The 8 Aboriginal Ways of Learning is a pedagogy framework for embedding Australian Indigenous perspectives into the classroom by emphasising Indigenous learning techniques across all subjects. The framework was developed by the James Cook University School of Indigenous Studies, in collaboration with the Western New South Wales Regional Aboriginal Education Team and DET staff in 2007-2009. It postulates that Indigenous perspectives and knowledges in the classroom are not about introducing “indigenised content”, but rather by practicing a pedagogy informed by “Indigenous processes” of knowledge transmission and identity.
The 8 way learning model explained at the Australian Indigenous College
The 8 Aboriginal Ways of Learning pedagogical framework highlighting the connections to axiology, ontology, epistemology and methodology.
The 8 Aboriginal Ways of Learning therefore is quite a unique approach linking pedagogy with with:
axiology – Indigenous ways of valuing, in particular particular cultural protocols, systems and processes
ontology – Indigenous ways of being, in particular cultural protocols of behaviour
epistemology -Indigenous ways of knowing, in particular cultural protocols such as initiation
methodology – Indigenous ways of doing, such knowledge transmission through storytelling
What makes the 8 Ways pedagogical framework so compelling in the Australian main school context is that it is informed by a large overlap between Indigenous and non-Indigenous learning processes. There is much common ground in the value to all students of teachers including:
Great teachers are storytellers that can also teach through narratives and songs. Sharing stories is also an important tool to connect with each other.
The visualisation of pathways of knowledge is also called concept mapping. According to the latest Hattie Effect Size update, there is good evidence that the development of learning maps is among the most effective teaching practice.
Six out of the seven learning styles are non-verbal and include the kinesthetic and interpersonal approaches. In the words of the 8 Ways pedagogical framework, “we see, think, act, make and share without words“.
Symbols and images:
symbols and images
Providing visual cues, including symbols and colour-codes in learning routines can significantly help students with hearing impairment and social communication difficulties such as Autism Spectrum Disorder. This point addresses that learning is often visual and supported by objects, images, symbols, signs, art and metaphors to explain concepts and content.
Great teachers make teaching content relevant by connecting it to the world in which their students live. This includes teaching lessons about the local environmental, highlighting traditional knowledge and connection to the land, including climate, fauna and flora, as well as the history about a place. The aspect of land links is also an important factor for training students’ ability of acute observations as required in Science and often best taught in nature. Land linkst is arguably an important part of teaching Sustainability, one of three cross-curriculum priorities in the Australian Curriculum.
Non-linear thinking can be translated as Critical and Creative Thinking, a key general capability to be developed in the Australian Curriculum. This is because lateral thinking or what we also call “thinking outside the box” is the foundation of innovation. In the words of the 8 Ways pedagogical framework, “we put different ideas together and create new knowledge“.
Deconstruct and reconstruct:
deconstruct and reconstruct
On the one hand this pedagogy describes the gradual release of responsibility instructional framework, where teachers first unpack new knowledge with the students by means of modelling and scaffolding, then encourage shared and individual practice. It also emphasises the importance of holistic knowledge, always anchoring new content in prior student knowledge, by “work[ing] from wholes to parts“.
The 8 Ways pedagogical framework is hardly radical, allows for broad practical applications in a wide range of local school contexts and does not prescribe any particular or commercial classroom materials and training requirements. It also sidesteps possible constraints in curriculum content choices, which prior to the introduction of the Australian Curriculum and in particular the cross-curriculum priority Aboriginal and Torres Strait Islander histories and cultures had the potential to constrain Indigenous education in mainstream schools and classes. It also recognises that every school community and every local Indigenous culture is different, and that one-size-fits-all prescriptions are problematic and limiting. Rather than being explicit about teaching content (e.g. Indigenous lesson units by commercial providers such as sharingculture.com), classroom and school management styles (e.g. Stronger Smarter developed by Chris Sara), particular teaching styles (e.g. Direct Instruction advocated by Noel Pearson), or conceptual frameworks for constructing individual teaching and learning episodes (e.g. Uncle Ernie’s framework), the 8 Aboriginal Ways of Learning approach promotes culturally sensitive and informed ways of teaching and learning practically anything in ways that benefit all students.
The challenges with such an open, inclusive, and non-commercial pedagogical framework are in professional adoption and meaningful translation and applications. The 8 Ways is only one of an ever growing number of culturally-informed pedagogical approaches advocated to Australian teachers, and while not necessarily in conflict with the others risks of being only superficially adopted and watered-down in practice to the point where it would make little difference to Indigenous students. Without any specific units, class material, applied recommendations in areas such as EAL/D, it will be easy for teachers to endorse it in theory but not in practice. This is even more likely within the non-commercial context of this framework, as it will not be actively marketed by consultants for professional development to schools.
The AITSL professional teacher standards 1.4 (strategies for teaching Aboriginal and Torres Strait Islander students) and 2.4 (understand and respect Aboriginal and Torres Strait Islander people to promote reconciliation between Indigenous and non-Indigenous Australians) frame much of the professional classroom practice in relation to Indigenous students, and teaching Indigenous perspectives and understanding. The 8 Ways approach has the potential to directly inform the teaching practice by offering a rich, culturally-informed framework to design teaching and learning episodes and activities. While not supporting this process with specific teaching material or recommendations for lessons on Indigenous people, culture, country/places which would inform teaching about the reconciliation process between Indigenous and non-Indigenous Australians, the 8 Ways framework supports meaningful cross-cultural dialogue and shared learning experiences.
Personally, I find the comprehensive nature and non-prescriptive approach of the eight interconnected pedagogies very appealing, because they can easily be to be applied across all curriculum areas. The are also an excellent starting point for discussing curriculum and pedagogy choices with the local Indigenous and non-Indigenous school community. This flexibility also ensures compatibility to work with any (future) version of the Australian Curriculum, across changing cross-curricular priorities, different whole-school approaches and communities, by offering pedagogical approaches that benefit all students and make real connections to local knowledges and practices.
Gifted and talented children are characterised by outstanding abilities and potential for high performance. The realisation of these talents however requires differentiated educational intervention and support. With 10% of the student population estimated to be gifted (Gagné, 2015), gifted students can be found in most classrooms. However, in the Australian school system, an estimated 50% of gifted students typically remain unidentified and underachieving, with up to 40% preliminary dropping out (Parliament of Victoria, Education and Training Committee, 2012).
This post provides information on how to identify gifted students, recognise their strengths and needs, and respond with responsive curriculum differentiation and teaching strategies.
Identifying Outstanding Student Potential
“High potential will not be realised if it is not identified or if it goes unrecognized” Merrotsy, P. (2015, p.256).
The identification of gifted students can be heavily biased by race, socio-economic background and gender (Bousnakis, et al. 2012; Coleman & Shah-Coltrane, 2015). Often, bright students who stand out as “teacher pleasers” are misidentified as gifted, while gifted students become either invisible or show challenging behaviours (Merrotsy, 2015). Gifted students are generally identified by performance in academic achievement tests (e.g. Scholastic Aptitude Test) and cognitive tests (i.e. WISC-V, Stanford-Binet 5). A more integrated approach, such as the ‘Coolabah Dynamic Assessment’ (see Resources GERRIC Module 4, Specialisation), is recommended to identify gifted underperformers (Bousnakis, et al. 2012).
