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Balance between fluency and understanding

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Balance between fluency and understanding

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Define Fluency for Kindergarten students

Consider how to develop/obtain fluency

How to avoid rote memorization

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We hear a lot about how American students lag behind their international peers academically, especially in subjects like math. In the most recent Program for International Student Assessment, commonly known as PISA, students in the United States ranked 26th out of 34 countries in mathematics. On the surface, it would seem that we're a nation of math dullards; simply no good at the subject. But a spate of new research suggests that we may be underestimating our students, especially the youngest ones, in terms of their ability to think about numbers.

A study published in the April issue of the American Educational Research Journal, for example, finds that kindergarten students learn more when they are exposed to challenging content such as advanced number concepts and even addition and subtraction. In turn, elementary school students who were taught more sophisticated math as kindergarteners made bigger gains in mathematics, reported the study's lead author, Amy Claessens of the University of Chicago.

Another study, published last year by Dr. Claessens with co-authors Mimi Engel and Maida Finch, concluded that as things stand, many children in kindergarten are being taught information they already know. The "vast majority" of kindergarteners have already mastered counting numbers and recognizing shapes before they set foot in the classroom, Dr. Claessens and her co-authors noted, yet kindergarten teachers report spending much of their math teaching time on these skills.

The students don't gain anything from going over familiar ground: In the article published this month, Dr. Claessens and her colleagues report that pupils do not benefit from basic content coverage, but that all the kindergarteners in the study, regardless of economic background or initial skill level, did benefit from exposure to more advanced content.

Discussions about how to improve learning for young children usually focus on the length of the whole school day or the number of students in classes, but rarely on what is taught during the hours school is in session. Increasing the time kindergarten teachers spend on more advanced math concepts may be a simpler and more cost-effective way to boost learning.

What about the play and the social interaction that is so important for young kids? The researchers note that time for such activities could easily be preserved by replacing instruction on basic math concepts with the teaching of more sophisticated ones - especially in light of the finding that students aren't benefiting from such basic coverage anyway. Kindergarteners could be tackling more challenging math ideas while still spending plenty of time in the blocks corner and the dress-up closet.

Young students are ready to learn more advanced math concepts, as long as they are presented in an engaging, developmentally appropriate way. The next time we lament the performance of older American students, we could think instead about how to improve the math instruction given to their younger brothers and sisters.
Read more on the latest in social science research about how we learn, like Designing Smarter Homework, on Annie Murphy Paul's Brilliant blog. Annie Murphy Paul is the author of "Origins," a book about the science of prenatal influences and "Brilliant: The New Science of Smart," to be published by Crown in 2015.

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Research on Children and Math: Underestimated and Unchallenged

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We hear a lot about how American students lag behind their international peers academically, especially in subjects like math. In the most recent Program for International Student Assessment, commonly known as PISA, students in the United States ranked 26th out of 34 countries in mathematics.

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Kindergarten teachers report spending much of their math instructional time teaching students basic counting skills and how to recognize geometric shapes-skills the students have already mastered before ever setting foot in the kindergarten classroom, new research finds. The findings reveal a misalignment between what the students are being taught and what they already know.

"This study is one of the first to raise the question: Is the content that teachers report teaching in kindergarten meeting the needs of the majority of their students?" Mimi Engel, assistant professor of public policy and education at Vanderbilt's Peabody College and lead author of the study, said.

"We looked at the data [from the Early Childhood Longitudinal Study-Kindergarten Cohort or ECLS-K], immersed ourselves in the literature, and we saw that it's been well-documented that the vast majority of kids can count once they start kindergarten," she continued. "About 95 percent of kids have mastered basic number skills-the numbers one through 10-both the language of counting and one-to-one correspondence. We also noticed that teachers spend a lot of time on these basic skills."

Engel and her collaborators reported their findings online in November in Educational Evaluation and Policy Analysis.

The study showed that the vast majority of students had mastered basic counting and shapes by the fall of kindergarten. In contrast, very few had mastered simple addition and subtraction.

Yet teachers reported spending the most classroom time-typically 13 days per month-on the skills the students had already mastered.

Devoting additional days per month to basic counting and shapes was negatively associated with end-of-kindergarten mathematics test scores. However, more time spent on teaching higher-level mathematics, such as place value, currency, ordinality, and addition and subtraction, was associated with an increase in math test scores at the end of kindergarten.

The findings showed that a large portion of the mathematics content taught during kindergarten may not meet the needs of many kindergarteners and that closer attention to children's knowledge and skills at school entry may be warranted. In fact, the increases shown by spending more time on more advanced math topics suggest that a relatively modest shift in classroom coverage would lead to small gains in mathematics achievement.

"This study doesn't help us get at why this [misalignment] occurred," Engel said. "However, there's another round soon to be released of the ECLS-K called the K:2011 that follows children who were in kindergarten in 2010-11. We're dying to get our hands on that data to see if policies like No Child Left Behind shifted content in the classroom."

Data from the ECLS-K was gathered by the National Center for Education Statistics to give a snapshot of what kindergarten looked like in 1998-99 for about 21,000 children nationally. These children were then followed through eighth grade. The data is rich in descriptive information on the children's status at entry to school, their transition into school and their progression through eighth grade. The longitudinal nature of the ECLS-K data enables researchers to study how a wide ranfge of family, school, community and individual factors are associated with school performance.

Engel's research uses the ECLS-K, other large-scale national databases and quantitative and qualitative means to investigate topics in education policy. The current research grew out of prior work on the effects of early academic achievement on later academic outcomes.

"Getting into it more from a policy perspective, there's a structure around reading and a huge knowledge base around early reading. But it's not clear that there's as much emphasis or knowledge around math in early elementary classrooms, though there is research being done to support more structure."

Engel collaborated on the study with Amy Claessens, assistant professor in the Harris Graduate School of Public Policy Studies at the University of Chicago, and Maida A. Finch, PhD'12, assistant professor in the Department of Education Specialties at Salisbury University.

The article is forthcoming in print.

The research was supported by the National Institute of Child Health and Human Development Grant #5R24HD051152- 07, the Oak Ridge Associated Universities Ralph E. Powe Junior Faculty Enhancement Award, the University of Chicago Population Research Center and Vanderbilt University.

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Most math being taught in kindergarten is old news to students

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Kindergarten teachers report spending much of their math instructional time teaching students basic counting skills and how to recognize geometric shapes-skills the students have already mastered before ever setting foot in the kindergarten classroom, new research finds. The findings reveal a misalignment between what the students are being taught and what they already know.

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Dev Psychol. Author manuscript; available in PMC 2009 Nov 25.

Published in final edited form as:

Abstract

Children's number competencies over 6 time points, from the beginning of kindergarten to the middle of 1st grade, were examined in relation to their mathematics achievement over 5 later time points, from the end of 1st grade to the end of 3rd grade. The relation between early number competence and mathematics achievement was strong and significant throughout the study period. A sequential process growth curve model showed that kindergarten number competence predicted rate of growth in mathematics achievement between 1st and 3rd grades as well as achievement level through 3rd grade. Further, rate of growth in early number competence predicted mathematics performance level in 3rd grade. Although low-income children performed more poorly than their middle-income counterparts in mathematics achievement and progressed at a slower rate, their performance and growth were mediated through relatively weak kindergarten number competence. Similarly, the better performance and faster growth of children who entered kindergarten at an older age were explained by kindergarten number competence. The findings show the importance of early number competence for setting children's learning trajectories in elementary school mathematics.

