# Teaching and learning fractions

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 ACMNA016recognise 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 ACMNA131Make 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.

1. 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.
2. 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).
3. 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

1. 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.

1. 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.

### Conclusion

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.

# 8 Aboriginal Ways of Learning – a review

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

Dr Karen Martin, a Noonuccal woman, NAIDOC Scholar of the Year 2008 and Associate Professor in the School of Education and Professional Studies at Griffith University unpacks how a culturally-informed pedagogy informed by means by speaking of ways of knowing, ways of doing, ways of being and ways of valuing”.

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:

Story sharing:

Story sharing

Great teachers are storytellers that can also teach through narratives and songs. Sharing stories is also an important tool to connect with each other.

Learning maps:

learning maps

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.

Non-verbal learning:

non-verbal

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:

non-linear

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“.

This pedagogy is an important aspect of social pedagogies that emphasise the importance of community engagement and authentic audiences to bolster student engagement. Arguably, community links goes one step further by applying learning for community benefit, or to paraphrase UNESCO to “empower disadvantaged communities through innovative education“.

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.

# Teaching and learning Maths: learning sequence catering for diversity

This post is addressing the Year 6 content strand ‘measurement and geometry’, substrand ‘using units of measurement’ and content descriptor ACMMG137solve problems involving the comparison of lengths and areas using appropriate units” (ACARA, 2017), which were discussed in the previous posts on Maths unit and lesson planning process, rubric construction, multiple representation of mathematical concepts, and using Math apps. The achievement standards are mapped to the proficiency strands and include:

• 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:

1. Perceiving and identifying the attributes ‘area’ and ‘length’
2. Comparing and ordering areas and lengths (non-standard units => standard units)
3. 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.

(The activities are detailed with examples in the post on multiple representations of mathematical concepts)

### 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

# Teaching and learning Maths: multiple representations of mathematical concepts

### Multiple representations

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.

Examples:

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

### References

• Ainsworth, S. (1999). The functions of multiple representations. Computers & Education, 33(2), 131-152.
• Australian Curriculum, Assessment and Reporting Authority. (2016). Home/ F-10 Curriculum/ Mathematics.
• 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.
• Walther, G., Heuvel-Panhuizen, M. V. D., Granzer, D., & Köller, O. (2012). Bildungsstandards für die Grundschule: Mathematik konkret. Humboldt-Universität zu Berlin, Institut zur Qualitätsentwicklung im Bildungswesen.

# Teaching and learning Maths: using Math apps

### 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

1. 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

1. 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

1. 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).

### References

• Australian Curriculum, Assessment and Reporting Authority. (2017). Home/ F-10 Curriculum/ Mathematics.
• 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
• Khan Academy. (2017). An uncommon approach to the Common Core.
• 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.
• Stokes, T. (2015). National Numeracy Study Mathletics and NAPLAN. 3P Learning Australia Pty Ltd.

# Teaching and learning Maths: unit and lesson planning process

### Purpose of mathematics planning

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

1. Relate teaching and learning goals to the Australian Curriculum (ACARA, 2016) relevant year-level descriptions, content and proficiency strands
2. Check year-level achievement standards and illustrations of graded work sample portfolios to inform assessment criteria guiding planning process
3. Develop challenging but achievable goals, considering the individual learning needs of all students based on diagnostic and formative assessments
4. 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)
5. 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)

### References

• Australian Curriculum, Assessment and Reporting Authority. (2017). Home/ F-10 Curriculum/ Mathematics.
• Anderson, L. W., Krathwohl, D. R., & Bloom, B. S. (2001). A taxonomy for learning, teaching, and assessing: A revision of Bloom’s taxonomy of educational objectives. Allyn & Bacon.
• Fisher, D., & Frey, N. (2007). Scaffolded Writing Instruction: Teaching with a Gradual-Release
Framework. Education Review//Reseñas Educativas.
• Gardner, H. (2006). Multiple intelligences: New horizons. Basic books.
• Queensland Curriculum and Assessment Authority. (2016). P–10 Mathematics Australian Curriculum and resources.

