I apprehend with some agitation the acknowledgment by Barry Garelick (“What It Takes to Actually Improve Math Education”) to Rick Hess’s account with Andrew Coulson (“The Case for Game-Based Math Learning”).
Garelick abominably sets up a straw-dog challenge amid acquirements conceptual ability or procedural ability in mathematics learning. There is bright affirmation that both are important and that they abutment anniversary other, including in one cardboard Garelick cites, and in the archetypal book, Conceptual and Procedural Knowledge, edited by James Hiebert, appear in 1986. The 2001 book, Adding It Up, accounting by the Committee on Mathematics Learning, accustomed by the National Analysis Council, broadly cited in our field, appear on bristles strands of algebraic proficiency. One of those is procedural fluency. These abstract underlie best abreast angle of mathematics education, including admonition to teachers. Suffice it to say, about no mathematics drillmaster would abjure abysmal access amid the two strands of knowledge, admitting there are differences of assessment on which should advance and when.
Consider the botheration of developing procedural and conceptual compassionate of division. An algorithm frequently taught, alike allowable in abounding accompaniment standards for mathematics, is continued division. About anyone who has completed a worksheet of twenty continued analysis problems can acquaint you that practicing procedures abandoned does not miraculously aftereffect in a conceptual compassionate of division. The more-complex-than-you-might-think concepts arranged calm with compassionate the algorithm accommodate the role of the divisor and the allotment in accepting the result, models for division, back analysis applies in analytic real-world problems, interpreting quotients that are bidding as decimals fractions, and the role of abode amount and our abject ten cardinal arrangement in accustomed out the analysis algorithm. These aspects of the concepts of analysis are underpinnings that any acceptance continuing in STEM apprenticeship and careers charge apperceive and be able to use.
The role of apprenticeship in allowance acceptance advance a multi-faceted appearance of analysis is clear. This apprenticeship is actual bent and the consistent ability useful. What ability that apprenticeship attending like? A affluent assignment (not a bad term) is answer that the long-division algorithm absolutely works for any two rational numbers. There are abounding means to do it, and abounding of these are attainable to youth. Comparing addition algorithms, invented or otherwise, is an important action for developing compassionate and a absolution of why either works all the time. This affectionate of allegory is a “generic” accomplishment account teaching. Back apprentice appoint in this affectionate of activity—justifying—they are accurately accommodating in a convenance that about every mathematician does: creating an argument, or proof, for a accustomed conjecture. (In this case, that the long-division algorithm works every time.) Back acceptance do this work, they are agreeable in accurate mathematics (another appellation Garelick disparages). And not surprisingly, they charge attack with the conceptual aspects of analysis I agenda above.
My colleagues and I accept accounting a book for teachers, Algebraic Argumentation in Average School, appear by Corwin, in which we accommodate an organized, accurate access to bringing this convenance of able mathematicians to the average academy classroom. It is based on a dozen years of analysis accomplishing aloof that. We anticipate it is account a apprehend for anyone who wants to breach chargeless of the conceptual/procedural tug of war.
Jennifer Knudsen is a chief mathematics drillmaster at TERC.
Last adapted July 21, 2021
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