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Article Excerpt
Abstract. Automaticity in math facts has been of considerable interest to special educators for decades. A review of the intervention literature suggests at least two common approaches to developing automaticity in facts. One is grounded in the use of strategies for teaching facts, the other emphasizes the use of timed practice drills. Recent research indicates that students might benefit from an integration of these two approaches. This experimental study contrasted an integrated approach (i.e., strategies and timed practice drills) with timed practice drills only for teaching multiplication facts. Participants were 58 fourth-grade students with a range of academic abilities. Fifteen of the students in the study had IEPs in math. Results indicated that both approaches were effective in helping students achieve automaticity in multiplication facts. However, students in the integrated approach generally performed better on posttest and maintenance test measures that assessed the application of facts to extended facts and approximation tasks. These results have implications for teaching a range of skills and concepts that are considered important to overall mathematical competence in the elementary grades.
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Information-processing theory supports the view that automaticity in math facts is fundamental to success in many areas of higher mathematics. Without the ability to retrieve facts directly or automatically, students are likely to experience a high cognitive load as they perform a range of complex tasks. The added processing demands resulting from inefficient methods such as counting (vs. direct retrieval) often lead to declarative and procedural errors (Cumming & Elkins, 1999; Goldman & Pellegrino, 1987; Hasselbring, Goin, & Bransford, 1988). Potential difficulties extend well beyond operations on whole numbers. Finding common multiples when adding fractions with unlike denominators or factoring algebraic equations are but two examples from secondary-school mathematics where automaticity in math facts can facilitate successful performance.
Advocates of contemporary approaches to mathematics, ones that tend to place more emphasis on conceptual understanding and problem solving than on computational skills, see an important place for automaticity in math facts. Isaacs and Carroll (1999), for example, note that automaticity is essential to estimation and mental computations. These skills, particularly the ability to perform mental computations (e.g., make approximations based on rounded numbers such as 10s and l00s), are central to the ongoing development of number sense. In an effort to reach consensus on the current state of K-12 mathematic education, Ball and colleagues (2005) also affirm the importance of automaticity in math facts.
Unfortunately, decades of research show that academically low-achieving students as well as those with learning disabilities (LD) exhibit considerable difficulty in developing automaticity in their facts. Difficulties and delays are apparent from the beginning of elementary school. These students fail to retrieve facts directly when presented in isolation or when embedded in tasks such as multidigit computations. Research on primary-grade students indicates that students with LD are more likely to rely on counting strategies than direct retrieval when working single-digit fact problems (Geary & Brown, 1991; Hanich, Jordan, Kaplan, & Dick, 2001; Hoard, Geary, & Hamson, 1999). These students also tend to make more retrieval and counting errors on simple addition problems than their non-LD peers.
Results of Goldman and her colleagues' research on second- through sixth-grade students with LD show that these students tend to rely heavily on counting over direct retrieval methods (Goldman, Pellegrino, & Mertz, 1988). When compared with non-LD peers, students with LD often employ suboptimal strategies when retrieving facts. For example, if given a problem like 2 + 9, students with LD generally do not employ the more sophisticated strategy of commuting and then adding the numbers in order to derive 11. Instead, they tend to count up from 2 to 11, a method that often results in an incorrect sum. Goldman et al. (1988) conclude that elementary students with LD are delayed in their ability to learn facts automatically, and suggest that this delay can be addressed through systematic practice. This finding is consistent with the results of other research (e.g., Geary, 1993), which suggest that interventions are necessary for students with LD in order to ensure that they can retrieve facts automatically by the end of elementary school.