# Teaching and learning Maths: constructing a rubric

### Purpose of a rubric

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

### Example

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.

 excellent satisfactory practice more! ‘Area’ conceptual understanding Excellent understanding, demonstrated in designing tangram shapes of equal area Homework: Solve expert puzzles You can define and explain ‘area’ but need more practice in applying your knowledge Homework: Watch tangram movie and play more tangram Your understanding of area needs more practice Homework: Review area movie ‘Area’ problems with simple units You are fluent in generalising any tangram puzzle in terms of parts and multiples of units Homework: Design a tangram puzzle for the class to solve next lesson You competently calculate basic areas as parts or multiples of tangram triangles. Practice applying this understanding to more creative tangram figures Homework: Create figures 1, 3 and 4 and write down the number of small triangles required for each animal head You can describe the shapes but need more practice to calculate how they relate to each other in terms of ‘area’ Homework: Complete worksheet by writing down the number of small triangles required for each shape ‘Area’ problems with metric units You are fluent in reframing geometric shapes in ways that allow you to calculate their area Homework: Work on area calculations for more complex shapes in this worksheet You can calculate areas of simple geometric forms by describing them as parts or multiples of rectangles. Work towards extending your understanding to complex shapes Homework: Complete area calculation worksheet You can measure the sides of geometric shapes but need more practice calculating their related ‘areas’ Homework: Review area movie and calculate these areas of shapes

# On the importance of being numerate

Numeracy, also called Quantitative literacy and Quantitative reasoning, is a relatively new term describing the applied facets of mathematics in daily life and modern society. The Australian Curriculum defines numeracy as “… the knowledge, skills, behaviours and dispositions … to use mathematics in a wide range of situations. It involves […] recognising and understanding the role of mathematics in the world and having the dispositions and capacities to use mathematical knowledge and skills purposefully” (Australian Curriculum, Assessment and Reporting Authority, 2016a).

Applied mathematics were used in certain professions since pre-historic times. The great ancient civilisations that built perfectly geometric pyramids, temples and planned cities on the river banks of the Nile, Euphrates and Tigris, and Indus-Sarasvati developed an advanced understanding of numbers. However, the conceptional leaps in our theoretical understanding of mathematical concepts, as for example those developed by Thales of Miletus and Pythagoras of Samos in the 6th century BCE, were not widely applied until the Renaissance. Only since the 17th century, did mathematics really start to inform all areas of human endeavour, including the Arts, Humanities, and Science, becoming an increasingly central tool to manage and model our affairs and the environment (Madison & Steen, 2007). Fast forward to the 21st century and mathematics and numeracy are at the core of how individuals and society operate, connecting everything in the ‘Internet of Everything’ (IoE) linking people, things, processes, and ‘Big Data’ (Karvalics, 2014). As a result, numeracy is fast becoming a central pillar of education, with school curricula all over the world being rewritten in its favour.

Initially, the term numeracy was introduced within the school context by two influential school reports in the United Kingdom (UK) in 1959 and 1982, defining numeracy skills relating to what students “can do” with mathematical knowledge, as opposed to what mathematics students “know”. Since the beginning of the new millennium, the OECD Program for International Student Assessment (PISA) started assessing numeracy skills internationally, thereby further promoting its prominence in curricula (Madison & Steen, 2007). This post outlines the importance of being numerate for the individual and society at large, and discusses the role that we can play as effective primary school teachers in developing critical numeracy skills in our students.

### The importance of numeracy skills for the individual

According to late Lynn Steen’s widely cited monograph (Vacher, 2016) on numeracy “Mathematics and democracy: The case for quantitative literacy” (Steen, 2001b), numeracy skills are required in all quantitative aspects of life, such as solving practical everyday problems and logical reasoning. This requires individuals to have the ability and inclination to draw on a range of mathematical concepts and tools, and to display a strong number and symbol sense (Goos, Dole, & Geiger, 2012). In contrast to formal mathematics that operate within an abstract world, numeracy is mathematical knowledge and skills applied to ‘real world’ situations. Numeracy skills shape the reality of 21st century citizens in very concrete ways:

