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Mathematics Clusters Reveal Strengths and Weaknesses in Adolescents’ Mathematical Competencies, Spatial Abilities, and Mathematics Attitudes

Associated data.

Pre-algebra mathematical competencies were assessed for a large and diverse sample of sixth graders ( n = 1,926), including whole number and fractions arithmetic, conceptual understanding of equality and fractions magnitudes, and the fractions number line. The goal was to determine if there were clusters of students with similar patterns of pre-algebra strengths and weaknesses and if variation between clusters was related to mathematics attitudes, anxiety, or for a subsample ( n = 342) some combination of intelligence, working memory, or spatial abilities. Critically, strengths and weaknesses were not uniform across the three identified clusters. Lower-performing students had pronounced deficits in their understanding of mathematical equality, fractions magnitudes, and the fractions number line. Higher-performing students had particular advantages in whole number and fractions arithmetic, and the fractions number line. Students could be reliably placed into clusters based on their mathematics self-efficacy and a combination of intelligence and spatial abilities. The results contribute to our understanding of key aspects of students’ mathematical development, highlight areas in need of intervention for at-risk students, and identify cognitive areas in which scaffolds might be incorporated into these interventions.

Introduction

Students’ mathematical competencies provide a gateway to well-paying mathematics-intensive careers and contribute to ease of coping with important life decisions ( Bynner, 1997 ; Reyna et al., 2009 ). Unfortunately, there is substantial variation in the extent to which students develop these competencies, differences that persist into adulthood ( Mamedova et al., 2017 ). Most of these studies have examined the relation between early mathematical competencies and later mathematics achievement or economic outcomes based on composite achievement scores ( Duncan et al., 2007 ; Ritchie & Bates, 2013 ). These studies confirm the importance of overall mathematical competencies but do not capture the variation existing within groups of students, especially those who have difficulties learning mathematics ( Bartelet et al., 2014 ; Geary et al., 2012 ; Vanbinst et al., 2015 ). Elementary school children have relative mathematical strengths as well as weaknesses ( Geary et al., 2007 , 2012 ), but much less is known about patterns of development in middle-school students.

Performance in middle school (grades 6 to 8, 11- to 13-year-olds) mathematics is critical because it lays the foundation for success in high school (grades 9 to 12, 14- to 18-year-olds) algebra, and provides an opportunity to identify and remediate deficits beforehand. Remedial interventions are typically focused on specific competencies, such as the fractions number line or mathematical word problems (e.g., Barbieri et al., 2020 ; Fuchs et al., 2020 ), and would be most beneficial if they targeted the most pronounced deficits of at-risk students. Accordingly, one goal here was to determine if lower-achieving students showed uniform deficits in pre-algebra competencies or strengths and weaknesses in these competencies. Identifying the latter would be useful for the development of focused remedial interventions. A second goal was to determine if clusters of students with different mathematical strengths and weaknesses differed in their mathematics attitudes, anxiety, and cognitive abilities. We were particularly interested in any relation between profiles of strengths and weaknesses and spatial abilities, as a consistent relation between the latter and mathematics is found but not fully understood ( Hawes & Ansari, 2020 ; Kell et al., 2013 ; Mix, 2019 ).

The study here is part of an on-going longitudinal project that is focused on individual differences in preparation for and success in high school algebra, as related (in part) to mathematics attitudes, anxiety, and spatial abilities. A novel feature of the study is the broad assessment of pre-algebra competencies and mathematics attitudes and anxiety in a large ( n = 1,926) and diverse group of sixth graders, and the integration of measures of intelligence, working memory, and spatial abilities of a subsample of them in seventh grade ( n = 342). We used state-of-the-art clustering techniques to identify groups of sixth graders with different pattens of strengths and weaknesses in pre-algebra arithmetic and multivariate pattern analyses to identify attitudinal and cognitive differences across these clusters.

Mathematical competencies

It has been shown that success in mathematics is dependent, in part, on the adequate development of prerequisite skills ( Geary et al., 2017 ; Lee & Bull, 2016). As noted, the larger study was designed to, among other things, identify the most important prerequisite competencies needed for later success in high school algebra and the selection of prerequisite measures was based on recommendations from the National Mathematics Advisory Panel ( NMAP, 2008 ) and results of cognitive studies of mathematical development (e.g., Alibali et al., 2007 ; Braithwaite et al., 2018 , 2019 ; Siegler et al., 2011 ). Competence with whole number arithmetic was identified by the NMAP as foundational to the preparation for algebra and is predictive of concurrent ( Tolar et al., 2009 ) and longitudinal algebraic outcomes ( Casey et al., 2017 ; Siegler et al., 2012 ). We thus included a measure of fluency with whole number arithmetic.

Fractions were highlighted by the NMAP (2008) and empirical studies support their importance for preparation for algebra. For instance, elementary students’ competence with fractions predicts their later competence in high school algebra, controlling domain-general abilities and family background ( Siegler et al., 2012 ). In the United States, students typically begin to learn fractions by fourth grade (sometimes earlier). Nevertheless, about half of them struggle with correctly ordering a series of fractions in eighth grade ( Martin et al., 2007 ), and many of them do not understand that the sum of 12/13 and 7/8 is closer to 2 than 19 or 21 ( Carpenter et al., 1981 ; Hecht & Vagi, 2010 ; Mazzocco & Devlin, 2008 ). In other words, there is substantive variation in middle school students’ understanding of fractions and this variation is likely to predict variation in later algebra outcomes. Thus, we included measures of fractions arithmetic and measures that assess students’ understanding of fractions magnitudes (e.g., the fractions number line).

Eventual competence with algebra is also dependent on an understanding of several core concepts ( Booth et al., 2014 ). In an analysis of algebra errors committed by high school students, Booth and colleagues identified several persistent categories of conceptual error that predicted end of year performance. Difficulties with mathematical equality, variables (e.g., a belief they represent a single value), and mathematical properties (e.g., confusing order of operation) were the strongest predictors of later algebra performance (see also Booth & Koedinger, 2008 ). We were not able to assess all of these and focused on equivalence, as students’ difficulties here have been extensively studied ( Knuth et al., 2005 ). Middle-school students who do not understand equivalence typically interpret the ‘=‘ as a signal that the preceding numbers need to be operated on, such as added, rather than an indicator of the relation between the values to the left and right of it. Students who have an operational conception of equivalence have difficulties solving problems presented in a non-standard format, such as 6 + 3 = _ - 1, and will often indicate that the ‘=‘ means that they should solve the problem ( Alibali et al., 2007 ; McNeil et al., 2011 ; McNeil et al., 2019 ). We included non-standard format equivalence problems to assess this misconception.

Mathematics attitudes and anxiety

Eventual success in algebra and beyond can also be influenced by confidence or efficacy about one’s abilities (i.e., a personal judgement of one’s ability to perform well in future endeavors, see Talsma et al., 2018 ) and by beliefs about the later usefulness or utility of mathematics (e.g., as related to future occupation; Eccles et al., 2016 ). Several meta-analyses have revealed small but reliable relations between students’ academic efficacy and their later grades or achievement ( Talsma et al., 2018 ; Valentine et al., 2004 ), however cause-and-effect are unclear. Older students’ and adults’ academic achievement contributes to subsequent attitudes, and these contribute to later achievement. Similar bidirectional effects are sometimes ( Gunderson et al., 2018 ) but not always ( Geary et al., 2019 ) found in younger students. In any event, Lauermann et al. (2017) found reciprocal relations between mathematics self-efficacy, utility beliefs, and math-intensive career plans throughout high school, which in turn predicted employment in a mathematics-intensive profession 15 years later.

