Num•ber Sense/Numb•er Sense
By Dr. Eugene Maier
Num·ber \'nem-ber\ n (1) a unit belonging to a mathematical system and subject to specified laws of succession, addition, and multiplication; esp : NATURAL NUMBER (2) an element of any of many mathematical systems obtained by extension or analogy with the natural number system
Numb \'nem\ adj numb·er; numb·est (1) devoid of sensation: DEADENED (2) devoid of emotion: INDIFFERENT
A former colleague and longtime friend called a few weeks ago. "There's a new book out that I think you'll find interesting." He was right, I did. As a matter of fact, I found it fascinating.
The Number Sense: How the Mind Creates Mathematics (Oxford University Press, 1997; 274 pp.) is written by Stanislas Dehaene, Research Affiliate at the Institute of Health and Medical Research in Paris, France. The book traces the rapid developments that are occurring in the field of mathematics cognition -- in the author's words, "the scientific inquiry into how the human brain gives rise to mathematics" -- fed by innovative experiments with infants, rapid advances in brain-imaging techniques, and the ongoing study of the effects of brain lesions, have led to a wealth of new discoveries. In a highly readable style, the author, himself a cognition researcher with a background in mathematics and neuropsychology, traces these developments, separating that which is known from that which is highly likely, while setting forth the conclusions he has reached and what they imply about the educational process.
The author maintains that the evidence is unmistakable that we are born with a deep, intuitive understanding of the counting numbers and that we find ways of adding and subtracting them without explicit instruction. This innate understanding apparently does not extend to other types of numbers, and to gain an intuitive understanding of these requires concrete experiences that lead to mental models. "To function in an intuitive mode," the author asserts, "our brain needs images."
Further, experimental evidence shows that the human brain has not evolved "for the purpose of formal calculations." Remembering multiplication facts and carrying out algorithmic procedures are not our brains' forte. To do this successfully we turn to verbatim memory - that is, memorization without meaning - at the expense of intuition and understanding. The danger is that we become "little calculating machines that compute but do not think."
Thus, the author suggests that we de-emphasize memorizing arithmetic tables and mastering paper-and-pencil algorithms. Instead, we should take advantage of our strength, which is our associative memory. This is what enables us to connect disparate data, use analogies to advantage, and apply knowledge in novel settings - all things that calculators don't do well. And above all, whatever we do in school, we should honor and nurture the vast amount of intuitive knowledge about numbers children bring to the educational process.
A few days before my friend's call, I had read the new "back-to-the-basics" standards adopted by the California State Board of Education, which emphasizes memorization and paper-and-pencil computations while limiting the use of electronic calculators. I was struck by the dissonance between this document and Dehaene's book. While the California standards promote memorization and drill as the basis of mathematical understanding, Dehaene is telling us that, while students can become proficient at such tasks by memorizing sequences of operations, they do so at the expense of understanding and creativity.
Dehaene suggests another course. He believes that the experimental evidence clearly shows that arithmetic algorithms are difficult for humans "to faithfully acquire and execute." Knowing that "we cannot hope to alter the architecture of the brain," he suggests we "adapt our teaching methods to the constraints of our biology." "Since," he continues, "arithmetic tables and calculation algorithms are, in a way, counternatural, I believe that we should seriously ponder the necessity of inculcating them in our children. Luckily, we now have an alternative -- the electronic calculator, which is cheap, omnipresent and infallible... I am convinced that by releasing children from the tedious and mechanical constraints of calculation, the calculator can help them to concentrate on meaning."
So, it appears, we mathematics educators have a choice. We can foster mathematics programs that build on the extensive num·ber sense children bring to the educational process and are compatible with the development of mathematical intuition and understanding, or we can foster programs that, each year, leave our students with a numb·er sense of mathematics than they had the year before.