Expanding Task Instructions May Increase Fractions Problem Difficulty for Students With Mathematics Learning Disability

Abstract

When a student struggles with a mathematics task, adults may rephrase or expand initial task instructions to clarify instructions or scaffold problem solving. Yet expanded instructions may not benefit all children, especially children with a mathematics learning disability (MLD). Here, we explore whether expanded instructions differentially affect fractions comparison performance for children with or without MLD. Fifth graders (N = 190) completed two consecutive sets of 24 fraction comparison items, each accompanied by initial or expanded instructions, respectively; and also completed vocabulary, spatial reasoning, verbal working memory, executive function, and number knowledge tasks. Results showed that fraction comparison performance was generally worse following expanded rather than initial instructions, particularly for difficult items or for children with MLD. Fixed ordered regressions showed that the strength of cognitive skills as predictors of performance varied depending on instructions format and MLD status, that the five cognitive predictors collectively accounted for more performance variation with initial compared to expanded instructions, and that vocabulary’s relative predictive strength as a single predictor increased when instructions were expanded, but only for children with mathematics difficulties. These findings support the notion that problem features differentially affect children with or without MLD and that not all children benefit from hearing expanded instructions for difficult mathematics tasks.

Keywords

fractions mathematics learning disability problem difficulty expanded task instructions

When a student appears to not understand a mathematics task, an adult may logically repeat, rephrase, or otherwise expand the initial task instructions—perhaps with a concrete example—intending to clarify the instructions or scaffold problem solving. Such expanded instructions or explanations may not benefit all children facing difficult problems. Children who have mathematics learning disability (MLD) may be especially prone to receive such expanded instructions for mathematics tasks simply because they err on mathematics tasks more frequently than their typically achieving (TA) peers or because adults make presumptions about their mathematics knowledge and difficulties. Therefore, it is important to evaluate whether children’s success following expanded instructions (as a follow up to initial instructions) varies as a function of their mathematics achievement level and to specifically evaluate whether expanded instructions benefit children who struggle with mathematics.

Here, we propose that expanded instructions, specifically expanded instructions for a fractions comparison task, will have variable effects on students’ success on the task, depending on their mathematics achievement levels, and that this variability is related to children’s executive function (EF) skills, particularly working memory. We focus on the contribution of children’s EF skills because of the well-documented relation between EF skills and mathematics performance and learning (e.g., Bull, Espy, & Wiebe, 2008; Clements, Sarama, & Germeroth, 2016), evidence for the role of working memory in MLD (Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007; Mazzocco & Kover, 2007; Murphy, Mazzocco, Hanich, & Early, 2007), and evidence that challenges faced by children with MLDs are at least partially linked to difficulty maintaining information in working memory (Bull & Scerif, 2001; Murphy et al., 2007). On one hand, expanded instructions may require all children to rely more heavily on their self-regulation or working memory skills due simply to the increase in information that comprises expanded instructions, information to which the child must attend and process. This increase in working memory load may thus hinder performance for all children, but especially for children who struggle with the mathematics task demands. On the other hand, all students may benefit from expanded instructions if the additional, expanded instructions reveal information that initial instructions failed to convey. However, the gains afforded from expanded instructions may be negligible for TA students if their performance is at ceiling levels; in this case, the benefits are limited to those students who perform below ceiling, and thus may improve performance. These alternatives illustrate how expanding task instructions may be differentially problematic or beneficial for children with different mathematics achievement levels.

In this study, we focus on instructions for a fractions task because of the frequency with which children experience difficulty with fractions, including but not exclusively children with MLDs (e.g., Hecht & Vagi, 2010; Mazzocco & Devlin, 2008). Given their persistent challenges, fractions tasks may be a particularly frequent target for expanded instructions and provide an ecologically valid situation under which to study such potential effects of expanded instructions. Our study is an exploratory, retrospective analysis of existing data from a completed longitudinal study on the trajectory and cognitive correlates of MLD. That larger study included a fraction magnitude comparison task with two sets of instructions administered sequentially to all participants at Grades 4 and 5. This design afforded comparing performance accuracy between the initial and expanded instructions, the latter of which involved an example. Measures of EF and working memory were also administered to all participants.

Analysis of these retrospective data was guided by two research questions. First, we asked whether expanded instructions influenced performance on a fractions magnitude comparisons task in general and whether this influence varied with students’ mathematics achievement level. Second, we asked whether EF skills are differentially relevant to performance on a fraction task presented with initial versus expanded task instructions. We include other cognitive skills associated with mathematics performance (e.g., vocabulary, visual perception, and number sense skills) and verbal comprehension ability to evaluate the specific predictive contribution of EF skills, and we examine which of these variables best predicts performance on a fractions magnitude comparison task presented with initial or expanded task instructions.

We predicted that expanded instructions would increase accuracy on task performance for students with low or average mathematics achievement (who have more age appropriate conceptual understanding of fractions than children with MLDs), relative to their performance following initial instructions. In contrast, we predicted that expanded instructions would selectively hinder performance of students with MLD, based on their limited conceptual grasp of fractions and their reported challenges maintaining information in working memory (Bull & Scerif, 2001; Murphy et al., 2007), and that a hindering effect will be partially accounted for by the relation between their EF and fractions comparison skills. Moreover, based on reported differences in rates of growth observed between children with MLD and children with low achievement who do not meet MLD criteria (Murphy et al., 2007), we predicted that the MLD and low mathematics achievement (LA) groups in the present study would show selective effects of expanded instructions at Grade 5, but not at Grade 4 (when fractions knowledge is still emerging for children in these two groups). Notwithstanding variation across local education agencies, fractions are typically introduced during Grade 3, yet many children continue to struggle with fractions throughout elementary school (Hansen et al., 2015; Hecht & Vagi, 2010) and beyond (Mazzocco & Devlin, 2008). Finally, we predicted that EF skills would account for more variation in performance under the expanded versus initial instructions, owing to greater cognitive demands associated with the former.

Method

Participants

Participants were drawn from a larger prospective longitudinal study of mathematics ability and disability (Mazzocco, Myers, Lewis, Hanich, & Murphy, 2013; Murphy et al., 2007) and were recruited from a socioeconomically diverse public school district. Schools with low rates of mobility (6.8% to 18.9%, M = 16.5%) and free or reduced lunch participation (1.58% to 29.04%, M = 16.5%) were enrolled, to minimize attrition. All English-proficient kindergartners attending those schools were invited to participate, and 249 (120 boys) enrolled, 86% of whom were White.

The sample for the present study was initially drawn from 201 participants (91 boys) who completed the fractions comparison task during Years 5 and 6 of the larger study, at which time most children were in Grades 4 and 5 and had no data missing for our variables of interest. We note that our fraction comparison task, the primary measure of interest, was based on the Grade 3 curriculum through which all participants were introduced to fractions. This is important because some children who had repeated a grade were in Grades 3 and 4, respectfully, at the time of the first and second annual assessments.

