Fostering At-Risk Kindergarten Children's Number Sense

A 9-month training experiment evaluated whether computer-assisted discovery learning of arithmetic regularities can facilitate kindergartners' fluency with the easiest sums. After a pretest, kindergartners with at least one risk factor (n = 28) were randomly assigned to either a structured add-0/1 training condition, which focused on recognizing the n + 0/0 + n = n and the n + 1/1 + n = the number-after-n rules, or an active control group. Using pretest fluency as the covariate, ANCOVAs revealed that the structured add-0/1 group significantly outperformed the control group on both practiced and unpracticed (transfer) n + 0/0 + n and n + 1/1 + n items at the delayed posttest and had significantly larger gains in mathematics achievement. Key instructional implications include: Early intervention that targets discovering rules for adding with 0 and 1 and family-specific developmental prerequisites is feasible and more effective than typical classroom instruction in promoting fluency with such basic sums. Such rules may be a critically important bridge between informal and formal mathematics. View correction statement:Errata ACKNOWLEDGMENTS The research reported herein was supported by a grant from the Institute of Education Science, U.S. Department of Education, through Grant R305K050082 ("Developing an Intervention to Foster Early Number Sense and Skill"). Preparation of this article also was supported, in part, by grants from the U.S. Department of Education (R305A080479: "Fostering Fluency with Basic Addition and Subtraction Facts") and (R305B100017: "UIUC Postdoctoral Research Training Program in Mathematics Education"). The opinions expressed are solely those of the authors and do not necessarily reflect the position, policy, or endorsement of the Institute of Education Science or Department of Education. Notes Unlike empirical induction (merely discovering a pattern) and like deductive reasoning, mathematical induction entails using what is known (the discovered pattern) and logical reasoning to create new knowledge. Unlike formal mathematical induction, its informal counterpart does not involves reasoning about all integers, formulating a logical proof, or require formal training. The successor principle and a number-after rule for mentally adding 1 are logically related but reverse operations. The former entails recognizing that 1 must be added to a number to obtain the number after it; the latter involves recognizing that the number after is the result of adding 1 to a number. A response bias determination was done separately for each session and based on all 20 items given in a session. A scorer scored the response to each item in one of five ways: possible inappropriate use of a strategy (resulting in an incorrect answer); possible appropriate use of a strategy (resulting in a correct answer); other incorrect estimates (error not attributable to the strategy); other correct estimates (correct response not attributable to the strategy); and use of concrete or abstract counting strategy or no response. In order to rise to the level of a response bias, four criteria had to be met: (a) total inappropriate uses is half or more of all estimation errors: aTOT ÷ (aTOT + cTOT) ≥ .50; (b) total inappropriate uses is more than 25% of all trials: aTOT ÷ 20 > .25 (MINIMUM of 6 trials); (c) total uses of strategy is half or more of all estimates: (aTOT + bTOT) ÷ (aTOT + bTOT + cTOT + dTOT) ≥ .50; and (d) total uses of strategy is half or more of all trials: (aTOT + bTOT) ÷ 20 ≥ .50 (MINIMUM of 10 trials). This straightforward scoring procedure yielded 99% interrater for agreement on whether a response bias was used and, if so, the specific strategy used in (28 participants × 2 test sessions × 2 sets/session) 112 possible cases. A more complete description of the scoring process is available upon request from the first author. Logically, skip counting by n is not possible if a child cannot start counting with n (instead of "one"). Skip counting also logically requires fluent number-after knowledge for continuing a skip count. For example, skip counting by two requires entering the count sequence at "two" and recognizing that the next number in the skip count is two numbers after two: after two is three and after three is four. A qualitative analysis of response biases further supported the feasibility of structured add-0/1 training. This analysis can be obtained by contacting the first author. For n + 1 items, the total number of repetitions would be 60 if practice with the related number-after n item and—as suggested by some (but not all) models of mental addition development—practice of a commuted 1 + n partner were counted. This is still relatively little practice. When simply scoring efficiency, the results for the practiced and unpracticed n + 0/0 + n were the same or similar as those produced by scoring in context. However, the significance level and effect size for the unpracticed n + 1/1 + n were substantially improved (p = .001 and d = 1.22), and the effect size for the unpracticed n + 1/1 + n was substantially increased (d = 1.34). The appreciable negative effect size for the unpracticed 4 + 2 may have been due to another a feature—the counting-by-twos practice. This practice may have enabled a couple of doubles participants to use skip counting by two (2, 4, 6) to efficiently determine the sum of 4 + 2. Clearly, though, such an effect needs to be replicated with a larger sample.