Why Children Have Difficulties Mastering the Basic Number Combinations and How to Help Them ena, a first grader, determines the sum of 6 + information (e.g., known facts and relationships) 5 by saying almost inaudibly, “Six,” and then, to logically determine (deduce) the answer of an Xwhile surreptitiously extending five fingers unknown combination under her desk one at a time, counting, “Seven, • Phase 3: Mastery—efficient (fast and accurate) eight, nine, ten, eleven.” Yolanda, a second grader, production of answers tackles 6 + 5 by mentally reasoning that if 5 + 5 is 10 and 6 is 1 more than 5, then 6 + 5 must be 1 more Educators generally agree that children should than 10, or 11. Zenith, a third grader, immediately master the basic number combinations—that is, and reliably answers, “Six plus five is eleven.” should achieve phase 3 as stated above (e.g., The three approaches just described illustrate NCTM 2000). For example, in Adding It Up: the three phases through which children typically Helping Children Learn Mathematics (Kilpatrick, progress in mastering the basic number combina- Swafford, and Findell 2001), the National Research tions—the single-digit addition and multiplication Council (NRC) concluded that attaining compu- combinations and their complementary subtraction tational fluency—the efficient, appropriate, and and division combinations: flexible application of single-digit and multidigit calculation skills—is an essential aspect of math- • Phase 1: Counting strategies—using object ematical proficiency. counting (e.g., with blocks, fingers, marks) or Considerable disagreement is found, however, verbal counting to determine an answer about how basic number combinations are learned, • Phase 2: Reasoning strategies—using known the causes of learning difficulties, and how best to help children achieve mastery. Although exagger- ated to illustrate the point, the vignettes that follow, By Arthur J. Baroody all based on actual people and events, illustrate the Art Baroody, [email protected], is a professor of curriculum and instruction in the College conventional wisdom on these issues or its practi- of Education at the University of Illinois in Champaign. He is interested in the development cal consequences. (The names have been changed of number and arithmetic concepts and skills among young children and those with learning difficulties. to protect the long-suffering.) This view is then contrasted with a radically different view (the 22 Teaching Children Mathematics / August 2006 Copyright © 2006 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. number-sense view), originally advanced by Wil- combinations in this time. When it was pointed out liam Brownell (1935) but only recently supported that mastering such combinations typically takes by substantial research. children considerably more time than ten days, the teacher revised her approach, saying, “Well, then, Vignette 1: Normal children can master the basic we’ll spend two days on the hard facts like the 9- number combinations quickly; those who cannot fact family.”) are mentally impaired, lazy, or otherwise at fault. Alan, a third grader, appeared at my office Vignette 2: Children are naturally unmindful of extremely anxious and cautious. His apprehension mathematics and need strong incentives to learn was not surprising, as he had just been classified it. as “learning disabled” and, at his worried parents’ Bridget’s fourth-grade teacher was dismayed insistence, had come to see the “doctor whose and frustrated that her new students had appar- specialty was learning problems.” His mother had ently forgotten most of the basic multiplication and informed me that Alan’s biggest problem was that division facts that they had studied the previous he could not master the basic facts. After an initial year. In an effort to motivate her students, Mrs. discussion to help him feel comfortable, we played Burnside lit a blowtorch and said menacingly, “You a series of mathematics games designed to create will learn the basic multiplication and division an enjoyable experience and to provide diagnostic facts, or you will get burned in my classroom!” information. The testing revealed, among other Given the prop, Bridget and her classmates took things, that Alan had mastered some of the single- the threat literally, not figuratively, as presumably digit multiplication combinations, namely, the n × 0, the teacher intended. n × 1, n × 2, and n × 5 combinations. Although not at the level expected by his teacher, his perfor- Vignette 3: Informal strategies are bad habits that mance was not seriously abnormal. (Alan’s class interfere with achieving mastery and must be pre- had spent just one day each on the 10-fact fami- vented or overcome. lies n × 0 to n × 9, and the teacher had expected Carol, a second grader, consistently won the everyone to master all 100 basic multiplication weekly ’Round the World game. (The game entails Teaching Children Mathematics / August 2006 23 having the students stand and form a line and assumptions about mastering the number combina- then asking each student in turn a question—for tions and mental-arithmetic expertise: example, “What is 8 + 5?” or “How much is 9 take away 4?” Participants sit down if they • Learning a basic number combination is a respond incorrectly or too slowly, until only one simple process of forming an association or child—the winner—is left.) Hoping to instruct and bond between an expression, such as 7 + 6 or motivate the other students, Carol’s teacher asked “seven plus six,” and its answer, 13 or “thirteen.” her, “What is your secret to winning? How do you This basic process requires neither conceptual respond accurately so quickly?” Carol responded understanding nor taking into account a child’s honestly, “I can count really fast.” Disappointed developmental readiness—his or her existing and dismayed by this response, the teacher wrote everyday or informal knowledge. As the teach- a note to Carol’s parents explaining that the girl ers in vignettes 1 and 4 assumed, forming a insisted on using “immature strategies” and was bond merely requires practice, a process that proud of it. After reading the note, Carol’s mother can be accomplished directly and in fairly short was furious with her and demanded an explana- order without counting or reasoning, through tion. The baffled girl responded, “But, Mom, I’m a flash-card drills and timed tests, for example. really, really fast counter. I am so fast, no one can • Children in general and those with learning dif- beat me.” ficulties in particular have little or no interest in learning mathematics. Therefore, teachers must Vignette 4: Memorizing basic facts by rote through overcome this reluctance either by profusely extensive drill and practice is the most efficient rewarding progress (e.g., with a sticker, smile, way to help children achieve mastery. candy bar, extra playtime, or a good grade) or, Darrell, a college senior, can still recall the if necessary, by resorting to punishment (e.g., a answers to the first row of basic division combina- frown, extra work, reduced playtime, or a fail- tions on a fifth-grade worksheet that he had to com- ing grade) or the threat of it (as the teacher in plete each day for a week until he could complete vignette 2 did). the whole worksheet in one minute. Unfortunately, • Mastery consists of a single process, namely, fact he cannot recall what the combinations themselves recall. (This assumption is made by the teacher were. and the mother in vignette 3.) Fact recall entails the automatic retrieval of the associated answer to an expression. This fact-retrieval component How Children Learn of the brain is independent of the conceptual and Basic Combinations reasoning components of the brain. The conventional wisdom and the number-sense view differ dramatically about the role of phases Number-sense view: Mastery that underlies 1 and 2 (counting and reasoning strategies) in computational fluency grows out of discovering achieving mastery and about the nature of phase 3 the numerous patterns and relationships that (mastery) itself. interconnect the basic combinations. According to the number-sense view, phases 1 Conventional wisdom: Mastery grows out of and 2 play an integral and necessary role in achiev- memorizing individual facts by rote through ing phase 3; mastery of basic number combinations repeated practice and reinforcement. is viewed as an outgrowth or consequence of num- Although many proponents of the conventional ber sense, which is defined as well-interconnected wisdom see little or no need for the counting and knowledge about numbers and how they operate or reasoning phases, other proponents of this perspec- interact. This perspective is based on the following tive at least view these preliminary phases as oppor- assumptions for which research support is growing: tunities to practice basic combinations or to imbue the basic combinations with meaning before they are • Achieving mastery of the basic number com- memorized. Even so, all proponents of the conven- binations efficiently and in a manner that pro- tional wisdom view agree that phases 1 and 2 are not motes computational fluency is probably more necessary for achieving the storehouse of facts that complicated than the simple associative-learn- is the basis of combination mastery. This conclusion ing process suggested by conventional wisdom, is the logical consequence of the following common for the reason that learning any large body of 24 Teaching Children Mathematics / August 2006 Figure 1 Vertical keeping-track method (based on Wynroth [1986]) Phase 1. Encourage children to summarize the results of their informal multiplication computations in a table. For example, suppose that a child needs to multiply 7 × 7. She could hold up seven fingers, count the fingers once (to represent one group of seven), and record the result (7) on the first line below the .