For analyses of group differences, MLD status was based on scores from a standardized mathematics assessment administered through Grade 6 (as part of the larger longitudinal study; Murphy et al., 2007). Sixteen children (10 boys; 15 White and one Black) whose scores fell at or below the 10th percentile (or within the 95% confidence intervals for this range) during at least four of their seven annual assessments (and who never scored in the TA range) were classified as MLD; another 26 children (15 boys; 19 White, six Black, and one Asian/Asian American) who scored within the 11th to 25th percentile were classified as having LA, and the remaining 117 children (55 boys; 104 White, four Black, four Asian/Asian American, three Hispanic, and two not specified) consistently scoring above the 25th percentile were classified as TA. Thus, 159 participants (80 boys) comprised the sample for group comparisons. Like the total sample, most (88%) of these participants were White. Importantly, six of the children (four boys) who repeated a grade were in the MLD group, one was in the LA group and one was in the inconsistent group (described subsequently). Therefore, data at Grade 5 are of particular interest in this study, as this is a time period when all children were beyond Grade 3.

Thirty-one additional participants (11 boys, one of whom repeated a grade) were excluded from any MLD-status group analyses because their mathematics calculation scores were too inconsistent across years of the study to meet the aforementioned MLD, LA, or TA criteria. However, these students were included in fixed-order regression analyses that excluded MLD status, excepting one child from the LA group who had data missing for measures of interest in these analyses. Therefore, 190 participants with all relevant data points were included in fixed-order regression analyses. Of these, 166 children were White, 11 were Black, eight were Asian/Asian American, three were Hispanic, and race was not specified for two participants.

Design and Materials

Each student completed a two-part magnitude comparison task as part of the larger study. The initial and expanded instructions were administered to all participants, in fixed order, for two reasons: First, presenting expanded instructions first could have a spillover effect on the more streamlined instructions. Second, moving from less- to more-elaborated instructions is a more ecologically valid sequence. [The precise instructions appear in the task description in the following section.] The primary outcome variable of interest was the difference in performance accuracy across the two subsets of fraction magnitude comparison problems, always measured within subjects.

To evaluate the potential contribution of EF skills to performance accuracy, we included a composite measure of EF and compared EF to other variables also known to correlate with mathematics achievement: vocabulary, visual perception, and number sense competencies. MLD status was based on test performance from the Test of Early Mathematics Ability (TEMA)-2 (Kindergarten to Grade 2; Ginsburg & Baroody, 1990) and the Woodcock Johnson Calculations subtest (Grades 3–6).

Fractions magnitude comparison

We used two subsets of a Symbolic Magnitude Comparison of Fractions task (SymMCoF, adapted from Mazzocco et al., 2013) to measure children’s fractions knowledge. We limited the assessment to subtests comprised of area model items only, with identical fraction (e.g., 1/2 compared to 1/4) per set and presentation formats. (A report on Arabic number vs. area model performance on the SymMCoF has been reported; Mazzocco et al., 2013.) Each item included in this study involved comparing two quantities depicted with area models and either circling the larger quantity (the one that represented the “larger part of a whole”) or placing an equal sign between two quantities to represent equivalence (“if the two amounts are the same”; e.g., $1 / 2 = 2 / 4$ ). There were four item pairs per page, and 12 pages per set of instructions.

Items were presented in a fixed order, following detailed directions and a practice session included to ensure that participants understood task demands (as reported in detail by Mazzocco et al., 2013). Area models comprised equivalent-sized rectangular units, some shaded in. Importantly, a warm-up trial was administered to convey that each area model represented a number and that size of the area’s units did not convey numerical value. Within the initial and follow-up item sets, the 48 area models comprised two identical sets of 24 pairs. Thus, the same area model fraction pair items were administered twice to allow us to measure reliability, which was good during both years of the study (Cronbach’s alpha = .968 and .956 at Grades 4 and 5, respectively). Time to complete the task was recorded by examiners.

Initial warm-up session

For each subset of items (presented with either the initial or expanded instructions), test trials were preceded by an extensive warm-up session, during which children were explicitly shown and taught each of several principles that they were then prompted to demonstrate. The initial warm-up trials were used to convey the purpose of the task and the representational systems used throughout, such that the aim of the task was to indicate which of two items represents the larger amount, not the physically larger item. The warm-up trial lasted approximately 5 min. Importantly, all children successfully completed the warm-up sessions.

Initial instructions

The initial instructions included asking children to determine the bigger amount of a whole, with clarification that a whole would be everything “inside the outer box,” the outline, which the examiner traced in its entirety with her pointing finger. Children were also instructed that it was possible for amounts depicted by area models to be equal, or to “show the same amount.” The procedure matched that of the warm-up trials, minus feedback.

Second warm-up session

The expanded instructions involved a cover story about a town’s water tank, presented with a picture of an opaque municipal tank. The examiner explained the importance of knowing how much water was in a tank and that each area model was “a tool used to show how much water is in the tank.” Children were to choose “which tool shows more water in the tank.” Although children were unlikely to have experience with measuring water in a municipal water tank, they were very likely to have experience with viewing levels of liquids in containers, such as beverages in a glass.

Procedure for both instructions

Throughout both sets of items, children marked their responses directly in a stimulus book, which included the standard task-specific prompt at the top of the page, identical to prompts used in the corresponding warm-up session. For the expanded instructions task only, a small illustration of the water tank appeared next to the prompt.

The entire test was untimed although the examiner recorded response times surreptitiously. Total administration required approximately 20 min, including about 11 min total test-taking across the two subsets of items and 9 min to review the instructions and practice portions of both sets of items. All students passed both warm-up trials.

Cognitive and achievement measures

To assess whether child characteristics differentially predicted fraction knowledge performance with expanded compared to initial instructions, we used several key variables identified in earlier studies as good predictors of later mathematics achievement (e.g., De Smedt, Verschaffel, & Ghesquière, 2009; Desoete, Ceulemans, De Weerdt, & Pieters, 2012). The achievement measures were used to assign MLD status described earlier in this section.

Mathematics achievement

The Woodcock–Johnson Psycho-Educational Battery–Revised (WJ-R; Woodcock & Johnson, 1990) is a widely used standardized achievement test. We used age-referenced standard scores from the Calculations subtest (Calc) to establish our grouping variable, MLD status, as reported earlier in the Participants section. WJ-R-Calc involves untimed paper-and-pencil arithmetic problems presented in order of increasing difficulty.

General cognitive ability: Vocabulary and spatial reasoning

We used the Wechsler Abbreviated Scale of Intelligence (WASI, Wechsler, 1999), a widely used standardized instrument, to obtain age-referenced standard subtest Vocabulary and Matrix Reasoning (MR) subtests scores. These scores were obtained at Grade 3 (~age 8 years). The WASI has high internal consistency (r = .95) for 8 year olds, and since Full-Scale Intelligence Quotient (FSIQ) scores are considered relatively stable beyond age 8 years, we did not repeat the WASI in subsequent years of the study. We excluded the FSIQ from analyses because including FSIQ as a covariate in previous studies of SymMCoF performance did not alter the pattern of results (Mazzocco et al., 2013), but we included each subtest score separately to evaluate their relative contribution to SymMCoF performance with each of the two types of instructions.

