate.

Group workMost MERGA members agree with the final report of the Numeracy Task Force (DfEE, 1998) in England, which drew on the National Numeracy Project findings to recommend whole class teaching supported by groupings working on tasks or problems. Factors such as teachers’ expectations were found to be more influential than whether the children worked in groups or not, and establishing the nature of appropriate interactions is vital.It may be that ability groups, or friendship groups, or mixed groups, or in fact individuals working on their own are most suitable, depending the nature of the task, the mix of students in the classrooms, and whether there will be advantages for specific types of classroom organisation. Here, it is not a matter of using groups or not, but of using them strategically. It may also make sense to start an activity individually and then move into groups to compare, explain and evaluate solutions, group appreciation of and interaction with the task, and perhaps produce a group response (see Mousley & Campbell, 2007). Mixed groups are likely to result in some cognitive conflict, a stimulus for mathematical discussion and consideration of new ways of thinking (Davis, Sumara & Kieren, 1996; Hiebert et al., 1997; Steffe, 1991; Wheatley, 1991). Gooding (1997) examined the roles and effects of Year 6 children working in groups, finding that learning from group tasks was retained long term and linked a number of interconnected ideas about the basic concept; but also found that the group dynamics in some boys’ groups hindered the participation and learning of some boys. Zevenbergen (1995) showed what many teachers know: that some low achievers are relatively passive in small-group work. Several MERGA researches have made the point that there is no clear answer as to which structure is most effective, and suggest

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that teachers need to consider the specific domains where children show strength and weakness. Further, through videotape analysis, researchers have found that learning outcomes vary for group members despite common engagement with a task. Another form of grouping, sometimes used effectively in the early years of schooling, is multi-age grouping. De Lemos (1996) undertook an evaluation of the Victorian First Steps pilot project, but the research did not provide evidence of improved mathematics outcomes in multi-age classrooms.The use of group work can be linked effectively with formative assessment as teachers can analyse students’ discussions to gauge levels of thinking and skills, misconceptions, and further teaching needs (Watson & Collis, 2001, Hunter, 2007).

MERGA recommends the use of flexible, purposeful grouping of students, as appropriate for the task and types of interaction sought, in mathematics classes.

Differentiated curriculaAs noted above, there is good reason to differentiate curriculum in the upper secondary area, and there is on-going concern that many students at this level are generally not all challenged at appropriate levels and could be better catered for. At primary and lower secondary levels, however, it seems possible to cater for a range of students with the whole class working on the same topic. Several models have been researched by MERGA members and found to be effective.

Open-ended tasksA significant amount of Australia research has been completed on the use of open questions from K–12 to incorporate modern investigative and problem solving approaches as well as directing student attention to valued skill and concept development. Open-ended tasks have main features:

They require more than remembering a fact or reproducing a skill. Pupils can learn by answering the questions, and the teacher learns about each pupil from

the attempt. There may be several acceptable answers. (Sullivan & Lilburn, 1997, p. 2)

Such tasks engage students in productive exploration (Christiansen & Walther, 1986), enhance motivation and students’ sense of control (Middleton, 1995), assist in building robust knowledge structures (English, 2007; Sinclair, 1990), and contribute to teachers’ appreciation of students’ mathematical and social development (Stephens & Sullivan, 1997). Students with various levels of competency respond effectively to open-ended tasks, as they provide different types of learning opportunities (English, 2006; Sullivan, Bourke & Scott, 1997; Sullivan, Clarke & Wallbridge, 1991). The extensive work of Sullivan, Mousley and Zevenbergen (e.g. Sullivan & Mousley, 1994; Sullivan, Zevenbergen & Mousley, 2006) is relevant here, as it focuses on whole class work with curriculum differentiation within open-ended tasks, supporting the range of learners in primary and secondary mathematics classrooms by using open-ended problems with appropriate “enabling” and “extending” prompts. Teacher education and professional development do not currently train teachers to cater for individual difference within whole-class teaching using these strategies. The researchers found that using such prompts enabled the establishment and retention across time of communities of inquiry (see below). Sullivan, Mousley and Zevenbergen propose a model for effective mathematics teaching that includes planning for sequenced tasks, enabling and extending prompts, normative interactions, and the development and use of mathematical community norms.Victoria’s Middle Years Numeracy Project included the use of open-ended tasks, and scoring rubrics were developed grading the quality of students’ mathematical responses to the tasks. These developments enabled open-ended tasks to be used to measure student performance in a large-scale study, and at the same time provided a means of checking consistency of teachers’ grading across schools (Siemon & Stephens, 2001).

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Communities of inquiryAnother way of coping with the need for differentiation within whole classes is the creation of “communities of inquiry”. Groves, Doig, and Splitter (2000) found that while there is much support for this notion among principals, teachers, and mathematics educators, it is rarely a feature of current teaching practice. They proposed that the common perception of the curriculum being fragmented and outcomes-focused does not encourage the development of such tenor in mathematics classrooms. Principals and mathematics educators rated the cognitive demands of typical lessons as low to very low, and noted that they are not challenging for most children; while teachers saw the cognitive demand as being determined by the tasks (Doig, Groves & Splitter, 2001; Groves, Doig & Splitter, 2000). The researchers later compared Australian and Japanese teaching in terms of cognitive demands of instructional tasks, and suggested that more attention in Australian classrooms needs to be given to conceptually focused, robust tasks that demand and support higher levels of mathematical thinking.

MERGA recommends that curriculum be differentiated at upper secondary levels but not in primary of junior secondary classrooms, except through the use of strategies that allow all students to engage fully with class tasks.

The use of technology

CalculatorsWhile ICT does not usually include calculators, there is a significant evidence base that these are important tools in the teaching, learning and doing mathematics. Following the significant Calculators in Primary Mathematics project (see, for example, Groves, 2004; Groves & Cheeseman, 1995; and Groves & Stacey, 1998), where giving all children in the early years of schooling free access to affordable hand-held computational power made a significant difference to both curriculum processes and content, other Australian studies have demonstrated that mathematical competence, number sense and problem solving skills are enhanced through good teaching with calculators. However, in many cases (see, for example, Sparrow & Swan, 1997) calculators are not being used well as a teaching aid because they are often not used in problem solving or mathematical investigations. There is also evidence that they are used differently in upper primary and lower secondary classrooms, and differently by pre-service and in-service teachers (Vale, 2007).The development of mathematical understanding with simple, scientific, and graphic calculator use has implications for curriculum development and curriculum processes as well as teacher’s professional development and initial training. Their inclusion in some Year 12 exams has had significant effects on senior secondary curriculum — a point to be noted if a national framework is developed. While opinions are divided on how much students should be able to use calculators, there is plenty of evidence to show positive benefits and that students respond well to their use. It is clear that even young children can make sensible decisions about whether to use metal or written computation or whether to use calculators, and that they can combine these approaches successfully — such as checking written or calculator solutions with rough mental estimation or checking mental calculations using another approach. In particular, this has been noted by Groves (2004), who summarised her findings as follows.

Children with long-term experience of calculators performed significantly better overall on the 24 computation items, with an item by item analysis revealing significantly better performance on the five items requiring a knowledge of place value for large numbers, subtraction with a negative answer, division with a remainder, and multiplication and division of money. These children also made more appropriate choices of calculating device and were better able to interpret their answers when using calculators, particularly where decimal answers were involved. (p. 1)

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Neill (2005) discusses the conceptions students bring to estimation, as well as ideas for extending the range of strategies that they may employ, which largely depend on good number sense. An online concept map on estimation is available on the Assessment Resource Banks at arb.nzcer.org.nz/nzcer3/supportmaterials/Maths/concept_map_estimation.php.

ComputersLikewise, Yelland (2000) provides case studies of classrooms where use of computers clearly enhanced children’s learning of mathematics. In one case, for example she reported that:

The research thus supported the use of scaffolded learning in the technological environment, so that learning experiences were not only more meaningful and engaging for young children, but also so that children may use metacognitive strategies more effectively and consistently. Furthermore, two of the most outstanding observations of these studies have been the high level of engagement that the children had with the tasks and their applied use of mathematical concepts well beyond that expected of them in traditional curriculum documents.

What we often see in classrooms today is almost the complete opposite of this. Computers are used to keep children who have completed their required work early or as a reward for good work in many cases. … not only does the computer work need to be embedded within the curriculum, but it also needs the teacher to take on an active role in the process, not only to scaffold learning but also to provide opportunities for children to brainstorm problem-solving strategies and to discuss and share solutions and techniques. (n.p.)

The report by Yelland (2000) prepared for DETYA/DEST, Teaching and learning with information and communication technologies (ICT) for numeracy in the early childhood and primary years of schooling, contains six recommendations that MERGA supports fully:

1. The use of ICT is a fundamental aspect of learning, and funding needs to be specifically targeted to its successful implementation in the schools and universities in Australia that provide teacher education programs.

