## 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?

## Motivation in mathematics

Motivation has an important role in all learning, including the development of mathematical skills (1). Motivation is also a term much used in everyday parlance, but what exactly do we mean when we talk about motivation?

The root of the word motivation is the Latin movere, to move. A well-known definition describes motivation as the energisation and direction of behaviour (2). The study of motivation could therefore be seen as an examination of what it is that moves individuals into action, and towards which end or ends their actions are directed. Research into motivation sees it as a multifaceted concept. For example, interest and self-concept, which we focus on in the iSeeNumbers project, are seen as different aspects of motivation. Furthermore, motivation is seen to have both cognitive and affective aspects, and so, we also examine how interest and self-concept are related to different emotions experienced in relation achievement activities and outcomes, or achievement emotions, as they are known.

INTEREST

The term interest refers to a psychological state that motivates and guides behaviour, and is characterised by increased attention and concentration, a desire for exploration, and mostly positive emotions (3). Interest is considered to be relational – in other words, there is always some object of interest, for example, an activity or content, towards which an individual experiences interest.

Interest is commonly differentiated into two different types or levels. Situational interest describes a more momentary experience of interest that may arise while engaged with some activity or task, whereas individual interest refers to a fairly stable, long-term ”interestedness” that is established over a longer period of time and repeated engagement (4).

It is often thought that interest in a school subject improves achievement, but research evidence for this claim has been quite mixed. Some studies have suggested that achievement, in fact, predicts interest, not vice versa (5), and yet others have found no relationship between interest and achievement either way (6). However, interest is known to be beneficial in many other ways. Interesting activity is experienced as rewarding, and engaging in such activity triggers positive energy (7). Interest supports effort (8) as well as persistence in the face of possible difficulties (4). It has also been found to buffer students against stress and experienced difficulty, which, in turn, have been linked with poorer performance in mathematics (9). It therefore appears important to awaken students’ interest in mathematics and support its development.

How, then, could interest in mathematics be supported in practice? Findings from research suggest that practising mathematical skills and making mathematical observations a part of the everyday life and activities at school might be one way. For example, students could count how many steps they need to take between the classroom and the playground, how many children are waiting in line, and so on (10). Providing a choice between different mathematics tasks might also be helpful.

SELF-CONCEPT

Academic self-concept (also called self-concept of ability, or competence perceptions) is a term used to describe students’ perception of their own abilities or competence in a specific subject, such as mathematics. Believing oneself to be competent is considered to be important for achievement, but also a valuable aim for education in itself (11).

In pre-school and during the first years of elementary school, students’ self-concept is typically quite positive. However, it tends to decrease over time, as students learn to differentiate between ability and effort, and evaluate their abilities more accurately (12). Self-concept and interest in a subject are usually connected, but self-concept is more strongly linked with achievement, whereas interest is related to effort and, in later school years, also course choice (13).

Different models describing the direction of the link between self-concept and achievement have been suggested (14). According to the self-enhancement model, believing that one is good at a given subject (i.e., having a positive self-concept) precedes achievement in that subject, whereas the skill-development model considers achievement to predict self-concept. In the reciprocal-effects model, positive self-concept and good achievement are seen to reinforce and support each other. Studies have so far mostly been conducted among older students. In these, reciprocal effects between self-concept and achievement have mostly been found, but the few studies examining these relationships in elementary school have given mixed results. This might be because the way they are connected may actually change during the elementary-school years, so that achievement predicts self-concept at first, and this then develops into reciprocal effects between the two (11). An increased understanding about the direction of the cause and effect is important, as it will assist in providing the right kind of support for students. The direction of these connections is one of the things we study in the iSeeNumbers project.

Feedback has been found to have an important role for students’ positive self-concept. Research has shown that instead of direct comparison between students, feedback should be individual and include what the student already knows or the skills they already possess. Furthermore, it should demonstrate how and in what things they can improve through practice (10). Emphasising the central role of effort and practice is important, as this can influence the reasons, or so-called attributions, with which students explain their successes and failures. For example, it is more beneficial if students attribute a lack of success to insufficient practice and effort, rather than to a lack of ability (i.e., “I need to work harder at this to learn it” instead of “I am stupid and will therefore never learn this”). However, it is vital that while the tasks should be challenging enough for the students, the level of difficulty should also be such that the student has a real chance to succeed with effort and practice. In the case of, for example, learning difficulties, emphasising the role of effort only could be counterproductive and further decrease motivation and weaken the student’s self-concept, unless support needed for developing their skills is also provided.

REFERENCES

1) 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.

2) Pintrich, P. R. (2003). A motivational science perspective on the role of student motivation in learning and teaching contexts. Journal of Educational Psychology, 95(4), 667–686.

3) Ainley, M. (2017). Interest: Knowns, unknowns, and basic processes. In P. A. O’Keefe & J. M. Harackiewicz (Eds.) The Science of Interest (pp. 3-24). Springer, Cham.

4) Hidi, S., & Renninger, K. A. (2006). The four-phase model of interest development. Educational Psychologist, 41(2), 111–127.

5) Garon-Carrier, G., Boivin, M., Guay, F., Kovas, Y., Dionne, G., Lemelin, J., et al. (2016). Intrinsic motivation and achievement in mathematics in elementary school: a longitudinal investigation of their association. Child Development, 87(1), 165–175.

6) Nuutila, K., Tuominen, H., Tapola, A., Vainikainen, M. P., & Niemivirta, M. (2018). Consistency, longitudinal stability, and predictions of elementary school students’ task interest, success expectancy, and performance in mathematics. Learning and Instruction, 56, 73–83.

7) Hidi, S. (2006). Interest: a unique motivational variable. Educational Research Review, 1(2), 69–82.

8) Arens, A. K., & Hasselhorn, M. (2015). Differentiation of competence and affect self-perceptions in elementary school students: extending empirical evidence. European Journal of Psychology of Education, 30(4), 405-419.

9) Rawlings, A.M., Tapola, A., & Niemivirta, M. (2021). Temperamental sensitivities differentially linked with interest, strain and effort appraisals. Frontiers in Psychology, 11:551806.

10) Mononen, R., Aunio, P., Väisänen, E., Korhonen, J., & Tapola, A. (2017). Matemaattiset oppimisvaikeudet. Jyväskylä: PS-kustannus.

11) Weidinger, A. F., Steinmayr, R., & Spinath, B. (2018). Changes in the relation between competence beliefs and achievement in math across elementary school years. Child Development, 89(2), e138-e156.

12) Nicholls, J. G. (1984). Achievement motivation: conceptions of ability, subjective experience, task choice, and performance. Psychological Review, 91(3), 328–346.

13) Schneider, R., Lotz, C., & Sparfeldt, J. R. (2018). Smart, confident, interested: Contributions of intelligence, self-concept, and interest to elementary school achievement. Learning and Individual Differences, 62, 23-35.

14) Marsh, H. W., & Craven, R. G. (2006). Reciprocal effects of self-concept and performance from a multidimensional perspective: Beyond seductive pleasure and unidimensional perspectives. Perspectives on Psychological Science, 1(2), 133-163.

Written by Anna Rawlings and Riikka Mononen (2021)