Introduction
In the recent years, most students have been performing relatively poor in mathematics in most of the schools. Consequently, much attention has been given to some of the major aspects that directly affect the performance of students. One of the core aspects in the recent focus on mathematics in schools has been the development of number sense among the students. Despite it continued growth in education literature, the term ‘number sense’ is not clear to most of the teachers. This paper gives the definition of number sense and numeracy and employs the constructivist approach in explaining the aspects that are essential in learning mathematics.
Definition/Description Number sense and numeracy
According to Anghileri (2000), number sense can be defined as “an awareness of relationship that enable [children] to interpret new problems in terms of relations of results they remember” (p. 1). More importantly, the author argues associates number sense with “[children’s] ability to make generalizations about the patterns and processes they have met and to link the information to their existing knowledge” (Anghileri, 2000, p. 1). So far, numeracy or arithmetic involved the study of multiplication tables and use of four principles of number (multiplication, division, addition, and subtraction). Hence, the discipline had long been considered as the limited algorithm to performing standards calculations.
Currently, focus on drill and practice exercises will not help children to succeed in a technologically advanced society. In this respect, a shift should be made to the logical connection between numbers and number-related calculations. In other words, teachers should encourage children to develop mental thinking, outline patterns, and foresee results. What is more important, they should develop meaningful connections, observe and develop patterns and practice their ‘feel’ for numbers (Anghileri, 2000, p. 2). Hence, number sense does not only involve proficiency in calculating operations, but in students’ ability to connect these calculations to their experience and background knowledge.
With regard to the above, numeracy is closely associated with number sense, but with a few differences. In particular, Anghileri (2000) insists, “numeracy goes beyond the requirement of teaching written calculation procedures to involve both mental calculation and estimation as efficient processes for calculating” (p. 2). Numeracy, therefore, implies the ability to understand number system, as well as confidence and proficiency in evaluating and solving problems related to numbers.
Judging from the above-presented definitions, teachers should realize that numeracy goes beyond number sense because it involves more important and advanced procedures in addition to standard calculations and mental process in which students are engaged. In this respect, Kammii and Lewis (1990) admit, “…children acquire number concept by constructing from the inside rather than by internalizing them from the outside” (p. 36). Hence, teachers should understand that children are able to build numerical relations through their ability to logically think.
Key Principles for Developing Number Sense
Teachers should ensure that their pupils develop a good number sense especially in their initial years in school since it is one of the most fundamental aspects in education. Thus, children with a good number sense have the ability to understand different numbers and are able to apply that knowledge in their everyday activities effectively (Reys et al., 2009, p. 134). In other words, they can provide a deep understanding of how other knowledge can be combined with number sense, as well as how it contributes to exploring other academic disciplines.
The National Council of Teachers of Mathematics (NCTM) has come up with several features that characterize good number sense in children. One of the characteristics is that such children have a good understanding of number meaning (Anghileri, 2000, p. 2). Specifically, the National Statement on Mathematics for Australian School considers it important to identify number sense as a core component of the school curriculum.
In this context, number sense refers to inventiveness and flexibility as basic approaches for calculating. Australian curriculum avoids overemphasis on computational procedures because they do not encourage children to apply to mental thinking (Anghileri, 2000). Additionally, the children recognize the relative magnitude of the numbers that they come across during their learning process. The third characteristic is that the children are able to develop a referent for most of the measures of the objects that they interact with in their everyday life as well as the situations in their immediate environment.
Number sense and numeracy can be understood through three major principles: recognition of patterns and relationships, links and connections between processes, and emphasis made on ‘relational understanding’ rather than ‘instrumental understanding’.
The first principle – recognition of patterns – implies knowledge of multiple representations (patterns) of numbers. In this respect, mathematics as the discipline studying patterns involves constructing, creating, and describing various modes and approaches to problem-solving skills. This dimension constitutes an inherent part of mathematical learning. The patterns can include relational attributes, geometric attributes or affective attributes (Reys et al., 2009, p. 139).
In addition, understanding of relationships are crucial because the shape the basis for building the so-called ‘benchmarks’ to refer to a group of numbers. In this context, number sense can be developed in a gradual way “…as a range of known facts and relationships among them are extended” (p. 7). For instance, the principle implies that children can understand the connection between subtraction, division, and counting. Apart from knowledge of attributes denoting shapes and figures, mathematical patterns also involve the awareness of links between operations (Anghileri, 2000, p. 6).
Children should be able to work out their own strategies for understanding the relationships between numbers, as well as between various counting operations. Finally, awareness of facility with numbers, as well as operations to problem solving should be applied effectively. In this respect, children should comprehend the connection between problem context and viable solutions strategies that can be applied to this particular situation.