2. There is a need for Australian research about the impact of ICT on learning, especially with regard to literacy and numeracy outcomes and the concept of multi-literacies that incorporate the use of ICT.

3. We need to identify relevant research questions which support the successful use of ICT in schools

4. Funding should support a variety of research methodologies which include quantitative and qualitative approaches as well as mixed-method research designs.

5. A longitudinal study to complement research in literacy and numeracy research would be beneficial

6. A set of exemplary teaching practices incorporating the use of ICT — such as those developed by DETYA’s Indigenous Education Branch in What works? — should be commissioned. (n.p.)

As with calculator use, the amount of emphasis put on the use of computers in teaching and learning mathematics varies between states, and even between schools. Again, the role of the teacher is a vital aspect (Baturo, Cooper, McRobbie & Kidman, 1999; McIntosh, Stacey, Tromp, and Lightfoot, 2000; Walta, 2000). The different levels of use in senior secondary levels are of major concern. Students in different states experience very different opportunities to learn how to use graphic calculators and computer algebra. University mathematics staff and employers do not know what to expect in terms of knowledge or skills in relation to standards of technology use for mathematics, and curriculum and textbook writers lack agreed guidelines. If Australian students were not to be disadvantaged in learning technological skills, it would be really useful to have common agreement on this matter and appropriate professional development organised at the national level.A significant study in technology use (i.e. computers, graphics calculators and the Internet) in secondary mathematics classrooms involved a survey of Queensland teachers undertaken by Goos

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and Bennison (2006). The researchers studied teachers’ confidence with and frequency of use of technologies, and analysed the role of variables known to be related to technology use, including access, teaching experience, attitudes, and professional development. They found that technology was used more in the upper secondary than middle years, but that around one half of the teachers surveyed reported that they never or rarely used computers with top Year 11 and 12 students. Goos and Bennison referred to previous research in Australia, New Zealand, and the USA that had found that mathematics teachers’ use of technology is dependant on pedagogical knowledge. They sample the majority of respondents reported good confidence levels. Access to computer laboratories for mathematics classes was difficult. They summarised the research be writing:

Confidence grows with teaching experience, but a more proactive approach to increasing teachers’ comfort with and use of technology needs to address issues of access to computers and graphics calculators, and professional development. … the findings presented here have clear implications for the resourcing of schools with respect to equipment and in-service education and point to the need for further research that investigates how and under what conditions teachers learn to effectively integrate technology into their practice. (pp. 13–14)

Forgasz (2006) undertook a 3-year study of issues associated with the use of computers in Victorian secondary schools, focusing in particular on gender and other equity factors. She found that the majority of teachers felt comfortable about, and did use, computers for teaching mathematics, and believed that computers helped students' mathematical learning, but that generally the teachers considered boys to be more confident and capable than girls with technology. Stacey and her colleagues at Melbourne University have undertaken extensive work on computer algebra systems over the last decade (CAS: see, for example, Ball 2004, 2006; Norton, Leigh-Lancaster, Jones & Evans, 2007; Pierce & Stacey, 2004). The URL for a website that gives only a taste of the many informative publications arising from this body of work, and that MERGA recommends to the writing team for evidence of the learning that takes place when well-informed teachers use CAS in mathematics classrooms, is http://extranet.edfac.unimelb.edu.au/DSME/CAS-CAT/publicationsCASCAT/Publications.shtml. It also recommends a pilot scheme that is currently running in New Zealand on the effective use of CAS calculators in junior secondary mathematics classes (Neill & Maguire, 2006). Early indications are that students are acquiring deeper understanding. While this is similar to the work of Pierce & Stacey (2002) from Victoria, that the latter is aimed at the senior level.Other research literature that illustrates effective learning with technology includes articles that focus on specific concepts. Brown (2007), for example, explored how students’ early notions of functions develop using extended tasks and discussion. A useful analysis of critical factors that support or hinder innovative teaching and learning with technology is provided by Goos and Benison (2007), whose current 3-year project is generating models of successful innovation in integrating technology into secondary school mathematics teaching. They summarise factors that influence uptake and implementation and show that it is not only well-resourced schools that are necessary, but also teachers with “productive beliefs about mathematics and the role of technology in mathematics learning” (p. 323).

MERGA recommends that the value of appropriate and well guided technology use for the teaching and learning of mathematics be recognised nationally and encouraged throughout the development of K-12 curriculum and assessment as well as targeted professional development.

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Primary teachers’ knowledgeThe review’s Discussion Paper asks about the depth of mathematical understanding required by primary teachers. Knowledge of the primary mathematics curriculum includes a good understanding of primary mathematics and how to teach it. Groves, Mousley and Forgasz (2004, 2005) summarised Australian research on the teaching and learning of Chance and data, Measurement, Number, and Space. Knowledge of this primary content and research findings about its teaching should be underpinned by adequate knowledge of Number, including natural numbers; divisibility, factorisation and primes; rational and irrational numbers; the nature of chance and expectation; the connection between probability and statistics; independent and dependent events; and the nature of a statistical investigation. The teaching of primary Geometry depends on strong foundational knowledge about regular polygons; tessellation, symmetries and transformations in the plane; reflection, rotation, translation and glide reflections and their combinations; creating and identifying finite and infinite patterns in the plane; Platonic solids and examples in art and nature; linear, quadratic, exponential and logarithmic functions; as well as graphs and networks. This depth of knowledge is not common amongst students who have studied mathematics only to Year 10, so MERGA suggests that a Year 12 mathematics subject should be an entry requirement for teacher education courses and that adult mathematics education should be one responsibility of teacher educators in universities.

MERGA recommends that a Year 12 mathematics subject should be an entry requirement for all teacher education courses and that improving students’ own mathematics knowledge should be a key responsibility of teacher educators in universities.

AlgebraJust as important as deep understanding of Number and geometry foundational knowledge is primary teachers’ awareness of the importance of algebraic thinking and secondary teachers’ knowledge of abstract and general concepts taught in many primary schools. Algebraic understandings have their beginnings in early number work and provide a solid footing for later mathematics and lifetime use. Algebra must be recognised nationally as a strand in the curriculum framework (perhaps as Mathematics Thinking). Currently this is not the case.

When children explore patterns and record what they have found, they are working algebraically. When students double quantities in cooking, draw a graph to represent number of pets, or summarise their findings about the weather and use these to make predictions, they are working algebraically. … it is most important to recognise that much of what is done already in primary school mathematics classes is algebra. (Herbert & Mousley, 1997, p. 130)

Consideration of the demands made on children’s capacities to abstract and generalise information in primary mathematics is essential in developing mathematical ways of thinking and working as well as helping children to make a smooth transition from arithmetic to algebra. The focus here needs to be on structure rather than specific examples (Boulton-Lewis, 2000; Chick, 2007; Mason, 2007; Mitchelmore & White, 2000, 2004a, 2004b; Mitchelmore, White, & McMaster, 2007; Warren & English, 2000). Such teaching is easy to start in pre-school settings (see, for instance methods taught to early childhood teachers by Papic and Mulligan, 2007), but is a well known shortcoming of primary mathematics teaching (Warren & English, 2000) — namely that students fail to abstract from their experiences with specific examples the mathematical structures that are necessary for them to make a later successful transition from arithmetic to algebra. This is a point that must be stressed in pre-service and in-service teacher education. Relevant strategies include:

focusing on looking for, describing, and making use of generalisable processes and structural properties of arithmetic as a conclusion of finding answers to specific problems;

encouraging the development of conceptual, relational understanding of processes; choice of resources including textbooks, worksheets, computer software, and curriculum

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outlines that present mathematics in a connected way; and substantial parts of lessons (such as the last third) being dedicated to discussion of links

between problems, processes used by different children, broad underpinning concepts, related mathematical and everyday knowledge, and practical applications.

At the middle school level, number work could better prepare students for formal algebra (Stacey & MacGregor, 1997). MacGregor and Stacey (1999) provide activities suitable for the middle years that highlight important features for classroom teaching together with samples of student work. Swafford and Langrall (2000) investigated Year 6 students’ use of equations to represent problem situations, demonstrating students’ capacity to generalise problem situations and use variables. Not only arithmetic can be used to develop children’s ability to think algebraically — other aspects of the primary curriculum develop the necessary ideas (see, for example, Outhred, 1993). Mousley (2003) identified reasons for a similar phenomenon observed in Year 6 classrooms:

typical primary lessons focusing on finding the answer to specific sets of problems and not on looking for, describing, and making use of generalisable processes and structural properties of arithmetic;

teachers’ own education (including primary and secondary schooling, but also pre-service teacher education and professional development) that has generally not encouraged conceptual, relational understanding of such processes and properties;

resources used by teachers, including textbooks, reproduced worksheets, computer software, and curriculum outlines that almost invariably present mathematics as discrete topics and examples; and

traditional lesson formats that do not allow a substantial part of the lesson (such as the last third) for discussion of links between problems, processes used by the children, underpinning concepts, and related ideas.

