The nature and significance of number sense

Mathematical competence is dependent on a secure understanding of basic number in both the short (Aubrey & Godfrey 2003; Aunio & Niemivirta 2010) and the long term (Aubrey et al. 2006; Aunola et al. 2004). For example, counting skills have been implicated in the arithmetical competence of students in Canada, England, Finland, Flanders, Taiwan and the USA respectively (LeFevre et al., 2006; Aubrey and Godfrey, 2003; Aunola et al., 2004; Desoete et al., 2012; Yang and Li, 2008; Jordan et al., 2007). Moreover, students who fail to acquire these basic but essential competences typically fail later (Gersten et al. 2005; Malofeeva et al. 2004). Importantly, appropriate interventions can prevent such children from remaining low achievers (Fuchs et al., 2010; Van Luit & Schopman 2000; Van Nes & Van Eerde 2010), which is important in light of research showing that number sense acquisition may be linked to parental education levels, particularly the mother’s (Ivrendi 2011) and especially for boys (Penner & Paret 2008), highlighting the influence of the child’s family’s SES in the development of number sense (Melhuish et al. 2008).

However, despite their significance, these basic competences have not been well defined (Griffin 2004; Faulkner 2009), typically because psychologists and mathematics educators operationalise them differently (Berch 2005). That said, our reading of the literature has identified three related perspectives (Andrews & Sayers 2015). The first, preverbal number sense (Butterworth 2005; Ivrendi 2011; Lipton and Spelke 2005), refers to those number insights that are innate to all humans and comprises an understanding of small quantities in ways that allow for comparison. For example, “6-month-olds can discriminate numerosities with a 1:2 but not a 2:3 ratio, whereas 10-month-old infants also succeed with the latter” (Feigenson et al. 2004, p. 307). The second, relational number sense, reflects a broad set of competences that, irrespective of a person’s age, enables that person to work with numbers confidently, accurately and flexibly (Reys et al. 1999), whether in the school, the workplace or some other aspect of the real world (Askew, 2015; Atweh et al., 2014; McIntosh et al., 1992). This leads us to the third perspective, the focus of our work, which connects the other two. Foundational number sense (FoNS) comprises those number-related understandings that require instruction and which typically occur during the first years of school (Ivrendi 2011; Jordan & Levine 2009). It is something “that children acquire or attain, rather than simply possess” (Robinson et al. 2002, p. 85) and differs significantly from the generic conceptualisation of number sense that has permeated Swedish mathematics education.

What is FoNS?

Defining FoNS involved an extensive and systematic review of several hundred peer-reviewed articles focused, typically, on the number-related skills research has shown to be necessary for children in kindergarten and year one. The intention was not to construct an extensive list of characteristic learning outcomes but a small set of simple to operationalise categories. This process yielded eight categories and placed, for example, rote counting to five and rote counting to ten, two narrow categories discussed by Howell and Kemp (2005), within a broad category of systematic counting. The eight FoNS categories are, in brief,

• Number recognition: Children recognise number symbols and know their vocabulary and meaning. They can identify a particular number symbol from a collection of number symbols and name a number when shown that symbol.
• Systematic counting: Children count to twenty and back, or count upwards and backwards from arbitrary starting points. They know that each number occupies a fixed position in the sequence of all numbers.
• Awareness of the relationship between number and quantity: Children understand the one-to-one correspondence between a number’s name and the quantity it represents, and that the last number in a count represents the total number of objects.
• Quantity discrimination: Children understand and compare magnitudes. They use words like bigger and smaller, understanding that eight is bigger than six but smaller than ten.
• An understanding of different representations of number: Children understand that numbers can be represented differently, including the number line, various manipulatives and fingers.
• Estimation: Children can estimate, whether the size of a set or an object. Estimation involves moving between representations; for example, placing a number on an empty number line.
• Simple arithmetic competence: Children transform small sets through addition and subtraction.
• Awareness of number patterns: Children extend and are able to identify missing numbers in simple number patterns. 

FoNS Category Definitions

These eight components are not unrelated, otherwise there is always the risk that children can count but not know that four is bigger than two.

