Three Steps to Mastering Multiplication Facts

Kling, Gina; Bay-Williams, Jennifer M.

doi:10.5951/teacchilmath.21.9.0548

Cited by 8 publications

(5 citation statements)

References 9 publications

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“…We draw on Ulrich's (personal communication) assertion that a child's spontaneous strategy can reveal their number sequence: counting-on reveals the least advanced sequence (INS), doubling reveals the intermediate sequence (TNS), and BAMT reveals the more advanced sequence (ENS). Importantly, this 'mapping' differs from a common expectation that doubling would be most closely related to multiplicative reasoning (Baroody, 2006;Kling & Bay-Williams, 2015;Verschaffel et al, 2007). In contrast to such an expectation, we hypothesize that children who spontaneously employ a BAMT strategy will reason differently and likely outperform students using a doubling strategy (and, of course, counting-on) when solving multiplicative tasks.…”

Section: Theoretical Frameworkmentioning

confidence: 65%

“…While we particularly stress the difference between counting-on and BAMT, we also intend to account for issues surrounding doubling as a commonly-used strategy. Using Carl's example, we emphasize how this strategy, which is often linked with multiplication (Kling & Bay-Williams, 2015), may mask operations on composite units that we distinguish from mDC.…”

Section: Qualitative Analysismentioning

confidence: 99%

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Children’s Spontaneous Additive Strategy Relates to Multiplicative Reasoning

Tzur

Johnson

Norton

et al. 2021

Cognition and Instruction

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We examine a hypothesis implied by Steffe's constructivist model of children's numerical reasoning: a child's spontaneous additive strategy may relate to a foundational form of multiplicative reasoning, termed multiplicative double counting (mDC). To this end, we mix quantitative and qualitative analyses of 31 fourth graders' responses during clinical, task-based interviews. All participants spontaneously used one of three additive strategies-counting-on, doubling, or break-apart-make-ten (BAMT)-to correctly solve an addition word problem (8 þ 7). We found between-group differences, with asymmetric association of those ordinal variables. We found counting-on to be mainly related to premultiplicative reasoning and BAMT to mDC reasoning. We discuss the theoretical significance and implications of this corroboration of Steffe's model. Children's spontaneous additive strategy relates to multiplicative reasoningPart of the attraction of qualitative, noncomparative methods … was that they promised to generate hypotheses that could then be tested. That promise has gone largely unmet. … The field needs to return more of its energies to conducting hypothesis-testing studies. (Kilpatrick, 2001, pp. 424-425) Children's development of multiplicative reasoning has long been a major concern for mathematics educators (

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Section: Theoretical Frameworkmentioning

confidence: 65%

Section: Qualitative Analysismentioning

confidence: 99%

Children’s Spontaneous Additive Strategy Relates to Multiplicative Reasoning

Tzur

Johnson

Norton

et al. 2021

Cognition and Instruction

View full text Add to dashboard Cite

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“…Rather, mathematical fluency and the ability to work facilely with numbers arises from unhurriedly acquiring conceptual knowledge not from anxiously enduring rapid-fire drilling (Boaler, 2014, 2018–2019). Many math educators question whether time-limited testing can even assess, much less develop, fluency (Clarke, Nelson, & Shanley, 2016); some directly deem it ineffectual (Kling & Bay-Williams, 2015), the source of the country’s lackluster mathematical achievement (Flynn, 2017; Pink, 2018), a ritualistic holdover from teachers’ own flawed education (McCloskey, 2014), driven by an erroneous conception of what mathematical skill acquisition is all about (Seeley, 2016). Consequently, the National Council of Teachers of Mathematics (2014) caution against the use of time-limited tests for acquiring or demonstrating mathematical fluency.…”