Overall, research in this area suggests that students fail to abstract from their primary school experiences the mathematical structures that are necessary for them to make a later successful transition from arithmetic to algebra.

MERGA recommends that algebraic thinking be recognised nationally as a strand in the curriculum framework from K to 12 and a major focus of mathematics teacher education.

3. Effective Teaching Practices

Features of effective mathematics pedagogyMERGA members agree with the points made in the Discussion Paper about the need for connected knowing, a focus on student learning, the need for a challenging curriculum, and high expectations of all students. Approaching mathematics in an investigative manner, using problem-solving as a basis for both teaching and learning, developing an understanding of the links between thinking and recording and talking mathematically, and relating mathematical ideas to real life are further key features of effective teaching.It is commonly felt that the current curriculum documents do little to support such approaches. There tends to be too much emphasis on what should be taught at specific levels rather than guiding principles for teachers, schools, systems, curriculum developers, textbook writers, and both pre-service and in-service professional development. At the 2007 conference of MERGA, desirable features of a useful and supportive curriculum framework that could be implemented nationally were identified. The following table elaborates the following principles related to consultation with a range of stakeholders, useful curriculum documents, recognition of and support for student diversity and development, recognition of and support for student teacher diversity and development, researched based pedagogy, and appropriate forms of assessment.

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Key idea ElaborationGuiding principles

Problem-solving: thinking, reasoning and working mathematically Mathematical thinking (including prediction, generalisation, abstraction, application

across contexts) Learning to think mathematically needs to underpin the whole

numeracy/mathematics agenda Exploring of relationships between useful applications and mathematical entities Focus on learning for the future; critical consumers of the world Written around big maths ideas A focus on connections between concepts Developed by leaders in the fields of mathematics curriculum, mathematics

learning and mathematics curriculumConsultation with a range of stakeholders

Stakeholders include including teachers, teacher educators, researchers, mathematicians & other tertiary, year 11-12 exam setters, employers, …

Sufficient time, resources and inclusive ways of working to develop a framework through strong consultation

Realistic and well-funded consultation timelines and processes A number iterations and involvement of a range of appropriate experts

Useful curriculum documents

A minimal framework, with scope for diversity and local relevance in interpretation Discretionary elements: needs to be strong enough to hold together, but flexible

enough to allow states to build on individual strengths A document that is only a benchmarking of skills is a waste of time Aspirational as well as specifying goals that the government commits to that all

students should achieve Emphasis on both what to teach and ways of approaching teaching Should not be too prescriptive—guidelines but must have examples of

implementationRecognises and supports student diversity and development

Consistent high expectations of students regardless of background High but achievable expectations, not lowest common denominator Attends to accessibility issues. Develops positive dispositions towards seeing, using and applying maths including

in other learning domains Open enough to cater for the diversity of children in any classroom Flexibility that allows teachers to match it with students’ needs and capabilities

Recognises and supports teacher diversity and development

Flexible enough to be adapted by experienced teachers but supportive of new teachers and those teaching out of their areas of expertise

Long-term support for professional learning, including materials, time, collaborative opportunities and partnerships, p.d., etc. for teachers — and especially novice teachers and ‘out of area’ teachers

The implementation is more important than the content. Teachers of mathematics need to be able to make connections across it and take

advantage of “teachable moments” Supporting the development of positive dispositions towards the learning of maths Should be accompanied by high levels of resources for professional development

Researched based

Fluid enough to adapt as research reveals further understandings Research on the content and pedagogical influences of learning Australian plus international research

Appropriate forms of assessment

Appropriate and varied, reflecting the above principles Diagnostic with feedback mechanisms Items able to measure ability to work mathematically Need to maintain States’ flexibility and accountability

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MERGA recommends that the features outlined above be used to underpin every national development in school mathematics and teacher preparation/support in the future.

One of the most recent papers that identified such features is that of Bill and Watson (2007), who noted that:

Contemporary statistics education research is remarkably consistent in relation to recommended pedagogy. Five key features of best teaching practice are identified.

1. Engage students with data and concepts – the “big ideas” of statistics – such as variation and distribution ….

2. Provide active learning opportunities … and authentic data analysis … with real or “messy” data sets and meaningful tasks in a context that students can understand and value.

3. Develop a culture and habits of enquiry and statistical process … use whole class discussion where students must construct arguments and justify their positions … Teachers should provide students with a working – not necessarily formal – statistical vocabulary.

4. Utilise technology tools that allow students to visualise and explore data by providing different representations of the same data set … and to move back-and-forth between the various representations of the data …

5. Use assessment that genuinely measures student learning and development and that accurately conveys to the student what is important …

(Paraphrased. Bill and Watson cited authors who have developed these propositions further.)

While these suggestions were made for the field of statistics education, these suggestions could be generalised to successful mathematics teaching in general.At a more general level, Mousley and Sullivan (1996) outlined six components of quality teaching, as shown in the following figure, and developed resources to support teacher development in these areas.

Building understanding refers to a role that the teacher assumes in order to convey some pre-determined meaning to students. It is recognition of particular understandings to be developed, and of strategies to achieve these ends by building on existing knowledge, using materials to explain and clarify concepts, choosing appropriate sequences, helping students to make connections, forming relationships, and knowing the meaning of terms. There is a strong inference of teacher decision, teacher direction, teacher explanations and teacher control.

Communicating relates to opportunities for talking, explaining, describing, listening, asking, clarifying, sharing, writing, reporting, and recording. It includes class organisation structures such as cooperative groups, and is characterised by an orientation of teacher and students to accepting communication as a two-way process.

Engaging refers to facilitating student involvement in their own learning. It includes engaging students in mathematical activities, and motivating students to learn. Teachers choose content and materials that are personally relevant for the students or based on real-world situations, and seek to make learning enjoyable.

Problem solving is when students work out for themselves how to perform mathematical tasks in such a way that it is the students' own work and they know that it is. It refers to activities such as risk-taking, challenging, exploring, investigating, thinking, asking, and posing.

Nurturing refers to the establishment of a relationship between the teacher and pupils. Teachers do not seek just to teach mathematics but to teach students as well. While no doubt teachers of mathematics endeavour to cater for the range of abilities in their class and to develop rapport with their students, nevertheless this is more a recognition that teaching and learning is a two-way

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process and that there is something natural in the expert/novice relationship which includes a nurturing component.

Organising for learning describes comments related to decisions made by the teacher about a specific focus for what would happen in the lesson and a commitment to pursuing that focus and to communicating the focus to the students. (Mousley & Sullivan, 1996, p. 76)

Muir (2007) also summarised features of successful mathematics pedagogy, where teachers:• Maintained a focus on and taught for conceptual understanding of important mathematical

ideas;• Used a variety of teaching approaches which foster connections between both different areas of

mathematics and previous mathematical experiences;• Encouraged purposeful discussion through the use of question types to probe and challenge

children’s thinking and reasoning and encouraging children to explain their mathematical thinking; and

• Possessed knowledge and awareness of conceptual connections between the areas which they taught of the primary mathematics curriculum and confidence in their own knowledge of mathematics. (p. 513).

Other useful synopses are provided by Jones, Tanner, and Treadaway (2000), and Clarke and Clarke (2002). A large ARC Linkage Project (Pegg, Panizzon, & Lynch, 2007) in NSW identified and explored the factors emanating from mathematics faculties leading to outstanding mathematics outcomes in junior secondary public education in NSW for students across the ability spectrum. Seven common themes were reported across the sites explored. These were the strong sense of team, staff qualifications and experience, teaching style, time on task, assessment practices, expectations of students, and teachers caring for students.In fact, many research projects by MERGA members have summarised evidence-based features of effective mathematics classrooms. It would be useful for DEST to sponsor some young researchers to find and draw this literature together so that the general findings could be used as a basis for a national drive to update teacher knowledge and qualifications as well as a comprehensive set of resources to support this work.

MERGA recommends that DEST to sponsor some young researchers to find and draw this literature together so that the general findings may be used as a basis for a national drive to update teacher knowledge and qualifications as well as a comprehensive set of resources to support this work.