• Mobility: With an urbanisation rate of close to 90%, most Australians are dependent on public transport or the road system to navigate from home to work and to access basic services such hospitals or shopping centres. To effectively use public transport, a number of numeracy skills are required, including the ability to read time tables, calculate the estimated time of arrival, transfer times, change platforms, and to understand the fare system often involving machines, swipe cards, PIN numbers and complex tariffs. Using the public road system involves even more complex numeracy skills, such as the ability to read street symbols (eg. traffic signs), monitor travel speed, estimate safe driving distance, and calculate optimal routes involving maps or tools such as a GPS. For most individuals, the desire for spatial mobility can extend beyond the city they live in, for example when planning an overseas trip, which would involve the comparison of complex fares including taxes, hotel rents, sequential check-in and transfer procedures, limits on luggage number and weight, as well as time-zone and currency related conversions.

Numeracy in action: public transport

• Health: The correlation between poor health literacy and poor health is well documented. Health literacy, can be defined as the degree to which an individual can make informed health-related decisions based on acquiring, processing and understanding basic health information and services (Mantwill, Monestel-Umaña, & Schulz, 2015). Numeracy plays a dominant part in health literacy, and includes aspects such as understanding nutrition information on food labels, interpreting clinical data (eg. blood sugar readings), refilling prescriptions and adjusting medications, and understanding probability in health risks. Numeracy skills are particularly important for patients with chronic illnesses that rely on self-management and self-administration of treatments (Rothman, Montori, Cherrington, & Pignone, 2008). More indirectly, health can be effected by poor self-esteem and depression which is often accompanied with poor literacy and numeracy skills.

Numeracy in action: healthy nutrition

• Wealth: The biggest impact of poor numeracy skills on personal wealth is arguably in the form of foregone or lower earnings reflected in typically higher unemployment or temporal employment rates and lower wages. While difficult to quantify, because of the close association between poor numeracy skills and other potentially controlling factors (eg. poor literacy, gender, race), there is a strong correlation between numeracy skills, employment rates, and wage distribution (Grinyer, 2005; Pro Bono Economics, 2014). Individuals with low numeracy skills are not only statistically more likely to earn less, but also face difficulties in controlling their household spending. Poor numeracy is adversely affecting individuals in managing spending when it comes to shopping, leisure, as well as more complex financial products such as mortgages and other forms of credit and debt, including any associated levels of interest (Graffeo, Polonio, & Bonini, 2015; Pro Bono Economics, 2014).

Numeracy in action: finance

• Decision making: Informed decision making plays a critical role in our ability to take control of our life, by helping us to realise opportunities and limi risks. Numeracy skills impact all aspects of logical thinking and strategic planning: from the accurate analysis of the present situation, the pursuit and acquisition of relevant missing information and knowledge, to the weighting of advantages and risks (Goos et al., 2012). Most informed decisions are based on a thorough evaluation of quantitative, spatial and probabilistic information, and require a high level of number and symbol sense, as well as the ability to project different scenarios along timelines into the future.

Numeracy in action: informed decision making

• Personal stability: As a result of the interplay of these factors within the wider social context that an individual operates in, poor numeracy skills can adversely impact self confidence and personal stability. Adults with poor numeracy skills are often characterised by impulsive and erratic behaviours, emotions, and a lack of self-regulation strategies. The development of numeracy skills directly contribute to growth in personal and social confidence. While the relationship between crime and numeracy is another area that is difficult to define in terms of statistical significance and causes and effect, the majority of adults in UK police custody are found to display substandard numeracy skills (Parsons & Bynner, 2005, figure 8, p.29).