Keywords: number competence, mathematics achievement, longitudinal studies

Poor achievement in mathematics is a major concern in U.S. schools. Advanced study in high school mathematics is a pillar for success in college science ( Sadler & Tai, 2007) and serves as a gateway for a wide range of vocations in science, technology, engineering, and mathematics disciplines. Unfortunately, many U.S. students do not develop the foundations for learning advanced mathematics. Mathematics difficulties are cumulative and worsen with time. For example, difficulties with whole numbers are obstacles to learning fractions, and difficulties with fractions lead to failure in algebra ( National Mathematics Advisory Panel, 2008). Mirroring U.S. national trends, recent data from the state of Delaware showed approximately 25% of students failed to meet state mathematics standards in third grade, with the percentage of failing students increasing each grade thereafter through ninth grade, where 49% failed ( Delaware Department of Education, 2007).

National statistics in the United States reveal persistent income disparities in mathematics achievement across grade levels, with low-income students scoring substantially lower than middle-income students ( National Assessment of Educational Progress, 2007). Yet, low-income youth who acquire strong mathematical skills by the end of middle school are much more likely to graduate from college than their peers who do not acquire such skills ( National Mathematics Advisory Panel, 2008).

Early Number Competence

Number Competence and Mathematics Difficulties

Weak general number competence is reflected in poorly developed counting procedures, slow fact retrieval, and inaccurate computation-all characteristics of children with mathematics difficulties ( Geary, Hamson, & Hoard, 2000; Jordan, Hanich, & Kaplan, 2003a, 2003b). Mastery of number combinations is tied to fundamental number knowledge ( Baroody & Rosu, 2006; Locuniak & Jordan, 2008). It is difficult to memorize arithmetic facts by rote without understanding how combinations relate to one another on a number line (e.g., 3 + 2, 2 + 3, 5 − 2, and 5 − 3). Accurate and efficient counting facilitates children's learning of number relations, leading to strong connections between a problem and its solution and reducing the need for rote memorization ( Siegler & Shrager, 1984). Developmental dyscalculia, a severe form of mathematics difficulties, is attributed to domain-specific cognitive impairments in number knowledge rather than domain-general deficits in memory, spatial processing, or language ( Butterworth & Reigosa, 2007). Children with dyscalculia show particular weaknesses in recognizing and comparing numbers as well as in counting and enumerating sets ( Landerl, Bevan, & Butterworth, 2004).

Malleability of Number Competencies

It has been argued that infants have a natural number sense or ability to represent numbers in a nonverbal manner. These preverbal systems (e.g., an object file system for the precise representation of small sets of objects and an analogue magnitude system for capturing approximate representations of large sets) develop without input or instruction ( Berch, 2005; Dehaene, 1997; Feigenson, Dehaene, & Spelke, 2004) and are shared by humans from differing cultural background as well as other species ( Gordon, 2004). They provide a foundation for acquiring number competencies involving words and symbols. Knowledge of the verbal or symbolic number system is heavily influenced by experience or the input the child receives ( Case & Griffin, 1990; Geary, 1995; Levine, Jordan, & Huttenlocher, 1992). Engaging young children in number activities and games (e.g., board games that emphasize one-to-one correspondences, counting, and number representations) is important for building number competence ( Baroody, 1987; Booth & Siegler, 2008; Copley, 2000; Gersten et al., 2005). Efforts to teach number competencies to high-risk kindergartners have resulted in significant gains on first-grade mathematics outcomes compared with control groups ( Griffin, Case, & Siegler, 1994).

Predictability of Early Number Competence

Although kindergarten number competence is clearly predictive of mathematics outcomes in relative proximity (e.g., kindergarten number competence to first-grade mathematics), it is possible that the relation fades and becomes less important with time. Because typical first-grade mathematics curricula focus on simple number concepts and calculation, measures of number competence and first-grade mathematics achievement are similar. In fact, Chard et al. (2005) used a number knowledge test as a marker for achievement in first grade, one that assesses the same constructs as the number measures that were used in kindergarten. Mathematics expectations, however, become more complex in elementary school. By the end of third grade, the point at which many U.S. children start taking state-mandated mathematics assessments, the curriculum layers Base 10 place value concepts, written computation with multidigit numbers, rational numbers, and so forth onto basic whole-number constructs ( Kilpatrick, Swafford, & Findell, 2001). Children are required to apply their number knowledge to solve novel problems. Although the elementary school mathematics curriculum diversifies, it still incorporates many elements of number competence. Algebra and even geometry (e.g., calculating perimeters and areas) depend on skill with numbers. Mazzocco and Thompson (2005) found that children who were identified with learning disabilities in mathematics also performed poorly on specific number competence items (e.g., magnitude judgments of one-digit numbers and mental addition) on a standardized early mathematics test in kindergarten.

Present Study

The aim of the present study was to examine the predictive relationship between early number competence and later mathematics achievement. It is the end piece of the Children's Math Project, a multiyear longitudinal investigation of children's mathematics difficulties. The present work extends previous studies ( Jordan et al., 2007; Mazzocco & Thompson, 2005) by examining number competence and growth over multiple time points in relation to later mathematics achievement and growth. Children were assessed in number competence six times, from the beginning of kindergarten through the middle of first grade. Our measure was designed to tap number skills that are relevant to the mathematics curriculum in primary school rather than more general cognitive abilities. It assessed competencies related to counting, number comparisons, and calculation, which are strongly related to one another ( Jordan et al., 2007). Then children's mathematics achievement was measured on five occasions, from the end of first grade through the end of third grade, on a standardized test assessing both written computation and applied problem solving. This design allowed us to use a sequential process growth curve model to examine level of and growth in number competence (kindergarten to early first grade) in relation to mathematics performance and growth from late first grade to late third grade. Recent studies (e.g., Fuller, Wright, Gesicki, & Kang, 2007) have emphasized the importance of understanding students' patterns of growth as well as their levels of proficiency.

We asked the following questions: (a) Does level of number competence in kindergarten predict level of mathematics performance after first grade? (b) Does level of kindergarten number competence predict rate of achievement in mathematics between first and third grades? (c) Does rate of growth in early number competence predict later mathematics level and growth? Because age (older children performed better than younger children), gender (boys performed better than girls), and income level (middle-income children performed better than low-income children) were significant predictors of number competence in previous studies (e.g., Jordan et al., 2006), we considered each of these background variables in the present analyses.