Executive function skills

We used two measures of EF, including the Contingency Naming Task (CNT; Anderson, Anderson, Northam, & Taylor, 2000). Consistent with other composite measures of EF (e.g., the Head–Toes–Knees–Shoulders task; McClelland et al., 2014), the CNT is used to assess cognitive flexibility, response inhibition, and overall performance efficiency under moderate and increasing working memory demands. The task requires naming the color or shape of individual stimuli (basic shapes with one embedded “inner” shape) according to rules that become progressively more challenging. The warm-up trials require merely naming the color or outer shape of each stimulus, whereas the test trials require switching between naming the stimulus color or shape depending on rules focused on one attribute (i.e., whether the inner and outer shapes within each stimulus match) or two attributes (i.e., whether these shapes match and arrow appear below a black arrow). Following an untimed practice trial that involves naming nine stimuli until a naming rule is mastered, the timed test trial that follows involves naming 27 stimuli as quickly as possible “without making mistakes.” Performance efficiency is measured in terms of a speed–accuracy trade-off index obtained by calculating [(1/RT)/(errors + 1)] × 100. Higher scores reflect better (more efficient) performance. Scores increase with age, but are not age adjusted. From ages 5 to 11 years, scores across trials and age groups range from approximately 0.20 to 2.0 (Mazzocco & Kover, 2007). The standardization report of the CNT does not include reliability data (Anderson et al., 2000), but a significant test–retest correlation has been reported for efficiency scores collected 2 years apart, r_s = .50, p < .001 (Mazzocco & Kover, 2007). In previous work, one- and two-attribute scores were found to show the greatest range in performance at Grades 3 and 5, respectively (Mazzocco & Kover, 2007). In this study, we used two-attribute efficiency scores at Grade 5 as a measure of concurrent EF efficiency potentially contributing to variation in Grade 5 SymMCoF performance.

We also used the Stanford Binet Fourth Edition (SBIV) Memory for Digits (MD) subtest (Thorndike, Hagen, & Sattler, 1986), which was administered at Grade 4. The task requires listening to an examiner recite digits at the rate of one per second and then repeating the string verbatim in either the same or the reversed order. In this study, we used the reversed digits total score as a measure of verbal working memory.

Number knowledge

During the first year of the overall longitudinal study, we obtained a “number knowledge” score from items predictive of later MLD (e.g., Desoete et al., 2012; Krajewski & Schneider, 2009; Mazzocco & Thompson, 2005). These items involved reading one- and two-digit numbers, adding two numbers, magnitude judgments (i.e., identifying which of two verbally presented numbers is larger), number constancy (e.g., recognizing that the quantity of a collection remains constant irrespective of the collection’s physical arrangement), and cardinality. We used the total score (0 to 6) at kindergarten to index school-entry number knowledge skills.

Procedure

Parent consent and child assent were obtained prior to testing following our institutional review board approved protocol. Most participants were tested in their school, in a private room with minimal interruption. Participants who elected to participate on nonschool days were tested at the principal investigator (PI)’s research lab.

Children were individually tested by one of three female examiners. The WJ-R Mathematics Calculations subtest and the fractions comparison task were administered during both years of the study (Grades 4 and 5). Except for the concurrent CNT and SBIV measures, the predictor variable assessments were administered prior to the years targeted for this study.

As aforementioned, the prospective nature of the longitudinal study prevented a priori group classification and thus required a fixed order of all tests administered. The testing battery was presented during two sessions, with the SymMCoF administered during the first session in both years. Although breaks were permitted, they were rarely requested, perhaps because testing sessions were limited to one 50-min class period.

Results

Primary Analyses

We ran two overall sets of planned analyses. We first present findings from the repeated measures ANOVAs to test contributions of MLD status (TA, LA, and MLD) to performance accuracy and review the post hoc analyses that followed. Second, we present results of fixed-order regression models used to test predictors of fractions knowledge as indexed by scores on the fractions comparison task, and compared model outcomes across TA, LA, and MLD groups.

Group comparisons

In view of the anticipated unequal sample sizes across levels of MLD status, we relied on the multivariate Wilk’s Lambda F values for ANOVAs and Bonferroni adjustments for multiple comparisons of within-subject variables. For comparisons across levels of MLD status, we use the Tamhane post hoc when inequality of variance emerged across groups.

Our initial analysis was a 3 (MLD Status) × 2 (Grade) × 2 (Format of Instructions: Initial vs. Expanded) × 2 (Gender) repeated measures analysis of variance (ANOVA), with repeated measures on grade and instructions format. We had no predictions concerning effects of gender, but included the variable because of widespread attention to gender differences in mathematics (Ceci, Williams, & Barnett, 2009). There was no main effect of gender, p = .25, $η_{p}^{2} = . 01$ , and no significant interactions involving gender, ps < .17. Therefore, gender was excluded from further analyses.

The main effect of instructions format was significant, F(1, 156) = 22.08, p < .00001, $η_{p}^{2} = . 12$ , with more accurate performance following the initial (70%, SD = .18) versus expanded task instructions (65%, SD = .21), Cohen’s d = 0.25, when collapsed across the two grades. As predicted, there were also significant main effects for grade, F(1, 156) = 34.18, p < .001, $η_{p}^{2} = . 18$ , and MLD status, F(2, 156) = 27.02, p < .001, $η_{p}^{2} = . 26$ , with greater accuracy at Grade 5; at both grades, TA children had greater rates of accuracy compared to children with MLD. The interaction between instructions and MLD status was not significant, p = .13, $η_{p}^{2} = . 03$ . There was also no interaction between grade, instructions format, and MLD status, p = .99.

The interaction was not significant, but we executed our planned comparison of MLD and LA groups only, in view of our primary research question concerning children with math difficulties at Grade 5 and because the disproportionate size of the TA group (78% of the sample) may have diminished effect sizes associated with MLD group differences. Children with MLD performed no differently from their LA peers—regardless of instructions format—at Grade 4 (ps >.18); at Grade 5, children with MLD were less accurate than children with LA regardless of instructions format (ps < .02), but especially when instructions were expanded (p < .004), Cohen’s d = 0.81. The within-group effect of instructions format was significant for only the MLD group, whose performance was more accurate with the initial versus expanded instructions, p = .02, Cohen’s d = 0.31 (see Table 1).

Table 1.

Group Means (and SDs) for Accuracy (Percent Correct) on Grade 5 MCoF Performance, as a Function of Instruction Format and Effect Sizes for Difference Scores.

