EDUCATION SEEMS to be plagued by false dichotomies. Until recently, when research and common sense gained the upper hand, the debate over how to teach beginning reading was character- ized by many as “phonics vs. meaning.” It turns out that, rather than a dichotomy, there is an inseparable connection between decoding–what one might call the skills part of reading–and comprehension. Fluent decoding, which for most children is best ensured by the direct and systematic teaching of phonics and lots of practice reading, is an indispensable condition of comprehension.
“Facts vs. higher order thinking” is another example of a false choice that we often encounter these days, as if thinking of any sort–high or low–could exist out- side of content knowledge. In mathematics education, this debate takes the form of “basic skills or concep- tual understanding.” This bogus dichotomy would seem to arise from a common misconception of math- ematics held by a segment of the public and the educa- tion community: that the demand for precision and fluency in the execution of basic skills in school math- ematics runs counter to the acquisition of conceptual understanding. The truth is that in mathematics, skills and understanding are completely intertwined. In most cases, the precision and fluency in the execution of the skills are the requisite vehicles to convey the conceptual understanding. There is not “conceptual understanding” and “problem-solving skill” on the one hand and “basic skills” on the other. Nor can one ac-quire the former without the latter.
It has been said that had Einstein been born at the time of the Stone Age, his genius might have enabled him to invent basic arithmetic but probably not much else. However, because he was born at the end of the 19th century–with all the techniques of advanced physics at his disposal–he created the theory of rela- tivity. And so it is with mathematics. Conceptual ad- vances are invariably built on the bedrock of tech- nique. Without the quadratic formula, for example, the theoretical development of polynomial equations and hence of algebra as a whole would have been very dif- ferent. The ability to sum a geometric series, some- thing routinely taught in Algebra II, is ultimately re- sponsible for the theory of power series, which lurks inside every calculator. And so on.