Look out for the following typical characteristics in gifted students:
Strong reasoning, knowledge retention and fast processing skills
Large vocabulary (sometimes multilingual) and advanced reading interests
Ask many questions and display broad knowledge and original, often unusual, thinking
Heightened emotional sensitivity, advanced ethical and existential reasoning
Discrepant achievement pattern across subjects and between school/after school activities
Question authority and can be uncooperative, stubborn, cynical and frustrated
Can be disorganised, absent minded, and show low interest in detail
Recognising Students Strengths and Needs
Giftedness is characterised by asynchronous development of chronological, mental and emotional age. Heightened intelligence is just one dimension of gifted children. Dabrowski further mentions the common heightened sensitivities and intense behaviours (Alias, et al. 2013).
Recognise the following strengths and needs of your GS :
Intellect – drive to want to learn vs. relentless questioning, the need to understand
Psychomotor – increased psychomotor awareness vs. need to engage hands, move body
Sensory – susceptibility to touch, sound, smell, light vs. overstimulation
Imagination – creativity, making connections vs. need to test unusual approaches
Emotions – feeling deeply, moral awareness vs. overwhelmed and existentialist angst
Common barriers to the intellectual and psychological well-being of gifted students include a lack of trust in the educational system and teachers, social pressure from family and peers to blend in (‘forced-choice dilemma’), and disengagement. Often gifted students abilities and needs are not recognised, or only within the context of special learning, social, emotional and behavioural difficulties (invisible and twice exceptional gifted students) (Merrotsy, 2015). The development of a positive, multidimensional self concept is often at the heart of gifted education, in order to develop self-efficacy, engagement and persistence (see Resources SENG webinars).
Educational Intervention Strategies
Appropriate educational intervention is required to support gifted students in developing their potential (Gagne´, 2015, Fig. 1). These include the provision of a challenging, enriched and differentiated curriculum, and a supportive learning environment. Maker’s (2005) updated recommendations on gifted education differentiate four dimensions of curriculum modifications:
Content – frame content in integrated, interdisciplinary ways organised around central ideas and the study of people and arts
Process – accelerated curriculum with emphasis on self-directed learning and discovery, variety and choice, metacognition and complex problem solving skills
Product – encourage working on real problems that require information transformation and results in unique products for real audiences
Environment – provide learning environments rich in resources, encouraging difference vs. conformity, independent vs. teacher-centred learning, physical and psychological flexibility
Online resources for teaching gifted and talented students in Australia
Six age-differentiated modules by the Gifted Education Research and Resource Centre, University of New South Wales, including on identification, social and emotional development, underachievement, curriculum differentiation and developing programs and provisions for gifted children
The Australian Association for the Education of the Gifted and Talented resource list for teachers including a link to the “Night of Notables”, a widely-used program catering for gifted children
Alias, A., Rahman, S., Majid, R. A., & Yassin, S. F. M. (2013). Dabrowski’s overexcitabilities profile among gifted students. Asian Social Science, 9(16), 120-
Bousnakis, M., Burns, T., Donnan, L., Hopper, S., Mugavero, G., & Rogers, K. B. (2011). Achievement Integrated Model: Interventions for Gifted Indigenous Underachievers. Giftedness From An Indigenous Perspective 11, 43-77
Coleman, M. R., & Shah-Coltrane, S. (2015). Children of Promise: Dr. James Gallagher’s Thoughts on Underrepresentation within Gifted Education. Journal for the Education of the Gifted, 38(1), 70-76.
Gagné, F. (2015). Academic talent development programs: a best practices model. Asia Pacific Education Review, 16(2), 281-295.
Aboriginal and Torres Strait Islander Histories and Cultures
‘Aboriginal and Torres Strait Islander histories and cultures‘ is one of three cross-curriculum priorities of the Australian Curriculum taught through the subjects disciplines. The cross-curriculum priorities were nominated and adopted by the Council of Education ministers translating the Melbourne Declaration on Educational Goals for Young Australians for the new Australian Curriculum. The aim of this cross-curriculum priority is to include Indigenous Australian perspectives and knowledge into all disciplines where relevant and applicable. On the one hand, this is to help Aboriginal and Torres Strait Islander students to see themselves better reflected in the national curriculum and become more engaged and empowered in their education. The other important objective is for all Australian students to participate in a process of reconciliation, by developing a deeper understanding of, and more respect for, Indigenous Australian Peoples, cultures, knowledges, beliefs and languages.
However, there are no content descriptor referencing this cross-curriculum priority in English (except for Year 8 ACELT1806), Mathematics, Science, Technologies, Health and Physical Education! In LOTE, there is an optional provision for separate first language, language revival and second language learner pathways for Aboriginal Languages and Torres Strait Islander Languages, and there are Options content descriptors in the Year 9-10 Work Studies. Objectively, the Indigenous cross-curriculum priority is therefore hardly a compulsory part of the core curriculum. However, within all subjects including Mathematics and Science most year levels provide at least one meaningful link and examples to the cross-curriculum priority in one or more elaborations of one or more content descriptors. These elaborations are optional, so teachers can choose whether or not to take up these opportunities to include the cross-curriculum priority in their teaching and learning units. In conclusion, by following the Australian Curriculum schools and classroom teachers are very much on their own in deciding whether or not to include the Indigenous cross-curriculum priority content into their lessons beyond the HaSS and Arts lessons.
The Australian Curriculum icon for Cross-curriculum priority Aboriginal and Torres Strait Islander histories and cultures
Embedding Aboriginal and Torres Strait Islander Perspectives (EATSIPS)
The ‘Embedding Aboriginal and Torres Strait Islander Perspectives’ (EATSIPS) is a framework initiated by the Queensland Government Reconciliation Action Plan in 2009-2012, with the aim to close the gap between non-Indigenous and Indigenous students’ achievements. The framework comprises three components:
classroom ethos, and
Explicit links to the national curriculum are drawn in four action areas:
curriculum and pedagogy
organisational environment, and
professional and personal accountabilities.
In appendix 2, EATS lists strategies for teachers to implement a culturally-appropriate curriculum, and to make the best use of opportunities towards embedding Indigenous knowledges and perspectives in the planning, delivery, assessment, moderation, reporting and evaluation processes. A particularly useful tool to develop measurable goals and gage success in terms of its implementation is a checklist with defined targets. For example, in the curriculum and pedagogy section, the vision includes:
culturally appropriate curriculum units connecting to the local area and histories, where possible making Indigenous knowledges and perspectives explicit
catering for all learning styles and backgrounds in curriculum delivery and pedagogy
celebrating local Indigenous stories, oral traditions and languages
critically reviewing teaching and learning resources (e.g. for authenticity, balanced representation, accuracy, exclusion of sacred content), and
sharing successes with the community
The Queensland Government EATSIPS framework
In conclusion, the EATSIPS framework provides schools and teachers with practical advice and guidelines towards implementing the opportunities that the Australian Curriculum cross-curriculum priority ‘Aboriginal and Torres Strait Islander histories and cultures‘ provides. EATSIPS further extends the curriculum by taking a more holistic approach towards developing culturally-appropriate personal, class and whole-school approaches towards teaching about and for Australian Indigenous Peoples.