Format	Mathematics achievement group			MLD vs. LA:
Format	TA (n = 117)	LA (n = 26)	MLD (n = 16)	Cohen’s d	p
Initial (I)	86 (0.14)	75 (0.19)	61 (0.18)	0.76	.01
Expanded (E)	85 (0.15)	72 (0.21)	55 (0.21)	0.81	.004
I–E: Cohen’s d (p)	0.06 (nsd)	0.15 (nsd)	0.31 (0.02)	—	—

Note. MCoF = Magnitude Comparison of Fractions; TA = typically achieving; LA = low mathematics achievement; MLD = mathematics learning disability.

Children with MLD performed significantly worse than children with LA in Grade 5 and significantly worse with expanded rather than initial instructions, but we remain cautious in attributing this effect to MLD, given the small sample and the low power of the interaction term. Our caution led us to rule out two potential confounds that may account for the ANOVA and planned comparison findings: Problem difficulty and children’s fatigue. Results of these unplanned exploratory post hoc analyses follow.

Problem difficulty

If problem difficulty accounts for the MLD group difference rather than MLD status per se, larger effect sizes would emerge for the whole sample when analyses are limited to more difficult items. Drawing from our earlier findings that fraction pairs with the same numerator lead to lower accuracy rates compared to fraction pairs with the same denominator, at least among fourth and fifth graders (Mazzocco et al., 2013), we used select fraction pairs as a proxy for item difficulty. We carried out a 3 (MLD Status) × 2 (Grade) × 2 (Instructions Format) × 2 (Item Type: Same Denominator vs. Same Numerator items) ANOVA. In addition to the anticipated effects of Grade, F(1, 156) = 5.93, p = .02, $η_{p}^{2} = . 04$ , and MLD status, F(2, 156) = 18.82, p < .00001, $η_{p}^{2} = . 19$ , there was a significant main effect of item type, F(1, 156) = 152.21, p < .00001, $η_{p}^{2} = . 494$ , with 97% versus 73% accuracy rates for same denominator and same numerator fractions (Cohen’s d = 1.31), respectively, validating the assumptions about item difficulty. Same denominator items were easier than same numerator items for children in all three groups (ps < .0001). More importantly, there was a significant Instructions Format × Item Type crossover interaction, F(1, 156) = 62.874, p < .00001, $η_{p}^{2} = . 287$ . On easier (same denominator) items, performance was more accurate with expanded (99%) compared to initial instructions (96%; p < .001), whereas performance on harder (same numerator) items was more accurate with initial (80%) versus expanded instructions (66%; p < .00001). The significant MLD Status × Instructions × Item Type interaction, F(2, 156) = 6.54, p = .002, $η_{p}^{2} = . 076$ was one of magnitude. That is, the crossover interaction described above was evident in all three groups of children, but was most discrepant in the MLD group (see Figure 1). With grade added to the model, the four-way interaction was not significant, p = .84, $η_{p}^{2} = . 002$ .

Figure 1.

Percent Correct on Fraction Knowledge Items as a Function of Item Difficulty (i.e., Item Type), Format of Instructions, and Participants’ Mathematics Achievement Level. TA = typically achieving; LA = low mathematics achievement; MLD = mathematics learning disability.

Fatigue effects

Since the expanded instructions always followed the initial instructions (to avoid contamination of the latter), children may have experienced fatigue by the end of the task, especially children with MLD who may have experienced greater task difficulty. If fatigue manifests as slower performance, it may be detected in a longer response time to complete the second portion of the SymMCoF relative to the first portion. Using the response time recorded by examiners during Grade 5 assessments, we used t tests to directly examine this possibility and found no significant group difference in response time between children with MLD (M = 131.00 s, SD = 74.24 s) and children with TA (M = 126.86 s, SD = 46.43 s) or LA (M = 144.59 s, SD = 46.25 s), ps >. 64. In addition, fatigue effects (if present) should lead to worse performance on cognitive tests that follow versus precede the SymMCoF, so we exploited the fact that we had administered the Stanford Binet MD subtest prior to the SymMCoF in Grade 4 and following the SymMCoF in Grade 5. Using a 2 (Order: Preceding or Following) × 3 (MLD Status) ANOVA on the SB-MD composite score to test this possibility, we found no significant order effect (p = .10) and, most importantly, no Order × Group interaction (p = .50), on digit span performance, failing to support indirect evidence of fatigue effects. Finally, since the effect of fatigue may be most evident on the reversed digit items, we repeated the ANOVA using only the reversed digit raw score as the outcome variable. Once again, there was no significant order effect, and no Order × Group interaction (ps > .86).

To summarize, neither fatigue nor problem difficulty accounts for the poorer performance of fifth graders with MLD on expanded compared to initial instructions, relative to their TA or LA peers. Investigating cognitive predictors may help to clarify the observed group differences.

Fixed-Order Regression Models

We planned regression analyses to evaluate the relative contribution of Grade 5 cognitive measures as predictors of SymMCoF performance under initial versus expanded instructions. Grade 5 performance was selected for these analyses because SymMCoF performance was more variable at Grade 5 than at Grade 4. The five predictors were continuous scores from measures of vocabulary, spatial reasoning (mental rotation [MR]), EF (Contingency Naming Test [CNT] and verbal working memory [MD, reversed]), and kindergarten number knowledge. We evaluated the contribution of each variable to predicting SymMCoF performance without accounting for the influence of the remaining predictors (i.e., entering it first in the regression model) and also its unique contribution when controlling for all remaining predictors (i.e., entering it last). For this reason, we used a fixed-order approach.

In our first of three sets of analyses, we included all participants who had data for all five predictors (N = 190), including children with MLD, LA, or TA and children with inconsistent MLD status whose data were excluded from the previous ANOVAs (see Table 2). Based on the recommendation from a reviewer, we repeated these sets of models twice, first with students who were not TA (i.e., children with LA, MLD, or inconsistent performance in math, combined; Table 3) and second with TA students only (see Table 4). We hereafter refer to the combined subsample as children with mathematics difficulties. Across all three of these parallel analyses, we ran five sets of models to systematically vary predictor order.

Table 2.

Fixed-Order Regression Analyses Predicting MCoF Performance for Full Sample (n = 190).

