The role of Subitising

Judy Sayers Stockholm University [email protected]

Trondheim November 2015

Vem jag är? ●

Teacher (lärare) in English primary school 4-11yrs. Specialist in mathematics, ICT & science.

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Senior lecturer and teacher educator at Northampton University – Taught trainee primary teachers in mathematics and all aspects of pedagogy. – Masters to teachers/practitioner research and professional development courses in all education.

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Researcher – PhD in how Primary teachers conceptualise whole class teaching phases of their mathematics lessons.

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Mathematics Traditions in Europe: Finland, England, Flemish Belgium, Hungary and Spain.

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Specialised my research into mathematics in the first years of formal schooling

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2yr post doc. at Stockholms universitet –August 2015 –

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Associate professor/universitetlektor in mathematics education (Sp. In F-3)

Subitisering Judy  Sayers  &  Anette de  Ron Subitisering är förmågan att omedelbart,   utan att räkna,  identifiera antalet objekt i en  liten mängd.  Det handlar alltså om en  typ av spontan och omedelbar antalsuppfattning.   Subitisering är en  viktig komponent i elevernas utveckling av taluppfattning.   I  den  här artikeln ger författarna några idéer om hur lärare kan arbeta med  subitisering för att elever ska kunna bygga upp inre talbilder,   en   betydelsefull grund för aritmetik. Sayers,  J., Andrews,  P.,  &  Björklund Boistrup,   L.  (2014).   The  role  of  conceptual  subitising  in  the  development  of  foundational  number  sense. In  T.  Meaney (Ed.),   A  Mathematics   Education  Perspective  on  early Mathematics   Learning  between  the  Poles  of  Instruction  and  Construction   (POEM).  Springer   publications   New  York.

● We will look at some of the simple and easily made

resources that can be used to support children’s exploration of structural pattern in early number, (including recent identification of Foundational Number Sense: Sayers et al. 2014; Andrews & Sayers,2014; Back et al. 2014), and can provide the foundations for later arithmetic success.

● What is it (subitising)? ● Why should we teach it? ● How can we teach it?

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What is it? Subitising (Perceptual subitising, Clements 1999) Definition: Subitising. Instantly recognising the number of objects in a small group, without counting. Example: when you can see that there are 5 coins without counting. verb: subitised, subitising (subitized US spelling) Psychologists: to perceive at a glance the number of items presented, the limit for humans being about seven.

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Subitising

● Young children spontaneously use the ability to recognise

and discriminate small numbers of objects (Klein and Starkey 1988). ● But some elementary/primary school children cannot

immediately name the number of pips showing on the dice.

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Conceptual Subitising ● Clements’ (1999) work, and others, have encouraged

teachers to attend to conceptual subitising. – Clements & Samara, 2009; Conderman et al. 2014) ● conceptual subitising:

– ability to recognise quickly and without counting by partitioning large groups into smaller groups ● Conceptual subitising-focused instruction related to

Foundational Number Sense (FoNS) component: numberrelated competences expected in first grade mathematics education (Back et al. 2014; Andrews & Sayers, 2014) – Eight components

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Conceptual Subitising ● Relates to how an individual identifies ‘a whole quantity

as the result of recognising smaller quantities… that make up the whole’ (Conderman et al. 2014, p29) ● Systematic management of perceptually subitised

numerosities to facilitate the management of larger numerosities (Obersteiner et al, 2013) ● Each number is subitised before

adding together mentally.

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Spatial Structuring ● Conceptual subitising (CS) can be construed as having

synonymity with the spatial structuring of numbers (Battista et al., 1998) ● Ability to recognise and manipulate numbers spatially

through the use of structured images of numbers, dice, ten frames, dominoes and fingers etc.

e.g.

● Thus conceptual subitising can be taught through

mathematical tasks that provide such images. (Clements, 2007, Mulligan et al. 2006, Penner-Wilger et al. 2007)

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Short history: on the one hand… ●

Counting did not imply a true understanding of number, but subitising did (Douglass, 1925)

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Many saw subitising as a prerequisite to counting.

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It was thought that where measurement focussed on the whole and counting focussed on the unit, only subitising focussed on the whole and the unit, thus underlying number ideas (Freeman, 1912)

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Carper (1942) agreed subitising was more accurate than counting and more effective in abstract situations.

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Klahr and Wallace (1976); Schaeffer, Eggleston, and Scott (1974) suggested notions of subitising was a more basic counting.

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Young children possess and spontaneously use subitising to represent the number contained in small sets and that subitising emerges before counting (Klein & Starkey, 1988).