Australian society is culturally and linguistically diverse, with languages other than English spoken in many homes and communities across the country. As a result, significant numbers of students enter the schooling system learning English as an additional language or dialect (EAL/D). Together with all other students, they are required to develop advanced language and literacy skills to fully participate in the curriculum and engage in increasingly higher-order thinking. The educational goals for Australian students as outlined in the Melbourne Declaration on Educational Goals for Young Australians (Barr et al. 2008) underline the role of teachers in addressing the needs of EAL/D students by requiring all schools to promote equity and excellence, and to empower all students to become successful learners as well as confident, creative, active and informed individuals. The Australian Council of TESOL Associations’ (ACTA) EAL/D elaborations of the Australian Professional Standards for Teachers highlight how pedagogies informed by the needs of EAL/D students play into all three domains of the teaching profession (ACTA, 2015).
Therefore, a review of recent theoretical and empirical research on pedagogical issues as relating to teaching EAL in Australian mainstream classrooms is essential to inform teachers with a better framework and best practices to address EAL/D students’ needs. In the following annotated bibliography I selected and reviewed five important journal articles on this topic:
Toni Dobinson, a lecturer with the School of Education at Curtin University, and Sylvia Buchori present a qualitative case study on selected Australian primary teachers’ knowledge and perspectives on catering for EAL/D students in mainstream classes. Reviewing relevant literature, the authors develop the argument that EAL/D students require specific guidance and support in academic subject- matter language acquisition. This includes structured implicit and explicit learning opportunities, appropriate “linguistically responsive” pedagogies (p.35), a multi-lingual mainstream literacy education inclusive of home languages, and teachers that serve language needs rather than act as “teachers of content” (p.36). The research part is based on interviews and class observations of four teachers and illustrates obstacles and well-intended but contra-productive pedagogical pitfalls. These include a lack of meaningful home language provision, strong beliefs on the benefits of monolingual classrooms, linguistically uninformed instruction, exclusive and deficit-focused ability grouping, and little explicit English language scaffolding. As solutions, the authors recommend teaching EAL/D- informed pedagogies to pre-service teachers and increased collaboration between mainstream ‘content teachers’ and specialist EAL/D teachers in the development of unit plans and differentiation strategies. The paper provides useful insights into the practical challenges faced by Australian primary school teachers in addressing EAL/D students’ needs. The authors demonstrate how specific knowledge on how to teach English as a new language is important and illustrate what can go wrong.
While not providing much practical pedagogical advice per se, their conclusion that a change in mindset away from content knowledge-only teachers towards discipline-knowledge and literacy teachers is convincing, and can inform our pedagogical practice in many Australian primary schools.
In this much-cited paper, Pauline Gibbons, Adjunct Professor at the University of New South Wales with extensive professional experience teaching and lecturing on EAL/D, makes the case for ‘high challenge, high support’ pedagogies for EAL/D students. She recommends combining an intellectually challenging curriculum with language scaffolding, which is essential to develop academic language and literacy across the curriculum. Based on collaborative research between university staff and primary school teachers in New South Wales, Gibbons sets out to define both the characteristics of intellectually challenging mainstream classrooms and the needs of EAL/D students. Challenging classrooms provide opportunities for students to engage in higher-order thinking with discipline-specific key ideas and concepts. This helps to transfer learned information to new contexts through inquiry-based learning, and to construct individual understanding through active participation and substantive conversations. For EAL/D students, information-transfer exercises, such as accessing and producing meaning from multiple sources of texts, are important but linguistically demanding. Gibbons offers pedagogical advice on how to create a supportive environment for EAL/D students, i.e.:
providing students with authentic contexts for collaborative inquiries and problem solving
explicit whole text-embedded teacher modelling of registers and genres, and
creating opportunities for EAL/D students to practice and contribute, because “[s]tudents learn […] about language in the context of using language” (p.171).
Gibbons provides a useful framework for teaching EAL/D in mainstream classes by highlighting the benefits of pedagogies that provide high cognitive challenges and high levels of differentiated support for all learners. Her approach to focus on EAL/D students’ potential provides a critical non-deficit perspective. While focusing on the bigger picture, however little practical advice and scaffolding strategies on EAL/D are forwarded.
Jennifer Hammond, Director of the former Centre for Language and Literacy, University of Technology Sydney, investigates the Australian Curriculum (AC) from the perspective of how EAL/D students’ needs are explicitly and implicitly recognised, and how specific EAL/D pedagogical approaches are positioned. Hammond first outlines the needs of EAL/D students in relation to the curriculum:
knowledge about language, literacy and language development, i.e. mastery of academic language registers and discipline-specific literacy,
intellectual challenge and ‘deep knowledge’ through high teacher expectations for all students,
planning and implementation of support programs providing required language scaffolding.
Hammond next summarises the hopes and concerns in the AC v3 for EAL/D students and teachers. Hopes are in the rejection of alternative/simplified curricula for EAL/D students, instead targeting equity through high intellectual challenge in mainstream education. A concern is that equity through challenge only works if teachers provide EAL/D students with targeted language and literacy support to access all areas of the curriculum, placing the onus on discipline teachers to also act as language teachers and scaffold for EAL/D students. This responsibility was not made explicit in AC v3, where the development of explicit knowledge about language is primarily placed in English. In the AC v8, literacy is more prominent and as a ‘general capability’ at the core of the national curriculum that needs to be addressed in all learning areas, making pedagogical knowledge around teaching language and literacy an important professional development area for many teachers.
Hammond’s critical review has been influential in emphasising literacy across all areas of the curriculum in later versions of the Australian Curriculum. It also sets the scene for discussing the roles teachers need to fill, and consequently the pedagogies teachers need to explicitly teach language and literacy skills in all learning areas.
This paper complements Hammond and Gibbons’ widely cited paper (2005) based on the same collaborative ESL scaffolding action research project. Michell, writing as Senior Education Officer with DET NSW, develops a contextual, multi-modal model of instructional scaffolding to address linguistic and cultural needs of EAL/D students in mainstream classrooms. The model is grounded in socio-cultural theories incorporating both intellectual (task-enabling support) and social semiotic (language-mediated co-regulation) perspectives, and is informed by the analysis of authentic classroom practice. The intellectual aspect involves instructional scaffolding along the zone of proximal development trajectory, managing task complexity and focus to cognitively challenge the student to learn, while providing the support required. Along this trajectory, support evolves from more explicit modelling towards guidance and allowing the student to take more control. The social semiotic aspect focuses on interactional dialogue providing students with the emotional support to fully participate and persevere, as well as opportunities to advance academic thinking and expression. The analysis of observed instructional scaffolding provides detailed insight into how lead teachers apply and inform the theoretical framework, including the resources they routinely draw on. The authors summarise the contextual pre-requisites for scaffolding.
