A new study by Robert Siegler, professor of cognitive psychology at Carnegie-Mellon University makes a lot of sense to me as a math teacher on the front lines. Carnegie-Mellon's summary states that
"fifth graders' understanding of fractions and division predicted high school students' knowledge of algebra and overall math achievement, even after statistically controlling for parents' education and income and for the children's own age, gender, I.Q., reading comprehension, working memory, and knowledge of whole number addition, subtraction, and multiplication."
Since I began teaching high school math, I have encountered intense fear of fractions in almost every student. Granted, I do not teach gifted or AP students, although I have tutored some. Students prefer decimal representations of numbers, especially when obtained by using their calculator. In Algebra, however, fractions are not only preferable, but they are also more exact if the number is a repeating, nonterminating decimal. (Compare the exactness of the fraction 1/3 to the decimal 0.33, for example.)
I have also found students often cannot do long division. This is an obstacle when we get to division of polynomials, which is based upon long division.
So the question occurs to me: Did the success of handheld calculators cripple our children's long-term math abilities?
Don't get me wrong. I love calculators. With graphing calculators I can teach higher-level concepts without the tedium of graphing and evaluating by hand. I no long need the eight-inch thick volume of logarithms, trig functions, and other mathematical tables that my parents gave me for Christmas when I was a senior in high school. Students today can work many more problems focusing on the "big picture" without getting bogged down in endless calculation. Even one tiny error in several pages of calculations used to doom us to a wrong answer.
However, there is no denying the fact that students often reach high school not having fully mastered basic math facts. Having bemoaned students' inability to multiply and divide, I recently found I had been overly-optimistic in my assessment. Many are not even fluent in addition and subtraction. Read on about my upsetting discovery.
This semester I start every class with a 5-minute practice of addition, subtraction, multiplication, and division on the website Xtramath.org. This is a wonderful tool for students to improve their fluency in basic math facts. I envisioned students zooming through addition and subtraction and only slowing down when they reached multiplication.
I was wrong. Alas, it is ten weeks into the semester, and only about one-fourth of my students are fully fluent in addition! While many "know" their addition facts, they have to consciously think about each simple problem. To succeed in more advanced math, all arithmetic facts must be full "automatized"; that is, one's brain must instantly comes up with the correct number with no conscious thought. It is the math equivalent of reading fluency which is so essential for long-term success in English.
Xtramath drills until a student's response time is nearly instantaneous. This is necessary to succeed, for instance, in factoring problems. Handheld calculators are of marginal use if you have to quickly judge whether the numbers in the expression 40x - 16 have a common factor. You have to know your division facts and divisibility rules. Not only that, but you have to know them without conscious thought. To solve multistep problems you cannot let your brain be hijacked by simple arithmetic. You need to hold the problem in your working memory while you do some quick mental testing of next steps. If your mind is laboring over arithmetic facts, you will quickly lose your bearings.
So now I am a believer. The depressing part is this: even if I can get my students fully fluent (i.e. automatic) in basic arithmetic facts, they still must master integer operations with all the sign rules, as well as recognize common squares and square roots. Then there are fractions. Fraction arithmetic must be re-taught since it was never completely learned. Where is there room for Algebra? And remember, this is not just my classroom, but hundreds of thousands of classrooms throughout the nation.
Now that we have cognitive scientists telling us this -- something that is common sense, actually -- American schools must do some fast adjusting, making elementary math much more rigorous and requiring mastery rather than familiarity.
Siebert cites research comparing the conceptual knowledge of American versus East Asian elementary math teachers. While the American teachers often had more years of schooling, they were frequently unable to answer such questions as: "When you divide two fractions, why do you multiply by the inverted divisor fraction?" They could teach the rote procedures but were unable to teach an understanding of the underlying logic. East Asian teachers were usually able to give several conceptual explanations in response to such questions.
For a summary of the Siegler's research, see Knowledge of fractions and long division predicts long term math success at the Association for Psychological Science website.