V.1 #5 Mathematics - Development and Support of Number Sense
As a construct, number sense is complex and difficult to operationalize. Its key components include counting, subitizing small quantities, discerning number patterns, estimating quantities, comparing numerical magnitudes, performing simple number transformations. Number sense should be regarded as an important instructional target, as it plays a vital role in developing lower and higher order mathematical thinking (e.g. acquisition of number combinations, fluency with operations) (Jordan, Kaplan, Olah, & Locuniak, 2006).
Examples of programs that some experts think can help students develop number sense are “Number Worlds” and “Numeracy Recovery” (Dowker, 2005; Gersten & Chard, 1999).
Number Worlds is a program (Griffin, 1997, 2000, 2004) for kindergarten, first grade, and second grade. It is based on five instructional principles.
Build upon children’s current knowledge by using multi-level activities that match children’s knowledge.
Follow the developmental progression of counting knowledge and knowledge of quantity, when selecting new learning targets.
Teach both computational fluency and conceptual understanding.
Use playful activities (e.g., board games) to give children lots of opportunity for communication, hands-on exploration, and problem solving.
Expose children to the major ways number is represented and talked about in developed societies. This includes group of objects, dot-set patterns, position on a line, position on a scale, and point on a dial.
According to Griffin and Case (1997), the effectiveness of Number Worlds, especially for at-risk children from low-income communities, has been supported in several evaluation studies. However, the program may present instructional problems when used with whole classes. While praising the program’s conceptual framework, Gersten, Jordan, and Flojo (2005) stress that the program’s learning activities are more easily used with small groups of children than with whole classes.
The Numeracy Recovery program (Dowker, 2001, 2005) aims to help six and seven year old children overcome their struggle with arithmetic. Specifically, it aims to help them learn basic counting procedures, counting principles, written symbols for numbers, place value in number operations, word problem solving, number fact retrieval, derived fact strategy use, arithmetical estimation, and translation between arithmetical problems presented in concrete, verbal, and numerical formats. Following are five examples of how Numeracy Recovery aligns concepts and activities.
Counting is taught by giving children counting practice and having them answer questions on cardinality and order-irrelevance about small and large sets, as well as repeated additions and subtractions coupled with “number before” and “number after” problems.
Acquisition of written symbolism for numbers is supported by sorting objects into groups of tens and extra units; the resulting numbers are recorded and read.
Word problem solving ability is taught through the systematic representation of problem situations. This can involve counters as well as questions to help students clarify the demands of the problem.
Number fact retrieval is taught through repeated presentations of basic addition and subtraction facts and through “number games.”
Derived fact strategies, essentially commutativity and the N+1 and N-1 principles, are demonstrated and explained with manipulative objects and number lines, and are practiced extensively with numerals.
Dowker (2001; 2005) states that the Numeracy Recovery program continues to undergo development and evaluation. The results have been positive as have teacher comments.
The nature of number sense is undoubtedly multifaceted. Some authorities (e.g. Reys, 1994; Verschaffel & De Corte, 1996) have asserted that it cannot be easily compartmentalized into instructional units, that activities specifically meant for its development are problematic. Rather, it should be viewed as a by-product of other learning.
In contrast, other authorities have argued that its complexity requires it to be taught systematically, through discrete activities and tasks (Berch, 2005; Dowker, 2005; Gersten, Jordan, & Flojo, 2005).
Clearly, the debate continues. So, what should teachers do for students who struggle with number sense? I recommend two general strategies. One—Use systematic instruction that aligns discrete, logically-related activities to particular concepts. Two—Monitor student progress, so adjustments can be made as soon as difficulties arise.
Berch, D. ( 2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38(4), 333-339.
Dowker, A. (2001). Numeracy Recovery: A pilot scheme for early intervention with young children with numeracy difficulties. Support for Learning, 16(1), 6-10.
Dowker, A. (2005). Early identification and intervention for students with mathematics difficulties. Journal of Learning Disabilities, 38(4), 324-332.
Gersten, R., & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. Journal of Special Education, 33(1), 18-28.
Gersten, R., Jordan, N., & Flojo, J. (2005). Early identification and interventions for students with mathematical difficulties. Journal of Learning Disabilities, 38(4), 293-304.
Griffin, S. (1997). Number Worlds: Grade one level. Durham, NH: Number Worlds Alliance Inc.
Griffin, S. (2000). Number Worlds: Preschool level. Durham, NH: Number Worlds Alliance Inc.
Griffin, S. (2004). Building number sense with Number Worlds: A mathematics program for young children. Early Childhood Research Quarterly, 19, 173-180.
Griffin, S., & Case, R. (1997). Re-thinking the primary school math curriculum: An approach based on cognitive science. Issues in Education, 3, 1-49.
Jordan, N., Kaplan, D., Olah, L., & Locuniak, M. (2006). Number sense growth in kindergarten: A longitudinal investigation of children at risk for mathematics difficulties. Child Development, 77(1), 153-175.
Reys, B. (1994). Promoting number sense in middle grades. Teaching Mathematics in the Middle School, 1, 114-120.
Verschaffel, L., & De Corte, E. (1996). Number and arithmetic. In A. Bishop, K. Clements, C. Keitel, J. Kirk Patrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 99-137). Dordrecht, The Netherlands: Kluwer.
Ioannis Agaliotis, Ph.D. is Assistant Professorn of Instructional Methodology for Students with Special Educational Needs in the Department of Educational and Social Policy of the University of Macedonia of Thessaloniki, Greece. Dr. Agaliotis is co-editor of the journal Insights on Learning Disabilities: From Prevailing Theories to Validated Practices, published by LDW®. He has presented at national and international conferences and has published articles and books on inclusive education, assessment and instruction for students with mild disabilities, mathematics for students with special needs, and academic and social support for students with learning disabilities.