Areas of specific concern

Indigenous studentsThe difference between the mathematical achievements of groups within society, and particularly Indigenous students, is of great concern to us. The TIMSS international comparisons of achievement, showed significant State and Territory differences, and Indigenous children scored significantly lower than non-Indigenous children (Lokan, Ford, and Greenwood, 1997). Even though only six countries outperformed Australian children overall, Australia has a long “tail” that correlates with socio-economic standards.The problem is not only Indigenous students’ engagement with mathematics, particularly in remote areas, but attraction to and retention of experienced teachers to rural and remote locations. We suggest that there needs to be more attention on mechanisms to ensure regular places in university programs for teacher education in mathematics, science and technology education in all states and territories. Nationally, we need to address barriers to teacher education and professional development participation, including the regional divide. With respect to Indigenous participation in teacher training, there is a need to raise the level of Indigenous support to encourage greater Indigenous participation in tertiary mathematics, and a national bank of appropriate resources is needed to encourage the effective preparation of all teacher education students for teaching

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Indigenous students. Distribution of a summary of practical strategies suggested in the DEST What Works? project would be a good starting point. A key theme established and researched in recent years by MERGA members is how positive relationships established between schools and communities can improve Aboriginal students’ learning (Goos et al, 2003; Howard & Perry, 2007). Goos et al undertook a major study, Home, School and Community Partnerships to Support Children’s Numeracy, that included a critical review of relevant literature; analysis of the range of Australian home, school and community programmes, strategies and practices; key principles for effective practices. The report includes recommendations that deserve national support and have implications for federal funding, particularly in relation to Indigenous communities and other rural communities.Similarly, the Building Community Capacity project (see Howard & Perry, 2007) is seeking to further the success of the Mathematics in Indigenous Contexts project and encourage its generalisation into other communities and contexts. Howard and Perry describe five challenges for schools and systems in relation to community capacity building for enhancing the education of Aboriginal students:

• mutual respect between the Aboriginal community and the school community;• mutual engagement with the community in developing learning approaches based upon

alternative and creative discourses;• evidence-based discourses to inform one’s understanding of learners and learning;• home-school-community alignment for enhancing student learning; and• personal and collective efficacy for community engagement. (p. 403)

The researchers present case studies of these challenges being confronted, and report how clear progress was achieved. They conclude that:

In the past, too much has been left to chance as well-meaning groups of people strived to improve the lot of Aboriginal people without Aboriginal people having a direct engagement in the process. The Mathematics in Indigenous Contexts project has provided a strong model for a shift in approach which does ensure that Aboriginal communities play a leading role in the development of their capacity. (p. 410)

Warren, Young, and deVries (2007) provide a very useful overview of research into language factors that may disadvantage Indigenous students in mathematics classes. These include:

a mismatch of conditions for learning for young Indigenous Australian children as they enter school;

tensions between policy and suggested strategies for Indigenous students; cultural differences and the difficulty of adjusting interactions and strategies for teaching

and learning in classrooms; differences between the discourses of the family and school; different languages and dialects of spoken English, as well as standard English being used in

school, with students learning new concepts in a new language.The point is made that insufficient attention is paid to such factors, especially for young children in the early years of schooling. Warren, Young, and deVries outline some implications for different ways of working.

Vulnerable studentsThere have been significant projects in states to identify students at risk, and some major intervention programmes aimed at assisting these children. These following projects and some outcomes were summarised by Groves, Mousley and Forgasz (2004), including:

ACT Assessment Program and Special Assistance Early Literacy and Numeracy Partnerships Early School Assessment Project Flying Start Literacy and Numeracy Development in the Middle Years of Schooling

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Lower Secondary Numeracy Project Making a Difference: Students at Educational Risk Mathematics Intervention Mathematics Recovery Year 2 Diagnostic Net (Qld.) Count Me In Too (NSW, Tas.) Early Numeracy Research Project (Vic.) Pattern and Structure Mathematics Awareness Project (NSW) Counting On: Transition 6–7

Further details of significant projects and some outcomes are supplied by van Kraayenoord, Elkins, Palmer, and Rickards, 2000) in the 4 volume Literacy, numeracy and students with disabilities report. The QuickSmart program (Pegg & Graham, 2005, 2007) is an intervention program directed at students below National Benchmarks in numeracy. The program has achieved considerable success with these students, including indigenous students, in a number of schools in the NT and NSW. The program seeks to improve automaticity in students’ responses, which is operationalised as students’ fluency and facility with basic academic facts and procedures in mathematics freeing cognitive resources for higher-order processing, using mathematical procedures and problem solving.Factors that are often affected by socio-cultural factors and that have been shown to impact on success in learning mathematics include language factors and classroom discourse (see Ellerton, Clements & Clarkson, 2000; Louden et al. 2000a, 2000b; Meaney, Fairhall, & Trinick, 2007; Menon, 1998; Pearn, 1994; Walshaw, & Anthony, 2007; Zevenbergen, 2000).

Early Intervention Programs and Assistance Programs Despite the best endeavours of school systems, school communities and classroom teachers, some children experience difficulty learning mathematics and are at risk of poor learning outcomes. This is a perennial problem, and a strategic systemic approach is necessary.The majority of students learn mathematics effectively in the context of the regular classroom, although this is dependent on the effectiveness of the teaching and learning environments children encounter. However, for a significant group of children the regular classroom program is not sufficient to ensure successful mathematical learning. These children have not yet received the type of experiences and opportunities necessary for them to construct the mathematical understandings needed to successfully engage with the mathematics curriculum. The term “vulnerable” is widely used in population studies (Hart, Brinkman, & Blackmore, 2003), and refers to children whose environments include risk factors that may lead to poor developmental outcomes. Some children arrive at school better prepared than others, so it is important that the first year of school prepares all children to learn mathematics successfully. However, by the end of the first year at school, Gervasoni (2004) found that some 40% of students are vulnerable in at least one aspect of number learning (Counting, Place Value, Addition and Subtraction, or Multiplication and Division). The number and combinations of domains for which children are vulnerable is diverse, highlighting the complexity involved in assisting these children and explainign why teachers with specialist knowledge are needed in schools.Such children may lose confidence in their ability or in the school environment as a place to which they belong, develop poor attitudes to learning and to school, and disengage from learning opportunities (Gervasoni, 2003). The gap between these children’s and other children’s knowledge grows, and the typical experiences provided by the classroom teacher may not enable each child to fully participate and benefit. Ginsburg (1997) concludes that “as mathematics becomes more complex, children with mathematics learning difficulties experience increasing amounts of failure, become increasingly confused, and lose whatever interest and motivation they started out with” (p.26). This outcome is exactly what all school communities must fight against.Therefore, it is important that schools are equipped to both identify and assist the children who are vulnerable in number learning before they experience failure. Indeed, prevention and intervention in

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early childhood is important for ensuring the educational success and general wellbeing of young people (Doig, McCrae, & Rowe, 2003; McCain & Mustard, 1999; Ochiltree & Moore, 2001; Shonkoff & Phillips, 2000). One example of an effective intervention program in mathematics is the Extending Mathematical Understanding Program (EMU, Gervasoni, 2004). This is an early intervention program designed for 6- and 7-year-old children who are vulnerable in aspects of number learning, designed in association with the Early Numeracy Research Project (Clarke et al., 2002. The EMU Program comprises daily 30-minute sessions for between 10 and 20 weeks. Specialist teachers work with groups of three students. In 2000, the effectiveness of both small group and individual program structures were trialled, and small groups were found to be most effective. This has both social and economic advantages over individualised programs, and Gervasoni (2004) found that small groups were more effective for most children than individual programs and the level of interaction between the teacher and a child is appropriate.The research associated with the development of the EMU program devised a process for identifying and for prioritising children for participation in the intervention program (Gervasoni, 2000), a process that has now been extended for use across the primary school (K-6). The research also identified common errors and difficulties in number learning that can be used as a focus for planning instruction and for teacher professional development. The EMU program is now used for intervention (prevention) at Grade 1, and as the basis of specialised assistance programs for children in all other primary grade levels. The specialist teachers customise learning experiences and focus children’s attention on key ideas. They help children develop language and provide manipulatives to support children’s thinking at critical moments, constantly encourage children to describe their thinking and strategies, build confidence and help children identify their strengths.Given the success of this program, in relation to vulnerable students, MERGA suggests the following approach.

Develop and implement policy whereby children are assessed by their classroom teachers early in the school year, to identify all those who are vulnerable in aspects of learning mathematics. Assessment and learning frameworks such as the ENRP Growth Point Framework (Clarke et al., 2002) or the Learning Framework in Number (Wright, Martland and Stafford, 2000) are useful for this purpose.

Implement a mathematics intervention program systemically for Grade 1 children who are vulnerable.

Use an intervention-style assistance program to enhance the learning of Grade 2-6 children who have not reached all the on the way growth points. This will be a longer and more complex process than assisting children in Grade 1.

Educate and employ mathematics intervention specialist teachers who can (1) implement intervention programs, (2) assess children’s mathematical knowledge, (3) assist classroom teachers to develop individual learning plans for vulnerable children, and (4) offer advice to classroom teachers concerning enhancing the mathematical learning of vulnerable children.

Implement professional development programs for classroom teachers that explicitly address the common difficulties children encounter in learning mathematics and effective strategies to enhance these children’s learning.