Teachers’ and students’ behave in culturally conditioned ways

Internationally, teachers and their students enact classroom culturally constructed routines in, often, unconscious ways (Oakland & Lub 2006; Santagata & Barbieri 2005). For example, US and Chinese mathematics teachers conform to culturally constructed and very different norms concerning their teaching strategies (Cai & Lester 2006; Huang & Cai 2011; Zhou et al. 2006), their management of public questions (Schleppenbach et al. 2009), their pedagogical content knowledge (An et al. 2004) and their beliefs about mathematics teaching and learning (Cai & Wang 2010; Correa et al. 2008). Put simply, research has identified a collective emphasis on memorisation before understanding in China and an individual emphasis on understanding before memorisation in the US. Such differences create very different learning expectations and opportunities for students. Even within Europe, where a casual observer might expect to see culturally similar traditions, distinct perspectives on mathematics teaching have been found in comparisons between England and Hungary (Andrews, 2007), Finland, Flanders and Hungary (Andrews & Sayers 2012), Finland and Flanders (Andrews, 2013), England, Flanders, Hungary and Spain (Andrews, 2007, 2009a, 2009b; Andrews and Sayers, 2013) and England and Germany (Kaiser 2002). However, while much is known about the teaching of mathematics in England generally, little can be said of Sweden and even less about the teaching of number in year one classrooms in either country.

Parents support their children’s learning in culturally conditioned ways

It is known that the role of family background in the construction of their children’s achievement and later earnings is significant (Björklund & Jäntti 2012; Tramonte & Willms 2010). However, little is known about the ways in which parents independently support their children’s learning. In the context of the United States, explicit parental support improves students’ overall mathematical achievement (Cai 2003) and their attitudes towards learning (Balli 1998). Also, where parents espouse the importance of mathematical mastery rather than just mathematical performance the more likely their children are to achieve mastery (Gutman 2006), particularly when female students receive encouragement from mothers with a positive attitude towards mathematics (Leaper et al. 2012). However, parents of low socio-economic status (SES), despite a desire to help, often struggle to know what to do (Drummond & Stipek 2004). From a comparative perspective, Chinese parents tend to check their children's homework while US parents provide additional resources, indicating cultural norms privileging hands-on and hands-off approaches to home-based interventions (Cai 2003). Also, Chinese parents have higher expectations of and spend more time helping their children than either non-Chinese American or non-Chinese Australian parents respectively (Chen et al. 1996; Cao et al. 2006). However, while highlighting broad cultural differences in the ways in which parents support their children, little is known, about the processes by which this occurs, particularly within the contexts of Sweden and England.

Project aims and methods

In light of the above, the FoNS project comprises four related aims concerning grade one pupils’ learning of basic number in Sweden and England. The first is to investigate how Swedish and English teachers conceptualise their teaching of number to year one pupils. What do they understand to be the key elements of number that children should learn? What approaches to they adopt when teaching such material? What do they believe is the role of parents in this process? The second is to investigate how Swedish and English teachers actually teach these key elements of number. What strategies do they employ? What forms of activity to do they privilege? How do they manage individual differences in children’s prior attainment? The third is to explore parents’ understanding of early number learning. How do they support their children’s learning? What do they believe is the responsibility of teachers? The fourth, is to support teachers and parents by identifying, categorising and disseminating FoNS-related best practice in the two countries. With these aims in mind, the the project team will undertake, over a five year period, a range data collection activity in England and Sweden. The activities will include

• Teacher Interviews. The project team will question teachers about how they teach basic number skills, how their schools support year one students learning of basic number skills, how they see parents being involved in children’s learning of basic number skills and so on.
• Teacher questionnaires. The project team will develop a questionnaire, based on the issues identified in the interviews and the research literature, to investigate at scale how teachers in the two countries understand FoNS, its teaching and the role of teachers and parents.
• Teacher observations. The project team will observe year one lessons in both countries with the aim of identifying different approaches to the teaching of basic number skills.
• Parent interviews. The project team will question parents in both countries about how they support their own children’s learning of basic number skills and what they believe are the responsibilities of teachers.
• Parent questionnaires. The project team will develop a questionnaire, based on the issues identified in teacher interviews, parent interviews and the research literature, to investigate at scale how parents help their children learn basic number skills.

Why the comparative dimension?

The comparative dimension is warranted in at least three ways. Firstly, England and Sweden share many cultural and educational similarities but have performed differently on international tests of mathematical achievement. This makes for an interesting and potentially illuminating comparison. Secondly, it is well-known that when researchers investigate a system other than their own, they acquire insights that allow them to evaluate more effectively the beliefs and practices found in their own system. Thirdly, in looking at two different systems’ perspectives on FoNS we increase the possibility of identifying practices and activities likely to prove productive in children’s learning.

References

An, S., Kulm, G., & Wu, Z. (2004). The pedagogical content knowledge of middle school, mathematics teachers in China and the U.S. Journal of Mathematics Teacher Education, 7(2), 145-172.

Andrews, P. (2007). The curricular importance of mathematics: A comparison of English and Hungarian teachers' espoused beliefs. Journal of Curriculum Studies, 39 (3), 317-338.