The paper contributes to the investigation of EAL/D pedagogies by providing a comprehensive and detailed model of scaffolding, informed and illustrated by classroom observations. The promoted approach of collective instructional EAL/D scaffolding is particularly informative and useful in the context of inclusive mainstream school settings in Australia. The paper complements the “network model of scaffolding” approach by Hammond and Gibbons (2005), which highlights scaffolding micro- and macro-level teacher choices.
Joel Windle and Jenny Miller, senior researchers at the Faculty of Education, Monash University, investigate the rate of implementation of recent EAL/D pedagogy frameworks by teachers in Victorian secondary schools that receive funding for low-literacy refugee-background (LLRB) students. They look at the explicit teaching of academic language, using students’ prior knowledge and the careful sequencing of learning through phases. Refugee students are an important category of EAL/D students who often lack literacy skills in their first language and have little prior experience of western education. The authors categorise EAL/D pedagogies into five broad categories each with examples of relevant strategies:
attention to comprehensible input
direct and explicit teaching of language
focus on metacognitive skills and strategies
focus on critical and creative skills.
The sixty-one teacher participants reported on their routine implementation of those strategies. Accordingly, teachers were more likely to engage in strategies that demanded an active role of themselves (teacher-focused activities) rather than providing students with opportunities to practise language through student inquiries and content generation. Also, scaffolding at the level of genre or text-type features is rarely implemented, in particularly in learning areas other than English. The authors conclude that teacher professional development activities need to focus more on building student autonomy through peer-supported practice, as well as on language and literacy scaffolding in learning areas other than English.
While the authors make LLRB students a focus of their inquiry, little insight is provided on pedagogies that might particularly benefit this EAL/D category. Instead, the paper investigates how teachers of these students draw on a range of general EAL/D strategies. Despite this limitation, the paper is included for its useful tabular overview of recent language and literacy strategies and the ranking by teachers.
The starting point for this annotated bibliography is Hammond’s 2012 review of the Australian Curriculum as the national framework and space in which EAL/D support in mainstream classrooms takes place. It informs on the legitimacy of EAL/D pedagogies and emphasises the importance of making teaching of language and literacy more explicit in all discipline areas. Indeed, changes to the curriculum since 2012 indicate that prescriptive content knowledge made space for more student-centred pedagogies and an emphasis on literacy, including phonics and phonemic awareness in English (Australian Curriculum, Assessment and Reporting Authority, 2016). These changes directly address her voiced concerns and needs of EAL/D students.
The curriculum developments are in line with Gibbons’ (2008) call for ‘high challenge, high support’ pedagogies that combine an intellectually challenging curriculum with language scaffolding for EAL/D students, and with Dobinson and Buchori’s (2016) advise for all subject area teachers to develop discipline literacy pedagogies and work more closely with EAL/D specialists. It is Windle and Miller (2012) that provide insights into EAL/D pedagogies and strategies practised by Australian teachers. Their tabular overview of recent EAL/D language and literacy strategies is a useful starting point to investigate EAL/D pedagogies in more detail, and it creates awareness around strategies currently undervalued in practice and possible reasons why.
Against this background and insight into the Australian landscape of EAL/D pedagogies, Michell and Sharpe’s (2005) detailed model of scaffolding, informed and illustrated by classroom observations, adds critical detail and practical examples. Their focus on collective instructional EAL/D scaffolding is particularly useful in the context of the inclusive mainstream classrooms in which teaching EAL/D takes place in Australia. However, it is important to note that their model is only one among other competing and complementing approaches, as highlighted by Hammond and Gibbons “network model of scaffolding” (2005) in the same volume.
Mastery of fractions is the foundation for many more advanced mathematical and logical reasoning skills, including proportional, probabilistic and algebraic thinking. The degree of early year fraction understanding often correlates with secondary school mathematical achievement (Siegler, Fazio, Bailey, & Zhou, 2013). At the same time, fractions present a wide range of teaching and learning challenges that have been the subject of educational research (Petit, Laird, Marsden, & Ebby, 2015).
In the first part of this post, issues surrounding the teaching and learning of common fractions are described and linked to teaching and learning strategies that can address these. In the second part, implications for the teaching and learning in diverse classrooms are investigated and addressed by the Universal Design for Learning (UDL) framework, with particular reference to opportunities that modern information and communication technology (ICT) can offer. Drawing on both parts, a logical sequence is developed detailing conceptual and procedural steps for teaching and learning the fraction equivalence concept.
Issues surrounding the teaching and learning of common fractions
In primary school, learners move from non-fractional, through early fractional and transitional strategies, to mastery in applying fractional knowledge to magnitude, unit fraction and benchmark reasoning, and in operations (OGAP, 2012). In the Australian Curriculum, teaching and learning of fractions starts in Year 1 with content descriptor ACMNA016 “recognise and describe one-half as one of two equal parts of a whole”, and it progresses to Year 6, where students are expected to have developed procedural fluency in all operations with fractions, decimals and percentages, with the capacity to solve authentic problems (ACARA, 2017).
Fractions, ratios and proportions are the most cognitively challenging concepts encountered in primary school mathematics (Booker, Bond, Sparrow, & Swan, 2015). For students, fractions often mark the transition from concrete to formal operational mathematical thinking (Siegler et al., 2013), where numbers do not anymore relate to whole objects, or the size, shape and arrangements of their parts, but instead to part-whole relationships between two quantities composed of equal parts of a whole (Pantziara & Philippou, 2012). One difficulty in expanding whole-number to rational-number thinking is that both share overlapping cerebral processing areas in the intraparietal sulcus of the prefrontal parietal cortex (Siegler et al., 2013). Additional difficulties are encountered with the notation system used to represent fractions (Brizuela, 2006). Explicit teaching of fraction notation is essential, since “one whole number written above another whole number, do not transparently communicate the meaning of fractions” (Gould, 2013. p.5). The relational action associated with the symbols is not an intrinsic property of the symbols. Learners first need to experience the symbols as an expression of the relational outcomes of actions that they have carried out or observed (Dörfler, 1991). Finally, there is the concept of changing units, where one whole can refer to both multiple objects or composite units within a single object; partition fractions or quantity fractions. Students need to learn to move between different representations, including multiple symbols referring to the same amount (Booker et al., 2015).
In teaching fractions, it is essential to explain and establish fraction terminology first, explicitly addressing language and conceptual misunderstandings that surround rational-number thinking. The links between terminology, symbology, notations and concepts such as whole-number and part-whole relationships must be established before moving on to mathematical operations involving fractions. Mastery requires that students develop both conceptual and procedural knowledge and understanding of fraction concepts (Pantziara & Philippou, 2012). Therefore, teachers need to value and at least initially prioritise deep conceptual understanding over automatic procedural skills (Booker et al., 2015).