Students with disabilitiesWith regard to students with disabilities, the Literacy, Numeracy and Students with Disabilities Project (Louden et al., 200a, 200b) concluded that there had been much more emphasis on literacy than on numeracy and that there is a dearth of literature on numeracy achievement amongst student with disabilities. We know from experience that teacher aides in classrooms have no or little numeracy training, yet take on much instruction. The 2002 Senate Inquiry into the Education of Students with Disabilities included details about visually impaired students learning mathematics, the use of Braille, the interaction of multiple disabilities, and different levels of handicap. A concern expressed and recognised by MERGA members is a trend to treat disability as an overarching

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generic condition and hence to minimise services that respond to specific types and levels of disability. We suggest that all teachers of mathematics need training in ways of identifying and building on students’ individual strengths.Hyde, Power and Zevenbergen (1999) studied the mathematical performance of a large number of Queensland students from Years 1 to 12 who required special educational assistance because of hearing loss, finding a need for teachers to focus on trigger words that sometimes prove difficult to discern. Markey (1997), in experimental work with deaf children learning about fractions, found that the children responded well to the use of games that focused on concept and language development.

Rural StudentsThere is currently significant research on mathematics education in rural and remote areas. Much of it is showing comparatively performance and difficulties such as attraction and retention of well-qualified teachers and the availability of professional development opportunities. Australian PISA 2003 results show that the mean score for Australia in scientific and mathematical literacy was well above the international mean, but the mean score for students in remote schools was well below the international mean.Panizzon and Pegg (2007) notes that this evidence is supported by national benchmark data, and their research illustrates some of the problems. They received survey responses from Responses were received from 547 secondary mathematics teachers from Government, Catholic and Independent systems, and problems included:

difficulty attracting teachers; many teachers teaching out of the area of their expertise and training; substantially greater level of perceived need for professional development but difficulties in

accessing it; comparatively poorer levels of resources and support such as suitably skilled personnel to

support ICT use; insufficient number of Indigenous assistants; and few opportunities to visit mathematics-related educational sites; little access to alternative/extension activities in mathematics teaching programs for gifted

and talented and for special needs students.We recommend the full Panizzon and Pegg paper to the writing team, as it identifies problems that need immediate national attention. There is plenty of evidence in further papers listed on the SIMERR website, http://www.une.edu.au/simerr/projects/projects.html that show similar findings with smaller state-based samples, but the Panizzon and Pegg paper covers most of the problems identified within each of the states. The website also reports on current projects where MERGA members are attending to some of these problems.

Student attitudes and motivationMany members of MERGA have undertaken research on affective factors, including Herrington, 1992; Watson & Chick, 2001a; 2001b. Vale (1999) undertook a study of children in Years 5 to 8, trialling strategies aimed at improving their engagement, attitudes, and achievement. Vale found that mathematics needs to:

be meaningful for young adolescents; cater for individual differences and ways of knowing; promote the explaining and writing of mathematical thinking; make use of team learning and develop team skills; engage students through the use of appropriate manipulatives and tools; use assessment that identifies student needs and enables appropriate modifications of

teaching programs.

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The teachers in the project found that other strategies that improved student attitudes and motivation included integrated learning and authentic applications, problem solving and investigations around good questions, and drawing upon students’ personal experiences.One point where students’ self concept and attitudes are at risk is during transition from primary to secondary schools. Fullarton (1998) identified four distinct groups of children:

a highly successful group who took transition in its stride; a poorly engaged and unmotivated group who continued to struggle; a group whose members were uncertain and not highly engaged in primary school, who

seemed to show higher levels of engagement in secondary school; and a group, unlikely to have been identified by their primary teachers as potentially having

problems at secondary school, who suffered declines in perceived control, engagement, coping skills, self-regulation and self-perception.

The first group is at risk of boredom and wasted time as teachers bring students from various feeder schools to a required standard. The second group continues to suffer from poor mathematical understanding as well as poor attitudes. The third group is well catered for. It is concern that Fullarton found that the fourth group is the largest.Teachers’ actions, and particularly encouragement, can have a positive influence on students’ development. Burnett (1999) studied the “self talk” of children in Years 3 to 7, finding teachers’ positive statements to be more influential than parents’ or peers’ comments and more influential than negative statements.It is clear from research evidence that many students’ problems are often related to poor language skills, and that that low achieving students experience difficulty relating to contextual word problems, rely on procedures and poorly-understood rules, and have not developed skills or had experience in rich mathematical thinking. Unfortunately, there is also often disconnection between realistic problems and the typical problem solving that is institutionalised in mathematics classrooms. Even when this is not the case, Lowrie (2007) found that the case-study participants assumed that what they knew about the real world was not useful or valid in solving school mathematics problems. In their work with student teachers, MERGA members are very aware of the anxiety levels of student teachers, teachers who are teaching mathematics “out of field”, and many primary teachers.The negative attitudes towards mathematics held by many primary pre-service teachers are well documented. Lack of confidence with subject matter knowledge has been proposed as one possible source of anxiety, and several recent Australasian studies have investigated, or attempted to remediate, this potential relationship (e.g., Southwell, White, Way & Perry, 2005; Wilson and Thornton, 2005a, 2005b).

In summary, it is clear that there has been a lot of research on ways of assisting students who may be disadvantaged in the learning of mathematics. These include Indigenous students, vulnerable early years and primary students, students with disabilities, rural students, and those who lack appropriate attitudes and motivation. There is no shortage of research-based evidence about intervention programming, even though there is evidence of Australia not providing well enough for these groups of students.

MERGA recommends that coherent and substantial programs be developed to attend to these areas of need, with nationally focused re-training of teachers of mathematics, provision of supporting personnel and resources especially through the early years and in rural and remote areas, and the provision of intervention-style assistance programs.

Classroom assessmentThere is great concern amongst MERGA members about broad-based state and national testing. We have no objections to sampling in order to assess standards and identify issues, but testing every child and the comparisons between children, teachers and schools that inevitably follows is

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counterproductive — especially when results are widely available. As a contribution to this submission, a MERGA member wrote:

Having recently returned from UK, I would like to contribute the following information.The impact of public data (comparison of schools in league table available on the internet) was the reason why in early March all classes I saw in Year 9 were doing practice papers and analysing practice papers for tests which would be held in mid-May. This applied even for teachers involved in a research project on implementing a maths program based on the Freudenthal Institute material. The researchers had expected and allowed for teachers to be using practice and past papers for at least 6 weeks. 

Such reports are not uncommon, and are supported by reports from UK colleagues. There is also evidence that pencil and paper tests, even if diagnostic tests are good indicators of mathematical knowledge or the ability to apply it. Classroom teachers report that they learn little from broad-based tests, and researchers are concerned about the ways that teaching-to-the-test has to narrow the taught curriculum.In particular, Australian citizens spend more time doing mental mathematics than using pencil of paper methods or calculators — and it has been found that the more time students spent doing mental maths, the better they use it and test for it (Paterson, 2004). As a result of an extensive Doctoral project, Peterson concluded that not enough mental maths is happening in secondary classrooms and recommends that students spend 15 minutes daily on mental maths. This has implications for lesson organisation and supporting resources such as software programs, but mostly for assessment practices. These do drive curriculum design! Further, there needs to be emphasis in teacher education, both at pre-service and in-service levels, on the importance of mental maths and ways of developing students’ appreciation, knowledge and skills in this area.

Significant developments in early years’ assessmentThere have been, however, recent improvements in teachers’ knowledge that resulted from major research projects on assessment. For example, there has been significant growth in school entry assessment. Major programs include the:

Early Numeracy Research Project (Victoria); Early School Assessment Project (New South Wales); and Early Years Assessment Project and School Entry Assessment program (South Australia).

Such programs have resulted in significant professional growth for teachers as they see evidence of children’s knowledge and map their development to a framework, see for themselves how young children think mathematically, consider how to use teaching approaches and strategies that draw on these skills, and consider how to help children to develop the next levels of skill and understanding. There is ample evidence that evaluation of children through one to one clinical interviews informs teacher’s work at this level of schooling (e.g. Pearn, 1996; Way, 1994).The discussion document asks for evidence that state and national assessment processes inform teaching practices at the classroom level. It is clear that formative observations of students as they work have an impact on teacher’s actions (see, for example, Mousley, 2003). This is often linked with questioning (Watson and Chick, 2001c). MERGA members have also researched how teachers encourage the use of intuitive strategies so that children’s meanings can be observed (Mulligan, 1990, Mulligan & Mitchelmore, 1997). A 5-year project of related studies with children in the first years of schooling examined their ability to use mathematical pattern and structure generalized across all mathematical content areas. Children in the lowest 20% on mathematical performance demonstrated, consistently, lack of structural thinking across mathematical tasks (Mulligan, Mitchelmore & Prescott, 2006). An assessment interview (PASA) identified levels of pattern and structure in the development of young children's mathematical thinking.

