Number Sense

We all know someone who is "good with numbers". Such people have a skill that I have simply chosen to call "number sense". Several authors refer to a similar concept using different terms. For example, John Allen Paolos, in his classic book Innumeracy, calls this skill "numeracy," a term first used in a 1959 report on education in the United Kingdom.

Number sense involves several informal aspects of quantitative reasoning.

• The ability to quickly estimate quantity and magnitude, to quickly determine roughly what sort of values the answer to a problem should have, or check whether an answer makes sense. For example, it is helpful to know that 847 + 1171 is around 2,000. For many real-life applications (e.g. knowing whether you have enough money to buy both a television and a computer) a rough estimate is often sufficient. Even if it isn't, knowing roughly what the answer should be can help ensure that you didn't make a mistake in calculation (it's really easy to punch a wrong key on a calculator). So, for example, if you did the problem and came up with an answer of 918 or 14,187, you would know that the answer is wrong, whereas if you came up with an answer of 2,018, you could be confident that your answer is likely correct.
• Related to estimation, having a sense of real-world quantity and magnitude. This sort of information isn't part of formal mathematics, but is important to help determine whether an answer to a real-life problem is reasonable or not. For example, is the population of the United States 300 thousand or 300 million or 300 billion? Is the distance from your house to the corner store 100 metres, 300 metres, 1 kilometre, or 3 kilometres?
• An understanding of the laws of arithmetic, geometry, and logic, not so much in being able to recite them, but in having an intuitive grasp of them that makes it possible to appraise and rearrange problems to help solve them quickly. For example, if you were asked to evaluate 64 + 57 + 36 + 43, by the commutative law of addition, you can rearrange the numbers in any order that you want. So, you might mentally rearrange the problem to (64 + 36) + (57 + 43). Each bracket sums to 100, so the result of the addition is 200.
• The ability to view mathematical concepts from multiple perspectives, so that, when confronted with a problem, you can use a perspective that you are comfortable with and that works well for the problem. Take, for example, the number of ways of defining at a circle:
  • The set of points equal to a given distance away from a given point.
  • The limit, as n approaches infinity of a n-sided regular polygon.
  • The conic section that is created with a cut parallel to the base of the cone.
  • The shape that encloses the greatest area for a given perimeter.
  • The figure that is not a straight line and whose curvature is constant at every point.
  Exercise: There are several other ways to define a circle. How many other ways can you come up with?
  If you understand all these ways of thinking about a circle, problems like "Colleen has 45 metres of fencing. She would like to use this fencing to enclose the greatest possible area. In what shape should she arrange the fencing?" or "The distance from the centre of a 215-sided regular polygon to a point on its edge is 29 centimetres. Estimate its area." are trivial. Unfortunately, most schools only teach one way of looking at concepts and don't encourage students to look for alternate representations of the same concept, so this is a skill not always learned by students.
• The ability to think critically about the use of mathematics in society (e.g. in the newspapers, by politicians, lenders, etc.); to be able to to ask whether the results make sense, whether the data were gathered and the calculations performed in a correct manner, to be able to validate the results for yourself, and so on.