Model	Dependent variable entered into model	Df	Initial instructions				Expanded instructions
Model	Dependent variable entered into model	Df	R ²	R² change	Change in F	p	R ²	R² change	Change in F	p
A-1	Vocabulary	1, 188	.207	.207	49.199	.000	.212	.212	50.655	.000
A-2	WASI MR	2, 187	.287	.079	20.834	.000	.266	.054	13.813	.000
A-3	CNT efficiency	3, 186	.313	.026	7.125	.008	.278	.012	3.083	.081
A-4	MD reversed	4, 185	.318	.005	1.333	.250	.279	.001	0.147	.702
A-5	Number Skills	5, 184	.364	.046	13.265	.000	.307	.028	7.418	.007
B-1	WASI MR	1, 188	.198	.198	46.429	.000	.162	.162	36.350	.000
B-2	CNT efficiency	2, 187	.253	.055	13.643	.000	.198	.036	8.321	.004
B-3	MD reversed	3, 186	.263	.010	2.573	.110	.201	.004	0.839	.361
B-4	Number Skills	4, 185	.321	.058	15.728	.000	.241	.040	9.632	.002
B-5	Vocabulary	5, 184	.364	.043	12.555	.001	.307	.066	17.538	.000
C-1	CNT efficiency	1, 188	.134	.134	29.168	.000	.096	.096	20.050	.000
C-2	MD reversed	2, 187	.169	.035	7.826	.006	.116	.020	4.165	.043
C-3	Number Skills	3, 186	.273	.104	26.605	.000	.194	.078	17.906	.000
C-4	Vocabulary	4, 185	.339	.066	18.520	.000	.286	.092	23.964	.000
C-5	WASI MR	5, 184	.364	.025	7.154	.008	.307	.021	5.512	.020
D-1	MD reversed	1, 188	.074	.074	15.080	.000	.046	.046	9.001	.003
D-2	Number Skills	2, 187	.240	.166	40.811	.000	.169	.124	27.828	.000
D-3	Vocabulary	3, 186	.326	.086	23.694	.000	.280	.111	28.558	.000
D-4	WASI MR	4, 185	.354	.028	8.147	.005	.303	.023	6.108	.014
D-5	CNT efficiency	5, 184	.364	.010	2.762	.098	.307	.004	1.069	.303
E-1	Number Skills	1, 188	.226	.226	54.993	.000	.163	.163	36.538	.000
E-2	Vocabulary	2, 187	.322	.096	26.343	.000	.279	.117	30.281	.000
E-3	WASI MR	3, 186	.353	.031	8.964	.003	.303	.023	6.262	.013
E-4	CNT efficiency	4, 185	.363	.010	2.923	.089	.307	.004	1.062	.304
E-5	MD reversed	5, 184	.364	.001	0.247	.620	.307	.000	0.015	.903

Note. MCoF = Magnitude Comparison of Fractions; WASI = Wechsler Abbreviated Scale of Intelligence; MR = Matrix Reasoning; CNT = Contingency Naming Test; MD reversed = Stanford Binet IV Memory for Digits reversed.

Table 3.

Fixed-Order Regression Analyses Predicting MCoF Performance for Children With Math Difficulties (n = 71).

Model	Dependent variable entered into model	df	Initial instructions				Expanded instructions
Model	Dependent variable entered into model	df	R ²	R² change	Change in F	p	R ²	R² change	Change in F	p
A-1	Vocabulary	1, 69	.246	.246	22.474	.000	.208	.208	18.173	.000
A-2	WASI MR	2, 68	.270	.025	2.292	.135	.214	.005	0.436	.511
A-3	CNT efficiency	3, 67	.283	.013	1.221	.273	.216	.003	0.228	.635
A-4	MD reversed	4, 66	.285	.001	0.126	.724	.225	.009	0.756	.388
A-5	Number Skills	5, 65	.350	.065	6.489	.013	.262	.037	3.265	.075
B-1	WASI MR	1, 69	.093	.093	7.043	.010	.044	.044	3.202	.078
B-2	CNT efficiency	2, 68	.150	.057	4.589	.036	.076	.031	2.299	.134
B-3	MD reversed	3, 67	.157	.007	0.568	.454	.078	.002	0.142	.708
B-4	Number Skills	4, 66	.254	.097	8.604	.005	.143	.065	5.004	.029
B-5	Vocabulary	5, 65	.350	.095	9.523	.003	.262	.120	10.531	.002
C-1	CNT efficiency	1, 69	.106	.106	8.146	.006	.055	.055	4.045	.048
C-2	MD reversed	2, 68	.124	.018	1.433	.235	.055	.000	0.003	.959
C-3	Number Skills	3, 67	.247	.122	10.888	.002	.137	.082	6.361	.014
C-4	Vocabulary	4, 66	.347	.101	10.200	.002	.261	.124	11.0884	.001
C-5	WASI MR	5, 65	.350	.002	0.226	.636	.262	.001	0.063	.802
D-1	MD reversed	1, 69	.063	.063	4.644	.035	.007	.007	0.508	.478
D-2	Number Skills	2, 68	.228	.165	14.535	.000	.121	.114	8.797	.004
D-3	Vocabulary	3, 67	.344	.116	11.869	.001	.260	.139	12.564	.001
D-4	WASI MR	4, 66	.347	.003	0.285	.595	.261	.001	0.085	.772
D-5	CNT efficiency	5, 65	.350	.003	0.257	.614	.262	.001	0.119	.731
E-1	Number Skills	1, 69	.221	.221	19.534	.000	.119	.119	9.336	.003
E-2	Vocabulary	2, 68	.344	.123	12.778	.001	.247	.127	11.493	.001
E-3	WASI MR	3, 67	.347	.003	0.317	.575	.247	.000	0.004	.952
E-4	CNT efficiency	4, 66	.350	.003	0.262	.611	.247	.000	0.008	.931
E-5	MD—reversed	5, 65	.350	.000	0.003	.959	.262	.015	1.362	.247

Table 4.

Fixed-Order Regression Analyses Predicting SymMCoF Performance for All Typically Achieving (n = 119).

Model	Dependent variable entered into model	df	Initial instructions				Expanded instructions
Model	Dependent variable entered into model	df	R ²	R² change	Change in F	p	R ²	R² change	Change in F	p
A-1	Vocabulary	1, 117	.088	.088	11.282	.001	.110	.110	14.439	.000
A-2	WASI MR	2, 116	.158	.070	9.670	.002	.164	.054	7.553	.007
A-3	CNT efficiency	3, 115	.190	.032	4.481	.036	.179	.015	2.067	.153
A-4	MD reversed	4, 114	.192	.003	0.393	.532	.180	.001	0.143	.706
A-5	Number Skills	5, 113	.220	.027	3.971	.049	.196	.015	2.177	.143
B-1	WASI MR	1, 117	.120	.120	15.976	.000	.108	.108	14.100	.000
B-2	CNT efficiency	2, 116	.166	.046	6.348	.013	.136	.028	3.762	.055
B-3	MD reversed	3, 115	.170	.004	0.623	.432	.138	.003	0.353	.554
B-4	Number Skills	4, 114	.201	.030	4.331	.040	.157	.019	2.523	.115
B-5	Vocabulary	5, 113	.220	.019	2.789	.098	.196	.039	5.431	.022
C-1	CNT efficiency	1, 117	.074	.074	9.291	.003	.049	.049	6.092	.015
C-2	MD reversed	2, 116	.084	.010	1.292	.258	.057	.007	0.875	.351
C-3	Number Skills	3, 115	.139	.055	7.400	.008	.095	.039	4.921	.028
C-4	Vocabulary	4, 114	.179	.040	5.534	.020	.161	.065	8.864	.004
C-5	WASI MR	5, 113	.220	.041	5.921	.017	.196	.035	4.911	.029
D-1	MD reversed	1, 117	.015	.015	1.823	.180	.011	.011	1.257	.264
D-2	Number Skills	2, 116	.106	.091	11.741	.001	.073	.063	7.840	.006
D-3	Vocabulary	3, 115	.160	.054	7.348	.008	.152	.078	10.608	.001
D-4	WASI MR	4, 114	.203	.044	6.224	.014	.188	.037	5.139	.025
D-5	CNT efficiency	5, 113	.220	.017	2.440	.121	.196	.007	1.043	.309
E-1	Number Skills	1, 117	.103	.103	13.377	.000	.071	.071	8.944	.003
E-2	Vocabulary	2, 116	.158	.056	7.695	.006	.151	.080	10.953	.001
E-3	WASI MR	3, 115	.202	.044	6.353	.013	.188	.037	5.220	.024
E-4	CNT efficiency	4, 114	.219	.017	2.461	.119	.195	.007	1.053	.307
E-5	MD reversed	5, 113	.220	.001	0.082	.775	.196	.000	0.015	.902