Teaching Facts Through Strategies
Brownell and Chazal's (1935) early work in math fact instruction initiated a debate over the best approach to teaching facts that has continued to the present day. Their work questions the traditional emphasis in schools on rote memorization, which, if done excessively, can reinforce students' use of immature methods for answering fact problems such as the counting-up strategy described above. Isaacs and Carroll (1999) echo this concern, emphasizing that students naturally develop strategies for learning math facts if given the opportunity. Research supporting the natural development of strategies may be found for addition and subtraction (Baroody & Ginsburg 1986; Carpenter & Moser 1984; Resnick 1983; Siegler & Jenkins 1989) as well as more recent work in the area of multiplication (Anghileri, 1989; Baroody, 1997; Clark & Kamii, 1996; Mulligan & Mitchelmore, 1997; Sherin & Fuson, 2005).
As a consequence of this research, a number of educators emphasize the use of explicit strategy instruction over traditional rote learning when teaching math facts. Methods vary from the use of visual displays such as ten frames and number lines (Thompson & Van de Walle 1984; Van de Walle, 2003) to more general techniques such as classroom discussions where students share fact strategies with their peers (Steinberg, 1985; Thornton, 1990; Thornton & Smith, 1988).
Specific recommendations for multiplication strategies vary considerably in the way fact strategies are linked to a broader ability to perform mental calculations. For example, some special education researchers (e.g., Miller, Strawser, & Mercer, 1996) stress basic rules for multiplication as they relate to math facts (e.g., multiplication by or 1, the commutative property). These rules are frequently taught in math classes. Others (e.g., Chambers, 1996; Garnett, 1992; Thornton, 1990) recommend a wider array of strategies and focus on patterns that are easier to learn. They suggest that doubles, times five, times nine patterns, and square numbers are easier for students to learn than facts such as 4 x 8 and 6 x 7. Van de Walle (2003) also makes this observation in his discussion of strategy instruction for facts.
The link between facts and mental calculations is more evident in recent discussions of number sense (author; Kilpatrick, Swafford, & Findell, 2001; Sowder, 1992). French (2005) shows how derived fact (e.g., 6 x 7 = 6 x 6 + 6) and counting backwards for 9s strategies (e.g., 8 x 9 = 8 x 10 - 8, 9 x 9 = 9 x 10 - 9) can be applied to calculations involving 2 x 1 digit numbers. For example, students with good number sense calculate 99 x 9 by converting the problem to 100 x 9 - 9. Strategies such as "split--add" for 8 x 4 involve splitting the problem into two smaller problems (i.e., 8 x 2 + 8 x 2) and then adding the products. A similar decomposition logic can be applied in the form of distributive multiplication (e.g., 27 x 2 = 25 x 2 + 2 x 2). Other strategies such as doubling and halving (e.g., 32 x 5 = 32 x 10 / 2) also expand a student's ability to compute mentally exact answers to multidigit multiplication problems. The extent to which French's prescriptions apply to academically low-achieving students is not clear.
Math educators argue that emphasis on strategies helps students organize facts into a coherent knowledge network (Isaacs & Carroll, 1999; Rathmell, 1978), thus facilitating long-term retention and direct recall. Strategic instruction facts can even include instruction on extended facts (e.g., 3 x 4 extends to 30 x 4). The link of facts to extended facts ostensibly helps students in estimation and mental computation tasks. Finally, Sherin and Fuson (2005) argue that strategic knowledge remains in many students even though they are able to answer multiplication fact problems within 3 seconds per fact parameters. Students may use a combination of strategies (i.e., "hybrids") when answering relatively difficult facts such as 6 x 7. Their research shows that for a fact such as this, students may rapidly employ a derived fact strategy.
Timed Practice Drills
Cumming and Elkins (1999) point out that many educators and researchers make the unwarranted assumption that strategies--either developed naturally or through explicit instruction--invariably lead to automaticity. However, research cited above (e.g., Geary, 1993; Goldman et al., 1988) indicates that students with LD do not develop sophisticated fact strategies naturally. Furthermore, empirical research on strategy instruction in math facts for students with LD is limited, and the results are mixed in terms of the effective development of automaticity (see Putnam, deBettencourt, & Leinhardt, 1990; Tournaki, 2003).
Timed practice drills as a method for...
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