Visual models are a central component in teaching fractions at all stages of conceptual development, rational-number thinking, procedural and operational problem solving (Petit et al., 2015). Provision of a variety of visual representations of identical fractions that differ in perceptual features, such as the location and shape of shaded areas (numerator), were demonstrated to be important in the development of a multi-dimensional understanding of fractions. However, it is important that teachers guide learners in developing the knowledge about how visual representations relate to the fraction concept (Rau, 2016).
There are three common visual fraction models: linear, area, and discrete. These can be taught using a variety of representations (e.g. rectangular and circular segments, arrays, object collections) and physical and virtual manipulatives. Recent research into cognitive numerical development highlights the importance of teaching students that fractions represent magnitudes that can be located on a number line. Number lines, where equal parts are defined by equal distance, can serve as a conceptual bridge between whole numbers, proper, improper and mixed fractions, decimals and percentages, and highlight the concepts of equivalence and continuous quantities of fractions (Booth & Newton, 2012; Siegler et al., 2013). Gould recommends focussing on the linear aspects of fraction models as the primary representation of fractions in younger years (2013). Nevertheless, traditional area models, where equal parts are defined by equal area, continue to play an important role in the conceptualisation of numerator and denominator, fraction division, the relationship between unit of measure and reference unit, and equivalence (Lamberg & Wiest, 2015; Booker et al., 2015). Discrete models or ‘set of objects’ arrays, where equal parts are defined by equal number of objects with countable sets and subsets of discrete entities, visualise the mapping of distinct countable sets onto numerators and denominators (Rapp, Bassok, DeWolf, & Holyoak, 2015) and help students to understand equipartitioning (Petit et al., 2015).
All three visual fraction models can be used in different learning modes, including group discussions (verbal, aural), kinesthetic activities, and even through music (Courey, Balogh, Siker, & Paik, 2012). Physical manipulatives are a valuable resource stimulating hands-on learning that can make abstract mathematical ideas more tangible (Petit et al., 2015). Access to a variety of representations and activities support students in building the foundations for solving complex questions and real problems that involve rational-number thinking which cannot be achieved by rote learning alone.
Learners need guidance and practice to expand their conceptual numerical understanding to include rational numbers (Petit et al., 2015). Procedural fluency and algorithmic operational problem-solving skills are best learned by moving back and forth between conceptual and procedural knowledge and practice. Individual students have different learning styles and learning preferences. Student diversity can be accommodated by empowering learners to make choices between different activities and task designs, including group, paired and individual work, different modalities and types of questions, resulting in increased motivation and persistence (Landrum & Landrum, 2016). A degree of choice of tasks, task sequence and stimulus can be introduced into the classroom through blended learning, where students engage part-time with online content and instructions using learning platforms such as Mathletics (see below). Blended learning also provides a degree of flexibility over time, place, path and pace, and can be implemented as station-rotation, flipped classroom, or flex model among others (Staker & Horn, 2012), depending on the opportunities and constraints of individual teaching and learning environments.
There is also a cultural dimension to how students learn mathematics in general and fractions in specific. Mathematics is a cultural construct with its own epistemology. It cannot simply be assumed to constitute a “universal language”. Indigenous Australian mathematician and head of the ‘Aboriginal & Torres Strait Islander Mathematics Alliance’ Chris Matthews developed a model for culturally-responsive mathematics that links students’ perceived reality with curriculum mathematics through a hermeneutic circle of abstraction and critical reflection based on practical problem-solving (Sarra, Matthews, Ewing, & Cooper, 2011).
It has long been argued that Indigenous Australian students prefer kinesthetic learning experiences with physical manipulatives, narrative learning, valuing group discussions and explicit guidance (Kitchenham, 2016). It is therefore important to link formal mathematical concepts to something concrete endowed with real meaning. In reference to the Maths as Storytelling (MAST) pedagogical approach (Queensland Studies Authority, 2011), the fraction concept could for example be learned by acting out, using groups of students to represent fractions in terms of varying parts of the class (e.g. boys vs girls), or perhaps more dynamically by connecting fractions with rhythm and dance (Campbell, 2014).
At the same time, it is important that students also learn that there are differences between everyday colloquial expressions and empirical understanding of fractions, such as in acts of sharing and distributing, and formal mathematical equivalents. Mathematical definitions are developed through theoretical or operative generalisation and abstraction and use symbols (verbal, iconic, geometric or algebraic) to describe the conditions or schemata of actions (Dörfler, 1991). Therefore, explicit teaching of the meaning behind the symbolic mathematical language through exposure to multiple representations and models is essential for student learning of mathematical concepts including rational-number concepts.
Providing a creative and active learning environment, offering choice and variation in learning activities, mathematical representations, and task and assessment modes, will foster student engagement and the development of a positive disposition to mathematics. Similar to the fraction understanding itself (Siegler et al., 2013), a positive mathematical self-belief is another key predictor of middle years students’ mathematics achievement (Dimarakis, Bobis, Way, & Anderson, 2014).
Implications for the teaching and learning in diverse classrooms
Australia is a multicultural country and home to the world’s oldest continuous cultures. Nearly half of the population have an overseas-born parent, 5% identify as Aboriginal and/or Torres Strait Islander, and 20% speak a language other than English at home (Australian Human Rights Commission, 2014; Australian Bureau of Statistics, 2016). This diversity translates to classrooms with diverse social, cultural, religious and linguistic approaches to learning (Shahaeian, 2014). The Australian-wide promotion of an inclusive education policy emphasises the right of students of all abilities to participate in all aspects of the mainstream education, adding an additional dimension of physical, sensory and intellectual diversity (Konza, 2008). According to the Australian Bureau of Statistics, 5% of all primary school-aged children have disabilities resulting in significant core-activity limitations and schooling restrictions (2012). At the other end of the ability spectrum are the 10% of gifted and talented students, often unidentified and significantly underachieving (Parliament of Victoria, Education and Training Committee, 2012).
It is therefore the legal, moral and professional obligation of teachers to embrace all learners in their diversity and make reasonable adjustments to facilitate their full participation towards achieving their best potential (Cologon, 2013; Poed, 2015). There are a number of models for responsive teaching that addresses all learning needs in diverse classrooms. The Universal Design for Learning (UDL) is a set of principles guiding teachers towards developing universally accessible learning environments and instructional practices (Flores, 2008). The fundamental idea is to make the curriculum delivery as accessible as possible to all students, limiting the need for additional modifications and individual support. The design focuses on providing equitable access to the curriculum by offering multiple means of representation, expression and action (Basham & Marino, 2013). Students are offered choice over curriculum content, learning activities and resources to best meet individual skill levels, learning preferences and interests. Assessments offer learners multiple ways of demonstrating acquired skills and knowledge. While UDL can cater for most students in the diverse classroom, preferential intervention and special provisions is given to small groups, including access to resources (e.g. teacher aide) materials (e.g. manipulatives) or equipments (e.g. calculator) for task completion, including additional time or accelerated curriculum, alternative input and response formats (Ashman, 2015). A third level of prevention and intervention offers short-term intensive and explicit instruction for individuals (Fuchs & Fuchs, 2001), for example explicit practice of mathematical terminology and symbols for new EAL/D students.
