## Number sense

## Number sense

## Number sense

Number sense is considered as an innate numeracy skill, which not only humans but also some animals, such as fish and birds have. As the name says, it refers to sense of numbers. Number sense enables us to estimate the number of cars on the parking lot without counting them on-by-one, or to tell whether there are more people queueing in one line or in the other one.

APPROXIMATE NUMBER SYSTEM (ANS)

It has been proposed that we use a system called approximate number system, ANS, when estimating magnitudes (e.g. object or dots) approximately, or making comparisons of numbers (1).

When testing what kind of number sense an individual has, researchers often use tasks, in which you need to compare a number of dots or number symbols. Those who have a good number sense often respond quicker and more accurately in these tasks compared to those with weaker number sense.

Look at the pictures.

Can you quickly estimate which of the pictures has more dots?

Number sense is considered as an innate numeracy skill, which not only humans but also some animals, such as fish and birds have. As the name says, it refers to sense of numbers. Number sense enables us to estimate the number of cars on the parking lot without counting them on-by-one, or to tell whether there are more people queueing in one line or in the other one.

APPROXIMATE NUMBER SYSTEM (ANS)

It has been proposed that we use a system called approximate number system, ANS, when estimating magnitudes (e.g. object or dots) approximately, or making comparisons of numbers (1).

When testing what kind of number sense an individual has, researchers often use tasks, in which you need to compare a number of dots or number symbols. Those who have a good number sense often respond quicker and more accurately in these tasks compared to those with weaker number sense.

Look at the pictures.

Can you quickly estimate which of the pictures has more dots?

Number sense is considered as an innate numeracy skill, which not only humans but also some animals, such as fish and birds have. As the name says, it refers to sense of numbers. Number sense enables us to estimate the number of cars on the parking lot without counting them on-by-one, or to tell whether there are more people queueing in one line or in the other one.

APPROXIMATE NUMBER SYSTEM (ANS)

It has been proposed that we use a system called approximate number system, ANS, when estimating magnitudes (e.g. object or dots) approximately, or making comparisons of numbers (1).

When testing what kind of number sense an individual has, researchers often use tasks, in which you need to compare a number of dots or number symbols. Those who have a good number sense often respond quicker and more accurately in these tasks compared to those with weaker number sense.

Look at the pictures.

Can you quickly estimate which of the pictures has more dots?

## Mathematical learning difficulties

In each classroom of 20 students, there are 4–5 students who struggle in their mathematics learning. It is important that these students are identified as early as possible, and they are provided with intensified support for their mathematics learning.

The prevalence of mathematical learning difficulties is around 15–20%. This number includes those with severe mathematical learning difficulties, namely developmental dyscalculia (5–7%), and in addition those with low performance in mathematics (10–15%) (1).

The latest International Classification of Diseases (ICD-11) diagnostic manual provides a description for Developmental learning disorder with impairment in mathematics, which can be related to developmental dyscalculia (2). It is a good description of the characteristics of developmental dyscalculia, and what needs to be taken into consideration (i.e., exclusion criteria) when considering if the child has developmental dyscalculia. ICD-11 is planned to be taken in use in 2022, and to replace the current ICD-10.

Those children who are considered as performing low in mathematics, often score a little better in mathematics achievement tests than children having developmental dyscalculia. Still, their difficulties are such that those interfere their learning of mathematics at school and coping in everyday life situations that require mathematical skills.

FACTORS AFFECTING MATHEMATICS LEARNING

There are several factors, which have shown to affect mathematics learning and to be possible reasons for having mathematical learning difficulties. A challenge of understanding the role of these different factors is that often the studies have looked the relations of only some of the factors, and different tests have been used to measure the same phenomenon, giving us sometimes mixed results.

Number sense

Several studies have shown that number sense seem to affect the learning of other mathematical skills, such as counting and arithmetic skills (e.g., 3, 4). In the developmental dyscalculia, the origin of the difficulties are suggested to relate to impaired development of brain mechanisms that are needed for processing and understanding of number sense (5). School-aged children with developmental dyscalculia often are characterized as having weaker number sense compared to their peers (6). More specifically, recent research shows that symbolic number sense may play a more important role in relation to mathematical learning difficulties than non symbolic number sense (7).