Note. SymMCoF = Symbolic Magnitude Comparison of Fractions; WASI = Wechsler Abbreviated Scale of Intelligence; MR = Matrix Reasoning; CNT = Contingency Naming Test; MD reversed = Stanford Binet IV Memory for Digits reversed.

Across the sets of analyses, similarities and differences were noted between the two conditions and between participant subsamples (see Tables 2 –4). A notable similarity in the findings is that the five predictors combined accounted for more variation in performance with the initial versus expanded instructions, regardless of participant sample. Specifically, when all 190 participants were included in the models, these predictors accounted for 36.4% versus 30.7% of SymMCoF performance, respectively. These R² values (and their difference between sets of instructions) decreased when only 119 participants from the TA group were included in the models and accounted for 22.0% and 19.6% of performance variance with the initial compared to expanded instructions, respectively. When participants were limited to children with mathematics difficulties, the predictors accounted for 35% versus 26.2% of SymMCoF performance variance with initial compared to expanded instructions, respectively. Thus, some of the findings reported for the full sample seem largely driven by children with mathematics difficulties (see Table 3). Results involving only the TA subgroups appear in Table 4.

Predictors of SymMCoF performance

Each of the five predictor variables individually accounted for some variance in SymMCoF performance under at least one of the two sets of SymMCoF items, for at least some participant groups. For the full participant group, three variables—Vocabulary, MR, and Number Skills—each accounted for more than half of the total variance when entered first in the model, accounting for 20.7%, 19.8%, and 22.6% of the variance with the initial instructions, respectively, ps < .0001 (total model variance = 36.4%), and 21.2%, 16.2%, or 16.3% of the variance when instructions were expanded, respectively, ps < .0001 (total model variance = 30.7%).

The pattern of findings differed when models were limited to either children with mathematics difficulty (MD) or children with TA. For analyses of children with MD, only Vocabulary and Number Knowledge each accounted for most of the variance when entered first, but when limited to items given with the initial instructions, the variance rates were 24.6% and 22.1%, respectively, ps < .0001 (total model variance = 35%). For items with expanded instructions, only Vocabulary accounted for most of the variance when entered first, 20.8%, p < .0001 (total model variance = 26.2%), although Number Knowledge accounted for 11.9% of the variance when entered first, p < .01. MR accounted for relatively little variance for items given with either the initial (9.3%, p < .02) or expanded instructions (4.4%, ns). For analyses of the TA group only, on problems presented with the initial instructions, only MR accounted for most of the variance when entered first, accounting for 12% of the variance, p < .0001 (total model variance = 22%). With the expanded instructions, Vocabulary and MR each accounted for most of the variance when entered first, with rates of 11% and 10.8%, respectively, ps < .0001 (total model variance = 19.6%). In summary, the relative predictive value of Vocabulary performance was strong regardless of what type of instructions were used, for all children, but its relative strength increased when instructions were expanded, but only for children with mathematics difficulties for whom the relative predictive value of number knowledge dropped considerably between the initial and expanded conditions.

The two remaining predictor variables were our key variables of interest: the CNT and Memory for Digits reversed (MD-R). They reached statistical significance (p < .0001) as a single predictor, but only in select analyses; moreover, neither reached the level of practical significance achieved by the three previous measures. That is, neither accounted for most of the variation captured by its respective total model. For children with MLD, the CNT and MD-R were significant single predictors of SymMCoF performance for items completed with the initial instructions, accounting for 10.6% and 6.3% of the variance, p < .01 and p < .05, respectively (where total model variance = 35%), but they both accounted for less variance when instructions were expanded (5.5% and 0.7%, p < .05 and p = .48, respectively, total model variance = 26.2%). For the TA group, the CNT as a single predictor accounted for 7.4% and 4.9% of the variance when the initial and expanded instructions were used, p < .01 and p < .02, respectively; but MD-R did not predict SymMCoF performance with either set of instructions, ps > 17. MD-R contributed very little if anything to the predictor models once CNT was included (i.e., Model C in Tables 3 and 4, Step C-2) and reached statistical significance only when all participants were included, when it then accounted for less than 4% of additional variance beyond the CNT, a general measure of EF that includes demands of working memory.

Discussion

In this study, we examined performance on initial and, later, expanded instructions on a fraction comparison task and found that fourth and fifth graders generally performed worse when fraction comparison instructions were expanded, compared to their initial performance. This was especially true for the more difficult fractions comparison items, and for children who met criteria for MLD—two factors that are confounded because children with MLD naturally had more difficulty with the task compared to their TA peers. Still, the interaction between problem difficulty and MLD status from our exploratory analyses suggests that whether expanded instructions improve or hinder performance on a mathematics task differs across children. Clearly, expanded instructions did not improve performance uniformly, but hindered performance for only some children, some of the time. These findings implicate the need to consider child cognitive and academic skills when evaluating potential benefits of expanded task instructions, which has implications for instruction and assessment.