Method

Participants

Participants were drawn from six schools in the same public school district in northern Delaware. The schools were selected because they served children from both low-income and middle-income families. Background characteristics of participants in kindergarten (where we set the intercept for number competence; n = 378) and third grade (where we set the intercept for children's mathematics achievement; n = 196) are presented in . In the kindergarten sample, 54% of the children were boys, 56% were minority, and 33% were low income. In the third-grade sample, 52% were boys, 45% were minority, and 23% were low income. For both samples, the mean age at the beginning of kindergarten was 5.5 years. The attrition rate was similar across the participating schools. All children were taught mathematics with the Math Trailblazers curriculum ( Teaching Integrated Mathematics and Science, 2004) throughout the course of the study, with the exception of 15 children who were taught with Investigations in Number, Data, and Space ( TERC, 1998) in second and third grades. The two curricula are similar in content and approach. The mean standard scores in mathematics achievement were about 100 in our third-grade sample (see ), which is also the mean of the national normative sample (on the Woodcock-Johnson III Achievement Tests; McGrew, Schrank, & Woodcock, 2007).

Table 1

Materials

Number competence

The number competency core battery was developed as a part of a larger longitudinal study of children's mathematics development (see Jordan et al., 2007). We used items and tasks that were given to children at all six time points in kindergarten and first grade. The 42-item measure has good internal reliability for kindergartners and first graders (coefficient alpha > .80 at each level, with an average reliability of .87, SD = .03). Administration and scoring procedures were standardized to reduce potential bias ( Jordan et al., 2006). Testers read instructions verbatim, and all scoring questions and ambiguities were discussed and resolved by the research team.

The test items tapped counting and number recognition, number comparisons, nonverbal calculation, story problems, and number combinations. The measure is not timed and typically takes about 25-30 min to administer. The subtasks were always presented to each child in the following order: counting and number recognition, number comparisons, nonverbal calculation, story problems, and number combinations. The total possible score on the measure was 42 points.

Counting and number recognition involved 10 items. Children were first asked to count to 10 and were given one point if they succeeded. Children were allowed to restart counting only once but were always allowed to self-correct. Children's knowledge of counting principles ( Geary, Hoard, & Hamson, 1999) was assessed after viewing a set of either five or nine alternating yellow and blue dots. A finger puppet told the child he or she was learning to count. The child was then asked to indicate whether the puppet counted "OK" or "not OK." Correct counting involved counting from left to right and right to left. Unusual but correct counts involved counting the yellow dots first and then the blue dots or counting the blue dots first and then the yellow dots. For incorrect counts, the puppet counted left to right but counted the first dot twice ("1, 2"). Children received a score of correct/incorrect for each of eight trials. (Children were screened for color blindness prior to the task, and no child failed the screen.) Finally, children were asked to name a visually presented number (i.e., 13).

Number comparisons, adapted from Griffin (2002), consisted of eight items. Given a number (e.g., 7), children were asked what number comes after that number and what number comes two numbers after that number. Given two numbers (e.g., 5 and 4), children were asked which number was bigger or which number was smaller. Children also were shown visual arrays of three numbers (e.g., 6, 2, and 5), each placed on the point of an equilateral triangle. They were then asked to identify which number was closer to the target number (e.g., 5) at the triangle's apex.

On nonverbal calculations the tester and child sat facing each other, with a box of 20 chips placed off to the side. The experimenter also had a box lid with an opening on the side. Four addition and four subtraction calculations were presented: 2 + 1, 4 + 3, 2 + 4, 3 + 2, 3 − 1, 7 − 3, 5 − 2, 6 − 4. The tester placed a number of chips on her mat (in a horizontal line) and told the child how many chips were on the mat. The examiner then covered the chips with the box lid. The researcher either added or removed chips (through the side opening) one by one and at the same time told the child how many chips were being added or taken away. For each item, the children were asked to indicate how many chips were left hiding under the box, either with chips or by stating the number word. Children's errors were corrected on the initial addition and subtraction calculations. The nonverbal calculation task was adapted from and validated by Levine et al. (1992).

Story problems involved four oral addition and four oral subtraction story problems, read one at a time. The calculations were the same as those used in nonverbal calculation. The addition problems were phrased "Sue has m pennies. Bill gives her n more pennies. How many pennies does Sue have now?" whereas the subtraction problems were phrased "Darrel has m pennies. Karen takes away n of his pennies. How many pennies does Darrel have now?"

Finally, four addition and four subtraction number combinations were given, with the same calculations that were presented on nonverbal calculation and story problems. The items were orally phrased as "How much is m and n?" and "How much is n take away m?"

Mathematics achievement

Children were given the Calculation and Applied Problems portions of the Woodcock-Johnson III for a composite mathematics achievement score (WJMath; McGrew et al., 2007). The WJMath is an individually administered, standardized diagnostic battery with strong reliability (coefficient alpha > .80 for ages 6-8). The WJMath has high content validity and is significantly correlated with performance on the third-grade math portion of the Delaware State Testing Program ( r = .77, p < .01). The Calculation (WJCalc) subtest measures the ability to perform computations in a conventional written format. The Applied Problems (WJApp) subtest requires the child to analyze and solve orally presented mathematics problems in various contexts. The correlation between the two subtests is .68 for ages 6-8. The WJMath measures are not timed and have a total administration time of about 30 min.

Procedure

Children were assessed longitudinally a total of 11 times between kindergarten and third grade. They were given the number measure on six occasions: four times in kindergarten (September, November, February, and April) and twice in first grade (September and November). Children's later achievement was assessed on the WJMath five times: April of first grade, September and April of second grade, and September and April of third grade. One of several trained graduate research assistants in school psychology assessed children individually in their schools. Each testing period lasted about a month, and schools were visited in approximately the same order over the various time points.

Results

Raw scores were used for all analyses. Correlations between WJMath and number competence at the various time points are presented in . All the correlations are positive and significant ( p < .01). On the 42-item number competence measure (in February of kindergarten, where we set the intercept in the subsequent growth curve analyses), a raw score of 18 was at the 25th percentile, a score of 22 was at the median, and a score of 29 was at the 75th percentile.

Table 2

Problems of Attrition

A common feature of longitudinal studies is loss of participants over time. Attrition gives rise to the usual problems of missing data in longitudinal studies. To assess this problem, we used a logistic regression analysis to examine the extent to which absence at the final time point is related to gender, income, minority status, and age. The results are shown in . Low-income and minority children were approximately 1.2 times more likely to be absent from the study at the last time point compared with their middle-income and majority children counterparts. These effects are statistically significant. Gender and age did not predict the odds of being absent from the study. The finding of differential attrition may limit the generalizability of the findings and should be kept in mind when interpreting the results.

Table 3

The most restrictive assumption that can be made regarding missing data in this context is that they are missing completely at random; that is, the missing data patterns are independent of both observed and missing data on all variables. A more realistic assumption regarding missing data is they are not missing at random, in which the missing data at any point in time t is due to observed responses at that time point and possibly previous time points. Statistical modeling for data not missing at random, however, requires a model for the process that led to missing data, and typically such modeling is not attempted.