Mathematical competencies are also correlated with mathematics anxiety, although cause-and-effect relations are again not fully understood ( Carey et al., 2016 ; Devine et al., 2017 ; Hill et al., 2016 ). Mathematics anxiety is apprehension or fear associated with thoughts about engagement in mathematical activities ( Dowker et al., 2016 ). One possibility is that anxiety undermines students’ mathematical performance by the intrusion of performance-related thoughts during problem solving that in turn reduce working memory capacity and increase problem-solving errors ( Ashcraft & Kirk, 2001 ; Maloney & Beilock, 2012 ). Mathematics anxiety can also result in an avoidance of mathematics and reduced opportunities to learn ( Hembree, 1990 ; Meece et al., 1990 ). Mathematics anxiety appears to be composed of several components that could influence mathematical development and performance in different ways. The core components include anxiety about mathematics learning generally (e.g., reading a math textbook) and anxiety during mathematics evaluations ( Baloglu & Koçak, 2006 ). The former could potentially result in a general avoidance of mathematics and the latter underperformance on high-stakes tests.

If mathematics attitudes or anxiety are associated with specific clusters of pre-algebra strengths and weaknesses, they could exacerbate or ameliorate at-risk students’ engagement in middle-school mathematics and thus their preparation for algebra. To assess if there is such a relation, we included measures of mathematics utility beliefs, self-efficacy, and anxiety as related to mathematics learning and evaluations. We also included a measure of students’ attitudes about English as a contrast to their mathematics attitudes, that is, to determine if the reported attitudes were specific to mathematics.

Cognitive mechanisms

The most consistent cognitive predictors of overall mathematics achievement or longitudinal gains in achievement are intelligence and components of executive function ( Deary et al., 2007 ; Geary, 2011 ; Geary et al., 2017 ; Van de Weijer-Bergsma et al., 2015 ). The latter includes updating or holding something in mind (often called working memory), inhibition of task-irrelevant information, and shifting from one task to another and then appropriately returning to the first ( Miyake et al., 2000 ). Of these, the most consistent predictor of mathematics achievement is updating or working memory ( Bull & Lee, 2014 ; Friso-van den Bos et al., 2013 ). These cognitive abilities are particularly important for mathematics learning because of the highly abstract nature of the material and the continual introduction of new material during schooling. Although the focus of the larger project is on spatial abilities, we included measures of working memory and intelligence as controls.

One goal, as noted, of the larger project is to examine the relation between spatial abilities and ease of learning spatial-related aspects of algebra (e.g., recognizing how common functions map to coordinate space) in high school. On the basis of this goal and a consistent finding of a relation between spatial and mathematical abilities ( Geer et al., 2019 ; Hawes & Ansari, 2020 ; Mix, 2019 ; Mix et al., 2016, 2017 ), we included three measures of spatial competence – visuospatial working memory, visuospatial attention, and mental rotation – in the study. We included three measures because the relative importance of different components of spatial ability could vary with the complexity and content of the mathematics assessments ( Bull et al, 2008 ; Geary et al., 2007 ). For instance, Gilligan et al. (2020) found that a short-term intervention that enhanced skill at rotating objects improved performance on missing item problems (e.g., 2 +  = 7), whereas enhancement of sensitivity to proportional relations improved accuracy in placing numerals on the number line.

Visuospatial short-term memory appears to be involved in an array of mathematical domains, especially in later grades, but these specific relations are not fully understood ( De Smedt et al., 2009 ; Li & Geary, 2017 ; Swanson et al., 2008 ). Visuospatial attention contributes to the ability to represent and compare the relative magnitude of numerals ( Longo & Lourenco, 2007 ; Zorzi et al., 2012 ) and is more strongly related to performance on the fractions number line than are other spatial abilities, such as visuospatial working memory ( Geary et al., 2020 ). The latter finding appears to be consistent with the results of Gilligan et al. (2020) ; specifically, focusing students’ attention on proportional distance improved number line performance. More complex spatial abilities (including mental rotation) are related to some aspects of mathematical abilities ( Mix & Cheng, 2012 ), and may become increasingly important as mathematical development shifts from number/arithmetic to algebra/geometry ( Casey et al., 1995 ; Kyttälä & Lehto, 2008 ).

Current study

The current study extends prior work with the inclusion of multiple mathematical competencies in a single assessment and in doing so enabled an examination of patterns of strengths and weaknesses in a large and diverse sample of lower- and higher-achieving students. The study also contributes to prior work by simultaneously identifying the attitudinal, anxiety, and cognitive strengths and weaknesses that emerge in different achievement clusters. More precisely, we used state-of-the-art clustering procedures to identify groups of students with different patterns of pre-algebra mathematical strengths and weaknesses and multivariate pattern analysis to identify attitudinal and cognitive predictors of cluster membership.

An assessment of whether intelligence, working memory, or one or several forms of spatial ability are associated with weaknesses in pre-algebra competencies could have implications for the development of remedial interventions. This is because interventions that include scaffolds that accommodate cognitive weaknesses are more successful in remediating at-risk students’ mathematical deficits than are interventions that do not include them (e.g., Fuchs et al., 2020 ). Information on the cognitive correlates of students with an at-risk profile of pre-algebra competencies would identify both areas of potential intervention and provide insights into the type of scaffold that might enhance remediation efforts. Similarly, the identification of attitudes or anxiety that predict cluster membership could indicate additional areas of remediation.

Participants

The original sample included 2,027 sixth graders who were assessed across two cohorts ( n s = 1,157, 870) conducted in collaboration with the Columbia Public Schools in Columbia, MO. Data were missing for 101 of these students (e.g., no information on student sex), leaving 1,926 of them for the analyses of mathematics performance and attitudes. The first cohort included approximately 86% of all sixth-grade students enrolled the district. The remaining 14% of students were absent the day of the assessment or unable to participate. The second cohort included 83% of invited sixth graders. The sampling for the two cohorts occurred in consecutive academic years and were otherwise identical, with the exception that one school was omitted for the second cohort because of over-representation of students from this school (from the first cohort) in the longitudinal sample. As part of an on-going longitudinal study, 342 students from the sixth-grade sample were tested again in seventh grade

Demographic information was not available for the sixth-grade sample but should be very similar to that of the district as a whole. For the district, sixth-graders were 61% white, 20% black, 5% Asian, 7% Hispanic, and 6% multiracial, and 43% qualified for free or reduced lunch. Demographic information was obtained through a parent survey ( n = 281) for the longitudinal sample. Eighty-eight percent of the students were non-Hispanic, 6% Hispanic or Latino, with the remaining unknown. The racial composition of the sample was 70% White, 14% Black, 3% Asian, 1% Native American, 10% multi-racial, and the remaining unknown. Self-reported annual household income was as follows: $0-$24,999 (12%); $25,000-$49,999 (18%); $50,000-$74,999 (12%); $75,000-$99,999 (22%); $100,000-$149,999 (19%); and $150,000+ (17%). Seventy-one percent of the students had at least one parent with a college degree. Sixteen percent of families received food assistance, and six percent housing assistance.

Mathematics Tests

The tests, detailed instructions, and raw data for the key analyses are available on OSF. The assessment included an Exponents and Radical Rules Test that is not reported here because of a high nonresponse rate, and an Equal Sign task due to a low number of trials. All tests were administered using paper-and-pencil for the first cohort, and the Equality Problems, Fractions Comparison Test, Fractions Number Line, and Academic Attitudes and Mathematics Anxiety were assessed using iPads for the second cohort. There were only 2 significant cohort effects (all other p’ s > .07), whereby students in the first cohort scored higher on the arithmetic fluency ( p = .002, d = .14) and fractions comparison test ( p = .0003, d = .17). Given the null effects for all other measures and the small difference for the arithmetic fluency and fractions comparison test, the two cohorts were combined.

Arithmetic fluency.

The test included 24 whole-number addition (e.g., 87 + 5), 24 subtraction (e.g., 35 – 8), and 24 multiplication (e.g., 48 × 2) problems. Students had 2 min to solve as many problems as possible. A composite arithmetic fluency score was based on the number correct across the three operations ( M = 18.94, SD = 6.25; α = .57). Despite the relatively low reliability, performance on this measure is more strongly correlated with mathematics ( r = .62) than reading achievement ( r = .32), indicating the measure shows both convergent and discriminant validity.

Equality problems.

