Can We Teach Number Sense?
Those who view number sense as an intrinsic ability will argue that the elementary components are genetically programmed, have a long evolutionary history, and develop spontaneously without explicit instruction as a young human interacts with the environment. However, most of these researchers do not view number sense as a fixed or immutable entity. Rather, they suggest that the neurocognitive systems supporting these elementary numerical abilities provide just the foundational structure needed for acquiring the expanded abilities cited by mathematics educators. And they recognize that both formal and informal instruction can enhance number sense development prior to entering school.
Berch (2005) notes that the abilities and skills associated with the expanded view of number sense cannot be isolated into special textbook chapters or instructional units, and that their development does not result from a set of activities designed specifically for this purpose. He agrees with those mathematics educators who contend that number sense constitutes a way of thinking that should permeate all aspects of mathematics teaching and learning. It may be more beneficial to view number sense as a byproduct of other learning than as a specific goal of direct instruction.
Gersten and Chard (1999) suggest that the innate qualities of number sense may be similar to phonemic awareness in reading development, especially for early experiences in arithmetic. Just as phonemic awareness is a prerequisite to learning phonics and becoming a successful reader, developing number sense is a prerequisite for succeeding in mathematics. They further propose that number sense is the missing component in the learning of early arithmetic facts, and explain the reason that rote drill and practice do not lead to significant improvement in mathematics ability.
Because Gersten and Chard (1999) believe that number sense is so critical to success in learning mathematics, they have identified five steppingstones that allow teachers to assess a child’s understanding of number sense. Their five levels are
 Level 1. Children have not yet developed number sense beyond their innate notions of numerosity. They have no sense of relative quantity and may not know the difference between “less than” and “more than” or “fewer” and “greater.”
 Level 2. Children are starting to acquire number sense. They can understand terms like “lots of,” “six,” and “nine,” and are beginning to understand the concepts of “less than” and “more than.” They also understand lesser or greater amounts but do not yet have basic computation skills.
 Level 3. Children fully understand “less than” and “more than.” They have a concept of computation and may use their fingers or objects to apply the “count up from one” strategy to solve problems. Errors occur when the child is calculating numbers higher than five, because this requires using the fingers of both hands.
 Level 4. Children are now relying on the “count up” or “counting on” process instead of the “counting all” process they used at the previous level. They understand the conceptual reality of numbers in that they do not have to count to five to know that five exists. Assuming they can count accurately, children at this level are able to solve any digit problem.
 Level 5. Children demonstrate retrieval strategies for solving problems. They have already automated addition facts and are acquiring basic subtraction facts.
References:
Berch, D. B. (2005, July/August). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38, 333–339.
Gersten, R., Chard, D., Jayanthi, M., & Baker, S. (2006). Experimental and quasiexperimental research on instructional approaches for teaching mathematics to students with learning disabilities: A research synthesis. Signal Hill, CA: Center on Instruction/RG Research Group.

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