A reasonable alternative in longitudinal studies is to assume that the missing data are missing at random, in which "missingness" (including attrition) at time t may be due to levels of responses at time t - 1, or perhaps earlier. Attrition in this study was found to be due primarily to children moving from the school district (rather than withdrawing from the study or being absent on the day of the testing), and there is no evidence that it resulted from the actual performance of the children at either time t or earlier time points. Thus, in the following growth curve analyses, missing data were handled by an imputation procedure based on full information maximum likelihood under the assumption that missing data are missing at random ( Little & Rubin, 2002). This procedure was developed by Arbuckle (1996) and implemented in the Mplus program ( Muthén & Muthén, 2007). It allows us to use all available data in the estimation of the sequential growth curve models discussed next.

Sequential Process Growth Curve Modeling

Number Competence and Woodcock-Johnson General Mathematics Achievement (WJMath)

A sequential process growth curve model was used to estimate the intercepts and slopes for number competence and mathematics achievement (WJMath). In keeping with the structural equation modeling specification of growth curve modeling ( Willett & Sayer, 1994), we refer to the intercepts and slopes as the growth factors. A sequential process growth curve model is a variant of multivariate growth curve modeling ( Kaplan, 2009; Willett & Sayer, 1994) that joins two conventional growth curve models but allows for the growth factors of one process to be predicted by the growth factors of the other, through the specification of regression relations among the growth parameters. In the present study, the sequential process growth curve model enables us to test whether the growth factors (i.e., intercepts and slopes) associated with WJMath can be predicted by the growth factors of number competence. All growth curve analyses used the Mplus software program ( Muthén & Muthén, 2007).

Two models are considered. Model 0 is the sequential process growth curve model without the addition of background predictors and serves as the baseline model. The path diagram in shows how the sequential process model is specified. The squares indicate the repeated measures of number competence (i.e., four times in kindergarten and twice in first grade), the rectangles indicate the repeated measures of mathematics achievement as measured by the WJMath (i.e., once in first grade, twice in second grade, and twice in third grade), and the circles represent the intercepts and slopes for number competence and the intercepts and slopes for WJMath. The manner in which the repeated measures provide estimates of the growth factors is based on the structural modeling approach to growth curve modeling (see, e.g., Kaplan, 2009; Willett & Sayer, 1994). Our primary interest is in the relationships among the growth factors of number competence and WJMath, and the single-headed arrows among the growth factors themselves indicate this.

Model 1 adds the background predictors of income (1 = low income, 0 = middle income), gender (1 = male, 0 = female), and kindergarten start age (centered around the sample mean). A path diagram of Model 1 is given in . In contrast to , shows the regressions of the growth factors on the background predictors (gender, income level, and kindergarten start age), indicated by the single-headed arrows from the predictors to the growth factors. Note also that Model 1 is a mediating effects model, in which the background predictors are hypothesized to influence the growth factors of WJMath only through the growth factors of kindergarten number competence.

presents the parameter estimates for the baseline Model 0 and for Model 1, which incorporates the background predictors of gender, income level, and kindergarten start age. On the basis of inspection of the empirical trajectories of number competence and WJMath, we specified a linear growth curve model for both models. The intercept for number competence was centered at the third number competence time point, corresponding to February of kindergarten, and the intercept for WJMath was set at the last time point, corresponding to April of third grade.

Typical measures of model fit are sensitive to the fact that structural models are being specifically constrained to yield estimates of growth. Moreover, in the context of this study, models to be compared are nonnested, in the sense that we will be adding additional variables to the model and not additional structural paths to the baseline model. Conventional measures of model fit, such as the likelihood ratio chi-square test and the root-mean-square error of approximation, are used for nested comparisons. In comparing improvement in the model when adding variables, we rely on the Bayesian information criterion (BIC). The BIC is a measure that balances the fit of the model with a penalty function for adding parameters to the model and can be used when comparing nonnested models. Among competing models, the model with the lower BIC value is recommended.

Baseline Model 0

Inspection of the first column of under Model 0 shows that the growth factors (i.e., intercepts and slopes) for both outcomes (i.e., number competence and WJMath) are statistically significant ( p < .05). Both number competence and WJMath show significant and positive growth over their respective periods. The observed and estimated mean raw scores for number competence over the six time points are displayed in and for WJMath over the five time points in .

In the interest of space, the standardized solutions are not shown. However, we report effect sizes in terms of standard deviation change in WJMath growth factors (intercepts and slopes) for a one standard deviation increase in number competence growth factors (intercepts and slopes). The means and standard deviations for the growth factor scores are intercept number competence, M = 23.26, SD = 6.01; slope number competence, M = 0.79, SD = 0.23; intercept WJMath, M = 46.56, SD = 6.83; and slope WJMath, M = 0.61, SD = 0.11.

The results under Model 0 of also show the regressions of the growth factors for WJMath on the growth factors of number competence. Higher average performance on number competence in kindergarten is associated with higher average performance on WJMath in third grade ( b = 0.96, z = 14.97, p < .05, CI[confidence interval].95 = 0.84 - 1.09). Thus, for a one standard deviation increase in number competence (corresponding to 6 points), there is a 5.3-point increase in WJMath in the third grade. Moreover, steeper rates of growth in number competence over kindergarten until the middle of first grade are associated with higher average performance on WJMath in third grade ( b = 3.53, z = 2.24, p < .05, CI.95 = 0.44 - 6.61). This result indicates that for a one standard deviation increase in the rate of growth in number competence (corresponding to 0.23 points), there is approximately a 1-point increase in WJMath in the third grade. We also observe that higher average performance on number competence in kindergarten is associated with steeper rates of growth in WJMath between first and third grades ( b = 0.01, z = 3.23, p < .05, CI.95 = 0.00 - 0.02). In terms of effect sizes, this result indicates that for a one standard deviation increase in number competence (corresponding to 6 points), there is a 0.03-point increase in the rate of growth in WJMath. Finally, rate of growth in number competence was not significantly related to rate of growth in WJMath.

Overall, we found modest (though significant) effects of higher levels of number competence with respect to growth over time in WJMath. The large effects for this baseline model were related to the intercepts; namely that higher levels of kindergarten number competence were predictive of statistically significant and substantively meaningful performance on WJMath in the third grade.

Model 1: Addition of background variables

Model 1 in shows the results of adding the background predictors of income, gender, and kindergarten start age in performance on number competence and WJMath. The results relative to Model 0 do not change appreciably. Of interest, however, is that low-income children demonstrated poorer average performance and poorer average growth in number competence in kindergarten compared with their middle-income counterparts. Older children showed higher average performance on number competence than younger children. There were no significant effects of gender. Adding the regression of number competence on background predictors yielded an improvement in the fit of the model as measured by the BIC. In total, 20% of the variance in number competence performance and 7% of the variance of the number competence growth can be accounted for by gender, income, and kindergarten start age. Regarding WJMath, approximately 66% of the variance in WJMath performance at the end of third grade and 10% of the variance in the WJMath growth can be accounted for by the number competence growth factors.