Students who struggle with mathematical equality have difficulties with problems in non-standard formats, such as 8 = __ + 2 – 3. Thus, we created a 10-item test in multiple choice format (4 options) to assess competence at solving such problems. 3.7% of the items were unanswered and scored as incorrect. A composite was the mean percent correct for the 10 items ( M = 79.99, SD = 23.89, α = .80).

Fractions arithmetic.

The items were sets of twelve addition (e.g., 1/3 + 1/6), twelve multiplication (e.g., 1/4 × 1/8), and 10 division (e.g., 2 ÷ 1/4) problems. Students had 1 min for each operation. The score was the number of correctly solved problems ( M = 10.33, SD = 6.91; α = .62). Despite the relatively low reliability, performance on this measure is more strongly correlated with mathematics ( r = .70) than reading achievement ( r = .41), indicating the measure shows both convergent and discriminant validity.

Fractions comparison.

For each of 48 pairs, students circled the larger of two fractions, within 90 sec. There were four item types that reflect common problem-solving errors and strategies (see Geary et al., 2013 ). In the first type the numerator is constant, but the denominator differs (e.g., 2/4 2/5), which assesses students’ understanding of the inverse relation between the value of the denominator and the quantity represented by the fraction. In the second type numerators have a ratio of 1.5 and denominators a ratio between 1.1 and 1.25 (e.g., 3/10 2/12), making identification of the larger magnitude easier using numerators (correct) than denominators (incorrect). Numerators and denominators in the third type are reversed (e.g., 5/6 6/5), which requires students to choose the fraction with the larger numerator and smaller denominator. The final type involves skill at using 1/2 as an anchor for estimating fraction values (e.g., 20/40 8/9). The foils are always close to one but contain smaller numerals than the 1/2 fraction. A composite was based on performance (correct – incorrect; M = 16.70, SD = 15.32) across the four item types ( α = .87).

Fractions number line.

Students sequentially placed 10 fractions (10/3, 1/19, 7/5, 9/2, 13/9, 4/7, 8/3, 7/2, 17/4, 11/4) that were centrally presented in large font onto the 0-to-5 number line ( Siegler et al., 2011 ), and 94.4% of the lines were completed within a 4 min time limit. The data for the remaining 5.6% of lines were estimated using a multiple imputations procedure in SAS (2014) . Individual items were scored as the absolute percent deviation between the placement and the correct location, which were averaged across the 10 items ( M = 20.50, SD = 14.09, α = .85), and then multiplied by −1 so that higher scores represent more accurate placements.

Academic attitudes and anxiety

The grade-appropriate mathematics and English language attitudes measures were from the Michigan Study of Adolescent and Adult Transitions ( http://garp.education.uci.edu/msalt.html ) and are designed to assess students’ self-evaluated efficacy and their beliefs about the long-term utility of competence in these areas ( Eccles & Wigfield, 2002 ; Meece et al., 1990 ). The mathematics attitudes measure included seven items on a 1-to-7 Likert scale; e.g., “How much do you like doing math?” rated from 1 (“a little”) to 7 (“a lot”), with six similar English items. We used three approaches to determine their factor structure, including an exploratory principle component analysis (PCA) with promax rotation ( SAS, 2014 ), as well as parallel and MAP analyses ( R Core Team, 2017 ).

Mathematics and English attitudes.

The MAP analysis suggested one component, but the eigenvalues from the PCA and the parallel analysis suggested two factors. The factor loadings were consistent with distinct utility (Items 1 to 4, inclusive) and self-efficacy (Items 5 to 7, inclusive) dimensions. The scores were the mean of the corresponding items (α = .77 for utility; 84 for self-efficacy). All of the procedures indicated a single factor for English Attitudes, which was scored as the mean of the six items (α = .85).

Mathematics anxiety.

The 10 items were adapted from Hopko, Mahadevan, Bare, and Hunt (2003) . Each item (e.g., “Taking an examination in a math course”) was rated on a 1 (induces low anxiety) to 5 (induces high anxiety) scale. All three analyses indicated two factors. The first was defined by five items that involved learning mathematics (e.g., “Having to use the tables in the back of a math book”). The second factor was defined by four items that involved some type of evaluation (e.g., “Taking an examination in a math course”), and one other item (i.e., “In general, how anxious are you about math?”). Composite scores were based on the mean of the five learning anxiety items (α = .78) and the five evaluation anxiety items (α = .84).

Standardized Measures

Intelligence..

Students in the longitudinal component were administered the Vocabulary and Matrix Reasoning subtests of the Wechsler Abbreviated Scale of Intelligence (WASI; Wechsler, 1999 ). Based on standard procedures, subscale scores were used to generate an estimated full-scale IQ. The intelligence of the longitudinal sample was average ( M = 105, SD = 13).

Achievement.

For the longitudinal cohort, mathematics and reading achievement was assessed with the age-appropriate Numerical Operations (NO) and Word Reading (WR) subtests from the Wechsler Individual Achievement Test–Third Edition ( Wechsler, 2009 ), respectively. The NO included basic arithmetic and continued through fractions, algebra, geometry and calculus. The WR assessed single word reading, beginning with 1-syllable words and progressing to more complex vowel, consonant, and morphology types. The mathematics ( M = 100, SD = 19) and reading ( M = 104, SD = 13) achievement of the students in the longitudinal sample was average.

Cognitive measures

The cognitive tasks were administered using iPads using customized programs developed through Inquisit by Millisecond ( https://www.millisecond.com ) or through Qualtrics ( https://www.qualtrics.com ). Manuals and detailed descriptions of these tasks are available at Open Science Framework ( https://osf.io/qwfk6/ ) and are all standard measures of working- and short-term memory, and spatial ability. A verbal memory task was assessed (described in the SOM ) but not reported here as we focused a-priori of working memory and spatial abilities.

Digit span.

Forward and backward digit span measures were administered, following Woods et al. (2011) . Digits were auditorily (using the iPad) and sequentially presented at 1s intervals, beginning with 3 digits for the forward task and 2 digits for the backward task. Students were asked to tap the digits on a circular display after the sequence in forward/backward order. Correct responses increased the sequence length for the following trial, and two consecutive errors reduced the sequence length. Each task terminated after 14 trials. The longest digit sequence correctly recalled before two consecutive errors at the same length was scored as the span ( M = 5.71, 4.59, SD = 1.12, 1.20 for the forward and backward, respectively).

Following Jaeggi et al. (2010) , a similar version of a single n-back task with letters was administered. A “target” letter and then a sequence of 20 stimulus letters was presented (all consonants; 6 are target; 14 are not; randomly determined), Students indicated whether the currently presented letter is a target by tapping a key within a 3,000 ms response period (500ms + 2,500ms blank ITI), otherwise withholding a response. Target letters were either the very first stimuli of the sequence (N = 0), the same as the one immediately preceding it (N=1), or as the one in the two (N = 2) or three (N=3) preceding trials. After several practice items, students began on level N = 0. Across five blocks, students either moved up, down, or remained at a level (<3 errors – move up; 3–5 errors – repeat level; >5 errors – move down). Feedback (% correct) was displayed after each block. The score ( M = 3.80, SD = 0.76) was calculated as (Hits – False Alarms) / (total blocks).

Spatial span.

The Corsi Block Tapping Task was administered ( Kessels, van Zandvoort, Postma, Kappelle, & de Haan, 2000 ). Students viewed a display of nine squares (appearing randomly arranged) that “lit up” in a predetermined sequence and tapped on the squares in the same order they were lit. Sequences started at two squares and increased up to nine squares. Students had two attempts at each sequence length. Correct responses were followed by an increase in sequence length, and two incorrect responses at a given level terminated the task. The score is the total number of correctly recalled sequences across the entire task ( M = 8.34, SD = 1.83).

Spatial ability.

The first spatial measure was the Judgment of Line Angle and Position Test (JLAP; Collaer, Reimers, & Manning, 2007 ). Here, students matched the angle of a single presented line to one of 15-line options in an array at the bottom of the screen. There were 20 test items sequentially presented. Stimuli were presented for a maximum of 10s, with the trial terminating when the student made a selection. The JLAP was scored as the number correct ( M = 13.33, SD = 3.03).