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Assessment of what children understand, as compared with what they can do, is not easy — partly because any mathematical understanding is relative, transitory, and bound to be partial. Teachers have to know what to look for and how to provoke exhibition of concepts. The research of Reynolds, Martin, and Groulx (1996) is relevant here. It demonstrated that while expert teachers were not aware of any particular indicators of understanding, their tacit knowledge enabled them to identify patterns of children’s behaviour that indicated growths of understanding. These researchers sourced teachers’ tacit knowledge using discussions about videos and seven “surface indicators of deeper understanding” in children were identified. In summary, these were:(a) changes in demeanor: changes in posture or facial expressions indicating excitement, obvious confidence, and willingness to share;(b) extending a concept: taking it further than expected;(c) making modifications: a shift from imitation to prideful ownership, action with new authority;(d) using a process in a different context: the capacity to see what they have not seen before;(e) using shortcuts: comprehending the bigger picture;(f) ability to explain: describing their thinking; and(g) ability to focus attention: persistent use of the new understandings.

(Reynolds, Martin & Groulx, 1996, p. 370, paraphrased)The most commonly used feature is the ability to explain (Mousley, 2003), so this suggests that use of videos in teacher education to train teachers to recognise the other features would be worthwhile.MERGA members wish to make the point that assessment should not be seen only as an activity of teachers. Self-awareness and assessment are important in learning mathematics, and metacognition is a useful basis for self-evaluation. Metacognition includes students’ declarative knowledge about their own understandings, self-regulatory monitoring and consequent decision-making, and self-knowledge about their performance and the usefulness of their new knowledge (Schoenfeld, 1995). Cheeseman and Clarke (2007) interviewed young about their mathematical thinking and asked them to reflect on their learning, finding that children could recall the impact that particular incidents had had, and many were able to give quite sophisticated accounts of their thinking and their learning at factual, procedural and conceptual levels.It is not always necessary to encourage children to verbalise their thinking in detail to provoke metacognition, as drawing their attention to the fact that they are thinking logically with a comment or a question can be just as effective (Williams, 1996). For instance, a teacher may use the prompt, “I think you are starting to see a pattern. Do you understand the reasons for it?” Encouraging the development and use of metacognitive thinking informally like this, or more directly, is a strategy that many authors have referred to in relation to supporting the growth of mathematical understanding (e.g., Mulligan, 1992; Schoenfeld, 1995; Siemon, 1993a, 1993b). Mousley (2003) discerned five complementary forms of metacognition used in the literature:

the expectation that children will reflect on and report solution processes in algorithmic situations or problem solving contexts (e.g. Ball, 1991b; Bell, 1993; Resek, 1990; Steffe, 1991);

encouraging self-regulatory reflection with questions such as “What if …?”, “How many ways …?” and “How do you know?” (e.g., Ashton, 1997; Gourgey, 1998; Mevarech, 1995);

deliberate training of students to report and discuss their own thoughts and behaviour (Johnson-Laird, 1983, Mevarech, 1995);

promoting cognitive conflict, where children realise the complexities of reality, and to test their mathematical understandings and skills against these (e.g., Davis, Sumara & Kieren, 1996; Hiebert et al., 1997; Steffe, 1991; Wheatley, 1991); and

use of an exploratory stage, when students become more aware of what they already know about a topic (Posner, Strike, Hewson & Gertzog, 1982; Von Glasersfeld, 1989).

These forms could be used as a focus for teacher education, as it takes concerted effort to integrate support and expectations for metacognitive practices into everyday instruction.

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In summary, there are many aspects of mathematical knowledge, skills and development that are not tested with pencil and paper tests. These include “mental maths”, investigations, and self-awareness of learning and the ability to use what has been learnt. While there is value in diagnostic testing, the current benchmark items do not test such aspects of mathematical achievement and tell teachers little that they do not already know about students. Most importantly, brad-based testing has the capacity to narrow curricula even further.

MERGA recommends broad-based sampling to assess standards and identify issues, banning comparisons between children, teachers and schools, and emphasis in teacher education ways of developing students’ appreciation, knowledge and skills in mental mathematics and investigations.

4. Teacher Education and Professional Development

The mathematics teaching standardsThe Discussion Paper acknowledges that AAMT has developed a set of Standards for Excellence in Teaching Mathematics in Australian Schools (AAMT, 2006). The research that led to this work (Clarke & Morony, 2002), and the advice and exemplars on the Standards website http://www.aamt.edu.au/standards/ illustrate effective practices and characteristics for both primary and secondary years of schooling.

MERGA recommends that these be made a basis nationally for the initial accreditation and higher-level reaccreditation of teachers of mathematics.

Characteristics of effective teachers and teacher developmentThe Discussion paper requests research evidence about the practices and characteristics of effective teachers of numeracy and mathematics schooling. Research evidence of this is far too expansive to summarise in a review submission. We draw the expert writers’ attention to the regular 4-yearly reviews of mathematics education research published by MERGA.The Early Numeracy Research Project identified the following characteristics of effective teachers and teaching of numeracy in the early years (McDonough, 2003; McDonough & Clarke, 2003).

Effective early numeracy teachers…

Mathematical focus

focus on important mathematical ideas make the mathematical focus clear to the children

Features of tasks structure purposeful tasks that enable different possibilities, strategies and products to emerge

choose tasks that engage children and maintain involvementMaterials, tools and representations

use a range of materials/representations/contexts for the same concept

Connections/ links use teachable moments as they occur make connections to mathematical ideas from previous lessons or

experiencesOrganisational style(s), teaching approaches

engage and focus children’s mathematical thinking through an introductory, whole group activity

choose from a variety of individual and group structures and teacher roles within the major part of the lesson

Learning use a range of question types to probe and challenge children’s thinking

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community and classroom interaction

and reasoning hold back from telling children everything encourage children to explain their mathematical thinking/ideas encourage children to listen and evaluate others’ mathematical

thinking/ideas, and help with methods and understanding listen attentively to individual children build on children’s mathematical ideas and strategies

Expectations have high but realistic mathematical expectations of all children promote and value effort, persistence and concentration

Reflection draw out key mathematical ideas during and/or towards the end of the lesson

after the lesson, reflect on children’s responses and learning, together with activities and lesson content

Assessment methods

collect data by observation and/or listening to children, taking notes as appropriate

use a variety of assessment methods modify planning as a result of assessment

Personal attributes of the teacher

believe that mathematics learning can and should be enjoyable are confident in their own knowledge of mathematics at the level they

are teaching show pride and pleasure in individuals’ success

These characteristics, of course, apply across all levels of mathematics teaching, including adult and tertiary classes. Further strategies that also apply not only to early years teaching are proposed and discussed in A good start to numeracy: Effective numeracy strategies from research and practice in early childhood (Doig, McCrae, & Rowe, 2003) as well as the many summative papers that are similar to The usefulness of value-added research in identifying effective schools (Young, 1999).However, it is clear from the research literature that effective teaching results in students developing a strong network of mathematical and everyday knowledge: referred to variously as “conceptual knowledge”, relational understanding” (from Skemp, 1976) and “connected knowing. It can be thought of as a connected web of knowledge, a network in which the relationships are as prominent as the discrete pieces of information. Students build an understanding of the relationships between concepts and processes that is linked to a wide network of ideas. There is significant research evidence to support this approach. Askew, Brown, Rhodes, Johnson, and Wiliam (1997) demonstrated the association between student achievement and teachers’ beliefs, with students in classrooms of teachers having “connectionist” orientations showing relatively high mean achievement gains. Ball (1993) showed how children in such classrooms develop a range of conceptual structures, then refine these into more generalisable understandings. It is important to note that Pesek and Kirshner (2000) found that teachers who aimed for both relational and instrumental understandings were less effective than those who focused on relational understanding only were. They concluded that “relational units appended to the existing classroom regimen may effectively be blocked from achieving the relational understanding sought” (p. 536).One strategy that has been researched and adopted widely, especially in the Netherlands and the UK for the teaching of addition and subtraction, is the empty number line. An important point about this tool to support the development of children’s mathematical thinking is that rather than using concrete materials that represent an adult view of place value, children are scaffolded as they work abstractly, so a shift to flexible and generalisable thinking is encouraged while emphasis is taken off specific representations. Some universities are providing pre-service and professional development to teachers of other curriculum areas to ensure they are aware of the numeracy demands of their subject areas. (See, for example Groves, 2001, where Deakin University’s Numeracy Across the Curriculum unit of study is described.)