Numeracy in action: self efficacy

### The importance of numeracy skills for society

How numeracy skills within a population impact a society at large is a relatively new research field with the first comprehensive study undertaken by the (US) National Council on Education and the Disciplines (NCED) at the beginning of our new millennium (Steen, 2001b). However, as the NCED Executive Director at the time pointed out, the potential impact of literacy and numeracy on society has been understood since WWI, quoting John Dewey (1931) in that “successful democracy is conceivable only when and where individuals are able to ‘think for themselves,’ ‘judge independently,’ and discriminate between good and bad information” (Orrill, 2001). In our data-driven information age, the level of numeracy of its citizens has significant implications for a society:

• Democracy: A democracy is based on citizens executing their rights to shape political decisions by forming opinions on a wide range of subjects. Historically unprecedented quality of (and access to) numerical information can strengthen the foundations of democratic by informing public discourse and civic decision making. However, if large parts of the society lack the ability to think numerically, to recognise how quantitative information and their presentation can inform and shape (manipulate) opinions on political, social, and environmental issues, they cannot fully participate (Orrill, 2001). As a result, poor numeracy skills across a population can pose a real danger to democratic societies by potentially strengthening populist movements and demagogues who master the art of manipulating data and discourse with misinformation.

Numeracy in action: informed voters

• Economy: In recognition of the increasing importance of numeracy on the national economy, some governments like in the UK started to commission studies investigating the economical costs associated with poor adult numeracy (Pro Bono Economics, 2014). First estimates go into the tens of billions of dollars and are calculated based on forgone or lower tax revenues, lower productivity of the workforce, and exchequer costs associated with benefit payments to jobseekers and jobless. As economies are becoming more global, competitive, and based on advanced knowledge and skills, the overall level of adult numeracy is increasingly defining the place of a society in the global market space.

Numeracy in action: economic growth

• Social: The impact of quantitative literacy levels on society extends well beyond the political system and national economy into the social realm. As broached above, there is relationship between poor numeracy on individual physical and mental health, which can negatively impact both health and criminal justice systems. While difficult to measure and quantify, the assumption can be made that misguided decisions based on ignored or misread numerical data will limit the prospects and prosperity of future generations and ultimately strain the social fabric of a society.

### Numeracy and our role as teachers

The mismatch of traditional mathematics curricula and the demands of today’s societies for applied quantitative literacy skills became apparent only in recent years. There is also a dawning realisation that numeracy cannot be taught alone within the mathematics classroom, but instead requires a dedicated approach across all learning areas (Steen, 2001a; Wade, 2001). “Numeracy is not just one among many subjects but an integral part of all subjects” (Quantitative Literacy Design Team, 2001, p.6).  In developing the Australian Curriculum, Australia seized the historic chance to highlight numeracy content and opportunities across all key learning areas (Australian Curriculum, Assessment and Reporting Authority, 2016a). It is now our role as teachers to develop and realise numeracy opportunities in the classroom.

The first step towards effectively teaching numeracy is developing a solid theoretical understanding of all numeracy components that can be applied within the classroom context. This can be informed by the “quantitative literacy elements” compiled by NCES (Quantitative Literacy Design Team, 2001, pp.8-9):

• developing confidence in estimating, calculating, interpreting, and presenting quantitative data
• developing appreciation of the role of numeracy in the ‘real world’, linking to technological progress, scientific inquiry
• developing competence in reading and analysing data, including creating an awareness for errors
• developing logical thinking and decision making based on evidence, evaluation, and risks and benefits assessment
• developing practical skills in approaching ‘real world’ problems with numerical tools
• developing number sense in understanding the meaning and relationships of numbers, units, mathematical operations in context
• developing symbol sense in understanding syntax and grammar of mathematical symbols

The next step is to investigate and highlight numeracy opportunities specific to key learning areas in the Australian Curriculum (Australian Curriculum, Assessment and Reporting Authority, 2016b):

• English: numeracy skills support reading comprehension and writing, in particular relating to document structure, systematic procedures, and the detection of assumptions in scientific texts
• Health and Physical Education: numeracy skills are required to understand time and unit measurements, statistics related to competitions and training, monitoring of health or performance related parameters, and developing team strategies
• Humanities and Social Science: numeracy skills assist in understanding data either from censuses, historical and archaeological records, and to read and communicate information in graphs and infographics. In Economics and Business, number sense and logical thinking can be developed by evaluating opportunities and risks
• Mathematics: provides the required tools and problem-solving strategies for numeracy skills
• Science: numeracy skills are developed by interpreting statistics (eg. laboratory experiments), probability and calculus (eg. rates of change, heredity). Chemistry and Physics provide great opportunities to develop and apply symbol and number sense
• Technologies: offer a range of numeracy applications including those related to geometry, computer algorithms, and database queries
• The Arts: are increasingly based on digital technology and editing tools. However, even traditional Music and Dance education offer numeracy aspects such as rhythm and balance, providing “embodied” numeracy opportunities

