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?
Emotions in mathematics
Mathematics is a school subject that students often associate with distinct emotions – some enjoy learning mathematics, others may feel bored in mathematics lessons, and yet others may even fear situations in which they have to deal with numbers. Prior research on achievement emotions (i.e., emotions that focus on learning activities or outcomes (1)) in mathematics has mainly been conducted among older students, with only a few exceptions of studies focusing on primary-school students (2, 3). We know that different emotions students experience in mathematics situations seem to be connected with how they perform in mathematics, often so that positive emotions are coupled with good performance, and vice versa (4). In learning situations, such as during a mathematics lesson, different emotions are aroused when students are working with a mathematics task (i.e., achievement activity) and when they successfully accomplish or sometimes fail in it (i.e., achievement outcome)(1).
Achievement emotions are categorised not only as positive or negative, but also further as activating or deactivating. Thus, for example, enjoyment would be considered a positive, activating emotion, whereas boredom would be a negative, deactivating emotion. In the iSeeNumbers project, we focus on three emotions important for mathematics learning, namely, anxiety, enjoyment, and boredom.
MATH ANXIETY
Math anxiety is defined as “a feeling of tension and anxiety that interferes with the manipulation of numbers and the solving of mathematical problems in … ordinary life and academic situations” (5). Around 11–17% of individuals experience high math anxiety (6). Among older students, math anxiety is often linked with lower performance in mathematics (7), but less is known about this relationship among younger students.
Chicken or egg?
Conflicting interpretations have been reported regarding the causal direction between math anxiety and performance. On the one hand, poor mathematics performance and negative memories associated with it have been found to lead to math anxiety (8). On the other hand, some studies have shown that mathematics anxiety can hinder mathematics performance (4). Math anxiety may affect mathematics performance either through avoidance of mathematics-related situations or, more directly, through interfering with the processing and retrieval of numerical information, both of which can lead to poor mathematics performance. In addition, according to one prominent view into the influence of math anxiety on mathematics performance, worries and distressing thoughts connected with math anxiety tax working memory resources that are needed for solving a mathematical task, hence leading to poorer mathematics performance (9). Finally, findings from some longitudinal studies point to a reciprocal relation where math anxiety and mathematics performance may influence each other over time, possibly leading to a vicious circle (10).
Are there gender differences?
It has been reported that as they approach adolescence and onwards, girls tend to experience higher levels of math anxiety than boys (11). However, a recent meta-analysis including studies with participants from primary school to university found no gender differences in the relation between math anxiety and mathematics performance (12).
The role of teachers and parents
Research has also shown that teachers’ beliefs and whether they feel anxious about mathematics may affect students’ level of math anxiety and performance, too. For example, Beilock et al. (13) found that if female teachers experienced math anxiety, girls in their classrooms believed that boys are good in mathematics and girls in reading, and this further affected girls' mathematics development negatively. This stereotypical belief that boys are better in mathematics than girls is, however, incorrect.
Further, some research also indicates a relation between parents’ and their children’s levels of experienced math anxiety (14). In this study, it was found that if a parent with math anxiety helped their child with their math homework often, the child’s math anxiety increased and led to weaker performance. The researchers concluded that the parent’s own math anxiety and worries about how their child would do in mathematics may have transferred to the child via the parent's emotional reactions, such as frustration and irritation, if the child had difficulties with doing mathematics tasks.
How can we support students with math anxiety?
If your student or child is experiencing math anxiety, you may try the following (15,16):
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time pressure in math tasks, quizzes, or tests may increase the level of math anxiety - let the student do the exam without such pressure
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let the student write about their feelings just before the exam - this has been shown to decrease negative emotion and improve performance
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try to make math tasks and activities motivating and interesting to the students - connect tasks to everyday situations
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encourage and support persistence in doing math tasks
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show that there often are many different ways to solve the problem, not only one
ENJOYMENT
The word “enjoyment” can describe pleasurable experiences in a broad range of situations or circumstances, from the pleasure brought about by working on an interesting task all the way to general enjoyment of life (17). In the project, we consider enjoyment as excitement, pleasure, and satisfaction experienced in relation to a learning activity (1) – more specifically, in the subject of mathematics.
Enjoyment is closely linked with interest, and the two often go together and promote each other (17). Interest motivates exploration and information-seeking, and enjoyment describes the pleasure and satisfaction derived from such activities and related achievements. For example, a child may be interested in working on a mathematics task, and delight in their skills and in learning something new. Enjoyment can also be felt in anticipation, for example, when a child is looking forward to and feeling excited about a future math lesson. Feeling confident about one’s ability and, hence, about the possibility of a successful outcome supports such anticipatory enjoyment, and enjoyment of learning more generally (18).
Enjoyment has been linked with mathematics achievement (3), but studies examining these relationships among young children and over time are still very few.
BOREDOM
Boredom, in turn, is defined as an unpleasant, negative emotional state that is characterised by a lack of interest and a desire to change, avoid, or withdraw from the activity or situation one is in (19). In a learning context, boredom is generally seen as maladaptive, since wanting to avoid or escape from an activity seen as boring may interfere with the student’s attention, and direct it towards something they perceive as more rewarding instead. Boredom, therefore, is considered to reduce the cognitive resources available for working on a task, impair engagement and persistence in an activity, and result in shallow information processing and a passive approach to learning. Boredom is, thus, largely seen as contributing to poorer achievement. Indeed, empirical findings show negative relations between boredom and attention, effort, and task performance.