Indirect effects

As noted earlier, Model 1 specified only indirect effects from the background predictors to the WJMath growth factors through the number competence growth factors. It is important to determine whether the process by which the background predictors influence WJMath is mediated only through number competence (pure mediation) or whether there are direct effects from the background predictors to WJMath over and above any indirect effect through number competence (partial mediation). One way that this can be determined is through an inspection of the modification indices and expected parameter changes (see Kaplan, 2009, for further explanation). An inspection of the modification indices and expected parameter changes associated with omitted direct effects (not shown) does not suggest that the model will be greatly improved if these direct effects are included. Thus, evidence supports a fully mediating model, and we now turn to the indirect effects.

presents the estimates for indirect effects of income and kindergarten start age for WJMath growth factors as mediated by the number competence growth factors. (Effects for gender were not significant in Model 1 and thus are not reported.) There were significant indirect effects for income and kindergarten start age on mathematics achievement growth factors by the number competence intercept only. Low-income children performed almost five points less than middle-income children on WJMath at the end of third grade ( b = −4.62, z = −7.48, p < .05, CI.95 = −5.83 - −3.41) and achieved at a slower rate ( b = −0.04, z = −2.98, p < .05, CI.95 = −0.07 - −0.02) as mediated through their level of number competence in kindergarten. The estimated means for number competence and WJMath at their respective time points are plotted for income in Figures 4A and 4B . The actual means, standard deviations, and range of scores, broken down by income level, are shown in . The standard deviations are roughly the same for the income groups, and the ranges reveal considerable overlap in the distributions.

Table 6

In terms of kindergarten start age, older children performed better on WJMath at the end of third grade ( b = 1.45, z = 4.36, p < .05, CI.95 = 0.80 - 2.10) and achieved at a faster rate ( b = 0.01, z = 2.68, p < .05, CI.95 = 0.00 - 0.02) owing to better number competence in kindergarten. In terms of effect sizes, these results indicate that each month older a child is when starting kindergarten confers a 1.45-point advantage on the WJMath as mediated through kindergarten number competence. Regarding achievement growth, the findings indicate that for a 1-month age difference, a child will achieve at a rate of approximately two thirds of a standard deviation on the WJMath as mediated through kindergarten number competence.

Number Competence and Components of Mathematics Achievement

To provide more information about the relation between level of early number competence and growth in later mathematics outcomes, we performed parallel sequential growth curve analyses on each of the two subareas on the WJMath: WJApp and WJCalc. In the interest of space, we do not present baseline model results. Instead, presents the parameter estimates for the model, which incorporates the background predictors. The intercept for number competence was centered at the third number sense time point, corresponding to February of kindergarten, and the intercept for both WJApp and WJCalc was set at the last time point, corresponding to April of third grade. The pattern of results for both applied problems and calculation was the same as the pattern for composite mathematics (WJMath). The only exception was that the number competence slope predicted the WJCalc intercept but not the WJApp intercept.

presents the estimates for indirect effects of income and kindergarten start age for WJApp and WJCalc as mediated by the number sense growth factors. Again, the pattern of results is the same as the one from the initial analysis with WJMath. The one exception was for kindergarten start age, in which there was an indirect effect through the number competence intercept for WJCalc slope but not for WJApp slope. The estimated means for WJApp at their respective time points are plotted for income in and for WJCalc in . The actual means, standard deviations, and range of scores, broken down by income level, are shown in .

Number Competence Subtask Analyses

A coefficient alpha of .70 or above is considered the minimum acceptable level for reliability of a test ( Reynolds, Livingston, & Willson, 2006). Two of our number subtasks reached this threshold: story problems (mean alpha coefficient of .73, SD = .08) and number combinations (mean alpha coefficient of .83, SD = .04). Thus, we performed separate sequential growth curve analyses on these subtasks. The results are presented in and . The intercept for both story problems and number combinations was centered at the third time point, corresponding to February of kindergarten, and the intercept for WJMath was set at the last time point, corresponding to April of third grade. (We do not present outcome math data broken down by WJCalc and WJApp because preliminary analyses showed not appreciable differences.)

The pattern of effects for story problems and number combinations is the same as the pattern seen in the initial analysis with general number competence. However, we did find that boys had a slightly faster rate of growth than girls on story problems but not on number combinations-an effect not seen on the analysis with the general number competence measure. The actual means, standard deviations, and range of scores, broken down by income level and problem type, are shown in .

Table 11

Discussion

Although kindergarten number competencies predict mathematics outcomes in first grade, few studies have examined whether the relation continues in subsequent grades, when the subject matter becomes more complex. Even fewer studies have examined early number competence in relation to rate of achievement in mathematics, even though growth rates are important for gauging children's progress under the U.S. No Child Left Behind Act ( Fuller et al., 2007).

Our findings provide compelling evidence that before first grade children develop foundational number competence that supports more complex mathematics. Higher levels of kindergarten number competence predicted statistically significant and substantively meaningful performance in composite mathematics achievement at the end of third grade. Level of kindergarten number competence was a modest but significant predictor of rate of achievement between first and third grades, and rate of growth in number competence was a modest but significant predictor of level of achievement at the end of third grade. Our findings did not vary appreciably when we examined kindergarten number competence in relation to elementary mathematics subareas of written calculation procedures and applied problem solving separately. Early understanding of number relations and operations provides support for learning complex calculation procedures involving larger numbers as well as problem solving in varied contexts. This is in keeping with arguments that conceptual knowledge and procedural skill are iterative, each prompting the learning of the other ( National Mathematics Advisory Panel, 2008; Rittle-Johnson, Siegler, & Alibali, 2001).

Consideration of background variables revealed poorer long-term mathematics outcomes (in general achievement and subareas) for children from low-income families than for those from middle-income families-a finding that extends previous work (e.g., Jordan, Huttenlocher, & Levine, 1992; Jordan et al., 2007; Siegler, in press). Low-income children performed more poorly than middle-income children in mathematics achievement at the end of third grade and achieved at a slower rate between first and third grades. More important, however, was the finding that the poor mathematics performance of low-income children in third grade was mediated by their relatively weak number competence in kindergarten. Low-income children enter kindergarten with fewer number-related experiences than their middle-income counterparts-at home, in their communities, and in their preschools ( Clements & Sarama, 2008)-and it is likely that early learning opportunities contributed to the substantial advantage of middle-income children in the present study. Because number competence is achievable for most children ( Ginsburg, Lee, & Boyd, 2008), the finding of its intermediate effect in predicting complex mathematics has practical relevance for teachers and families.

The better performance and faster growth (between first and third grades) in mathematics achievement of older children also was explained by kindergarten number competence. It is not clear, however, why starting kindergarten at an older age confers a modest but reliable advantage over younger children in number competence. Possibly, some younger children enter kindergarten with less mature behavior or weak attention, which in turn makes them less available for learning. Further research is needed to investigate this issue and to verify similar age effects in other samples of children. There were no meaningful gender effects in our study.

It could be argued that our findings reflect general cognitive competence, rather than number competence per se. However, Jordan, Glutting, and Ramineni (2008) found a divergent association between number competence at the end of kindergarten and reading performance at the end of third grade ( r = .29), as opposed to a convergent association between number competence at the end of kindergarten and mathematics performance at the end of third grade ( r = .63). Moreover, Locuniak and Jordan (2008) found that early number skills are uniquely predictive of mathematics fluency when general verbal and spatial factors were considered.