Our findings make a small but important contribution to several distinct areas of research. First, the study contributes to ongoing debates about the role of EFs in mathematics achievement and performance. Contrary to our prediction, language skills were more predictive of fractions comparison performance compared to EF or even numerical skills, when the task involved expanded instructions. This finding does not refute the relation between EF and mathematics, but supports the notion of multiple cognitive skills supporting mathematics problem solving. We speculate that for children with MLD, their EF and number knowledge skills may have been insufficient to overcome the demands associated with expanded instructions, especially if the verbal demands of expanded instructions reduced available cognitive resources available to support EF or numerical processing contributions to the task. Regardless of the mechanism underlying these findings, our results suggest that children with different mathematics achievement levels engage different cognitive processes during problem solving and that the nature of their cognitive engagement also differs with the nature of the task instructions. Second, our findings are consistent with the broader notion that problem features—including their instructions—affect performance differentially for children with or without MLD (e.g., Powell, Fuchs, Fuchs, Cirino, & Fletcher, 2009) and contribute to this literature by showing that this pattern extends to fractions magnitude comparison performance. Third, our findings illustrate the importance of considering individual differences in educational contexts and the importance of potential child by instruction interactions in mathematics (e.g., Connor et al., 2017; Fuchs et al., 2014). That is, the questions we addressed and those that we propose for further study are not simply whether children with MLD fare better or worse than their peers when instructions are expanded, but rather how the consequences of expanding instructions vary with child cognitive characteristics. Despite growth in the “science of the individual” (Rose, Rouhani, & Fischer, 2013), child by instruction characteristics are often neglected in education research and, more importantly, may be neglected in the classroom. This omission may constrain advances in the both research and practice.

Finally, the study makes a small contribution to the inconsistent body of findings among studies of the consequences of concrete “real world examples” in mathematics instruction (e.g., Bottge, 1999; McNeil, Uttal, Jarvin, & Sternberg, 2009; Uttal, Scudder, & DeLoache, 1997). Although we did not systematically vary only the contextualized nature of the expanded instructions, our expanded instructions included a “real world context” which—at a minimum, did not enhance performance—but we cannot conclude whether the example or the expanded nature of the instructions themselves drove our findings. Still, our findings suggest that child characteristics may account for some of the inconsistency in these studies of contextualized instructions.

Potential Sources of Effects From Expanded Instructions

On the fractions knowledge task used in this study, the expanded instructions did not facilitate performance for children in the MLD group, but instead had a mild to moderate hindering effect most observable in the MLD group (see Table 1). Still, it was not the case that all children with MLD showed this effect. The small but significant MLD status findings are weaker than the item difficulty results. This pattern of results raises the question of whether MLD status, or problem difficulty, is a key determiner of expanded instructions effectiveness. These two constructs overlap—some fractions comparison problems may be more difficult for children with MLD than for children with typical math achievement, and children with MLD will have difficulty on more items compared to their TA pairs. The only instance in which contextualized instructions improved performance was when items were so easy that accuracy approached ceiling levels (e.g., >95%).

Children may draw upon different skills when solving problems under different conditions, and thus, child characteristics in relevant cognitive domains may contribute to performance levels differentially. The fixed-order regression models support this notion, because they revealed differences in which cognitive skills accounted for variation in performance with initial versus expanded instructions. That vocabulary accounted for much variation in fractions knowledge performance under both conditions is not surprising given the detailed verbal instructions used in each condition. Three remaining variables—MR, Number Skills, and CNT (EF skills)—also accounted for performance variation, consistent with models of pathways to mathematics performance (LeFevre et al., 2010) and to fractions performance specifically (Hansen et al., 2015). EF skills may contribute to potential hindering effects from expanded instructions, but they were not the strongest predictor of performance on the fractions task for any group.

Limitations

Our findings are limited to the nature of test items and conditions included in our study, by our relatively small sample of children with MLD and by the fixed order in which conditions were administered. First, our study focused only on fraction items; however, we note that fractions are particularly relevant to studies of expanded task instructions because of the pervasive nature of fractions difficulty (Gabriel et al., 2013; Hecht, Vagi, & Torgesen, 2007) and the frequent reliance on elaborated instructions when presenting fractions to young students. Second, we examined only one set of initial and expanded instructions, so we cannot generalize from these findings. Third, we could not fully delineate unique contributions of MLD status and problem difficulty, since these factors are confounded. Fourth, the limited sample size prevented us from testing meditational models of the roles of EF and number sense, but we nevertheless found some support for the importance of building and testing such models. Fifth, although we combined the MLD, LA and inconsistent groups in our regression models, this violates multiple reports of qualitatively important differences between these groups that are lost when the groups are combined (e.g., Geary et al., 2007; Murphy et al., 2007), so we acknowledge that the regressions for the “mathematics difficulties” group are heterogeneous.

Finally, our report of negative effects of the expanded instructions may be an artifact of order effects, which we could address only indirectly by evaluating potential fatigue effects. Still, this notion is countered by the similar performance levels among children with TA or LA across both sets of items, and by the benefits children with MLD experienced when expanded instructions were paired with the very easiest (“same denominator”) SymMCoF items. Children from all three groups required only about 2 min to complete the expanded condition section of the SymMCoF, and there were no group differences in response time to complete the task. But the reasons for this short amount of time may differ across groups. For highly able students, the problems were not difficult. Students who were over-challenged may have simply given up, taking less time on the task simply due to not exerting any effort. Moreover, children performed as well on the digit span task (MD-R) regardless of whether it was administered prior to, or following, the entire SymMCoF. Collectively, these findings diminish our concern about possible fatigue effects, but do not rule out whether order effects had undetected consequences.

Conclusions and Future Directions

The aim of this research was to explore the notion that expanded task instructions have differential effects on children’s mathematics performance for children with MLD. We found that whether expanded instructions benefit or hinder fractions comparison performance depends on MLD status and problem difficulty. In view of the limitations of our study, we cannot generalize the findings to other types of instructions, to other types of mathematics tasks, nor to children of all ages. Still, this exploratory study suggests that instructional effects vary across children and task difficulty. It would be a gross overgeneralization to conclude that children with deficient mathematics performance never benefit from expanded instructions, given the limitations of our study. But it would be unfortunate to assume that children with MLD always or often benefit from expanded instructions, when, at least in this study, they rarely did.

Footnotes

Acknowledgements

This study is a secondary analysis of data collected with support from NIH RO1HD 34061. We acknowledge the dedicated support of the original Project Coordinator, Gwen F. Myers, the Baltimore County Public School students and parents who participated in this study and the faculty and staff that supported the original work.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Drs. Herold and Mazzocco received partial support for this work from University of Minnesota funds awarded to MM; Drs. Bock and Mazzocco also received partial support from Award 2016-078 from the Heising-Simons Foundation, supported the Development and Research in Early Mathematics Education (DREME) Network of which MM is a member.

ORCID iD

Michèle M. M. Mazzocco

References

Anderson

Northam

Taylor

H. G.

(2000). Standardization of the Contingency Naming Test (CNT) for school-aged children: A measure of reactive flexibility. Clinical Neuropsychological Assessment, 1, 247–273. doi:10.1076/chin.8.4.231.13509

Bottge

B. A.

(1999). Effects of contextualized math instruction on problem solving of average and below-average achieving students. The Journal of Special Education, 33, 81–92. doi:10.1177/002246699903300202

Bull

Espy

K. A.

Wiebe

S. A.

(2008). Short-term memory, working memory, and executive functioning in preschoolers: Longitudinal predictors of mathematical achievement at age 7 years. Developmental Neuropsychology, 33, 205–228. doi:10.1080/87565640801982312

Bull

Scerif

(2001). Executive functioning as a predictor of children’s mathematics ability: Inhibition, switching, and working memory. Developmental Neuropsychology, 19, 273–293.