The second measure used was the Mental Rotation Task (MRT-A; Peters et al., 1995 ). On each of 24 trials, students viewed 3D images of 10 connected cubes. On each trial, there was one target and four response options, requiring students to select the two options that were the same figure, only rotated to various degrees. After four practice problems, students completed two blocks of 12 problems each with a 3 min per block time limit. The MRT was scored as the number of problems on which both correct responses were chosen ( M = 8.75, SD = 4.13).

During a 45 min assessment, groups of 14 to 32 students were administered the mathematics tests and academic attitudes measures in their classroom (Age : M = 146.88 months , SD = 4.51). For the longitudinal component, tasks were individually administered in 45 min sessions in a quiet location in their school. The cognitive measures were administered in the fall semester (Age: M = 152.63 months, SD = 4.45) and the achievement and attitudes measures in the spring (Age: M = 156.79 months, SD = 4.41). Tasks were administered in a fixed order across participants. Parents provided informed written consent, and assent was obtained from adolescents for all assessments. The University of Missouri Institutional Review Board (IRB; Approval # 2002634, “Algebraic Learning and Cognition”) approved all methods included in this study.

Individual continuous mathematics measures were standardized ( M = 0, SD = 1), and the attitudes and anxiety measures were standardized to M = 100 ( SD = 15). Hierarchical cluster analyses (Ward’s method) were used to identify subsets of students with different math competence profiles. The stability of the cluster solution was validated using the Bootstrapped Jaccard similarity coefficient (shortened to J ; see Hennig, 2007 for details), with values above 0.6 interpreted as acceptable. Between-subjects ANOVAs (and Bayes Factor ANOVA equivalents) were used to test differences across clusters. A “leave-one-out” (LOO) cross-validation approach was also used for testing cluster differences in each mathematics measure. For example, to test cluster differences between performance on the fractions number line, the cluster analysis was re-estimated using every measure except for the fractions number line. This ensures that cluster differences on a given test reflects a novel, out-of-sample test, thus avoiding issues of circularity.

For the sixth-grade attitudes and anxiety measures and cognitive measures from the longitudinal component, multivariate pattern analyses (MVPA; logistic regression classifier, L2 penalty, C = 0.5) using the scikit-learn module in python was used to predict students’ identified math cluster. A similar LOO cross-validation was used for model training and testing to ensure each students’ prediction was a novel out-of-sample case left out of model training. Classifier accuracy is reported here, with chance being 33.3% (three possible cluster labels). Model significance was tested through permutation testing, which entails randomly shuffling the cluster labels of students and re-training the classifier 500 times on the shuffled data. Significance is reported as the proportion of times the shuffled data accuracies exceeded the true observed accuracy.

To further elucidate the relative contributions of each specific predictor, a Bayesian model comparison approach was used ( Rouder, Engelhardt, McCabe, & Morey, 2016 ). Cluster labels were used as the dependent measure, with the six cognitive measures and IQ, or the five attitude/anxiety measures, as predictors. To assess evidence for the inclusion/exclusion of a certain predictor (e.g., digit span), ratios of Bayes Factors (BFs) were taken from the best model that included a certain predictor (e.g., digit span, MRT, and JLAP) to the model that excluded it (only MRT and JLAP). All analyses were performed using R (v.3.5.1) and python (v.3.7.4). Main statistical interpretations are based on Bayesian statistics ( Bayes Factor package using default priors), where BFs < 1 indicate evidence for the null hypothesis, 1–3 weak/ambiguous evidence, and > 3 as evidence for the alternative hypothesis ( Kass & Raftery, 1995 ). As a supplement, frequentist statistics are also presented alongside each analysis, as using multiple estimates can often be informative in model evaluation ( Valentine, Buchanan, Scofield, & Beauchamp, 2019 ).

Correlations among the sixth-grade mathematics variables and the cognitive, attitudes, and achievement measures are shown in Table 1 ; correlations among the seventh-grade cognitive and achievement measures are in the SOM and data used in the analyses are available in OSF ( https://osf.io/wqjv7/ ).

Correlations between the mathematics variables and cognitive and attitudes variables.

Note. correlations > |.11| are significant, p < .05.

Table 2 presents descriptive statistics for the sixth-grade and longitudinal sample variables; code for the full statistical analyses are available in the SOM. The clustering analysis was applied to the standardized scores for individual mathematics measures. Two main clusters were first identified, segregating students into high ( n = 1,429; J = .758) and low ( n = 497; J = .608) groups. Figure 1A shows a radar chart for the two main clusters (high and low), where values closer to the center indicate lower performance and values closer to the perimeter of the circle indicate higher performance. For all five mathematics measures, clear high-low differences emerged ( BF 10 ’s ≥ 1.3e +98 , p ’s < .001), and were maintained when using the left-out validation test measures ( BF 10 ’s ≥ 2.2e +65 , p ’s < .001). More interestingly, from inspection of the cluster solution (visual inspection of the cluster dendrogram; see supplemental materials ) and examining cluster stability, it was apparent that the high group was further divided into two higher subgroups, creating a total of three groups overall. For the remainder of the analyses, we discuss these as low ( n = 497; J = .683), medium ( n = 1,001; J = .649), and high ( n = 428; J = .692) groups.

A) Radar chart of sixth grade mathematics measures from the two (high, low) cluster solution; lower scores are near the center and higher near the periphery. Clear high (dashed) and low (solid) groups are evident across all measures. 1 = Fractions comparison; 2 = Fractions number line; 3 = Arithmetic fluency; 4 = Equality problems; 5 = Fractions arithmetic. B-F) Boxplots depicting the median and first and third quartiles across the three clusters for each mathematics measure. To the left of each boxplot is a density plot showing the distribution of scores.

Descriptive statistics for the low, medium, and high groups

Note. The individual mathematics variables are centered ( M = 0, SD = 1), and the intelligence and numerical operations are standardized ( M = 100, SD = 15). Effect sizes (Cohen’s d ) are presented in two ways. First, to highlight deficits in the lowest groups, Cohen’s d is calculated comparing the low group versus the medium and high group together (one vs. rest approach). Likewise, the high group advantages highlight the relative differences between the high group and both the low and medium groups.

When analyzing differences among the three clusters, we observed strong main effects across all five measures; Arithmetic fluency, BF 10 = 6.8e +218 , F (2,1923) = 676.35, p < .001; Equality problems, BF 10 = 3.3e +428 , F (2,1923) = 1746.59, p < .001; Fractions arithmetic, BF 10 = 7.8e +376 , F (2,1923) = 1431.61, p < .001; Fractions comparison, BF 10 = 3.1e +180 , F (2,1923) = 532.09, p < .001; and, Fractions number line, BF 10 = 9.7e +234 , F (2,1923) = 741.08, p < .001. Post-hoc independent-samples t -tests (Tukey’s HSD) revealed significant differences across all clusters and measures ( p ’s < .001; Low < Medium < High). While students composing the clusters showed the same graded pattern among measures, their strengths and weaknesses were not uniform. As shown in Table 1 , the low group (as compared to the medium and high groups combined) had especially pronounced deficits for the equality problems, fractions comparison, and the fractions number line. Higher performing students (as compared to the medium and low groups combined) had especially high performance in fractions arithmetic, arithmetic fluency, and the fractions number line. Figure 1B – 1F shows mean differences across clusters for the mathematics measures. Importantly, we validated these patterns of strengths and weaknesses and overall cluster differences using a LOO cross-validation approach, and main effects for all measures remained strong ( BF 10 ’s ≥ 2.9e +66 , p ’s < .001).