Why are early number competencies important to later mathematics functioning? Understanding of numbers and number relations makes formal mathematics more accessible. It helps children make sense of the count sequence, develop a mental number line, and reason about numbers ( Baroody, 1985; Siegler & Booth, 2004; Siegler & Robinson, 1982). Counting knowledge allows children to count on or up from addends to solve novel number combinations-a key arithmetic strategy in early elementary school ( Geary, Hoard, Byrd-Craven, & DeSoto, 2004). Reasoning about numbers and number relationships also helps children develop strategies for solving complex word problems ( Kilpatrick et al., 2001).

Our findings demonstrate the importance of kindergarten number competence for setting children's developmental learning trajectories in elementary school mathematics. Theories of development with atypical populations (e.g., specific language impairment) indicate that timing of skill acquisition plays a critical role in development ( Karmiloff-Smith, 1998). A seemingly circumscribed problem early on can trigger more problems as development progresses ( Karmiloff-Smith, 1998; Thomas & Karmiloff-Smith, 2003). If children leave kindergarten with weak number competencies, especially with respect to operational knowledge and skills, they will enter first grade at a disadvantage and may never catch up to children who started with good number competencies. Children who start first grade ahead are likely to advance their mathematics learning, whereas children who are behind are still acquiring basic skills. In the reading area, Juel (1988) found that children who entered first grade with little or no phonemic awareness-an early reading-related skill-almost always became poor readers, a finding also supported more recently by Schatschneider, Fletcher, Francis, Carlson, and Foorman (2004), who demonstrated the importance of letter-sound knowledge to reading. Although many children in the Juel study made steady progress in phonemic skills during first grade, they still entered second grade with less phonemic awareness than children who became average or good readers and continued to show reading delays into fourth grade. The early reading difficulties created a vicious cycle, such that children who had poor reading skills in first grade began to dislike and avoid reading; avoidance of reading then led to poor vocabulary and conceptual development and so on. Such a cycle, if not interrupted, results in permanent gaps between good and poor readers. In a similar vein, early weaknesses in number competence may lead to a cascade of mathematics failure in school.

Number competencies, especially with respect to addition and subtraction, are closely tied to the mathematics curriculum in elementary school (in contrast to general cognitive competencies, such as working memory or spatial facility), and thus it is not altogether surprising that they are predictive over time. Similarly, kindergarten skills that are closely tied to the reading curriculum are more predictive of reading than are general language or perceptual skills ( Schatschneider et al., 2004). However, it should be kept in mind that children enter kindergarten with large individual differences in number competence and varied growth trajectories; some children with initial delays in number skills are responsive to typical kindergarten experiences and catch up to their peers, whereas others have intractable weaknesses ( Jordan et al., 2006).

In conclusion, early number competence is important for setting children's learning trajectories in mathematics. Our study demonstrates a robust and sustained predictive relation between number competence (addition and subtraction number combinations and story problems in particular) and later mathematics outcomes, in terms of level of performance and, to a lesser extent, rate of development. Although the strength of the relation between early number competence and mathematics achievement beyond third grade was not tested, recent work suggests the connection is lasting ( Duncan et al., 2008). The effects of weak number competence, if not addressed, are likely to be felt throughout the school years.

Acknowledgments

This work is supported by National Institute of Child Health and Human Development Grant R01HD036672. We wish to thank the participating children and teachers for their extremely generous cooperation over the course of this 4-year project.

Footnotes

¹Although competencies related to the verbal or symbolic number system also have been referred to as number sense, the term is defined differently in different fields of research. Thus we use number competencies to stress that this knowledge can be learned and taught.

Contributor Information

Nancy C. Jordan, School of Education, University of Delaware.

David Kaplan, Department of Educational Psychology, University of Wisconsin-Madison.

Chaitanya Ramineni, School of Education, University of Delaware.

Maria N. Locuniak, School of Education, University of Delaware.

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# data[].attributes.attachment_link.title
Early Math Matters: Kindergarten Number Competence and Later Mathematics Outcomes

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Children's number competencies over 6 time points, from the beginning of kindergarten to the middle of 1st grade, were examined in relation to their mathematics achievement over 5 later time points, from the end of 1st grade to the end of 3rd ...

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Many students with disabilities lack fluency in solving mathematical calculations using basic math facts ( Garnett, 1992). Early difficulties in mathematics are often characterized by deficits in the ability to quickly recall basic math facts as well as delays in applying computation strategies ( Jordan & Montani, 1997). These deficits in the automatic recall of basic math facts limit students' future success in higher order problem solving skills. With more emphasis on meaningful problem solving in math classrooms, the speed and accuracy with which computation problems are solved is important ( Meltzer et al., 1996). As such, students would benefit if teachers focused on increasing math fact automaticity.

Several strategies have been evaluated to develop more fluent recall of basic math facts, including goal setting and performance feedback. Research suggests that performance feedback has a greater impact on student achievement when performance data are visually displayed in the form of a graph and shown to students than when the data are simply recorded ( Fuchs & Fuchs, 1986). A number of studies have demonstrated the benefits of providing students with graphic displays of their performance data when this type of feedback was implemented alone or as part of an intervention package (e.g., Carr & Punzo,1993; DiGangi, Maag, & Rutherford,1991; Gunter, Miller, & Venn, 2003; McDougall & Brady, 1998; Sutherland & Snyder, 2007).

For example, Codding, Lewandowski, and Eckert (2005) demonstrated that performance feedback and student-selected goal setting were effective in increasing math fluency in students diagnosed with Attention-Deficit/Hyperactivity Disorder (ADHD). The participants also graphed their own performance data as part of the performance feedback. Therefore, students received both verbal and visual feedback related to goals that they set and those set by their teachers. In light of these findings, Gunter and Denny (2004) expressed concern that data graphing is not widespread in classrooms, perhaps due to teachers perceiving the task of graphing as cumbersome. The authors suggested that teachers should involve students as active participants in evaluating their own performance to reduce the time and effort required.

Self-recording academic and social behavior performance data has been found to improve behavior in the absence of other intervention components (e.g., Studswell & Moxley, 1984) and provides the student with immediate feedback ( Moxley, 1998). In Gunter et al. (2003), for example, a student with disabilities was taught to use computer software to graph her reading performance data. The authors suggested that further research on this strategy was warranted and that the best use of computerized self-graphing procedures might be to increase fluency of skills students already have in their repertoire.

As such, the purpose of this study was to evaluate the impact of having students with mild disabilities participate in their own evaluative procedures by graphing their rate of correctly written solutions to basic mathematic calculations and comparing their performance to a goal set by the teacher. Based on the literature reviewed, one might expect students to benefit academically from the graphic display of their performance data. Furthermore, their participation could reduce the burden of daily graphing for teachers and result in more frequent use of this strategy.