Boredom is thought to arise when achievement activities are perceived as not important and lacking in subjective value. Also, both experiencing a task as not sufficiently challenging, as well as perceiving it as exceeding one’s capabilities, tend to give rise to boredom (19). In research, boredom has been found to be one of the most commonly experienced emotions in learning contexts, among students of different ages and educational levels, from secondary school to university (19). However, boredom has so far been studied less than some other emotions (for example, anxiety), and this is especially the case among younger children. The few studies that have been conducted tend to suggest that young students experience little boredom, but also that boredom may begin to increase already in elementary school (3). This development and the reasons behind it is one of the things we study in the iSeeNumbers project. One possible mechanism is that young students’ self-concept is generally quite positive, which, in turn, is usually connected with high levels of enjoyment. However, self-concept tends to become less positive over time, as students become better at accurately assessing their ability, for example, in mathematics. Students may then begin to experience mathematics tasks as exceeding their capabilities, and this, in turn, may make mathematics less enjoyable and result in boredom. Supporting students’ positive self-concept even as mathematics tasks get more difficult might, in this case, be helpful.
REFERENCES
1) Pekrun, R., Frenzel, A. C., Goetz, T., & Perry, R. P. (2007). The control-value theory of achievement emotions: An integrative approach to emotions in education. In Emotion in education (pp. 13–36). Academic Press.
2) Lichtenfeld, S., Pekrun, R., Stupnisky, R. H., Reiss, K., & Murayama, K. (2012). Measuring students’ emotions in the early years: The Achievement Emotions Questionnaire-Elementary School (AEQ-ES). Learning and Individual Differences, 22(2), 190–201.
3) Raccanello, D., Brondino, M., Moè, A., Stupnisky, R., & Lichtenfeld, S. (2019). Enjoyment, boredom, anxiety in elementary schools in two domains: Relations with achievement. The Journal of Experimental Education, 87(3), 449-469.
4) Carey, E., Hill, F., Devine, A., & Szücs, D. (2015). The chicken or the egg? The direction of the relationship between mathematics anxiety and mathematics performance. Frontiers in Psychology, 6, 1987.
5) Richardson, F. C., & Suinn, R. M. (1972). The mathematics anxiety rating scale: Psychometric data. Journal of Counseling Psychology, 19(6), 551–554.
6) Devine, A., Hill, F., Carey, E., & Szűcs, D. (2018). Cognitive and emotional math problems largely dissociate: Prevalence of developmental dyscalculia and mathematics anxiety. Journal of Educational Psychology, 110(3), 431–444.
7) Skagerlund, K., Östergren, R., Västfjäll, D., & Träff, U. (2019). How does mathematics anxiety impair mathematical abilities? Investigating the link between math anxiety, working memory, and number processing. PLOS ONE, 14(1), e0211283.
8) Wang, L. (2020). Mediation relationships among gender, spatial ability, math anxiety, and math achievement. Educational Psychology Review, 32(1), 1–15.
9) Ashcraft, M. H. (2002). Math anxiety: Personal, educational, and cognitive consequences. Current Directions in Psychological Science, 11(5), 181–185.
10) Aldrup, K., Klusmann, U., & Luedtke, O. (2020). Reciprocal associations between students’ mathematics anxiety and achievement: Can teacher sensitivity make a difference? Journal of Educational Psychology, 112(4), 735–750.
11) Dowker, A., Sarkar, A., & Looi, C. Y. (2016). Mathematics anxiety: What have we learned in 60 years? Frontiers in Psychology, 7.
12) Zhang, J., Zhao, N., & Kong, Q. P. (2019). The relationship between math anxiety and math performance: A meta-analytic investigation. Frontiers in Psychology, 10.
13) Beilock, S. L., Gunderson, E. A., Ramirez, G., & Levine, S. C. (2010). Female teachers’ math anxiety affects girls’ math achievement. Proceedings of the National Academy of Sciences of the United States of America, 107(5), 1860–1863.
14) Maloney, E. A., Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2015). Intergenerational effects of parents’ math anxiety on children’s math achievement and anxiety. Psychological Science, 26(9), 1480–1488.
15) Beilock, S. L., & Willingham, D. T. (2014). Math anxiety: Can teachers help students reduce it? Ask the cognitive scientist. American Educator, 38(2), 28–32.
16) Mononen, R., Aunio, P., Väisänen, E., Korhonen, J., & Tapola, A. (2017). Matemaattiset oppimisvaikeudet. Jyväskylä: PS-kustannus.
17) Ainley, M., & Hidi, S. (2014). Interest and enjoyment. In R. Pekrun and L. Linnenbrink-Garcia (Eds.) International handbook of emotions in education (pp. 205-227). London, UK: Routledge.
18) Pekrun, R. & Perry, R.P. (2014). Control-value theory of achievement emotions. In R.
Pekrun and L. Linnenbrink-Garcia (Eds.) International handbook of emotions in education (pp. 120–141). London, UK: Routledge.
19) Goetz, T. & Hall, N.C. (2014). Academic boredom. In R. Pekrun and L. Linnenbrink-Garcia (Eds.) International handbook of emotions in education (pp. 311–330). London, UK: Routledge.
Written by Riikka Mononen and Anna Rawlings (2021)