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However…

Clements  (1999)

Counterarguments exist… ● Beckmann (1924) found that younger children used

counting rather than subitising. ● Others agreed children develop subitising later, as a short

cut to counting (Beckwith & Restle, 1966; Brownell, 1928; Silverman & Rose, 1980) ● In this form subitising is a form of rapid counting

(Gelman & Gallistel, 1978). ● Misconception of type of subitising…

Why should we teach it? ● A Powerful Tool ●

Conceptual Subitising has been linked positively to a variety of counting skills including speed and efficiency, as well as cardinality

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It underpins children’s understanding of: – – – –

the equivalence of different decompositions partitions of numbers commutativity of addition part-whole knowledge E.g. 9+7=16 because 5+5=10, 4+2=6, and 10+6=16. (Van  Nes &  Doorman,  2011)

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Importantly, poor performance on both perceptual and conceptual subitising may be linked to later mathematical difficulties.

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Furthermore, evidence from maths ed. research… ● Clement concluded that subitising can play an important

role in the development of basic mathematical skills including addition and subtraction skills. ● There are a number of intervention programmes that

specifically incorporate subitising to support low-attaining 3rd-4th graders (8-10 yr olds). ● Maths recovery (Wright et al, 2006; Munn & Reason,

2007) ● Count me in etc…

Further evidence…

● Recent research of the literature resulted in the

development of a framework. The framework’s components relate to the key literature/research fundamental core of ‘number sense’ at this early stage of number development. ● Our research (Sayers & Andrews, 2015; Andrews &

Sayers, 2014; Sayers & Andrews, 2014) ● What does ‘number sense’ relate to?

Sayers, MND, Stockholm

What is Number Sense? ● Although a traditional emphasis in early childhood

classrooms (Casey et al., 2004) and key component in curricula it remained elusive (Gersten et al.; Griffin, 2004)

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Research approach ● Using an inductive approach, we (Andrews & Sayers, 2015)

reviewed the literature in four fields of study: – Mathematics Education – Psychology – Special Education – Early Years ● Using constant comparison analysis (Strauss & Corbin,

1998) of the literature identified three distinct concepts: – Preverbal – Foundational – Applied

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Number Sense: Three conceptions Preverbal number sense  refers to those elements  of number sense  that are innate to all  humans  and  comprises an  understanding of small  quantities in  ways that allow for   comparison.   Foundational  number  sense  (FoNS),   concerns  the  number-­‐ related  understandings  that  require  instruction  and  typically   occur  during  the  first  year  of  schooling  – this  is  the  focus   of   our   work.   Applied  number  sense refers  to  the  “basic  number  sense   which  is  required  by  all  adults  regardless  of  their  occupation   and  whose  acquisition  by  all  students  should  be  a  major  goal   of  compulsory  education”  (McIntosh  et  al.,  1992:  3).   Sayers,  MND,  S tockholm

Eight Categories of FoNS are: 1. Number recognition 2. Systematic counting 3. Awareness of the relationship between number

and quantity 4. Quantity discrimination 5. An understanding of different representations of

number 6. Estimation 7. Simple Arithmetic competence 8. Awareness of number patterns

(Andrews & Sayers, 2014) Sayers, MND, Stockholm

Eight Categories of FoNS are: 1. Number recognition

Conceptual subitising possibilities

2. Systematic counting 3. Awareness of the relationship between number and

quantity 4. Quantity discrimination

Conceptual subitising possibilities Conceptual subitising possibilities

5. An understanding of different representations of

number

Conceptual subitising possibilities

6. Estimation 7. Simple Arithmetic competence

Conceptual subitising possibilities

8. Awareness of number patterns

Conceptual subitising possibilities

(Andrews & Sayers, 2014) Sayers, MND, Stockholm

Evaluating the framework Analyses based on case study evidence from several European countries ● Comparing sequence in England and Hungary (7 Cat.)

(Back et al. 2013) ● Comparing grade one teaching of number in England,

Sweden and Hungary (Andrews & Sayers, 2014) ● Comparing the teaching of conceptual subitising in Hungary

and Sweden (Sayers et al. 2014) ● In the process of comparing the use of the number line in

Hungary, Poland and Russia (CERME. 9. Andrews et al. 2015)

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Implications of FoNS Substantial implications are reported throughout the literature: ● Poorly developed number sense is implicated in later

mathematical failures (Jordan et al. 2009; Gersten et al., 2005) ● Research has shown that the better a child’s number sense

is – the higher their achievements will be later on both short and long term (Aubrey & Godfrey, 2003; Aunio & Niemivirta, 2010) – reported in several countries. ● Have I convinced you???