Our role as teachers is to employ effective numeracy teaching strategies. In recent years, a number of studies in Australia and overseas evaluated teacher knowledge and classroom culture to define the most successful numeracy teaching approaches (Stephens & Australian Council for Educational Research, 2009). Mathematical Pedagogical Content Knowledge (MPCK) is the ability of teachers to make mathematical content accessible to students by building on prior knowledge and skills to bridge knowledge gaps. MPCK was found to be the most important variable that defines effective numeracy teachers. A number of “scaffolding practices” can support numeracy learning, including teaching strategies such as excavating, collaborating, probing, orienting (Stephens & Australian Council for Educational Research, 2009, table 5.1, p.31).

In terms of classroom organisation, a UK study by Askew, Brown, Rhodes, Johnson and Wiliam (1997) concludes that the most effective teachers of numeracy are ‘connectionist teachers’; teachers who use children’s prior knowledge and approaches and employ teaching strategies that emphasise making practical connections with mathematical concepts. All this suggests that numeracy is best taught by combining ‘constructivist learning’, where students build new knowledge on top of prior knowledge through exploration, with elements of pedagogically informed ‘direct teaching’, where teachers provide high-level questioning, guidance, and probing. This is supported by recent studies that conclude that the approach of combining ‘direct teaching’ with instructional interactions between teacher and students are most effective in teaching mathematical ideas, terminology and procedures, while cultivating student engagement and content relevancy (Stephens & Australian Council for Educational Research, 2009). Instructional interactions include discussions that encourage students to explain their thinking, share approaches to problem solving, and transfer existing skills to new contexts.

Finally, there are the practical teachings tools and activities to consider. Research suggests that while ‘ability grouping’ students in general can be problematic, task-specific work in small mixed-ability groups are likely to benefit low-ability and average-ability students (Council of Australian Governments & Human Capital Working Group, 2008). As for finding activities that are appropriate to the year level and target specific aspects of numeracy, professional journals such as the “Australian Primary Mathematics Classroom” (eg. Muir, 2012, on number sense; Hurrell, 2013  on measurements), dedicated numeracy content websites such Numeracy Continuum (New South Wales, Department of Education, 2016), and professional content forums such as Australian Curriculum Lessons (Australian Curriculum Lessons, 2016) all provide great points of departure.

### Conclusion

The primary school students of today will grow up in and shape a world that is increasingly defined by digital data and processes that will require solid numeracy skills to master. It has become increasingly evident over the last decades that poor quantitative literacy skills have direct and substantial implications for individuals and the society at large. As a result, teaching numeracy skills becomes a priority for schools everywhere. While based on formal mathematics, numeracy is much more than Maths. Numeracy is applied quantitative knowledge and skills and expands into all key learning areas. It is a new “language” or literacy that teachers need to teach within and outside the mathematics classroom. To quote late Lynn Steen: “numeracy will thrive […] because it is the natural tool for comprehending information in the computer age. As variables and equations created the mathematical language of science, so digital data are creating a new language of information technology” (2001b, p.111).

While this might seem a daunting task to some classroom teachers, at least in Australia can build on the new Australian Curriculum that emphasises numeracy across all key learning areas. There is also a growing number of resources providing profession digital content that can inform our teaching strategies and support our lesson planning. Numeracy is now considered so important in Primary Schools that other learning areas are being reduced and combined to achieve a “laser-like focus on literacy and numeracy” (Education Minister Christopher Pyne, 2015, cited in The Australian (Bita, 2015).