Utilisation of ICT, including augmented and alternative communication devices that can support students with physical impairments, has great potential to help addressing all individual learning needs in a diverse classroom (Blum & Parete, 2015). Modern teaching and learning devices such as the iPad have been designed with disabilities in mind and can be easily configured to support the visually, hearing and physically impaired (Apple Inc., 2016). The iPad provides quick and simple access to a wide range of mathematics apps. Preliminary research highlights the potential of using iPads in primary school Mathematics classrooms to motivate and engage students (Hilton, 2016). Mathematics teaching and learning software, such as Mathletics developed by the Australian company 3P Learning provides teachers with tools to custom-design learning sequences for any topic in alignment with the Australian Curriculum, even activities with year level and content descriptors, lesson plans and ebooks (3P Learning, 2016). Australian schools that use Mathletics are performing significantly better in NAPLAN numeracy tests irrespective of socio-economic and regional status (Stokes, 2015).
The reported positive outcomes for all students, including students with learning support needs as well as gifted and talented students, could be a result of the combination of design features in the software:
student-led design that encourages learning at individual pace and at multiple difficulty levels (easier, core, harder)
instant and encouraging feedback to learners highlighting mistakes and solutions without teacher intervention
powerful formative assessment capabilities allowing teachers to monitor student progress and to identify learning gaps
tools that allow teachers to develop individual student learning pathways
app and web-based access allows Mathletics to be used as a flipped classroom tool and assign individual homework
gamified character in modules including class, school and world challenges (LIVE Mathletics)
Apps can also provide virtual manipulatives that enable more creative work with objects. For fractions, the educational graphing calculator GeoGebra is discussed below for building fraction bar models (Cooper, 2014).
As powerful as some apps and technology can be, ICT should only complement the teaching and learning of mathematics side by side with explicit teaching and multi-modal activities that encourage verbal and written communication, group discussions and the use of physical manipulatives that encourage kinesthetic learning. Also, apps are not always designed in alignment with UDL and can include barriers for students with disabilities (Smith & Harvey, 2014). Particularly in intervention instruction, it is advised to make use of both virtual and physical manipulatives to teach fractions (Westenskow & Moyer-Packenham, 2016).
Teaching and learning steps for acquisition of the equivalence concept
Fraction equivalence is one of the most important mathematical ideas introduced in primary school and know to cause difficulties for many students (Pantziara & Philippou, 2012). The big idea behind teaching equivalent fractions is for students to understand that fractions of a given size can have an infinite number of different names and corresponding symbols, and to develop efficient procedures for finding equivalent fractions. Finding equivalent fractions enables students to compare, order and operate with fractions (Petit et al., 2015).
The curriculum is the starting point for the design of teaching and learning units by defining the learning objectives and expected outcomes for each year level. The Australian Curriculum (AC) follows a spiral-based approach that gradually builds mastery of skills and concepts by sequentially increasing the cognitive demands (Lupton, 2013). Equivalence is introduced in the AC v8.3 in Year 4, where students are expected to “recognise common equivalent fractions in familiar contexts and make connections between fraction and decimal notations up to two decimal places”. In Year 5, equivalence of fractions is not specifically addressed but students are expected to develop the capacity to “... order decimals and unit fractions and locate them on number lines. They add and subtract fractions with the same denominator”. The equivalence concept is expanded in Year 6, where students are expected to “connect fractions, decimals and percentages as different representations of the same number”, more specifically detailed in content descriptor ACMNA131 “Make connections between equivalent fractions, decimals and percentages”. Full mastery of equivalence of fractions is not expected until Year 8 (ACARA, 2017).
In the learning continuum encountered in diverse classrooms, it is critical to develop an understanding of the sequence of teaching and learning steps of mathematical concepts and establish prior understanding of conceptual knowledge and procedural skills in all students.
Step One starts with diagnostic assessment to establish existing foundational knowledge of common fractions, notation conventions, the relation between fractions to whole numbers, including proper/improper fractions and mixed numbers. Explicit teaching and practice of terminology and revisiting previously learned concepts might be required to establish critical conceptual understanding without which any further learning would be only procedural and rely on rote learning.
Step Two explores new concepts and terminology by making use of physical manipulatives and encouraging student discussion. One example would be having students folding paper rectangles that have been vertically subdivided into equal, partially-shaded parts lengthwise in two, three, four bars of equal thickness The shaded fraction remains the same while the total number of equal parts as outlined by the creases increases. Students count shaded and unshaded parts and discuss equivalence (Booker et al., 2015, p.184).
Step Three elaborates and reinforces equivalence fractions through multiple representations working from the visual-concrete towards the symbolic-abstract. The activities help to develop procedural fluency, the accurate, efficient and flexible use of mathematical skills in renaming equivalent fractions (Petit et al., 2015). Fraction games, ideally focusing on equivalent fraction grouping, are employed using material (Booker et al., 2015) or online virtual resources (e.g. Math Playground Triplets). A “fractional clothesline” can be used to establish the magnitude of fractions, sort and locate equivalent fractions, improper fractions and mixed numbers (Heitschmidt, n.d.). This activity involves kinesthetic and visual learning, and can encourage verbal learning through student discussions. It also serves as a formative assessment tool. Number lines illustrate the big idea that equivalent fractions share the same value (Petit et al., 2015) and are highly recommended as a representation that can conceptually bridge whole-number and rational-number thinking (Booth & Newton, 2012; Gould, 2013).
Fraction clothesline example
Step Four integrates the acquired procedural knowledge and conceptual knowledge by looking for patterns and developing rules, progressing from concrete presentations towards symbolic presentations and abstract algorithms. The focus is on finding the next, rather than any equivalent fraction, making use of “fraction bars” as graphical representations. Fraction bars can be build using Lego blocks and extended by educational dynamic mathematics software (Cooper, 2014). Alternatively, an innovative lesson sequence works with stacks of papers of different thickness (Brousseau, Brousseau, & Warfield, 2014).
Example for Lego fraction bars that can be used to investigate equivalent fractions.
Step Five extends the learned knowledge and understanding of equivalent fractions to real-world scenarios. This includes investigating the relationships between alternative representations of fractions (e.g. decimals, percentages) in wide variety of cross-curriculum contexts (e.g. Science, Economics and Business, Music). At this stage, a summative assessment of learning is important to evaluate the achieved mastery of the concept.
Quality teaching is based on proficient subject-matter and pedagogical knowledge. Teachers need to understand the full spectrum of individual challenges and potential barriers that students can face with cognitively challenging mathematical concepts such as rational-number thinking. It is important to invest the time to allow students to gain deep conceptual understanding before moving on towards procedural fluency. This will require well-sequenced teaching and learning steps, supported by multiple representations, modes and questions, working from physical and visual towards more symbolic and abstract problem-solving activities. Both hands-on manipulatives and appropriate use of ICT can support the learning process, especially at both ends of the ability spectrum. Offering variety and choice will help to engage all learners and establish students’ confidence and positive dispositions towards mathematics.
students are to understand and describe properties of surface area and length,
develop fluency in measuring using metric units,
solve authentic problems, and
be able to explain shape transformations
A short learning sequence of comparison of lengths and areas – major steps
Booker et al. detail the conceptual and procedural steps required to master length and area (2015). Applied toACMMG137, these include three major steps:
Perceiving and identifying the attributes ‘area’ and ‘length’
Comparing and ordering areas and lengths (non-standard units => standard units)
Measuring areas and lengths (non-standard units => standard units), including covering surfaces without leaving gaps
This sequence is introduced using multiple representations, progressing from hands-on experiences with manipulatives towards abstract logical thinking and transformation tasks (see examples).