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Other evidence from Bruner (1966) distinguished three modes of representation in learning:

Enactive

Iconic

Symbolic

Bruner  &  Kenney  (1966)  

2.5 Numeracy Sub-categories: ● Recognize and describe includes being able to identify situations involving figures, units and geometric figures found in play, games, subject-related situations in work, civic and social life. It involves identifying relevant problems and analysing and formulating them in an appropriate manner. ● Apply and process involves being able to choose strategies for problem solving. It involves using appropriate units of measurement and levels of precision, carrying out calculations, retrieving information from tables and diagrams, drawing and describing geometric figures, processing and comparing information from different sources. ● Communicate means being able to express numerical processes and results in a variety of ways. Communicate also means being able to substantiate choices, communicate work processes and present results involving numbers. ● Reflect and assess means interpreting results, evaluating validity and reflecting on effects. It involves using results as basis for a conclusion or an action. Sayers, Stockholm

Competence aims - competence aims Year 2 (Norge, 2012) Numbers ●

The aims of the studies are to enable pupils to

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count to 100, divide and compose amounts up to 10, put together and divide groups of ten up to 100, and divide double-digit numbers in to tens and ones

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use the real number line for calculations and demonstrate the magnitude of numbers

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make estimates of amounts, count, compare numbers and express number magnitudes in varied ways to develop, use and converse about varied counting strategies for addition and subtraction of double-digit numbers, and evaluate how reasonable the answer is

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double and halve, recognise, talk about and further develop structures in simple number patterns.

Sayers, Andrews & Björklund Boistrup/MND, Stockholm

How can we teach it? Hur kan vi undervisar subitising? ACTIVITY ● Firstly – can you subitise? ● Try out, with a partner, using seven counters, then nine

counters. ● How can you make the task different and why? ● Let’s look at the resources on the tables…

Subitising slides Judy Sayers

Subitising

Let’s see how you do…

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Easy?

15+3

12+4 Composition…   leading   to  decomposition

www.sugarmaths.net/Subitising/presentation/PDF.pdf

Counters ● Different structures…

Mulligan,  J.,  &  Mitchelmore,  M.  (2009).   Awareness  of  pattern  and  structure  in  early   mathematical  development.  Mathematics   Education  Research  Journal,  21(2),  33-­49. Mulligan,  J.  T.,  &  Mitchelmore,  M.  C.   (1997).  Young  children's  intuitive  models  of   multiplication  and  division.  Journal  for   Research  in  Mathematics  Education,  309-­ 330.

Modelling manipulation of number quantities:

● Addition, subtraction, multiplication,

division.

● Cardinally – matching/one-to-one

correspondence, eggs in cups, ● chocolates in a box etc.

● Transition to… ● Ordinality – measurement and the

number line.

Next steps, or first steps? ● Using this ability to subitise to develop calculation is what

other countries already do ‘explicitly’ in their curriculum.

http://www.mathematik.uni-­‐marburg.de/~thormae/lectures/ti1/code/abacus/soroban.html

Ansari & Nosworthy (& others in Canada) ● Continued wealth of evidence reports that children’s ability

to understand numerical magnitude (quantity) is a critical building block of early maths skills. ● Numerous studies have shown that children who are faster

and more accurate at comparing which of two numbers is larger are also those that perform better on standardized measures of math achievement, such as tests of arithmetic achievement. For a review see De Smedt et. al 2013. There is evidence showing that both the ability to compare numbers that are presented as symbols (e.g. the Arabic Numerals; 1,2,3…) and non-symbolically (e.g. clouds of dots or collections of objects) is related to individual differences in children’s present and future maths skills. For example: Sayers, MND, Stockholm

De Smedt et al. (2011) Numeracy screener on magnitude.

Chinn  (2010)

Not so far away from multiplicative thinking

● An example from the US on visualising number

combinations (conceptual subitising)… ● https://www.teachingchannel.org/videos/visualizing-

number-combinations

(6mins)

Numicon

The  Primary  1,2  guide  to   Numicon  in  7  minutes   https://www.youtube.com/watc h?v=EIGN3ekzpjc Adding  2x2  digit  numbers   https://www.youtube.com/ watch?v=0lj3Yg1TbA8 http://www.numicon.co.nz/resources.html

Moving  on  from  counting… ● Counting  should  be  short  term  check  on  subitising/conceptual  

subitising  moving  on  to  skip  counting  and  groups  e.g.  number   line:

Support  individuals  with  quantities  alongside   number  line   Cardinality ● Manipulating quantities

with: ● rods of cubes ● Bundles of straws ● Base 10 rods etc.