# Developing lexical precision through semantic gradients

Lexical selection serves the purpose of isolating and identifying the single most appropriate word or item from a cohort of semantically related words. Semantic gradients are an excellent teaching and learning device to broaden and deepen students’ vocabulary, by developing awareness of the subtle connotations and differences between semantically connected or overlapping words. The fundamental idea of semantic gradients is to develop a continuum of related words between two gradable opposites such as antonyms. All the words are sequenced by order of degree through minimising semantic distances.

The idea for semantic gradients was developed by Camille Blachowicz in 1986, in her investigation of alternatives to vocabulary notebooks. Blachowicz developed a five-point strategy that stood the test of time:

1. Prior knowledge activation
2. Development of predictive connections
4. Refining word meaning using cues
5. Applying learned words in writing, additional reading

Blachowicz initially compared existing innovative strategies such as:

• exclusion brainstorming – students discuss if stimulus words are likely, unlikely to appear in certain field or genre
• knowledge rating – students rank words by difficulty or prior knowledge, e.g. can define, can make predictions, previously encountered, unknown
• semantic mapping – students connect related words, explaining relationships
• semantic feature analysis – tabular characterisation of words across different dimensions, e.g. shared features
• concept ladder – relationships of focus word to other words and concepts, e.g. flute is made of wood, used to make music, …
• predict-o-gram – cloze procedure in which students predict how author will use words from a word bank

It is easy to see how Blachowicz combined these ideas to developed the semantic gradient teaching and learning approach.

Original semantic gradient example by Camille Blachowicz (1986)

More recently, Scott Greenwood and Kevin Flanigan (2007)  picked up the idea of semantic gradients and combined it with context clues for each word, including:

• formal definitions
• antonyms
• synonyms
• inferences in full sentences

The authors also offer cloze activities where students consider “the best word for the job”.

In general words can be arranged along the semantic gradient as an array creating a horizontal “bridge”, or as a vertical “ladder”. A graphic organiser that combines the two approaches in a diagonal can be downloaded here.

The activity is best practiced in small groups, followed by a class-wide comparison of results. This is to encourage discussion around the subtle differences in meaning between the semantically related words.

There are a number of scaffolding approaches, such as providing students with a word bank of related words, establishing the antonyms and most neutral word to be placed in the centre, colouring the words using a colour gradient.

Semantic gradient with colour cues

Semantic gradients are best practiced in an authentic and meaningful context. While this could be choosing “the best verb for the job” in a narrative writing activity in English, the teaching and learning device is also ideal to explicitly teach academic language and connect abstract concepts in other key learning areas.

In Science students can be asked to sequence words describing temperature (e.g. freezing to boiling), rock-forming minerals by density (e.g. Calcite to Pyrite ) or hardness (e.g. Talc to Quartz), alkaline and acid solutions by pH value.

In Maths, Brook Giordarno presented how semantic gradients can be used to practices the terms describing different angles (e.g. acute, right, obtuse, straight).

Other examples could be to sort and name polyhedral by number of vertices, edges, faces and diagonals.

In Humanities, semantic gradients can for example be used to explore power relationships (e.g. Queen, Prime Minister, Premier, Lord Mayor, Mayor, Councillor, …)

History stimulus example: rank in a medieval estate-based society

Below is a step-by-step instruction for how to set up a semantic gradient activity adapted from Reading Rockets:

1. Create multi-ability groups of maximum four students
2. Select a pair of gradable opposites, avoiding complementary pairs such as ‘on/off’
3. Generate at least five synonyms for each antonym
4. Arrange each set of synonyms from most to least extreme
5. Combine both sets of synonyms from most to least extreme, with the least extreme words in the middle, and the most extreme words on each end
6. Discuss choices with a peers. Use reference sources to help settle any disputes.
7. Make adjustments to your arrangement based on your discussion.

### References:

• Blachowicz, C. L. (1986). Making connections: Alternatives to the vocabulary notebook. Journal of Reading, 29(7), 643-649.
• Greenwood, S. C., & Flanigan, K. (2007). Overlapping vocabulary and comprehension: Context clues complement semantic gradients. The Reading Teacher, 61(3), 249-254.