Attitudes and mathematics anxiety

Next, we tested how self-reported attitudes and anxiety measures related to the clear differences found in the objective mathematics measures. We found strong evidence for a main effect of cluster for mathematics utility, BF 10 = 2.0e +7 , F (2,1922) = 22.54, p < .001, and an even larger effect for mathematics efficacy, BF 10 = 2.8e +50 , F (2,1922) = 130.91, p < .001. In a graded fashion, attitudes about mathematics utility and efficacy increased across cluster groups ( p ’s ≤ .001; Low > Medium > High; Figure 2A – 2C ). This was not the case, however, with attitudes about English. While still significant using frequentist statistics [ F (2,1922) = 5.87, p = .003], Bayesian evidence revealed a severely dampened effect of English attitudes ( BF 10 = 2.1) compared to the strong effects found in mathematics attitudes. Both mathematics anxiety measures showed significant cluster effects: Evaluation, BF 10 = 5.0e +7 , F (2,1922) = 23.47, p < .001; Learning, BF 10 = 2.0e +35 , F (2,1922) = 91.68, p < .001. Post-hoc comparisons showed that in both measures, there were decreases in anxiety from the low- to high-cluster groups ( p ’s ≤ .004; High < Medium < Low), which are visualized in Figure 2D – 2E .

Boxplots illustrating the median and first and third quartiles across the three clusters for A-C) mathematics and English attitudes, and D-E) mathematics anxiety. To the left of each boxplot is a density plot showing the distribution of scores.

As an alternative to the mass univariate analyses presented above, MVPA classifiers were used to predict students’ cluster labels from the multivariate relationships of the five attitudes and anxiety measures. This is an especially useful approach when analyzing factors that are external to the clustering analysis. Using a LOO cross-validated logistic regression classifier, a model using these five measures was able to predict cluster membership with 55.0% accuracy (see Figure 3 ). Permutation testing confirmed this to be above chance accuracy, p < .001 (with chance performance at 33.3%). Bayesian model comparisons clearly identified mathematics efficacy as the measure contributing the most to cluster differences (see Table 3 ). BF 10 ’s for mathematics utility, anxiety for evaluation, anxiety for learning, and English attitudes were all < 1 (evidence for the null hypothesis) compared to models including mathematics efficacy. Indeed, the best overall model was for efficacy alone, BF 10 = 445.44 against an intercept only model. Comparing the model including all five predictors to one excluding efficacy showed the importance of efficacy ( BF 10 = 30.52) while controlling for all other predictors.

Accuracy of classifying students into their respective clusters (with standard error of the mean) using the variables noted on the x -axis (see text). The darker gray for the attitudes and anxiety indicates this model was trained using the sixth-grade sample. All other models were trained using the longitudinal data. All predictors indicates that the model was trained using the six cognitive measures, IQ, Numerical Operations, Word Reading, and the attitudes and anxiety measures. Cog. = the working memory and spatial measures; NO = Numerical Operations; WR = Word Reading; Att./Anx. = mathematics attitudes, anxiety, and English attitudes. The dashed horizontal bar indicates chance performance at 33.33%. All models were significantly above chance performance from permutation testing.

Bayes Factors for individual and combined predictors

Note . Bayes Factors for a given predictor, controlling for the contributions of each other predictor in the model

Cognitive Measures

The last set of analyses sought to test the prediction that students’ cognitive performance in seventh grade was related to and can predict mathematics competence from the end of the previous year. Univariate analyses showed strong main effects of cluster across all cognitive variables: Digit span forward, BF 10 = 2.4e +6 , F (2,339) = 20.56, p < .001; Digit span backward, BF 10 = 8.7e +8 , F (2,339) = 27.69, p < .001; N-back, BF 10 = 8.3e +8 , F (2,339) = 27.61, p < .001; Spatial span, BF 10 = 1.2e +7 , F (2,339) = 22.45, p < .001; Judgment of Line Angle and Position, BF 10 = 1.1e +6 , F (2,339) = 19.58, p < .001; Mental Rotations Test, BF 10 = 2.3e +6 , F (2,339) = 20.55, p < .001. Intelligence was additionally assessed in the longitudinal component, BF 10 = 4.1e +24 , F (2,339) = 75.98, p < .001. With intelligence showing strong effects across the different clusters, we re-assessed the cognitive differences while controlling for intelligence, and all predictors retained their effects ( BF 10 ’s ranging from 13.15 to 347, p ’s ≤ .05) with the exception of the Mental Rotations Test ( BF 10 = 0.421, p = .089). As seen in Figure 4A - ​ -4I, 4I , graded effects of cognitive differences are evident, similar to the attitudes and anxiety measures, (all p ’s ≤ .007; Low < Medium < High).

Boxplots illustrating the median and first and third quartiles across the three clusters for A-F) the six cognitive measures, and G-I) intelligence, Numerical Operations, and Word Reading. To the left of each boxplot is a density plot showing the distribution of scores.

Next, a similar logistic regression classifier was trained and validated to predict cluster membership, this time solely using the six cognitive measures. We were able to predict cluster membership with a classification accuracy of 53.51%, which was significantly above chance, p < .001. As strong IQ differences were seen across clusters, IQ scores were added to the cognitive measures, which improved classification accuracy to 56.4%. While post-hoc comparisons may have indicated larger deficits and strengths in measures of working memory before controlling for IQ, Bayesian model comparisons ( Table 2 ) showed that individually, both spatial span and performance on the JLAP contributed the most to cluster differences ( BF 10 ’s of 3.62 and 4.06, respectively). However, the overall best model was one that included JLAP, spatial span, as well as IQ. Thus, a combination of measures resulted in above chance accuracies, above the contribution of any single predictor. Spatial span with JLAP ( BF 10 = 54.45) appeared to contribute more than either spatial span with IQ ( BF 10 = 24.10), or JLAP with IQ (39.23).

For completeness, Numerical Operations and Word Reading scores in seventh grade were also considered. Numerical operations showed large across-cluster differences in the expected direction (Low < Medium < High), BF 10 = 9.1e +42 , F (2,339) = 146.48, p < .001, as did Word Reading scores, BF 10 = 5.7e +18 , F (2,339) = 56.82, p < .001. Indeed, adding Numerical Operations scores to the cognitive predictors and IQ increased cluster classification accuracy to 66.7%, p < .001, and to a lesser extent with the addition of Word Reading scores, 58.2%, p < .001 (see Figure 3 ). Notably, adding the sixth-grade attitudes/anxiety variables to the model including the cognitive measures, IQ, Numerical Operations, and Word Reading minimally improved classification accuracy (67.5%, p < .001) compared to the alternative models. However, these results show that later cognitive performance of students can reliably predict across-grade mathematical competencies, underscoring the importance of mathematical differences in its relation across different domains.

The primary contributions of the current study include a broad assessment of pre-algebra competencies and the identification of clusters of students with similar patterns of strengths and weaknesses in these competencies, as well as identifying the attitudes and cognitive abilities that best discriminate the clusters. Moreover, we applied state-of-the-art clustering techniques and multivariate pattern analyses to the performance of students in a larger and more diverse sample than is typical in these types of studies. As the reader knows, the use of these methods resulted in the identification of three clusters of students with different patterns of pre-algebra strengths and weaknesses. The results make theoretical and practical contributions to our understanding of mathematical development, but the focus here is on the lowest performing students This is because their patterns of strengths and weaknesses provide useful information on where to best focus intervention efforts. The cognitive abilities that best classify students into groups provide potentially useful information on the types of scaffolds that might be included in any such interventions.

Mathematics competencies

The lower-performing students had deficits across mathematical areas, but critically these were not uniform, in keeping with studies of younger students who have difficulties with mathematics learning ( Geary et al., 2012 ). In support of recent work by Siegler and colleagues ( Braithwaite et al., 2019 ; Siegler et al., 2011 ) and McNeil and colleagues (e.g., McNeil et al., 2019 ), the lower-performing students had particularly large deficits in their understanding of fractions magnitudes and mathematical equality. If early deficits in these areas predict later performance in algebra – a planned assessment for the current project once data collection is completed – controlling other factors (e.g., intelligence, parental education), then they could be used as a quick screening measure to identify at-risk students before they enter middle school. This would not mean that these are their only deficits but would indicate a broader assessment of their strengths and weaknesses is warranted. Whatever their performance in other areas, targeted interventions in these domains would be an important component of better preparing them for success in algebra. This is because an understanding of fractions and mathematical equality have been found to predict later performance in algebra ( Booth et al., 2014 ; Booth & Koedinger, 2008 ; Siegler et al., 2012 ).