Activities to aid the learning sequence
The steps are mapped to a range activities that cater for diverse classrooms in alignment with the framework of Universal Design of Learning (UDL) (Fuchs & Fuchs, 2001):
Students cut their own tangram puzzle (with or without template) and explore how small shapes can create larger shapes
Students order tangram shapes by area and perimeter and establish base units: smallest shape (small triangle) as area unit, side of small square and hypotenuse of small triangle as length units
Students colour tangram pieces and puzzle range of objects (with and without colour, line clues), exploring how larger geometric shapes can be covered by smaller and making statistical observations on the number of units within each shape and corresponding perimeter. Non-standard units are measured and used for calculations.
Adjustments for a child with learning difficulties
Student with very limited English knowledge (e.g. EAL/D beginning phase). ACARA provides detailed annotated content descriptors (ACARA, 2014). The language and cultural considerations are specifically addressed by keeping discussion relevant to the tasks, offering alternatives to ‘word problems’ in both activities and assessment (as highlighted in the rubric design). Teaching strategy considerations are followed by explicitly teaching the vocabulary, making explicit links between terminology, symbols and visual representations (e.g. by pausing explanatory movie and writing out and illustrating on the whiteboard using colours (e.g. area = blue, equal sides = green, hypotenuse = red, labelling the count of units). The EAL/D student is provided with opportunities to develop cognitive academic language proficiency through mixed-ability group work. All content knowledge can be demonstrated by the student using physical manipulatives, charts and algorithms.
Adjustments for a child with advanced abilities
Children with advanced abilities can only develop their potential if provisions are made to deliver a challenging, enriched and differentiated curriculum, and a supportive learning environment
(Gagné, 2015). Maker’s updated recommendations on the four dimensions of curriculum modifications (2005) are applied as follows:
Content – content is framed in an interdisciplinary way, using tangram that connects to Japanese culture and art
Process – design emphasises self-directed learning, choice, variety and discovery of underlying patterns by offering a range of tangram puzzle options at multiple levels of difficulty to be explored in abstract terms (i.e. sorting by ratio of area to perimeter)
Product – high-ability students are encouraged to work on expert puzzles and transform learned concept knowledge by designing their own tangrams with constraints (e.g. tangrams with identical perimeter, sequence reduced by one length unit, …) and present their products to the class
Environment -high-ability students are provided access to spreadsheet software (e.g. for statistical observations, to graph relationships between area and perimeter) and allowed time to work independently
Battista, M. T. (2007). The development of geometric and spatial thinking. In Lester, F.K.Jr. (Eds) Second handbook of research on mathematics teaching and learning, Volume 2. National Council of Teachers of Mathematics, 843-908.
Booker, G., Bond, D., Sparrow, L., & Swan, P. (2015). Teaching primary mathematics. Fifth edition. Pearson Higher Education AU.
Fuchs, L. S., & Fuchs, D. (2001). Principles for the prevention and intervention of mathematics difficulties. Learning Disabilities Research & Practice, 16(2), 85-95.
Gagné, F. (2015). Academic talent development programs: a best practices model. Asia Pacific Education Review, 16(2), 281-295.
Benefits of apps to the Maths teaching and learning process
With the widespread introduction of mobile learning technology to Australian classrooms (i.e. iPads), an unprecedented development of educational software (apps) takes aim to complement traditional teaching. The potential benefits of apps need to be critically appraised for their pedagogical content, learning-area specific knowledge and technological requirements and ease of implementation (Handal, Campbell, Cavanagh, & Petocz, 2016). The emerging research suggests that the use of iPads in primary school Mathematics classrooms has great potential to develop and maintain positive student attitudes (Hilton, 2016) and support self-paced learning. However, research also points out that individual apps can have both supportive and inhibitive consequences on students’ learning performance and efficiency, depending on the student, prior instruction and the phase in the learning and teaching cycle (Moyer-Packenham, 2016).
Examples of three Math apps
Mathletics by 3P Learning Australia, Sydney. Mathletics is the most widely used app in Australian primary schools with comprehensive modules that complement for the K-12 Maths curriculum. (see more detail below)
Screenshot of Live Mathletics challenge
Khan Academy, Mountain View, California.
Khan Academy started out as a content provider of free educational movies and since evolved into student-centred learning app with a strong focus on Maths, with recent initiatives towards more international curriculum alignments (Khan Academy, 2017).
Khan Academy Maths opening page
LÜK-App by Westermann Gruppe, Braunschweig, Germany.
German curriculum-aligned quality app with a unique gamified approach towards learning, including all areas of Maths covered in primary schools (no German knowledge required)
LÜK app Maths task example
Detailed description of Mathletics
Mathletics software is developed in Sydney since 2004 and is marketing itself by stating that Australian schools that use Mathletics are performing significantly better in NAPLAN tests, irrespective of their socio-economic and regional status (Stokes, 2015). While running as an app, Mathletics is more of a comprehensive cloud-based educational platform offering school and class management tools, individual student learning pathways, global online competitions, and professional teacher training courses. The author has been using this app with his daughter throughout F-Year 3 and is particularly impressed with the pedagogical quality that went into the sequential buildup of mathematical concepts, the comprehensive content and close alignment with the Australian Curriculum (Australian Curriculum, Assessment and Reporting Authority, 2017), the quality of technological implementation and support. It is one of the few Math apps that combines declarative, procedural and conceptual knowledges (Larkin, 2015).
Handal, B., Campbell, C., Cavanagh, M., & Petocz, P. (2016). Characterising the perceived value of mathematics educational apps in preservice teachers. Mathematics Education Research Journal, 28(1), 199-221.
Hilton, A. (2016). Engaging Primary School Students in Mathematics: Can iPads Make a Difference?. International Journal of Science and Mathematics Education, 1-21. DOI 10.1007/s10763-016-9771-5
Larkin, K. (2015). “An App! An App! My Kingdom for An App”: An 18-Month Quest to Determine Whether Apps Support Mathematical Knowledge Building. In Digital Games and Mathematics Learning (pp. 251-276). Springer Netherlands.
Moyer-Packenham, P. S., Bullock, E. K., Shumway, J. F., Tucker, S. I., Watts, C. M., Westenskow, A., … & Jordan, K. (2016). The role of affordances in children’s learning performance and efficiency when using virtual manipulative mathematics touch-screen apps. Mathematics Education Research Journal, 28(1), 79-105.