Method

Participants

The participants were three female students, all of whom had been identified with a mild disability. Shannon was an 8-year, 6-month old student enrolled in the second grade. Because she had previously been retained in kindergarten and was unable to meet regular promotion criteria at the end of the first grade, she was placed in the second grade for the subsequent academic school year. Shannon had a diagnosis of ADHD and was receiving special education services as a student with mild intellectual disabilities. Her full-scale intelligence quotient (IQ) was 72, as measured by the Wechsler Intelligence Scale for Children - Fourth Edition (WISC-IV). Shannon's academic skills were found to be commensurate with her cognitive ability. She participated in a non-standard administration of her state achievement testing at the end of her first-grade academic school year. According to the results of this standardized assessment, Shannon's math skills did not meet the standards set for first-grade students. A weakness was noted in her ability to recall addition facts with sums to 18.

The second participant was a 7-year, 8-month old student who had been retained in the first grade. According to an initial psychoeducational evaluation, Mandy exhibited a significant developmental delay in the area of communication with a standard score of 69 as measured by the Test of Early Language Development, Third Edition (TELD-3). Mandy's cognitive skills were found to be within normal limits as measured by the Developmental Assessment of Young Children (DAY-C). However, she was identified as a student with a significant developmental delay/learning disability, and she received specialized support in the areas of reading and math in her special education program. According to the results of a non-standard administration of the state achievement test, Mandy's math skills met the standard set for first-grade students. She demonstrated the ability to recall addition and subtraction problems with sums to 18, but she required more time than was allowed during a standard test administration.

The third participant, Dianna, was a 7-year, 2-month old student, retained in the first grade. She was identified with right hemiparesis, causing weakness on the right side of her body. Although this condition did not impede her ability to write and type, it did seem to affect her endurance with both of these motor activities; that is, she reported often that she was physically tired from writing and typing. This participant had a diagnosis of ADHD but was not receiving medication for the condition. Based on the results of an initial psychological evaluation, Dianna exhibited a significant developmental delay in the areas of communication and motor skills. Her cognitive skills were found to be within normal limits as measured by the Battelle Developmental Inventory (BDI). She received special education services for students with learning disabilities at her school; additionally, she received private physical and occupational therapy in the home. Dianna participated in a non-standard administration of the state achievement test. According to the results of this assessment, Dianna's overall performance in math met the standards for first grade; however, she did not meet the standard for number sense and numeration. Dianna was able to recall the sums to addition facts up to 18 if provided additional time as well as paper and pencil with which to draw pictures to represent those facts.

Setting

Sessions were conducted in a special education resource classroom located in a public K-3 elementary school in the southeastern United States. The classroom was equipped with two student computers (standard Macintosh desktop computers), individual student desks, and a table used for small group instruction. All sessions were conducted during each participant's math instruction period. In addition to the participants, one other student was present; this student worked on academic skills and activities similar to those of the participants. The teacher worked with one participant at a time while the other students in the room (2 participants and 1 non-participant) were working independently.

Materials

During each session, students were administered sheets of 21.6 x 27.9 cm math worksheets consisting of 40 one-digit by one-digit numbers with sums from 0 to18. Problems were arranged in rows across columns. Only addition facts mastered at an accuracy level of 80% or greater were included on the worksheets used to evaluate fluency of correctly calculated digits. To determine this accuracy level, each participant had the opportunity to answer calculations at least five times prior to the inclusion of any item on the fluency evaluation worksheet, with at least four of those five opportunities resulting in the correct response. Worksheets were constructed each day to ensure a random presentation of the problems by using the Basic Facts Worksheet Factory available from Schoolhouse Technologies at http://www.schoolhousetechnologies.com/. This program is downloadable at no cost. Problems were added to the problem bank to be included on math worksheets if students demonstrated 80% accuracy in their calculations of new material during separate practice sessions. The number of problems remained the same, but the problems changed each day.

Response Measurement

After each student completed all problems on her assigned worksheet, the teacher checked each problem and circled any digit(s) that was calculated incorrectly; she then counted the total number of correct digits. All teacher-evaluated worksheets were maintained as permanent product records. Correct digits per minute was the dependent measure and was calculated by dividing the total number of correct digits by the total time required of the student to complete the worksheet. Single-digit numeral reversals were counted as correct responses as long as the intended numeral represented the correct digit.

Baseline

Participants were timed using a digital timer while completing their math facts worksheets. Participants were instructed to begin working only after their teacher provided the direction, "Please begin." The timer was started simultaneously with this direction. Each participant was instructed to raise her hand and put her pencil down as soon as all of the problems on the worksheet were completed. The students were timed until they finished. As soon as a participant completed the assigned worksheet, the teacher recorded the time on the back of that participant's worksheet. The teacher then immediately scored the worksheet, calculated the rate of correct digits per minute for each participant, and entered the performance data for each participant in a Microsoft Excel ® spreadsheet. Neither verbal nor graphic feedback was provided to students during these sessions.

Self-Graphing with Goal Setting

An aim line (i.e., goal for each practice session across the entire academic year) was calculated for each participant prior to student participation in development of graphic displays of their performance data. The starting goal was determined by taking each participant's average rate of correct digits during baseline. The goal for the end of the academic year was then established based upon rates suggested by Binder, Haughton, and Bateman (2002) for single-digit addition problems (70 to 100 correct digits per minute). Given the learning needs of the students, the lower parameter of the expected range was used.

Each student's aim line was superimposed onto her graph. To do this, all possible dates during which instruction would take place throughout the course of the school year were entered in cells on a row of a Microsoft Excel ® spreadsheet. Data from the baseline sessions were entered into the cells below each corresponding date. Next, a graph was produced on the spreadsheet that contained the graphed baseline data points and the remainder of an empty graph with spaces for data points for each date of the school year to be graphed. The created aim line was placed on the part of the graph for which data were not entered. It started from a point representing the average of the baseline data points at the beginning of the first intervention phase to the level representing 70 correct digits per minute at the end of the graph area. Additional details about these procedures can be found in Gunter, Miller, Venn, Thomas, and House (2002).

Each participant's graph was saved as a Microsoft Excel ® file. Not only can Excel ® be used to produce graphs suitable for single-subject designs ( Carr & Buckholder, 1998), it also provides the user with several options for customizing the graph. Graphs can be customized by selecting line and marker colors, as well as by selecting background pictures. Background pictures were selected by each participant.

Prior to the first intervention session, each participant received training on how to use the Excel ® software program to enter rate data to plot points on her graph. During this training, the teacher explained the purpose of the aim line to each participant. Participants were encouraged to meet or exceed the aim line by increasing their daily correct digits per minute score. Students were taught to open their Excel ® programs, which were saved in individual student folders on the computer's desktop, enter the spreadsheet, find the empty cell corresponding with the date, enter the correct digits per minute score provided by the teacher, and press the "Enter" key to see the data point automatically entered onto the graph. The need to alternate between chart and spreadsheet screens was eliminated by embedding each participant's graph within her spreadsheet,

During the intervention, the procedures employed for the completion of the math worksheets were identical to those of baseline. However, at the completion of the worksheets, students were informed of their performance and allowed to graph their scores using the procedures taught on the training day. The teacher was available to assist each participant if necessary, as well as to ensure that the correct rate scores were entered under the correct days. Verbal praise was provided to participants when the data point was correctly entered and additionally if the data point met or exceed the aim line. Participants also were allowed to change the background picture on the graph if all data points for the week were entered correctly.