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A further important issue for mathematics teacher education is for teachers to develop a sense of identity as a person who works mathematically and to appreciate the importance of mathematics knowledge and skills that they possess and use. Opportunities to develop positive approaches by working mathematically in the context of mathematics teacher education is essential (Owens, in press). MERGA members are not aware, though, of any teachers of other subjects being trained to operate in accordance with AAMT’s standards and this is of concern.Vital aspects of pre-service teacher education are the practicum and other forms of direct professional experience. It is of concern that while national guidelines suggest increasing the number of days spent in schools, and funding to universities has been increased, there is no recognition of (a) the costs involved in arranging for these placements, (b) the difficulty of finding enough schools teachers who are willing to host and supervise the students, and (c) the need to support students financially to gain experience in regional, rural and remote areas. The shortage of well-qualified mathematics teachers, who could demonstrate how to instill a love of mathematics into their students, is a compounding factor here.We wish to attract the writing team’s attention to papers and submissions from the International Commission on Mathematical Instruction (ICMI) Study, The Professional Education and Development of Teachers of Mathematics. The study will be of professional education and development of mathematics teachers around the world, including curriculum of mathematics teacher preparation as well as in-service mathematics teacher education practice and policy. The study had three phases: (a) the dissemination of a discussion document (ICMI, 2004); (b) a conference in May 2005; and (c) publication of the Study Volume – a report of the study’s achievements, products and results — by Springer in 2008.We also refer the writing team to the mapping of research into in-service professional development undertaken by McRae, Ainsworth, Groves, Rowland & Zbar (2001), who distinguished and discussed 4 models of professional teacher learning. They were:

Centralised training, with experts circulating among schools; Centralised short courses, with a slightly higher level of awareness of individual student

needs, with some use of train-the-trainer and “cascade” models. Whole schools’ pupil-free days with local control; and School-based work on the learning culture approach, with schools and teachers working

together to improve knowledge of pedagogy.This series has grown over time in the amount of grass-roots control, including identification of needs, attention to local conditions, and a focus on grass-roots support. A current trend is towards lesson study.

Lesson studyOne strategy for on-going professional development is lesson study. Much pedagogical content knowledge (how to teach particular concepts) in Japan comes from the study of specific lessons. Typically, groups of teachers in a school undertake self-guided study of particular lessons (kenkyuu jugyou or lesson research) where they study ways of teaching one particular lesson for several hours a week over many weeks. It is interesting that lessons studied are called jugyou kenkyuu (research lesson). This form of professional development is so common that a Japanese teacher quoted by Stigler & Hiebert (1999) claimed, “You won’t find a school without research lessons” (p. 111).The teachers from different levels discuss the basic concepts to be taught, and any points of interest such as those outlined for division and multiplication above. They discuss in detail the language and materials and activities it is possible to use to convey such concepts, appropriate ordering of ideas and useful connections that can be made, common misconceptions and mistakes and how best to make the most of these as part of the normal teaching and learning process, and even which numbers should be used in particular problems. Competing ideas about such details are raised by teachers as they study the lesson (usually videotaped, and sometimes with a transcription available) and teachers earn recognition for publishing a description of the lesson and a summary of different opinions about points arising during these discussions. The focus in planning lessons is consideration of the best ways for students to move from current development and where they are

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expected to be at the end of the teaching period. For me, this seemed similar to the “learning trajectory” concept portrayed by Cobb & McClain (1999) who proposed that teachers should actively consider an “instructional sequence (that) takes the form of a conjectured learning trajectory that culminates with the mathematical ideas that constitute our overall instructional intent” (p. 24).Lesson study seems to echo many of the characteristics of Japanese mathematics classrooms, including:

intensive, longer-term focus on objects of study; co-operative efforts to share knowledge and ideas; willingness to put forward thoughts in the knowledge that all can learn from their analysis; recognition that even the less-experienced can make worthwhile contributions and that

mistakes are a good source of learning; a sense of wanting the whole group to move forward; intrinsic motivation and relatively

autonomous work on a shared problem; and most importantly, and both mathematics lessons and this form of teacher development share a focus on the

development of deep understanding about the object of study.Isoda, Miyakawa, Stephens, and Ohara (2006) produced a major account of Japanese lesson study in mathematics: Its impact, diversity and potential for educational improvement. We commend this text to the Review’s expert writers because it explains how lesson study is generated, supported and endorsed in Japan and shows how it has been used as a long-term key to improved practice and outcomes. School systems, teachers, publishers, universities and supervisors value and make room for this collaborative effort. In particular, there is a discussion of how policy documents have introduced new themes or priorities that then became the subject of lesson study in schools.Japanese teachers whom we have contact with are surprised that Australian Departments of Education, re-registration Institutes, and schools do not require or support lesson study. One teacher put it, “How do teachers continue to learn about teaching?” Combined with the power of lesson study is the notion that professional learning needs to be on-going (i.e. over a period of months or even years) to allow for true reflective practice and changes in student outcomes to be seen. The value of the professional learning is teachers working in their own classes with their own students. This makes the professional learning immediately relevant, and combined with reflective practice it appears that changes in teacher’s beliefs and attitudes about the teaching of mathematics can occur.

Other teacher researchFrid and Sparrow (2007) point out the importance of teachers’ taking ownership over their own professional development.

Mechanisms for growth and change must ask teachers to act as their own change agents, while gently challenging ideas and fostering critical reflection upon ideas and experiences. Thus, “coming to know” as a professional is based upon ownership of ideas and related teaching practices, a form of professional empowerment. From an empowerment perspective, professional development is an educative process in which teachers make meaningful and thoughtful choices about their practices. (p. 297)

This point is also made well by Pritchard and Bonne (2007), who partnered teachers as reflective practitioners formulating their own interpretive frames through making teachers’ views explicit and exploring the complexities of their work in mathematics classrooms.

Gould (2005) has identified the need to reduce the “gap” that exists between research and practice in classrooms. Approaches that encourage teachers to carry out their own research in the context of their own classrooms, with the support of researchers, serve to validate their perspectives and enable greater insights into the complexities of teaching and learning. … Traditional views about the relationships of knowledge and practice, and the roles of teachers in educational change are

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challenged, … resulting in increased learning for both partners, and significant contributions to the knowledge base of teaching. (p. 621)

Pritchard and Bonne (2007) explored methodologies that enabled teacher researchers to gain insights into their current teaching practice when analysing aspects of their numeracy lessons. They provide evidence of the teachers having opportunities to develop a sense of ownership and control but that this was hindered by traditional relationships between and positioning of teachers and researchers with discrete roles. They took this into account in later stages of the project, with some success, so their report provides valuable guidelines for people involved in similar endeavours:

Support for further research that includes the teachers’ perspectives in the analysis of teaching practice is vital. To allow teachers to develop the research skills necessary to contribute their perspective in a meaningful and rigorous manner, teacher researchers need to be provided with:

· sufficient release time to examine their practice in depth, and to attend research meetings,· access to experienced researchers for support and guidance,· research forums for discussing ideas with other teacher researchers, and· interest and encouragement from management and colleagues within their schools.

Research questions that originate from teachers themselves can contribute to a closer alignment between research and practice. To enable them to have authentic ownership of research questions, involvement in the earliest stages of a research project needs to be encouraged. Teacher initiation of such proposals could be promoted by the inclusion of a research component into teachers’ job descriptions. Consideration also needs to be given to methods that enable teachers to have maximum ownership of processes throughout. (p. 629)

In fact, there are many reports of teacher-researcher activity in MERGA publications, and in most cases academics acted as facilitators. Most recently, these have focused on improving features of teaching and learning such as the following.

Questioning: e.g., Bonne & Pritchard (2007) explored types of questions with teachers and their purposes.

Argumentation: e.g. Brown & Redmond (2007) found that middle years teachers underwent longer-term changes in their teaching behaviours as a result of using collective argumentation to scaffold the teaching-learning relationship.

Teachers’ mathematical and pedagogical content knowledge: e.g. Cheeseman (2007) reported a teacher’s 4-year self-development journey.

Teachers learning to notice important teaching moments during classroom interactions, e.g. Davies & Walker (2005) who supported a group a teachers as they learned to note and make the most of critical incidents, and then Davis and Walker (2007) working with the teachers on their capacity to listen and respond appropriately to students’ comments and ideas.

Investigation and inquiry-based teaching: e.g. Goos, Dole and Makar (2007), who teachers observed teachers’ opportunities and hindrances in pursuing investigative teaching and assessment approaches; Makar (2007) whose teachers worked on developing their expertise, confidence, and commitment to teaching mathematics through inquiry.

With reference to teacher change, Rogers (2007) examined the teacher-researcher relationships with professional learning, classroom practices, teacher beliefs, and change in teacher attitudes. She researched a Victorian teacher who is part of a project in the Australian Government Quality Teacher Programme, a federal flagship initiative for supporting quality teaching and school leadership, recording changes in the teacher’s beliefs and attitudes resulting from project participation. Rogers found that when the teacher used new ideas and gained evidence of positive change in student learning outcomes, there was potential for permanent change — but student outcomes needed to be positive before a change in teacher’s beliefs and attitudes was observed. This demonstrates that professional development is a complex phenomenon and that most teachers care primarily about evidence of students’ progress and secondarily about professional advice from others. Rogers (2006) also reports that it is teachers acting as learners that provide effective modeling for their students’ own learning. Coupled with the opportunity to work with professional partners (i.e. other teachers) and even an external critical friend such as a mathematics educator,

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teachers remain at the centre of the learning, and hence retain ownership of their own professional development. The risk in assuming that that teachers’ professional development will inevitably have a quick impact on students’ learning is also suggested by a research report of Watson, Beswick, Brown, and Callingham (2007).