Section: Further Researchmentioning

confidence: 99%

“…Rather, mathematical fluency and the ability to work facilely with numbers arises from unhurriedly acquiring conceptual knowledge not from anxiously enduring rapid-fire drilling (Boaler, 2014(Boaler, , 2018(Boaler, -2019. Many math educators question whether time-limited testing can even assess, much less develop, fluency (Clarke, Nelson, & Shanley, 2016); some directly deem it ineffectual (Kling & Bay-Williams, 2015), the source of the country's lackluster mathematical achievement (Flynn, 2017;Pink, 2018) Turning to second language instruction, the debate over time-limited versus untimed tests in second language instruction dates as far back as any of the topics we have covered in this article (Carroll & Sapon, 1959, as cited in Carroll, 1990. Even in those earliest days, second language instructors questioned rote drilling as a means for developing fluency (Crawford & Leitzell, 1930, as cited in Nadal, 1950, and researchers demonstrated that fluency was independent of speed (Carroll, 1958).…”

Section: Further Researchmentioning

confidence: 99%

Four empirically based reasons not to administer time-limited tests.

Gernsbacher

Soicher

Becker‐Blease

2020

Translational Issues in Psychological Science

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For more than a century, measurement experts have distinguished between time-limited tests and untimed power tests, which are administered without time limits or with time limits so generous all students are assured of completing all items. On untimed power tests, students can differ in their propensity to correctly respond to every item, and items should differ in how many correct responses they elicit. However, differences among students' speed of responding do not confound untimed power tests; therefore, untimed power tests ensure more accurate assessment. In this article, we present four empirically-based reasons to administer untimed power tests rather than time-limited tests in educational settings. (1) Time-limited tests are less valid; students' test-taking pace is not a valid reflection of their knowledge and mastery. (2) Time-limited tests are less reliable; estimates of time-limited tests' reliability are artificially inflated due to artefactual consistency in students' rate of work rather than authentic consistency in students' level of knowledge. (3) Time-limited tests are less inclusive; time-limited tests exclude students with documented disabilities who, because they are legally allowed additional test-taking time, are often literally excluded from test-taking classrooms. (4) Time-limited tests are less equitable; in addition to excluding students with documented disabilities, time-limited tests can also impede students who are learning English, students from under-represented backgrounds, students who are older than average, and students with disabilities who encounter barriers (e.g., stigma and financial expense) in obtaining disability documentation and legally-mandated accommodations. We conclude by offering recommendations for avoiding time-limited testing in higher educational assessment.

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“…However, when the response time is constrained to ensure that participants use retrieval rather than strategies to associate the multiplication problem and its answer [ 11 ] multiplication facts can be learnt the same way that word-pairs are presented in other studies of sleep-related memory consolidation. Indeed, research suggests that rote memorization techniques are frequently used by children to learn multiplication problems as part of a range of strategies [ 12 ]. This raises the potential of leveraging sleep-associated benefits to improve children's multiplication fact knowledge.…”

Section: Introductionmentioning

confidence: 99%

Positive impact of sleep on recall of multiplication facts

Spiller,

Gilmore

2023

R. Soc. open sci.

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This study tested the hypothesis that learning complex multiplication problems (e.g. 8 × 23 = 184) prior to sleep would benefit recall in adult participants compared with learning the problems prior to a period of wakefulness. This study used a within-participant design where all participants learnt complex multiplication problems in two conditions separated by one week. In one condition, learning was before bed (sleep learning condition) and in the other condition learning was in the morning (wake learning condition). In each condition, recall was tested approximately 10.5 h later. Data were collected online from 77 participants. In the subset of the sample with greater than or equal to 60% accuracy at the initial learning session ( n = 37), the sleep learning condition participants had better recall compared with the wake learning condition. This equated to a moderate effect size, Cohen's d = 0.51. Regardless of initial levels of learning ( n = 70) the same beneficial effect of sleep on recall was found with a small effect size, Cohen's d = 0.33. This study has identified a beneficial effect of learning prior to sleep on recall of complex multiplication problems compared with learning these problems during the daytime. Future research should explore whether similar effects are observed with children learning simple multiplication facts.

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Three Steps to Mastering Multiplication Facts

Abstract: First understand what fluency is, then use these games and a sequence of strategies to help your students develop facility and confidence.

Cited by 8 publications

References 9 publications

Children’s Spontaneous Additive Strategy Relates to Multiplicative Reasoning

Children’s Spontaneous Additive Strategy Relates to Multiplicative Reasoning

Four empirically based reasons not to administer time-limited tests.

Positive impact of sleep on recall of multiplication facts

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