Individualized Modifications

If a participant's progress was below the aim line for three consecutive sessions, instruction was adjusted. With this data-based decision making rule in place, the teacher made modifications in instruction as needed for Dianna. The decision was made to adjust instruction by reducing the number of practice problems from 40 to 20. This was done based on the teacher's observation that Dianna became fatigued before completing the 40-problem worksheet. She completed the remaining 20 problems later in the day but only graphed her performance for the first 20 problems. This resulted in an initial performance increase. However, three consecutive data points were below the aim line by the 29 th session. Another instructional decision was made to continue to present the practice problems in shorter sets as well as to provide an additional verbal reminder not to touch and count every single practice problem, which the teacher observed Dianna doing. The teacher hypothesized that the physical act of touching each problem was actually slowing the addition process. In addition, Dianna was provided 1 min of free time each day her data point for correct calculations met or exceeded the aim line. Although her performance continued to improve, she still did not consistently meet the aim line. Following the 37 th session, in addition to all of the instructional changes previously described, Dianna was allowed to work in an individualized setting to reduce distractions. This modification resulted in an immediate increase in rate, but was followed by three additional data points below the aim line. After the 41 st session, the number of problems in each practice session was reduced from 20 to 10 in addition to the other interventions. She completed the remaining 10 problems during a timed practice session later in the day but only graphed her correct responses for the first 10 problems. Again, this modification was made based on observations that Dianna became fatigued when completing the worksheets. However, it should be noted that, for the purposes of the evaluation, the results of the two daily practice sessions were combined to calculate the rate of correct digits per minute for all 20 problems. This was done to better compare the data from this phase to those in the previous phases (during which Dianna completed 20 problems).

Return to Baseline

Data entered during the second baseline condition were recorded by the teacher and seen only by the teacher. Each participant's graph was modified so she did not see data from the withdrawal period when intervention was reinstated.

Experimental Design

An ABAB design was simultaneously implemented for 2 of the 3 participants to evaluate the effect of the intervention. The intention was to use the same design for Dianna; however, an AB design resulted due to the need to continually modify the intervention to improve the outcomes.

Procedural Integrity and Inter-rater Agreement

In order to verify that the researcher was faithfully following the procedures as written, a second observer collected intervention fidelity data. A special education teacher served as the second observer. This observer collected procedural integrity data during 20% of baseline sessions and 20% of the intervention sessions using a procedures checklist. Prior to the first session during which procedural integrity was assessed, the second observer was taught the various steps that were implemented. This observer participated in role-play sessions with the teacher to ensure that there were no questions regarding any of the steps followed. Data were collected on teacher behaviors such as accuracy of time monitoring and calculation of rate, as well as participant behavior such as entering the correct rate score into the computer (the checklist protocol is available from the second author). To compute the percentage of procedural fidelity, the total number of observed teacher behaviors was divided by the total number of behaviors possible. The mean procedural fidelity across observations was 99% (range, 92% to100%).

Inter-rater agreement was assessed on 25% of each student's worksheets distributed evenly across conditions. Worksheets were independently scored by a second special education teacher trained in scoring procedures by the primary scorer prior to the onset of the study. Duplicated worksheets contained no information regarding the participant's name, date, or condition during which the worksheet was completed. Digit-by-digit agreement was assessed by counting the number of digits agreed upon as correct and incorrect and dividing the number of agreements by the number of agreements plus disagreements and then multiplying by 100. Mean interobserver agreement across participants and conditions was 99.4% (range, 93% to100%).

Results

The effects of the intervention on the students' correct digits per minute are shown in and . Shannon and Mandy showed a decreasing trend in performance during baseline (). During the two phases of self-graphing with goal setting, Shannon and Mandy's performance met or exceeded the aim line on 25 of 30 (83%) and 27 of 30 (90%) occasions respectively.

Dianna's average rate of correct digits during the first baseline phase was 13 (). Although there were several data points above the aim line when the intervention was implemented, her performance did not consistently improve until multiple modifications were made to the intervention. During the final phase, Dianna's rate scores increased to an average of 38 digits per minute.

Conclusions and Recommendations for Practitioners and Teachers

In part, the results support and extend earlier findings, indicating that performance feedback (in the form of self-graphing) and goal setting can improve students' fluency (e.g., Carr & Punzo, 1993; DiGangi et al., 1991; Gunter, et al, 2003; McDougall & Brady, 1998; Sutherland & Snyder, 2007). Anecdotally, the participants demonstrated genuine excitement when their data points met or exceeded the aim line and wanted the teacher to observe their progress. The students also expressed disappointment when they were denied the opportunity to graph their daily data during the reversal.

These results suggest that students with and without disabilities may benefit from the use of computerized applications that allow them to participate in their assessment procedures. Once teachers have been trained to use the computer generated graphics, little time is needed to allow student participation. Graphs serve as a powerful visual means to document student progress on an ongoing basis and can be shared with parents and administrators. In addition, teachers can determine instantly if their instruction is working and make changes if needed. The intervention was less successful for Dianna, who required a number of modifications to the intervention. This case highlights the importance of using student performance to guide program development and intervention ( Fuchs & Fuchs, 1986).

Although Dianna knew the answers to her math facts fluently when questioned orally, she appeared overly focused on forming her numbers perfectly when required to provide written responses. This delayed the completion of her practice problems. In addition, she appeared to become easily fatigued and often complained that her hand hurt, perhaps due to her right hemiparesis and fine motor delays. She was often distracted to the point that she had difficulty remaining seated and appeared to have difficulty remaining focused on the correct location on the worksheet. For example, she would often forget which problem she was working on and would lose valuable time finding her place. The additional intervention components were intended to improve her progress relative to her aim line; however, the various modifications make it difficult to determine what was responsible for the improvements in her performance. Additional research is needed to identify which students would benefit the most from self-graphing and goal setting.

The teacher's use of verbal praise when students' data met or exceeded the daily aim also may have contributed to the outcomes for all students. This factor should be controlled in future evaluations to better understand the sole impact of students graphing their own performance data. The following specific guidelines are recommended for practitioners and teachers:

Use technology available in almost every classroom.

Familiarize yourself with Microsoft Excel ® or other available graphing software.

Have a paraprofessional, student worker or parent volunteer create an Excel ® file for each academic subject for each student prior to beginning the intervention.

Have students record only data that they will use for self-monitoring.

Establish a specific time to evaluate graphic displays of performance.

Even if self-graphing alone does not positively impact student performance, it provides teachers with a graphic representation of the data to assist in program modifications.

Given the theoretical relationship between undesirable social behavior and academic responding (c.f., Gunter & Denny, 1998) and the academic achievement deficits of students identified with conditions such as ADHD and emotional and behavioral disorders ( Kauffman, 2001), it seems prudent to intervene with unobtrusive methods as early as possible to prevent development of both the aberrant social and academic behaviors.

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Effects of Self-Graphing and Goal Setting on the Math Fact Fluency of Students with Disabilities

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We evaluated the impact of goal setting and students' participation in graphing their own performance data on the rate of math fact calculations. Participants were 3 students with mild disabilities in the first and second grades; 2 of the 3 students were ...