Ceci

S. J.

Williams

W. M.

Barnett

S. M.

(2009). Women’s underrepresentation in science: Sociocultural and biological considerations. Psychological Bulletin, 135, 218–261. doi:10.1037/a0014412

Clements

D. H.

Sarama

Germeroth

(2016). Learning executive function and early mathematics: Directions of causal relations. Early Childhood Research Quarterly, 36, 79–90.

Connor

C. M.

Mazzocco

M. M. M.

Kurtz

Crowe

Tighe

Wood

Morrison

(2017). Using assessment to individualize early mathematics instruction. Journal of School Psychology, 66, 97–113.

De Smedt

Verschaffel

Ghesquière

(2009). The predictive value of numerical magnitude comparison for individual differences in mathematics achievement. Journal of Experimental Child Psychology, 103, 469–479. doi:10.1016/j.jecp.2009.01.010

Desoete

Ceulemans

De Weerdt

Pieters

(2012). Can we predict mathematical learning disabilities from symbolic and non-symbolic comparison tasks in kindergarten? Findings from a longitudinal study. British Journal of Educational Psychology, 82, 64–81. doi:10.1348/2044-8279.002002

10.

Fuchs

L. S.

Schumacher

R. F.

Sterba

S. K.

Long

Namkung

Malone

. . . Changas

(2014). Does working memory moderate the effects of fraction intervention? An aptitude–treatment interaction. Journal of Educational Psychology, 106, 499–514. doi:10.1037/a0034341

11.

Gabriel

Coché

Szucs

Carette

Rey

Content

(2013). A componential view of children’s difficulties in learning fractions. Frontiers in Psychology, 4, Article 715. doi:10.3389/fpsyg.2013.00715

12.

Geary

D. C.

Hoard

M. K.

Byrd-Craven

Nugent

Numtee

(2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Development, 78, 1343–1359.

13.

Geary

D. C.

Hoard

M. K.

Nugent

(2012). Independent contributions of the central executive, intelligence, and in-class attentive behavior to developmental change in the strategies used to solve addition problems. Journal of Experimental Child Psychology, 113, 49–65. doi:10.1016/j.jecp.2012.03.003

14.

Ginsburg

Baroody

(1990). Test of early mathematics ability (2nd ed.). Austin, TX: Pro-Ed.

15.

Hansen

Jordan

N. C.

Fernandez

Siegler

R. S.

Fuchs

Gersten

Micklos

(2015). General and math-specific predictors of sixth-graders’ knowledge of fractions. Cognitive Development, 35, 34–49.

16.

Hecht

S. A.

Vagi

K. J.

(2010). Sources of group and individual differences in emerging fraction skills. Journal of Educational Psychology, 102, 843–859. doi:10.1037/a0019824

17.

Hecht

S. A.

Vagi

K. J.

Torgesen

J. K.

(2007). Fraction skills and proportional reasoning. In Berch

D. B.

Mazzocco

M. M.

(Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 121–132). Baltimore, MD: Paul H. Brookes.

18.

Krajewski

Schneider

(2009). Exploring the impact of phonological awareness, visual-spatial working memory, and preschool quantity-number competencies on mathematics achievement in elementary school: Findings from a 3-year longitudinal study. Journal of Experimental Child Psychology, 103, 516–531. doi:10.1016/j.jecp.2009.03.009

19.

LeFevre

J.-A.

Fast

Skwarchuk

S.-L.

Smith-Chant

B. L.

Bisanz

Kamawar

Penner-Wilger

(2010). Pathways to mathematics: Longitudinal predictors of performance. Child Development, 81, 1753–1767. doi:10.1111/j.1467-8624.2010.01508.x

20.

Mazzocco

M. M.

Devlin

K. T.

(2008). Parts and “holes”: Gaps in rational number sense among children with vs. without mathematical learning disabilities. Developmental Science, 11, 681–691. doi:10.1111/j.1467-7687.2008.00717.x

21.

Mazzocco

M. M.

Kover

S. T.

(2007). A longitudinal assessment of executive function skills and their association with math performance. Child Neuropsychology, 13, 18–45. doi:10.1080/09297040600611346

22.

Mazzocco

M. M.

Myers

G. F.

Lewis

K. E.

Hanich

L. B.

Murphy

M. M.

(2013). Limited knowledge of fraction representations differentiates middle school students with mathematics learning disability (dyscalculia) versus low mathematics achievement. Journal of Experimental Child Psychology, 115, 371–387. doi:10.1016/j.jecp.2013.01.005

23.

Mazzocco

M. M.

Thompson

R. E.

(2005). Kindergarten predictors of math learning disability. Learning Disabilities Research & Practice, 20, 142–155. doi:10.1111/j.1540-5826.2005.00129.x

24.

McClelland

M. M.

Cameron

C. E.

Duncan

Bowles

R. P.

Acock

A. C.

Miao

Pratt

M. E.

(2014). Predictors of early growth in academic achievement: The head-toes-knees-shoulders task. Frontiers in Psychology, 5, Article 599.

25.

McNeil

N. M.

Uttal

D. H.

Jarvin

Sternberg

R. J.

(2009). Should you show me the money? Concrete objects both hurt and help performance on mathematics problems. Learning and Instruction, 19, 171–184. doi:10.1016/j.learninstruc.2008.03.005

26.

Murphy

M. M.

Mazzocco

M. M.

Hanich

L. B.

Early

M. C.

(2007). Cognitive characteristics of children with mathematics learning disability (MLD) vary as a function of the cutoff criterion used to define MLD. Journal of Learning Disabilities, 40, 458–478. doi:10.1177/00222194070400050901

27.

Powell

S. R.

Fuchs

L. S.

Fuchs

Cirino

P. T.

Fletcher

J. M.

(2009). Do word-problem features differentially affect problem difficulty as a function of students’ mathematics difficulty with and without reading difficulty? Journal of Learning Disabilities, 42, 99–110.

28.

Rose

L. T.

Rouhani

Fischer

K. W.

(2013). The science of the individual. Mind, Brain, and Education, 7, 152–158. doi:10.1111/mbe.12021

29.

Thorndike

R. L.

Hagen

E. P.

Sattler

J. M.

(1986). Stanford-Binet Intelligence Scale (4th ed.). Chicago, IL: Riverside Publishing Company.

30.

Uttal

D. H.

Scudder

K. V.

DeLoache

J. S.

(1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18, 37–54. doi:10.1016/S0193-3973(97)90013-7

31.

Wechsler

(1999). Wechsler abbreviated scale of intelligence. San Antonio, TX: The Psychological Corporation.

32.

Woodcock

R. W.

Johnson

M. B.

(1990). Woodcock Johnson Psycho-Educational Battery–Revised. Allen, TX: DLM Teaching Resource.