The fraction deficits of these lower-performing students were evident with the simple comparison of the magnitudes of two fractions and in terms of their skill at situating fractions magnitudes on the number line. A strikingly ( d = 2.95) poor understanding of mathematic equivalence also stood out among the lower-performing students. The magnitude of these students’ deficits here was roughly double their relative deficits on the fraction measures and more than double their relative cognitive deficits. We did not have information on how they were approaching these problems, but the results are in keeping with a rigid operational view of the ‘=‘ ( Alibali et al., 2007 ; McNeil et al., 2011 ; McNeil et al., 2019 ). Whatever they were doing, the results here add substantial weight to the argument that mathematical equivalence is a key part of students’ conceptual understanding of mathematics and that a poor understanding of this concept slows mathematical development.

The higher-achieving students had particularly large advantages in computational fluency with whole number and fractions arithmetic, that is, larger advantages than might be expected based on their cognitive performance and intelligence. Fluency is dependent on the memorization of basic facts and procedures and for fractions arithmetic is dependent on a conceptual understanding of these procedures (e.g., that multiplying proper fractions results in smaller magnitudes; Braithwaite et al., 2019 ; Braithwaite et al., 2018 ). A relatively better understanding of fractions is also indicated by their strong performance on the fractions number line that in turn may have been facilitated by their ability to transform fractions (e.g., 17/4 = 4 1/4; Siegler et al. 2011 ). These advantages in number development and computational fluency are very likely to result in later advantages in learning algebra above and beyond the contributions of cognitive abilities (NMAP, 2008; Siegler et al., 2012 ). Moreover, these results suggest that achieving computational fluency is an important part of mathematical development, likely because fluency will reduce the working memory demands associated with solving more complex problems in which basic facts and procedures are embedded ( Geary & Widaman, 1992 ; Sweller, van Merriënboer, & Paas, 2019 ).

The important point is that the results from our large and diverse sample are consistent with previous studies and confirm that a poor understanding of fractions magnitudes and mathematical equivalence are core deficits of low-achieving students. A unique contribution here is that these deficits are evident above and beyond their more general mathematics deficits. The pattern of results shown here is useful not only in identifying and targeting specific deficits, but in identifying areas in which higher-performing students especially excel. The latter was most evident on the measures of fluency of solving whole number and fractions problems, which was highlighted by the NMAP (2008) as foundational to preparation to algebra. The implication is that any interventions that focus on at-risk students’ conceptual understanding of fractions and mathematical equivalence should not do so at the expense of building procedural fluency. The relative importance of these different pre-algebra skills in predicting later algebra outcomes is not currently known but will be assessed in the on-going longitudinal component of the study.

Students’ performance in mathematics, whether it was toward the low- or high-end, had little relation to their attitudes about English. The result indicates that students were differentiating between English and mathematics and that difficulties with mathematics might not be associated with negative attitudes about schooling more broadly, at least for sixth graders. In keeping with many previous studies, the lower-performing students had a lower mathematics self-efficacy, lower beliefs in the utility of mathematics, and higher levels of mathematics anxiety for both learning and evaluation than did their higher-performing peers ( Ashcraft & Kirk, 2001 ; Carey et al., 2016 ; Dowker et al., 2016 ; Hill et al., 2016 ; Maloney & Beilock, 2012 ). However, when controlling for these various attitudes and anxiety measures instead of analyzing them individually, mathematics self-efficacy stood out in terms of predicting cluster membership. A striking result of this is the weaker relation between mathematical competencies and all other attitudes and anxiety once controlling for self-efficacy, in support of results found by Devine et al. (2017) , that is, that many students struggling with mathematics do not show particularly high levels of mathematics anxiety.

The disconnect between the lower-performing students’ attitudes and their actual competencies is illustrated by a contrast of the differences between them and their typically-achieving (middle cluster) peers. Differences in mathematics ranged from d = 1.19 to 2.95 ( M = 1.7), but group differences for mathematics self-efficacy, the best predictor of cluster membership, showed a Cohen’s d of 0.75 ( Cohen, 1988 ). The latter is less than half the size of their actual mathematics deficits, with even starker disconnects for the anxiety measures. These patterns are consistent with the broader Kruger-Dunning effect, whereby lower-performing individuals significantly over-estimate their relative competencies in many domains ( Kruger & Dunning, 1999 ). The reasons for the disconnect between actual performance and estimates of relative performance are widely debated and include statistical regression, poor meta-cognition, and insensitivity to the commission of errors, among others ( Gignac & Zajenkowski, 2020 ; Jansen, Rafferty, & Griffiths, 2021 ; McIntosh, Fowler, Lyu, & Della Sala, 2019 ). The Kruger-Dunning effect is not generally integrated within studies of children’s academic development, but our results suggest that such an integration might be fruitful.

It is not clear what is underlying the effect in this sample, but one potential contributing factor is that these students are receiving inaccurate feedback about their mathematical development and thus might not fully appreciate the difficulties they will face in high school algebra. At the same time, whatever is maintaining their self-efficacy and keeping their anxiety in check could serve as a protective mechanism as they move forward in their mathematics education.

Cognitive performance

Students in the lowest performing group had general deficits in mathematics that were related to intelligence, but above and beyond the effects of intelligence, we observed reliable cognitive differences across clusters. Students in the lowest-performing group had an achievement and cognitive profile that is very similar to that of students identified as having mathematical learning difficulties in the broader literature ( Geary et al., 2007 ; Murphy et al., 2007 ). More precisely, their cognitive profile was characterized by lower intelligence and working memory span (e.g., n-back) than students in other groups, as well as spatial deficits. Interestingly, through Bayesian model comparisons, even though digit span and n-back showed significant main effects, their contribution was mitigated when the effects of other cognitive factors were taken-into-account. This is surprising, as Geary et al. (2012) found especially poor working memory for fifth-graders with a mathematical learning disability, as is commonly found ( Bull & Lee, 2014 ; Mazzocco & Kover, 2007 ; McLean & Hitch, 1999 ). We included a more extensive assessment of cognitive abilities than in most previous studies, including various spatial abilities and spatial working memory. The inclusion of these measures is the most likely explanation for the mitigated contributions of standard working memory measures (e.g., digit span) in differentiating lower-achieving students from their higher achieving peers ( Miyake et al., 2001 ).

The deficits of the lower-performing group, and differences across clusters broadly, were best explained through the combination of intelligence and specific spatial abilities; specifically, visuospatial working memory and visuospatial attention (JLAP). The relationship between mathematics achievement and spatial processing has long been of interest (see Bishop, 1980 for review), and multiple lines of evidence have implicated visuospatial processing in the development of mathematics skills ( Geary, 2004 ; Geer et al., 2019 ; Hawes & Ansari, 2020 ; Mix, 2019 ; Mix et al., 2016, 2017 ; Reuhkala, 2001 ). This might be due to the use of the visuospatial system for representing numerical and mathematical knowledge and implementing certain types of problem-solving strategies, but this is still uncertain ( Hawes & Ansari, 2020 ; Mix, 2019 ).

In any case, Li and Geary (2017) found that the relation between visuospatial working memory and mathematics achievement increases across grades. These types of patterns do not mean there is a casual relationship but if there is (see Gilligan et al., 2020 ), additional deficits in mathematics areas that are influenced by spatial abilities may emerge as students move through middle school and into high school. The results also suggest that incorporating more spatial information into remediation programs could be helpful for struggling students. These could involve short spatial activities that might prime visuospatial attention before the presentation of the mathematical information, as was done by Gilligan and colleagues. Instruction on the use of visuospatial representations to aid in mathematical problem solving, such as word problems, is often helpful ( Hord & Xin, 2013 ). And interventions to improve students’ fractions knowledge and their understanding of equivalence often have spatial components built into them ( Barbieri et al., 2020 ; McNeil et al., 2019 ).