Executive functions

While doing a mathematical task, a child needs to be able to regulate his actions: to plan, assess and alter his actions, if needed. For example, carrying out tasks of counting and mental arithmetic require moment-to-moment monitoring, processing, and maintenance of task-relevant information (8). Executive functions is often used as an umbrella term, including different cognitive processes: working memory, inhibition and cognitive flexibility (9). The role of working memory (WM) in relation to mathematics performance and development has been studied extensively in recent years (10). WM has been found both to be related to and to be a significant predictor of mathematical achievement (11). Especially visual-spatial WM functioning seems to have a significant role in how children perform in mathematics. On the one hand, those children who have weak visual-spatial WM skills have been found to perform weak in mathematics, both in arithmetic fluency and word problem solving. On the other hand, children having mathematical learning difficulties have been found to have weaker working memory skills overall compared to their age peers. Not only in visual-spatial WM functioning but also in phonological and central executive WM functioning.

Language

Many early mathematics tasks require using and understanding language. For instance, to count proficiently a child needs to know number words (12). For transcoding between number words and number symbols, a child has to understand the meaning of the number word and the rules that govern the structure for number words. The nature of the interrelations between language skills and mathematics skills is not yet clear, because the studies concerning language and mathematics skills may have operationalised language sub skills differently (i.e., using different types of tasks to measure one skill), and investigated the relation to a variety of different mathematical skills.

Based on the research, there is evidence that early language skills, namely, oral language (i.e., vocabulary, and understanding grammatical rules and structure of language), phonological awareness (i.e., differentiating and manipulating meaningful segments of a spoken language, for instance, being able to blend and delete parts of words) and print knowledge (i.e., knowledge of letter names and sounds, words, and basic conventions about books and print), have a strong relation to early mathematics development (e.g., 13,14,15).

Motivation and emotions

Motivation has an important role in all learning, also in the development of mathematical skills (16). In general it has been found that having strong mathematical skills is related to having strong motivation in learning of mathematics. However, there has been also identified those students who have both good knowledge in mathematics and cognitive skills, but whose poor motivation and negative emotions towards mathematics have become obstacles for their learning (17). Regarding the group of students having mathematical learning difficulties, it has been suggested that already in lower grades these students may have weaker perceived competence on learning of mathematics (e.g., I am not good at mathematics), and less interest in learning of mathematics (18).

Some students may feel negative feelings and anxiety towards mathematics. Math anxiety is more common in upper grades, however, children already in lower grades have been found to have math anxiety. There is evidence that math anxiety has a negative influence on math performance. One possible explanation for this is that math anxiety takes resources from those cognitive functions that are needed in solving a mathematics problem (17). For example, a student with math anxiety may think and worry about his performance and a poor outcome in the task while trying to solve the mathematics problem, thus not able to use his cognitive resources (e.g., working memory) in full capacity.

Learning environment

Children come from families of dissimilar socio-economic background. Studies that have investigated and compared the early mathematics performance between children from low-income and middle-income families have revealed that children from low-income families often lag behind and make less progress in early mathematics skills than their peers (e.g., 19, 20).

Education providers have the responsibility in providing all students with high quality mathematics instruction in a motivating learning environment. Teachers have a big role in this, although available resources and learning conditions affect successful implementation as well. Lack of availability of education may lead to weak performance in mathematics. However, that cannot be considered as the reason for developmental dyscalculia (ICD-11) itself.

IN OUR PROJECT

In our project we are interested in the interplay and role of different skills, motivation and well-being for mathematics development. Children's executive functions (e.g., inhibition and working memory), language skills (e.g., grammar and vocabulary), motivation and emotions in mathematics learning are assessed with various tasks during their three first school years.

REFERENCES

(1) Geary, D. (2011). Consequences, characteristics, and causes of mathematical learning disabilities and persistent low achievement in mathematics. Journal of Developmental & Behavioral Pediatrics, 33, 250–263.

