Teaching and learning Maths: using Maths 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 Maths 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

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 Maths apps that combines declarative, procedural and conceptual knowledges (Larkin, 2015).


  • 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: multiple representations of 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.


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.
  • 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: 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


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 and tangram 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

Structuring slides of associated lesson


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 Maths

Blooms taxonomy applied to teaching and learning Maths (Resource can be downloaded for free on Tes Global Ltd)


  • 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.

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.
nutrition fact label

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).
interest amortisation chart

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.
decision making

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.


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).


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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
  3. Contextual reading
  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.


  • 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.