The issue that remains to be determined is whether there is an interaction between student’s spatial abilities and the use of spatial interventions in learning key mathematical concepts. It could be that student’s with strong spatial abilities do not need the same level of spatial scaffolding as their peers which in turn could lead to an underestimate of the importance of these scaffolds in general classroom settings.

Limitations

The correlational nature of the data precludes strong causal statements, and even though all mathematics, attitudinal, and cognitive measures were validated during analysis, the results are largely descriptive. Moreover, students were administered many tasks in an attempt to achieve assessment breadth, but one potential cost is fatigue during the assessments that might have resulted in less-than-optimal performance. Despite these limitations, the assessment of a very large and diverse sample of adolescents across core pre-algebra mathematical competencies and academic attitudes and anxiety provided a broader assessment of mathematical strengths and weaknesses for different subgroups of adolescents than is typical in this literature. Against a background of relatively poor achievement, lower-performing students have an especially poor understanding of mathematical equality and do not fully understand fractions comparisons and how fractions magnitudes map onto the number line. The contribution of the spatial measures to the classification of students into clusters suggests that direct spatial interventions (e.g., Gilligan et al., 2020 ) or the use spatial scaffolds might be useful for the remediation of lower-achieving students’ difficulties with equality and fractions.

Supplementary Material

Supplementary information, acknowledgements:.

The study was supported by grants DRL-1659133 from the National Science Foundation and R01 HD087231 from the Eunice Kennedy Shriver National Institute of Child Health and Human Development and We thank Dana Hibbard of Columbia Public Schools, as well as the staff, faculty, students and families of Columbia Public Schools for their assistance with the logistics of the study. We thank Kristin Balentine, Mandar Bhoyar, Amanda Campbell, Maria Ceriotti, Felicia Chu, Anastasia Compton, Danielle Cooper, Alexis Currie, Kaitlynn Dutzy, Amanda Evans, James Farley, Amy Jordan, Bradley Lance, Kate Leach, Joshua McEwen, Kelly Mebruer, Heather Miller, Natalie Miller, Sarah Peterson, Nicole Reimer, Laura Roider, Brandon Ryffe, Logan Schmidt, Jonathan Thacker, Zehra Unal, and Melissa Willoughby for their assistance with data collection and processing.

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Strengths, weaknesses of students’ math abilities

Each year, the Office of Superintendent of Public Instruction analyzes where students, as a group, have trouble in math on the WASL. Here's a sampling from...

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Each year, the Office of Superintendent of Public Instruction analyzes where students, as a group, have trouble in math on the WASL. Here’s a sampling from this year’s analysis: FOURTH GRADE Strengths: • Algebraic sense, such as understanding how to write an equation to determine how many barrettes two girls have, if one has four in her hair and another has three. • Locating points on a grid, recognizing reflections and lines of symmetry (geometric sense). • Converting minutes to hours, or minutes to seconds, or hours to minutes. Weaknesses: • Understanding relative values of whole numbers and fractions, such as knowing that 36,700 is less than 37,600, and determining what’s larger: 2/3 or 3/5. • Comparing and interpreting information in a chart. • Organizing data for a given purpose, such as in a chart, or to support a point of view. SEVENTH GRADE Strengths: • Explaining or describing mathematical information. • Solving single-variable, one- and two-step equations. • Identifying what information is needed, and what isn’t, to solve a problem. Weaknesses: • Figuring out a method to solve a problem and justifying the results. • Determining percents. • Taking data that’s in one form and putting it in another, such as making a chart or graph. 10TH GRADE Strengths: • Understanding ratio, percent, proportion. • Ability to interpret tables and graphs. • Converting units of measurement. Weaknesses: • Analyzing complex information in a table or chart to draw conclusions about what the data say. • “Great” difficulty figuring out how perimeter, area, surface area or volume changes when, for example, the side of a rectangle changes. Confusing circumference of circle with area of circle. • Determining the number of possible outcomes in a probability question. Source: Office of Superintendent of Public Instruction

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2017 Detecting strengths and weaknesses in learning mathematics through a model classifying mathematical skills.pdf

Through a review of the literature on mathematical learning disabilities (MLD) and low achievement in mathematics (LA) we have proposed a model classifying mathematical skills involved in learning mathematics into four domains (Core number, Memory, Reasoning, and Visual-spatial). In this paper we present a new experimental computer-based battery of mathematical tasks designed to elicit abilities from each domain, that was administered to a sample of 165 typical population 5th and 6th grade students (MLD = 9 and LA = 17). Explanatory and con rmatory factor analysis were conducted on the data obtained, together with K-means cluster analysis. Results indicated strong evidence for supporting the solidity of the model, and clustered the population into six distinguishable performance groups with the MLD and LA students distributed within ve of the clusters. These ndings support the hypothesis that di culties in learning mathematics can have multiple origins and provide a means for sketching students’ mathematical learning pro les.

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Intellectual Mathematics

How to write a history of mathematics essay

This is a guide for students writing a substantial course essay or bachelors thesis in the history of mathematics.

The essence of a good essay is that it shows independent and critical thought. You do not want to write yet another account of some topic that has already been covered many times before. Your goal should not be to write an encyclopaedia-style article that strings together various facts that one can find in standard sources. Your goal should not be to simply retell in your own words a story that has already been told many times before in various books. Such essays do not demonstrate thought, and therefore it is impossible to earn a good grade this way.

So you want to look for ways of framing your essay that give you opportunity for thought. The following is a basic taxonomy of some typical ways in which this can be done.

Critique. A good rule of thumb is: if you want a good grade you should, in your essay, disagree with and argue against at least one statement in the secondary literature. This is probably easier than you might think; errors and inaccuracies are very common, especially in general and popular books on the history of mathematics. When doing research for your essay, it is a good idea to focus on a small question and try to find out what many different secondary sources say about it. Once you have understood the topic well, you will most likely find that some of the weaker secondary sources are very superficial and quite possibly downright wrong. You want to make note of such shortcomings in the literature and cite and explain what is wrong about them in your essay, and why their errors are significant in terms of a proper understanding of the matter.

The point, of course, is not that finding errors in other people’s work is an end in itself. The point, rather, is that if you want to get anywhere in history it is essential to read all texts with a critical eye. It is therefore a good exercise to train yourself to look for errors in the literature, not because collecting errors is interesting in itself but because if you believe everything you read you will never get anywhere in this world, especially as far as history is concerned.

Maybe what you really wanted to do was simply to learn some nice things about the topic and write them up in your essay as a way of organising what you learned when reading about it. That is a fine goal, and certainly history is largely about satisfying our curiosities in this way. However, when it comes to grading it is difficult to tell whether you have truly thought something through and understood it, or whether you are simply paraphrasing someone else who has done so. Therefore such essays cannot generally earn a very good grade. But if you do this kind of work it will not be difficult for you to use the understanding you develop to find flaws in the secondary literature, and this will give a much more concrete demonstration of your understanding. So while developing your understanding was the true goal, critiquing other works will often be the best way to make your understanding evident to the person grading your essay.

For many examples of how one might write a critique, see my book reviews categorised as “critical.”

Debate. A simple way of putting yourself in a critical mindset is to engage with an existing debate in the secondary literature. There are many instances where historians disagree and offer competing interpretations, often in quite heated debates. Picking such a topic will steer you away from the temptation to simply accumulate information and facts. Instead you will be forced to critically weigh the evidence and the arguments on both sides. Probably you will find yourself on one side or the other, and it will hopefully come quite naturally to you to contribute your own argument for your favoured side and your own replies to the arguments of the opposing side.

Some sample “debate” topics are: Did Euclid know “algebra”? Did Copernicus secretly borrow from Islamic predecessors? “Myths” in the historiography of Egyptian mathematics? Was Galileo a product of his social context? How did Leibniz view the foundations of infinitesimals?