The representation of mathematical concepts and objects plays an important discipline-specific role. Doing Maths relies on using representations for otherwise inaccessible mathematical objects. The concept of multiple representations (MR) has been introduced to teaching and learning of mathematics in the 1980’s (i.e. Janvier, 1987). Some primary school curricula (e.g. Germany) highlight MR as a key mathematical idea (Leitidee) (Walther, Heuvel-Panhuizen, Granzer, & Köller, 2012), while the Australian Curriculum (v8.2) includes specific references to some year-level proficiency standards (ACARA, 2016). This could reflect that different mathematical content domains apply particular kinds of representations (Dreher & Kuntze, 2015).
Benefits and difficulties
Research emphasises both the importance of MR to developing mathematical understanding and the difficulties that can be faced by learners (Ainsworth, 1999). Multiple representations can make all facets of mathematical objects visible. The ability to move between different representations is key to develop multi-faceted conceptual mathematical thinking and problem solving skills (Dreher & Kuntze, 2015). The difficulty with MR is that no single representation of a mathematical object is self-explanatory. Each representation requires understanding of how this representation is to be interpreted mathematically, and how it is connected to corresponding other representations of the object. These connections must be made explicit and require learning that engages higher cognitive levels. Interpreting individual representations, making connections between MR of corresponding mathematical objects, and changing between MR can present significant obstacle to learners (Ainsworth, 1999).
Sequencing the introduction of multiple representations
Booker, Bond, Sparrow & Swan (2015) highlight the importance of gradually sequencing the introduction of MR from the concrete to the abstract over time and identify the functions that MR can serve in developing mathematical understanding.
One such sequence is illustrated for content domain ‘geometry’ (compare ACMMG137) by applying the five ways of working (Battista, 2007).
Step 1: Visualisation of spatial arrangements – Students are provided with the following A4 template and are asked to cut out Tangram pieces along the blue lines and arrange them in one row by size.
A4 tangram template for students to cut out
Step 2: Development of verbal and written communication skills – Students are asked to discuss and describe their size order using explicitly taught concepts of ‘area’ and the small triangle as ‘1 unit’.
Tangram pieces sorted by size
Step 3: Symbolic representation through drawing and model making – Students are asked to colour their tangram pieces and puzzle the objects of projected image below (rotation, transformation)
Example colours for student tangrams
Step 4: Concrete and abstract logical thinking – Students are asked to create a column chart of the number of units (triangles) within each shape (colour). Students are allowed to cut one set of shapes into triangles (working in pairs).
Column chart depicting number of triangle units for each (coloured) tangram piece
Step 5: Application of geometrical concepts and knowledge – Students are asked to investigate how many different parallelograms they can form and the number of units required. Next, they measure and calculate the base unit and apply multiplication to calculate the areas.
Smallest possible parallelogram consisting of 2 small triangle units
2 units, 2 x 8 cm2 = 16 cm2
Largest possible parallelogram consisting of 16 small triangle units
16 units, 16 x 8 cm2 = 128 cm2
Ainsworth, S. (1999). The functions of multiple representations. Computers & Education, 33(2), 131-152.
Booker, G., Bond, D., Sparrow, L., & Swan, P. (2015). Teaching primary mathematics. Fifth edition. Pearson Higher Education AU.
Battista, M. T. (2007). The development of geometric and spatial thinking. In Lester, F.K.Jr. (Eds) Second handbook of research on mathematics teaching and learning, Volume 2. National Council of Teachers of Mathematics, 843-908.
Dreher, A., & Kuntze, S. (2015). Teachers’ professional knowledge and noticing: The case of multiple representations in the mathematics classroom. Educational Studies in Mathematics, 88(1), 89-114.
Janvier, C. E. (1987). Problems of representation in the teaching and learning of mathematics. Centre Interdisciplinaire de Recherche sur l’Apprentissage et le Développement en Education, Université du Quebec, Montréal. Lawrence Erlbaum Associates.
A rubric is a tabular set of criteria for assessing student knowledge, performance or products, informing the teaching and learning practice. Each line details criteria that are being assessed, each column the expected or achieved quality of learning (depth of understanding, extent of knowledge and sophistication of skill) by the student.
Rubrics are an assessment and reporting tool used to make expectations explicit to students, identify areas that require practice, and for self-assessment purposes (State of Victoria, Department of Education and Training, 2013). Rubrics are used to report learning outcomes to students, parents and carers, and can guide them towards flipped-classroom activities to improve individual results.
Key points in constructing a rubric
Formal grade achievements follow the five letter ratings, where ‘C’ indicates that a student is performing at the standard expected of students in that year group (ACARA, 2012).
Descriptors can be adapted and simplified for formative assessment purposes. The teacher selects aspects that are being assessed (criteria) and describes how achievements will be measured. ‘SMART’ criteria (O’Neill, 2000) (‘S’ – specific, ‘M’ – measurable, ‘A’ – attainable and agreed, ‘R’ – relevant to curriculum, ‘T’ – time-bound which means year-level appropriate) and Bloom’s taxonomy (Anderson, Krathwohl, & Bloom, 2001) can guide this process. Rubrics need to be designed and written in a language accessible to students, parents and carers.
Setting SMART goals for your students
This is an example for a 3-criteria, 3-descriptor rubric Year 6 lesson based on content descriptor ACMMG137 “solve problems involving the comparison of lengths and areas using appropriate units“. It is designed for formative teacher assessment, and to provide students with feedback on how they currently meet expectations and what differentiated homework tasks will help them to improve results.
‘Area’ conceptual understanding
Excellent understanding, demonstrated in designing tangram shapes of equal area
Unit and lesson planning are critical steps in the teaching and learning cycle among assessment, programming, implementation, evaluation and reflection. The objective of the planning process is to provide all students with appropriate learning experiences that meet the demands of the curriculum in terms of expected learning outcomes.
Major steps in the planning process
Relate teaching and learning goals to the Australian Curriculum (ACARA, 2016) relevant year-level descriptions, content and proficiency strands
Check year-level achievement standards and illustrations of graded work sample portfolios to inform assessment criteria guiding planning process
Develop challenging but achievable goals, considering the individual learning needs of all students based on diagnostic and formative assessments
Design sequence of activities, instructional scaffolding and learning extensions that build on existing student knowledge following the ‘gradual release of responsibility’ model (Fisher & Frey, 2007)
Evaluate achieved learning outcomes to inform subsequent lesson planning and to ensure that all students are on a trajectory to achieve best possible outcomes
Personal reflection on the process
The described back-mapping approach makes teaching and learning goals explicit and central to the planning process. By making learning intentions and expected outcomes explicit to the students at the beginning of each lesson and reviewing both at the end, students can develop a clear understanding of expectations and a reflective practice.
Planning is essential to deliver effective lessons that engage all students with appropriate learning activities. These can be informed by Bloom’s taxonomy of learning (Anderson, Krathwohl, & Bloom, 2001), as well as Gardner’s multiple intelligences (Gardner, 2006) to cater for the full spectrum of abilities with group work, targeted teacher aide support, differentiated homework and modifications to assessments.
Blooms taxonomy applied to teaching and learning Maths (Resource can be downloaded for free on Tes Global Ltd)