When Edge.org invited scientists to write about scientific terms which ought to be more widely known, mathematician Keith Devlin chose the term "number sense." I agree with him that it is an important concept to know. But he and I probably disagree as to what can be done when a child lacks it.
Different people define "number sense" rather differently. Dr. Devlin defines it as "fluidity and flexibility with numbers, a sense of what numbers mean, and an ability to use mental mathematics to negotiate the world and make comparisons." That is not a bad definition. But I would define it a little differently. I like Wikipedia's definition: "an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations." The key word here is INTUITIVE. It is an important adjective because it distinguishes it from "mathematical reasoning," which requires conscious thinking. To do a geometry proof requires mathematical reasoning. On the other hand, number sense is like our five senses. It is natural, subconscious and instinctive. It is typically something that children have, which is not a result of explicit teaching. It doesn't come by thinking. Rather, it goes by feel.
Dr. Devlin ends his essay by saying that parents should ensure that their children have acquired number sense by the time they graduate from high school. This seems to suggest an underlying assumption--that number sense can be taught or aquired if one does not have it in the beginning. This may be too optimistic an assumption. I am a certified teacher with over 12 years of math teaching experience. I have also tutored mostly middle school children for about 10 years. Through my tutoring experience, I have come across students who have an extraordinarily weak number sense. These students do not have an intuitive feel of the meaning of a number. If you ask them which one is bigger, 1/3 or 1/2, they may answer that 1/3 is bigger. They typically have much trouble memorizing the multiplication table. And they have difficulty understanding and operating with fractions. Because of this, they also have difficulty with divisions, decimals, percentages, ratios, etc. Thus, a lack of number sense would have wide and serious implications.
For those students lacking in number sense, extra effort would only help marginally. The same can be said about tutoring or extra parental help. This has greatly troubled me as a tutor. First, it is very sad for the child. Second, as a learning specialist and professional tutor, I would like to show results. But improvements do not come easily, regardless of the amount of effort. The parents may or may not be understanding. I tutored some minority child last summer. The child attends an independent school in the neighborhood. The mother was quite open in sharing her observations. She told me that while she is very good with numbers, her husband has always had problems with math. She understands the genetic component of the problem.
I have students with similar difficulties once in a while. Thankfully, extreme cases of lack of number are rare. But I tend to have one case every year. American culture has a certain "can do" attitude. It seems to many Americans that anything is possible, if only one applies oneself. This is not true. And we cannot properly deal with the problem unless we are willing to recognize reality. The truth is that number sense is mostly driven by genetics. Dr. Melissa Libertus of Johns Hopkins University has done extensive research with the relationship between primitive number sense and elementary math aptitude. Because she studied young children between ages 3 and 5, it would be safe to assume that what was observed had to do with innate traits and abilities. As a teacher and professional tutor, I have no doubt that one can always improve through effort and guidance by a learning specialist. For many years, I have been making the case that intelligence can be both taught and learned. On the flip side of th