(2) ICD-11: The 11th Revision of the International Classification of Diseases (ICD-11). https://icd.who.int/browse11/l-m/en

(3) 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, 64–81.

(4) Xenidou-Dervou, I., De Smedt, B., van der Schoot, M., & van Lieshout, E. C. D. M. (2013). Individual differences in kindergarten math achievement: The integrative roles of approximation skills and working memory. Learning and Individual Differences, 28, 119–129.

(5) Price, G. R., & Ansari, D. (2013). Dyscalculia: Characteristics, causes, and treatments. Numeracy, 6(1), Article 2.

(6) Mazzocco, M. M. M., Feigenson, L., & Halberda, J. (2011b). Impaired acuity of the approximate number system underlies mathematics learning disability (dyscalculia). Child Development, 82(4), 1224–1237.

(7) Schwenk, C., Sasanguie, D., Kuhn, J.-T., Kempe, S. Doebler, P. & Holling, H. (2017). Non-symbolic magnitude processing in children with mathematical difficulties: A meta-analysis. Research in Developmental Disabilities, 64, 152–167.

(8) Baddeley, A. D., & Logie, R. H. (1999). Working memory. The multiplecomponent model. In A. Miyake & P. Shah (Eds.) Models of working memory. Mechanisms of active maintenance and executive control (pp. 28–61). Cambridge, UK: Cambridge University Press.

(9) Miyake, A., Friedman, N. P., Emerson, M. J., Witzki, A. H., Howertwer, A. & Wager, T. D. (2000). The unity and diversity of executive functions and their contributions to complex “frontal lobe” tasks: a latent variable analysis. Cognitive Psychology, 41, 49–100.

(10) Mazzocco, M. M. M., & Räsänen, P. (2013). Contributions of longitudinal studies to evolving definitions and knowledge of developmental dyscalculia. Trends in Neuroscience and Education, 2, 65–73.

(11) Hornung, C., Schiltz, C., Brunner, M., & Martin, R. (2014). Predicting first-grade mathematics achievement: the contributions of domain-general cognitive abilities, nonverbal number sense, and early number competence. Frontiers in Psychology, 5, 1–18.

(12) Cowan, R., Donlan, C., Newton, E. J., & Lloyd, D. (2005). Number skills and knowledge in children with specific language impairment. Journal of Educational Psychology, 97(4), 732–744.

(13) LeFevre, J.-A., Fasr, L., Skwarchuk, S.-L., Smith-Chant, B. L., Bisanz, J., Kamawar, D., & Penner, Wilger, M. (2010). Pathways to mathematics: Longitudinal predictors of performance. Child Development, 81(6), 1753–1767.

(14) Praet, M., Titeca, D., Ceulemans, A., & Desoete, A. (2013). Language in the prediction of arithmetics in kindergarten and grade 1. Learning and Individual Differences, 27, 90–96.

(15) Purpura, D. J., & Ganley, C. M. (2014). Working memory and language: Skillsspecific or domain-general relations to mathematics? Journal of Experimental Child Psychology, 122, 104–121.

(16) Denissen, J. J. A., Zarrett, N. R. & Eccles, J. (2007). I like to do it, I’m able, and I know I am: Longitudinal couplings between domain-specific achievement, self-concept, and interest. Child Development, 78, 430–447.

(17) Chang, H. & Beilock, S. L. (2016). The math anxiety-math performance link and its relation to individual and environmental factors: A review of current behavioral and psychophysiological research. Current Opinion in Behavioral Sciences, 10, 33–38.

(18) Mononen, R., Aunio, P., Väisänen, E., Korhonen, J. & Tapola, A. (2017). Matemaattiset oppimivaikeudet [Mathematical learning difficulties]. PS-kustannus, Jyväskylä, Finland.

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

(20) Siegler, R., & Ramani, G. B. (2008). Playing linear numerical board games promotes low-income children’s numerical development. Developmental Science, 11(5), 655–661.

Written by Riikka Mononen (2019)