Operations can be modelled in different ways using different resources.

● Decomposition and

composition of numbers using a variety of materials provides images to visualise and develop that shift to mental calculation

Singapore methods of teaching… •

The Singapore method of teaching early mathematics is to encourage and teach explicit representations. Such as the picture shows.

They work on counting and number bonds for over half a year before expecting children to engage in addition subtraction.

Singapore bar method

Cuisenaire rods

http://nrich.maths.org/public/leg.php?code=-­297&cl=2&cldcmpid=4561

8gy +  5gy =  ?gy.

!

!

The  summary  was  that  the  thirteen   was  a  result  of  ten  plus  three.

8  +  3 =  ?

8  +  2  +  1  =  11 and 8  +  3  =  11 ! Sayers, MND, Stockholm

Movie clips here to show lesson sequences

Sayers, Andrews & Björklund Boistrup/MND, Stockholm

Kerstin (Sweden)

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!

!

Sayers, Andrews & Björklund Boistrup/MND, Stockholm

References: ●

Back, J., Sayers, J., & Andrews, P. (2014). The development of foundational number sense in England and Hungary: A case study comparison. In B. Ubuz, Ç. Haser & M. A. Mariotti (Eds.), Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education (pp. 1835-1844). Ankara: Middle East Technical University on behalf of ERME.

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Andrews, P., & Sayers, J. (2014). Foundational number sense: A framework for analysing early number-related teaching. In Proceedings of the ninth Matematikdidaktiska Forskningsseminariet (MADIF 9). Umeå.

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Andrews, P., & Sayers, J. (2014). Identifying Opportunities for Grade One Children to Acquire Foundational Number Sense: Developing a Framework for Cross Cultural Classroom Analyses. Early Childhood Education Journal, 1-11.

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Andrews, P., Sayers, J., & Marschall, G. (2015). Developing Foundational number sense: number line examples from Hungary, Poland and Russia Paper presented at the Ninth Congress of European Research in Mathematics Education (CERME9), Prague. Berch, D. (2005). Making sense of number sense. Journal of Learning Disabilities, 38(4), 333339.

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Casey, B., Kersh, J., & Young, J. (2004). Storytelling sagas: an effective medium for teaching early childhood mathematics. Early Childhood Research Quarterly, 19(1) p167-172. Howell, S., & Kemp, C. (2005). Defining early number sense: A participatory Australian study. Educational Psychology, 25, 555-571 Sayers, J., Andrews, P., & Björklund Boistrup, L. (2014). The role of conceptual subitising in the development of foundational number sense. In T. Meaney (Ed.), A Mathematics Education Perspective on early Mathematics Learning between the Poles of Instruction and Construction (POEM). Malmö: Malmö Högskolan. Sayers, J. & de Ron, A. (2015). Subitising. Nämnaren Nr 1 – p81 (in Press). Strauss, A., & Corbin, J. (1998). Basics of qualitative research: techniques and procedures for developing grounded theory. London: Sage. Sayers,  MND,  S tockholm

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Atteveldt, N. & Ansari, D. (2014) How symbols transform brain function: a review in memory of Leo Blomert. Trends in Neuroscience and Education, 3, 44-49.

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Lyons, I.M., Ansari, D. & Beilock, S.L. (2015) Qualitatively different coding of symbolic and nonsymbolic numbers in the human brain. Human Brain Mapping, 26, 475-488.

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Matejko, A. & Ansari, D. (2015) Drawing Connections Between White Matter and Numerical and Mathematical Cognition: A Literature Review. Neuroscience & Biobehavioral Reviews, 48C, 35-52.

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Vogel, S.E., Goffin, C. & Ansari, D. (2014) Developmental specialization of the left parietal cortex for the semantic representation of Arabic numerals: An fMRAdaptation study. Developmental Cognitive Neuroscience, 12C, 61-73.

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Vogel, S.E., Remark, A. & Ansari, D. (2014) Differential processing of symbolic numerical magnitude and order in 1st grade children. Journal of Experimental Child Psychology, 129, 26-39.

Sayers, Andrews & Björklund Boistrup/MND, Stockholm

Tack!

E-mail [email protected] for subitising resource pack

Sayers,  Andrews  &  B jörklund   Boistrup/MND,  S tockholm