The practicumOne concern shared amongst pre-service teacher educators is the difficulty of translating theory into their teaching practice. Bobis (2007) provides a case study of a practice-based component of a primary teacher education program to gain insight into the type of experiences that assist beginning teachers to make this transition. A feature of the case was co-teaching, with colleagues working together at all phases of the teaching process, including planning and evaluation. Over the extended practicum experience, student teachers:

used more higher-order questions, integrating these throughout lesson plans improved directionality and purpose of learning activities developed more proficiency in requiring children to explain their strategies and to

communicate their reasoning orally or in writing explicitly addressed children’s misconceptions increasingly encouraged detailed explanations of concepts that prospective teachers

generally find more difficult to relate to children. Bobis proposed the following suggestions for assisting the translation of theory to practice

Alternating university-based tutorials with school-based practice; Using situated learning contexts to provide a secure environment in which to rehearse

pedagogical strategies and develop effective “scripts”; Co-teaching, to give student teachers opportunities to learn from each other and support for

experimentation, and to explore what to teach, how to teach it and how students learn best; and

Embedded formative assessment to allow shortcomings in planning and teaching to be addressed immediately. (p. 69, paraphrased)

Cavanagh and Prescott (2007) described some of the major constraints for pre-service teacher’s learning, and from experience we know that these are also a barriers to development in their first few years of teaching, particularly when a formal probationary period is used. The barriers noted during interviews included:

The dangers of trying something new:During the practicum, opportunities to re-imagine other forms of teaching mathematics are limited, largely because the pre-service teachers tend to focus almost exclusively on the technical aspects of teaching, especially classroom management and organisation. They plan lessons that are often tightly structured and predominantly teacher-centred because they believe that such an approach is more likely to discourage student misbehaviour. (p. 185)

Focus on the self:During lessons, they are more concerned with monitoring their own actions than attending to students, and often fail to notice whether any significant student learning is taking place. (p. 185)

Disjunction between theory and practice:… alignment between the community of the university methods course and that of the practicum school is severely restricted. (p. 185) … some pre-service teachers believed that the reform teaching approaches encouraged by university staff were more useful for high achieving students than the predominantly low-ability classes they were usually required to teach. (186)

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Limited modelling reifies the dominant hegemony:The pre-service teachers often recalled how the classroom practices of their supervising teachers fitted well with memories of their own time as high school mathematics students. The student-teachers reported that most of the mathematics lessons observed during their practicum followed a familiar pattern: reviewing the work from the previous day, some teacher exposition of new material, worked examples on the board, and individual seat work for pupils to practise new skills and procedures. The most common description reported by the pre-service teachers was one of “chalk and talk” lessons where pupils completed many textbook exercises, working predominantly on their own. (p. 186)

… they expected to find themselves in mathematics faculties much like those they experienced during their practicum: ones where traditional teaching approaches were the norm. (p. 187)

A common theme among the pre-service teachers was that when they started teaching in the subsequent year, they did not believe they would have much support from other mathematics staff members because most of their colleagues would not be accustomed to a reform style of teaching and therefore could not offer practical advice on how to implement it in the classroom. (p. 188)

Competing expectations:The expectations of university lecturers also had some influence on the pre-service teachers’ pedagogy during their practicum, but these were often dismissed as unworkable in the “real world” of the classroom. (p. 186)

… student-teachers also reported that they found it difficult to depart too far from the style of the supervising teacher because the pupils reacted against any change from the traditional classroom routines to which they had become accustomed. (p. 187)

… students, too, had certain expectations about the kind of lessons they would receive when they arrived for class and that they “expect a certain style of teaching in mathematics”, which typically meant a traditional approach. Thus, the pre-service teachers thought that it might be difficult to overcome their students’ demand for instrumental rules and procedures and teach for relational understanding using an investigative or discovery style. (p. 187)

Lack of “ownership” over lessons taught:All of the student-teachers in our study commented favourably on the fact that they would no longer have to contend with the difficulties associated with teaching classes that essentially belonged to another teacher. (p. 187)

In contrast to the university lecturers, the practicum supervising teachers were far more influential in shaping the participants’ teaching styles. … This was most apparent when the student-teacher devised a lesson plan focusing on group work or activity-based learning but the supervising teacher insisted that the plan be changed to a more teacher-centred method of delivery. (p. 186)

As Frid and Sparrow (2007) wrote, A problem facing pre-service mathematics teacher education is the challenge of preparing teachers to “break the cycle of tradition” of mathematics teaching and learning practices that centre on memorisation of facts, and practice of pre-set meaningless procedures, which promote a view of mathematics as lacking creativity, imagination, or critical thought [… yet …] Breaking the cycle of tradition will not occur unless graduate teachers are able to put into practice the non-traditional mathematics curriculum and pedagogical beliefs, ideas, and skills they developed in their pre-service programs.” (p. 295-6).

The practicum becomes an issue for hard to staff areas — remote, rural and lower SES areas in particular. It is not only difficult to place students in such areas (because they have on-going accommodation, casual jobs and relationships near city universities) but for academics to liaise with schools and the student teachers during placements.

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Distance also affects recent graduates’ ability to get on-going support and mentoring from university departments. Steketee and McNaught (2007) found that video conferencing helped overcome this latter barrier, as they supported newly graduated teachers with mathematics teaching and learning. All three schools concerned had identified that this lack of experienced teachers on staff, and small staff numbers, limited the capacity of the school to offer a mentoring program onSite — and indeed the project showed that they did not plan well and hence benefited from the support.Unfortunately there is little incentive for teachers to undertake significant programs of professional development. It is not recognised in rates of remuneration, is valued equally with short p.d. activities for re-registration, and is not likely to be undertaken by the teachers who are fixed in their ways and who need it most. HECS is not tax deductible and CSP places are limited, while full fee courses do not attract teachers. However, we recognise significant funding that has had positive outcomes in early years and middle years’ programs, resulting in much of the research and development reported above.

In summary, there is a lot that is known about effective professional development, and it is clear that what is needed is not the short courses that are used in many state-based programs and “curriculum days”. What is needed is longer-term engagement where teachers and researchers work together on improving planning, teaching and assessment practices in ways that will lead to consistent, on-going improvement of learning outcomes as well as teachers’ knowledge of mathematical content and how it is best learned. It seems that national and state authorities, teachers and teacher educators could work much more closely to this end. Hopefully, this review will be lead to some practical outcomes in this respect.

MERA recommends that this review, its findings, and its recommendations be taken seriously enough at national and state levels to result in a significant nation-wide program to improve teachers’ knowledge of mathematics and their understanding of evidence about how it is best taught and learned.

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Key idea ElaborationGuiding principles

Problem-solving: thinking, reasoning and working mathematically Mathematical thinking (including prediction, generalisation, abstraction, application

across contexts) Learning to think mathematically needs to underpin the whole

numeracy/mathematics agenda Exploring of relationships between useful applications and mathematical entities Focus on learning for the future; critical consumers of the world Written around big maths ideas A focus on connections between concepts Developed by leaders in the fields of mathematics curriculum, mathematics

learning and mathematics curriculumConsultation with a range of stakeholders

Stakeholders include including teachers, teacher educators, researchers, mathematicians & other tertiary, year 11-12 exam setters, employers, …

Sufficient time, resources and inclusive ways of working to develop a framework through strong consultation

Realistic and well-funded consultation timelines and processes A number iterations and involvement of a range of appropriate experts

Useful curriculum documents

A minimal framework, with scope for diversity and local relevance in interpretation Discretionary elements: needs to be strong enough to hold together, but flexible

enough to allow states to build on individual strengths A document that is only a benchmarking of skills is a waste of time Aspirational as well as specifying goals that the government commits to that all

students should achieve Emphasis on both what to teach and ways of approaching teaching Should not be too prescriptive—guidelines but must have examples of

implementationRecognises and supports student diversity and development

Consistent high expectations of students regardless of background High but achievable expectations, not lowest common denominator Attends to accessibility issues. Develops positive dispositions towards seeing, using and applying maths including

in other learning domains Open enough to cater for the diversity of children in any classroom Flexibility that allows teachers to match it with students’ needs and capabilities

Recognises and supports teacher diversity and development

Flexible enough to be adapted by experienced teachers but supportive of new teachers and those teaching out of their areas of expertise

Long-term support for professional learning, including materials, time, collaborative opportunities and partnerships, p.d., etc. for teachers — and especially novice teachers and ‘out of area’ teachers

The implementation is more important than the content. Teachers of mathematics need to be able to make connections across it and take

advantage of “teachable moments” Supporting the development of positive dispositions towards the learning of maths Should be accompanied by high levels of resources for professional development

Researched based

Fluid enough to adapt as research reveals further understandings Research on the content and pedagogical influences of learning Australian plus international research

Appropriate forms of assessment

Appropriate and varied, reflecting the above principles Diagnostic with feedback mechanisms Items able to measure ability to work mathematically Need to maintain States’ flexibility and accountability

MERGA recommends that the features outlined above be used to underpin every national development in school mathematics and teacher preparation/support in the future.