Andrews, P. (2007). Negotiating meaning in cross-national studies of mathematics teaching: Kissing frogs to find princes. Comparative Education, 43 (4), 489-509.

Andrews, P. (2009a). Comparative studies of mathematics teachers’ observable learning objectives: validating low inference codes. Educational Studies in Mathematics, 71(2), 97-122.

Andrews, P. (2009b). Mathematics teachers' didactic strategies: Examining the comparative potential of low inference generic descriptors. Comparative Education Review, 53(4), 559-581.

Andrews, P., & Sayers, J. (2012). Teaching linear equations: Case studies from Finland, Flanders and Hungary. The Journal of Mathematical Behavior, 31(4), 476-488.

Andrews, P., & Sayers, J. (2013). Comparative studies of mathematics teaching: Does the means of analysis determine the outcome? ZDM: The International Journal on Mathematics Education, 45(1), 133-144.

Andrews, P., & Sayers, J. (2015). Identifying opportunities for grade one children to acquire foundational number sense: developing a framework for cross cultural classroom analyses. Early Childhood Education Journal, 43(4), 257-267.

Askew, M. (2015). Numeracy for the 21st century: a commentary. ZDM, 47(4), 707-712.

Atweh, B., Bose, A., Graven, M., Subramanian, J., & Venkat, H. (2014). Teaching numeracy in pre-school and early grades in low-income countries. Bonn: Deutsche Gesellschaft für Internationale Zusammenarbeit.

Aubrey, C., & Godfrey, R. (2003). The development of children's early numeracy through key stage 1. British Educational Research Journal, 29(6), 821-840.

Aubrey, C., Dahl, S., & Godfrey, R. (2006). Early mathematics development and later achievement: Further evidence. Mathematics Education Research Journal, 18(1), 27-46.

Aunio, P., & Niemivirta, M. (2010). Predicting children's mathematical performance in grade one by early numeracy. Learning and Individual Differences, 20(5), 427-435.

Balli, S. (1998). When mom and dad help: Student reflections on parent involvement with homework. Journal of Research and Development in Education, 31(1), 142-146.

Berch, D. (2005). Making sense of number sense. Journal of Learning Disabilities, 38(4), 333-339.

Björklund, A., & Jäntti, M. (2012). How important is family background for labor-economic outcomes? Labour Economics, 19(4), 465-474.

Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46(1), 3-18.

Cai, J. (2003). Investigating parental roles in students' learning of mathematics from a cross-national perspective. Mathematics Education Research Journal, 15(2), 87-106.

Cai, J., & Lester, F. (2005). Solution representations and pedagogical representations in Chinese and U.S. mathematics classrooms. Journal of Mathematical Behavior, 24(3-4), 221-237.

Cai, J., & Wang, T. (2010). Conceptions of effective mathematics teaching within a cultural context: perspectives of teachers from China and the United States. Journal of Mathematics Teacher Education, 13(3), 265-287.

Cao, Z., Bishop, A., & Forgasz, H. (2006). Perceived Parental Influence on Mathematics Learning: A Comparison Among Students in China and Australia. Educational Studies in Mathematics, 64(1), 85-106.

Chen, C., Lee, S., & Stevenson, H. W. (1996). Academic achievement and motivation of Chinese students: A cross-national perspective. In S. Lau (Ed.), Growing up the Chinese way: Chinese child and adolescent development (pp. 69-92). Hong Kong: The Chinese University Press.

Correa, C., Perry, M., Sims, L., Miller, K., & Fang, G. (2008). Connected and culturally embedded beliefs: Chinese and US teachers talk about how their students best learn mathematics. Teaching and Teacher Education, 24(1), 140-153.

Desoete, A., Ceulemans, A., De Weerdt, F., & Pieters, S. (2012). Can we predict mathematical learning disabilities from symbolic and non-symbolic comparison tasks in kindergarten? Findings from a longitudinal study. British Journal of Educational Psychology, 82(1), 64-81.

Drummond, K., & Stipek, D. (2004). Low-income parents' beliefs about their role in children's academic learning. The Elementary School Journal, 104(3), 197-213.

Faulkner, V. (2009). The components of number sense. Teaching Exceptional Children, 41(5), 24-30.

Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307-314.

Fuchs, L., Powell, S., Seethaler, P. et al. (2010). The effects of strategic counting instruction, with and without deliberate practice, on number combination skill among students with mathematics difficulties. Learning and Individual Differences, 20(2), 89-100.

Gersten, R., Jordan, N., & Flojo, J. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38(4), 293-304.

