BASIC SKILLS VERSUS CONCEPTUAL UNDERSTANDING A Bogus Dichotomy in Mathematics Education BY H. WU DUCATION SEEMS to be plagued by false quire the former without the latter. E dichotomies. Until recently, when research and It has been said that had Einstein been born at the common sense gained the upper hand, the debate time of the Stone Age, his genius might have enabled over how to teach beginning reading was character- him to invent basic arithmetic but probably not much ized by many as “phonics vs. meaning.” It turns out else. However, because he was born at the end of the that, rather than a dichotomy, there is an inseparable 19th century—with all the techniques of advanced connection between decoding—what one might call physics at his disposal—he created the theory of rela- the skills part of reading—and comprehension. Fluent tivity. And so it is with mathematics. Conceptual ad- decoding, which for most children is best ensured by vances are invariably built on the bedrock of tech- the direct and systematic teaching of phonics and lots nique. Without the quadratic formula, for example, the of practice reading, is an indispensable condition of theoretical development of polynomial equations and comprehension. hence of algebra as a whole would have been very dif- “Facts vs. higher order thinking” is another example ferent. The ability to sum a geometric series, some- of a false choice that we often encounter these days, as thing routinely taught in Algebra II, is ultimately re- if thinking of any sort—high or low—could exist out- sponsible for the theory of power series, which lurks side of content knowledge. In mathematics education, inside every calculator. And so on. this debate takes the form of “basic skills or concep- The analogue of the same phenomenon in the artis- tual understanding.” This bogus dichotomy would tic domain is even more transparent. A violinist who seem to arise from a common misconception of math- still worries about fingering positions cannot hope to ematics held by a segment of the public and the educa- impress with the beauty of tone or the elegance of tion community: that the demand for precision and phrasing, and an opera singer without the requisite fluency in the execution of basic skills in school math- high notes would try in vain to stir our souls with sear- ematics runs counter to the acquisition of conceptual ing passion. In good art as in good mathematics, tech- understanding. The truth is that in mathematics, skills nique and conception go hand in hand. and understanding are completely intertwined. In The desire to achieve understanding in a technical most cases, the precision and fluency in the execution subject such as mathematics while minimizing the of the skills are the requisite vehicles to convey the component of skills is a most human one. There are conceptual understanding. There is not “conceptual situations where efforts to this effect are called for understanding” and “problem-solving skill” on the one and, indeed, brilliantly executed. One can think of the hand and “basic skills” on the other. Nor can one ac- classics of Courant and Robbins (What Is Mathemat- ics?)1 and Hilbert and Cohn-Vossen (Geometry and the Hung-Hsi Wu is professor of mathematics at the Uni- Imagination).2 In the context of school mathematics, versity of California at Berkeley. He has written nu- however, such a desire cannot be indulged without merous articles on mathematics education, and is doing great harm to students’ education. There are also the co-author with R.K. Sachs of General Relativ- many reasons. Sometimes a simple skill is absolutely ity for Mathematicians. indispensable for the understanding of more sophisti- 1 AMERICAN EDUCATOR/AMERICAN FEDERATION OF TEACHERS FALL 1999 cated processes. For example, the familiar long divi- It is good to start with simple fractions that children sion of one number by another provides the key ingre- can visualize, and they should do many such problems, dient to understanding why fractions are repeating until they have a firm grasp of what they are doing decimals. Or, the fact that the arithmetic of ordinary when they divide fractions. But we should not make fractions (adding, multiplying, reducing to lowest students feel that the only problems they can do are terms, etc.) develops the necessary pattern for under- those they can visualize. We should explain to them standing rational algebraic expressions. At other times, that of course they cannot draw a picture of X\.m ÷ CZ\zm; it is the fluency in executing a basic skill that is essen- it is doubtful that anyone can. But this does not mean tial for further progress in the course of one’s mathe- they cannot do the problem! Or that more complex matics education. The automaticity in putting a skill to problems like this one are not essential. use frees up mental energy to focus on the more rigor- An analogy to addition may be helpful. When chil- ous demands of a complicated problem. Such is the dren were first learning to add, perhaps they counted case with the need to know the multiplication table out three blocks and then counted out four blocks and (for single-digit numbers) before attempting to tackle joined them to get seven blocks. But we didn’t tell the standard multiplication algorithm, a fact we will them that, when faced with the problem 1,272 + 846, demonstrate in due course. Finally, when a skill is by- their only choice was to gather up hundreds of blocks passed in favor of a conceptual approach, the resulting or draw hundreds of dots on their paper and count conceptual understanding often is too superficial. This them. Nor did we tell them the problem was too diffi- happens with almost all current attempts at facilitating cult for them or not important. No, we told them there the teaching of fractions.3 was a mathematical route to the answer. And not a Let us illustrate the last statement with the example “rote, meaningless” one, but a procedure based on of the division of fractions. Recall the familiar method simple but sound mathematical principles. And we of “invert and multiply”: taught it to them. a ÷=×c d a And so we can do with fractions. From the intuitive b d c b to the abstract, and from primitive skills to sophisticated Nowadays, “invert and multiply” has become almost ones, such is the normal progression in mathematics. synonymous with rote learning. Among recent at- The way to approach the division of non-simple frac- tempts to inject conceptual understanding into this tions is not to bypass “invert and multiply,” but to con- topic, the following approach is not untypical.4 front it. We begin by asking what it means to say a frac- tion Rather than relying on algorithms, where memorization of rules is the focus, the Mathland approach relies heavily x on active thinking. To solve problems such as Z\v ÷ Z\x, stu- y dents need to be able to verbalize the question: How equals many halves are there in one-fourth? This kind of fluency a c enables students to use their own logical and visual think- ÷ ing skills to really know what the solution (Z\x) means in b d , relation to the problem. How many halves are there in and realize that perhaps we have not fully come to Z\v? There is one-half of Z\x in Z\v. terms with the meaning of the division of whole num- Many pictures go with the explanation because it is bers. Children are taught, for example, that 2443=8 easy to represent one-half, one-fourth, etc., by squares. means that if you “divide 24 objects into 3 equal por- Three pages down (p. 132), the “invert and multiply” tions, each portion would have 8 objects.” However, algorithm is introduced and students are urged to “see such a grouping of the 24 objects shows that it is if the answers you get by using [the algorithm] match 8+8+8, which is therefore the same as 338. So in this up with answers you got earlier this week.... Allow case, “2443=8” means exactly that 24=338. This rea- plenty of time to experiment with the standard algo- soning turns out to be general, in the sense that if we rithm, then ask [students] to choose one problem that analyze any other example, say 80416=5, then repeat- they worked with both ways and write about how the ing the preceding reasoning leads to a similar conclu- two solution methods compare.” The problems sug- sion that it is the same as 80=5316. Along this line, gested for practice are all of the type fifth-graders should have no trouble understanding B\n ÷ 16, Z\x ÷ Z\n, 3 ÷ Z\n, etc. With conceptual under- that, in general, for whole numbers m, n and k, the standing thus restored—or so it seems—the mathemat- statement ical exposition on the division of fractions comes to an m4n=k end. If only simple fractions such as those given above says exactly the same thing as are involved, the preceding approach emphasizing the 3 visual aspect of division is for the most part adequate. m=n k. The worm in the apple is the need to deal with divi- This then provides an abstract point of view to under- sion problems when the fractions are not at all simple. stand division in terms of multiplication. It is common For example, what do the above brand of logical and to express this interpretation of division as “division is visual thinking skills have to say about X\.m ÷ CZ\zm? Noth- the inverse operation of multiplication.” ing, of course.