The second principle – recognition of links and connection between processes – is heavily discussed by Reys et al. (2009), Anghileri (2000), and Thompson (1999). In this respect, Anghileri focuses on mathematics not as a discipline exploring relations between numbers, but on as process involving logical and mental processes seeking to understand the relations between numbers going beyond mere arithmetic. To support the idea, Reys et al. (2009) refers to the mental computation process, which is essential for developing children’s number sense and numeracy. Similar to Reys et al. (2009), Thompson (1999) considers the mental process which goes ahead of the computation.
The final principle – relational understanding as a substitute of instrumental understanding – is closely related to mental computation. According to Skemp (1987), relational understanding lies in learning new concepts and establishing connections between those. This type of understanding is far more important than instrumental understanding where the primary concern is connected with giving correct answers on specific questions, which does not imply developing mental thinking.
Important Learning Related to Number Sense
Following the constructivist approach in the development of number sense among students, the learning process should meet some specific goals. They should be involved actively in the development of complex, abstract and powerful mathematical structures than the ones that they possessed when they joined the class. This is portrayed in their increased ability to solve a variety of meaning mathematical problems. Students should not only become self-motivated but also autonomous in all forms of mathematical activity (Sparrow, 1994, p. 7). In addition, the students should be able to make sense of different mathematical patterns and applications besides being able to communicate effectively in meaningful mathematical language.
Teachers should embrace on giving constructivist instructions to their students as far as the development of number sense is concerned. They should understand that their primary role in the classroom is not only to guide but also to support the students’ invention or rather the viable mathematical ideas as opposed to imposing the already established ways of solving mathematical problems (Kamii & Lewis, 1990, p. 36).
Their instructions should be geared towards the development of the student’s personal mathematical ideas and approaches to different mathematical problems (Clements & Battista, 1990, p. 34). They should encourage students to apply their own methods in solving problems both within the classroom setting and in other areas where they are expected to portray a great deal of number sense and numeracy as well. Teachers should not give much weight to the conventional/traditional approaches of solving mathematical problems.
Apart from constructivist principles, it is strongly suggested that teachers apply to mathematics patterns. This of particular concern to the model introduced by Frid (2004), which implies three elements of mathematical representation – enactive, iconic, and symbolic. As soon as these patterns are conceived, children will be able to construct meaningful patterns beyond standards calculations.
Subitizing is another principle that students should learn. It implies specific attention to developing students’ ability to carry out calculations at a glance without written counting (Reys et al., 2009). However, written counting and recording is also essential tool for developing logical and mental skills. All these abilities should be equally practiced and developed during the classrooms.
In order to encourage learning in diverse classrooms in terms of cognition, psychological and emotional perception, as well as social background, teachers should take greater responsibility for each student in the classroom and create a positive learning environment through establishing safe and comfortable classrooms, stimulating students to learn mathematics, enhancing transparent communication to avoid confusion, and, finally, introducing a ground for developing critical thinking skills (Reys et al., 2009). All these strategies should be accomplished with regard unique needs of individuals with various levels of attention, motivation, and skills.
In order to encourage students to develop the above-presented principles and skills, specific activities should be implemented. In particular, teachers should encourage children to carry all computations in their minds, using no technology and equipment. They can also introduce activities through which children can understand the links between numbers and other concepts. For instance, it is possible to count objects with regard to their specific features (color, shape, and other characteristics).
Conclusion
Number sense and numeracy are essential as far as the development and application of mathematical knowledge are concerned. The development of mathematical knowledge occurs through the student’s natural ability to think. According to the constructivist approach, the teacher’s main role is to help the students to construct a network of numerical relationships. Generally, there are two main principles that should guide teachers in their work. They should help students come up with their own ways of solving problems rather than showing them how to do it. Finally, they should also encourage the students to think and share their ideas as opposed to reinforcing answers on them.
References
Anghileri, J. (2000). Teaching number sense. London: Continuum.
Clements, D.H., & Battista, M. T. (1990). Constructivist Learning and Teaching. Arithmetic Teacher, 38(1), 34-35.
Frid, S. (2004). To Boldly Go Where No One Has Gone Before. A Journal of Mathematics Association of Western Australia (Inc.), 15(3), 1-9.
Kamii, C., & Lewis, B. A. (1990). Construction and First Grade Arithmetic. The Arithmetic Teacher, 38(1), 36.
Kaminski, E. (2002). Promoting Mathematical Understanding: Number Sense in Action. Mathematics Education Research Journal, 14(2), 133-149.
Reys, R., Lindquist, M., Lambdin, D., & Smith, N. (2009). Helping Children Learn Mathematics (9th Ed.). New York: John Wiley & Sons.
Skemp, R. R. (1987). The Psychology of Learning Mathematics. US: Routledge.
Sparrow, L. (1994). Children as Starting Points for Mathematical Activity in the Primary School. Cross Section, 6(3), 7-10.
Thompson, I. (1999). Getting Your Head around Mental Calculation. In I. Thompson (Ed.). Issues in Teaching Numeracy in Primary Schools. Buckingham: Open University Press. pp. 145-156.