Griffin, S. (2004). Building number sense with Number Worlds: A mathematics program for young children. Early Childhood Research Quarterly, 19(1), 173-180.

Gutman, L. (2006). How student and parent goal orientations and classroom goal structures influence the math achievement of African Americans during the high school transition. Contemporary Educational Psychology, 31(1), 44-63.

Howell, S., & Kemp, C. (2005). Defining early number sense: A participatory Australian study. Educational Psychology, 25(5), 555-571.

Huang, R., & Cai, J. (2011). Pedagogical representations to teach linear relations in Chinese and U.S. classrooms: Parallel or hierarchical? The Journal of Mathematical Behavior, 30(2), 149-165.

Ivrendi, A. (2011). Influence of self-regulation on the development of children’s number sense. Early Childhood Education Journal, 39(4), 239-247.

Jordan, N., & Levine, S. (2009). Socioeconomic variation, number competence, and mathematics learning difficulties in young children. Developmental Disabilities Research Reviews, 15(1), 60-68.

Jordan, N., Kaplan, D., Locuniak, M., & Ramineni, C. (2007). Predicting first-grade math achievement from developmental number sense trajectories. Learning Disabilities Research & Practice, 22(1), 36-46.

Kaiser, G. (2002). Educational philosophies and their influence on mathematics education – an ethnographic study in English and German mathematics classrooms. ZDM, 34(6), 241-257.

Leaper, C., Farkas, T., & Brown, C. (2012). Adolescent girls’ experiences and gender-related beliefs in relation to their motivation in math/science and English. Journal of Youth and Adolescence, 41(3), 268-282.

LeFevre, J.-A., Smith-Chant, B., Fast, L., Skwarchuk, S.-L., Sargla, E., Arnup, J. et al. (2006). What counts as knowing? The development of conceptual and procedural knowledge of counting from kindergarten through Grade 2. Journal of Experimental Child Psychology, 93(4), 285-303.

Lipton, J., & Spelke, E. (2005). Preschool children's mapping of number words to nonsymbolic numerosities. Child Development, 76(5), 978-988.

Malofeeva, E., Day, J., Saco, X., Young, L., & Ciancio, D. (2004). Construction and evaluation of a number sense test with Head Start children. Journal of Educational Psychology, 96(4), 648-659.

McIntosh, A., Reys, B., & Reys, R. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12(3), 2-8.

Melhuish, E., Sylva, K., Sammons, P., Siraj-Blatchford, I., Taggart, B., Phan, M., & Malin, A. (2008). Preschool influences on mathematics achievement. Science, 321, 1161-1162.

Oakland, T., & Lub, L. (2006). Temperament styles of children from the People's Republic of China and the United States. School Psychology International, 27(2), 192-208.

Penner, A., & Paret, M. (2008). Gender differences in mathematics achievement: Exploring the early grades and the extremes. Social Science Research, 37(1), 239-253.

Reys, R., Reys, B., Emanuelsson, G., Johansson, B., McIntosh, A., & Yang, D. C. (1999). Assessing number sense of students in Australia, Sweden, Taiwan, and the United States. School Science and Mathematics, 99(2), 61-70.

Robinson, C., Menchetti, B., & Torgesen. J. (2002). Toward a two-factor theory of one type of mathematics disabilities. Learning Disabilities Research & Practice, 17(2), 81-89.

Santagata, R., & Barbieri, A. (2005). Mathematics teaching in Italy: A cross-cultural video analysis. Mathematical Thinking and Learning, 7 (4), 291-312.

Schleppenbach, M., Perry, M., Miller, K. F., Sims, L., & Fang, G. (2007). The answer is only the beginning: Extended discourse in Chinese and U.S. mathematics classrooms. Journal of Educational Psychology, 99(2), 380-396.

Tramonte, L., & Willms, J. D. (2010). Cultural capital and its effects on education outcomes. Economics of Education Review, 29(2), 200-213.

Van Luit, J., & Schopman, E. (2000). Improving early numeracy of young children with special educational needs. Remedial and Special Education, 21(1), 27-40.

Van Nes, F., & Van Eerde, D. (2010). Spatial structuring and the development of number sense: A case study of young children working with blocks. The Journal of Mathematical Behavior, 29(2), 145-159.

Yang, D.-C., & Li, M.-N. (2008). An investigation of 3rd-grade Taiwanese students' performance in number sense. Educational Studies, 34(5), 443 – 455.

Zhou, Z., Peverly, S. T., & Xin, T. (2006). Knowing and teaching fractions: A cross-cultural study of American and Chinese mathematics teachers. Contemporary Educational Psychology, 31(4), 438-457.