PA Coalition for World Class Math

    The Word Versus Does Not Belong Between "Basic Skills" and "Conceptual Understanding"

Disagreements about math education are frequently mischaracterized as arguments about "basic skills" versus "conceptual understanding." The PA Coalition for World Class Math strongly believes that both basic skills and conceptual photounderstanding are crucial to a strong math foundation. Liping Ma's seminal book, Knowing and Teaching Elementary Mathematics, casts considerable doubt on the premise that a profound understanding of elementary math is possible in the absence of strong procedural skills. It also establishes that other countries have, indeed, learned to impart both superior procedural skills and conceptual understanding. That is our goal. We believe it is the only reasonable goal for mathematics education in our state and our country if we are to maintain our current status as a leader in science, medicine, mathematics, and technology. 

Hung-Hsi Wu, a professor of mathematics at the University of California, Berkeley, contributed an article to the AFT's Publication American Educator titled: "Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education." We recommend reading the entire paper; here are two brief excerpts:

 “Facts vs. higher order thinking” is [an] example of a false choice that we often encounter these days, as if thinking of any sort—high or low—could exist outside of content knowledge. In mathematics education, this debate takes the form of “basic skills or conceptual understanding.” This bogus dichotomy would seem to arise from a common misconception of mathematics held by a segment of the public and the education community: that the demand for precision and fluency in the execution of basic skills in school mathematics 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 acquire the former without the latter. . . .

"Sometimes a simple skill is absolutely indispensable for the understanding of more sophisticated processes. For example, the familiar long division of one number by another provides the key ingredient to understanding why fractions are repeating decimals. Or, the fact that the arithmetic of ordinary fractions (adding, multiplying, reducing to lowest terms, etc.) develops the necessary pattern for understanding rational algebraic expressions. At other times, it is the fluency in executing a basic skill that is essential for further progress in the course of one’s mathematics education. The automaticity in putting a skill to use frees up mental energy to focus on the more rigorous demands of a complicated problem. Such is the case with the need to know the multiplication table (for single-digit numbers) before attempting to tackle the standard multiplication algorithm, a fact we will demonstrate in due course. Finally, when a skill is bypassed in favor of a conceptual approach, the resulting conceptual understanding often is too superficial. This happens with almost all current attempts at facilitating the teaching of fractions."

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