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.
https://en.wikipedia.org/wiki/Pythagoras

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.

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).

References

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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.

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.