Compare & contrast. The compare & contrast essay is a less confrontational sibling of the debate essay. It too deals with divergent interpretations in the secondary literature, but instead of trying to “pick the winner” it celebrates the diversity of approaches. By thoughtfully comparing different points of view, it raises new questions and illuminates new angles that were not evident when each standpoint was considered in isolation. In this way, it brings out more clearly the strengths and weaknesses, and the assumptions and implications, of each point of view.

When you are writing a compare & contrast essay you are wearing two (or more) “hats.” One moment you empathise with one viewpoint, the next moment with the other. You play out a dialog in your mind: How would one side reply to the arguments and evidence that are key from the other point of view, and vice versa? What can the two learn from each other? In what ways, if any, are they irreconcilable? Can their differences be accounted for in terms of the authors’ motivations and goals, their social context, or some other way?

Following the compare & contrast model is a relatively straightforward recipe for generating reflections of your own. It is almost always applicable: all you need is two alternate accounts of the same historical development. It could be for instance two different mathematical interpretations, two perspectives emphasising different contexts, or two biographies of the same person.

The compare & contrast approach is therefore a great choice if you want to spend most of your research time reading and learning fairly broadly about a particular topic. Unlike the critique or debate approaches, which requires you to survey the literature for weak spots and zero in for pinpoint attacks, it allows you to take in and engage with the latest and best works of scholarship in a big-picture way. The potential danger, on the other hand, is that it may come dangerously close to merely survey or summarise the works of others. They way to avoid this danger is to always emphasise the dialog between the different points of view, rather than the views themselves. Nevertheless, if you are very ambitious you may want to complement a compare & contrast essay with elements of critique or debate.

Verify or disprove. People often appeal to history to justify certain conclusions. They give arguments of the form: “History works like this, so therefore [important conclusions].” Often such accounts allude briefly to specific historical examples without discussing them in any detail. Do the historical facts of the matter bear out the author’s point, or did he distort and misrepresent history to serve his own ends? Such a question is a good starting point for an essay. It leads you to focus your essay on a specific question and to structure your essay as an analytical argument. It also affords you ample opportunity for independent thought without unreasonable demands on originality: your own contribution lies not in new discoveries but in comparing established scholarly works from a new point of view. Thus it is similar to a compare & contrast essay, with the two works being compared being on the one hand the theoretical work making general claims about history, and on the other hand detailed studies of the historical episodes in question.

Sample topics of this type are: Are there revolutions in mathematics in the sense of Kuhn ? Or does mathematics work according to the model of Kitcher ? Or that of Lakatos or Crowe ? Does the historical development of mathematical concepts mirror the stages of the learning process of students learning the subject today, in the manner suggested by Sfard or Sierpinska ? Was Kant’s account of the nature of geometrical knowledge discredited by the discovery of non-Euclidean geometry?

Cross-section. Another way of combining existing scholarship in such a way as to afford scope for independent thought is to ask “cross-sectional” questions, such as comparing different approaches to a particular mathematical idea in different cultures or different time periods. Again, a compare & contrast type of analysis gives you the opportunity to show that you have engaged with the material at a deeper and more reflective level than merely recounting existing scholarship.

Dig. There are still many sources and issues in the history of mathematics that have yet to be investigated thoroughly by anyone. In such cases you can make valuable and original contributions without any of the above bells and whistles by simply being the first to really study something in depth. It is of course splendid if you can do this, but there are a number of downsides: (1) you will be studying something small and obscure, while the above approaches allow you to tackle any big and fascinating question you are interested in; (2) it often requires foreign language skills; (3) finding a suitable topic is hard, since you must locate an obscure work and master all the related secondary literature so that you can make a case that it has been insufficiently studied.

In practice you may need someone to do (3) for you. I have some suggestions which go with the themes of 17th-century mathematics covered in my history of mathematics book . It would be interesting to study for instance 18th-century calculus textbooks (see e.g. the bibliography in this paper ) in light of these issues, especially the conflict between geometric and analytic approaches. If you know Latin there are many more neglected works, such as the first book on integral calculus, Gabriele Manfredi’s De constructione aequationum differentialium primi gradus (1707), or Henry Savile’s Praelectiones tresdecim in principium Elementorum Euclidis , 1621, or many other works listed in a bibliography by Schüling .

Expose. A variant of the dig essay is to look into certain mathematical details and write a clear exposition of them. Since historical mathematics is often hard to read, being able to explain its essence in a clear and insightful way is often an accomplishment in itself that shows considerable independent thought. This shares some of the drawbacks of the dig essay, except it is much easier to find a topic, even an important one. History is full of important mathematics in need of clear exposition. But the reason for this points to another drawback of this essay type: it’s hard. You need to know your mathematics very well to pull this off, but the rewards are great if you do.

Whichever of the above approaches you take you want to make it very clear and explicit in your essay what parts of it reflect your own thinking and how your discussion goes beyond existing literature. If this is not completely clear from the essay itself, consider adding a note to the grader detailing these things. If you do not make it clear when something is your own contribution the grader will have to assume that it is not, which will not be good for your grade.

Here’s another way of looking at it. This table is a schematic overview of different ways in which your essay can add something to the literature:

The table shows the state of the literature before and after your research project has been carried out.

A Describe project starts from a chaos of isolated bits of information and analyses it so as to impose order and organisation on it. You are like an explorer going into unknown jungles. You find exotic, unknown things. You record the riches of this strange new world and start organise it into a systematic taxonomy.

You need an exotic “jungle” for this project to work. In the history of mathematics, this could mean obscure works or sources that have virtually never been studied, or mathematical arguments that have never been elucidated or explained in accessible form.

An Explain project is suitable when others have done the exploration and descriptions of fact, but left why-questions unanswered. First Darwin and other naturalists went to all the corners of the world and gathered and recorded all the exotic species they could find. That was the Describe phase. Darwin then used that mass of information to formulate and test his hypothesis of the origin of species. That was the Explain phase.

Many areas of the history of mathematics have been thoroughly Described but never Explained.

What if you find that someone has done the Explain already? If you think the Explain is incomplete, you can Critique it. If you think the Explain is great you can Extend it: do the same thing but to a different but similar body of data. That way you get to work with the stimulating work that appealed to you, but you also add something of your own.

Likewise if you find two or more Explains that are all above Critique in your opinion. Then you can do a Compare & Contrast, or a Synthesise. This way you get to work with the interesting works but also show your independent contribution by drawing out aspects and connections that were not prominent in the originals.

See also History of mathematics literature guide .

Why Math Is More Difficult for Some Students

• M.Ed., Education Administration, University of Georgia
• B.A., History, Armstrong State University

In 2005, Gallup conducted a poll that asked students to name the school subject that they considered to be the most difficult. Not surprisingly, mathematics came out on top of the difficulty chart. So what is it about math that makes it difficult? Have you ever wondered?

Dictionary.com defines the word difficult as:

“...not easily or readily done; requiring much labor, skill, or planning to be performed successfully.”

This definition gets to the crux of the problem when it comes to math, specifically the statement that a difficult task is one that is not “readily” done. The thing that makes math difficult for many students is that it takes patience and persistence. For many students, math is not something that comes intuitively or automatically - it takes plenty of effort. It is a subject that sometimes requires students to devote lots and lots of time and energy.

This means, for many, the problem has little to do with brainpower; it is mostly a matter of staying power. And since students don't make their own timelines when it comes to "getting it," they can run out of time as the teacher moves on to the next topic.

Math and Brain Types

But there is also an element of brain-style in the big picture, according to many scientists. There will always be opposing views on any topic, and the process of human learning is subject to ongoing debate, just like any other topic. But many theorists believe that people are wired with different math comprehension skills.

According to some brain science scholars, logical, left-brain thinkers tend to understand things in sequential bits, while artistic, intuitive, right-brainers  are more global. They take in a lot of information at one time and let it "sink in." So left-brain dominant students may grasp concepts quickly while right-brain dominant students don’t. To the right brain dominant student, that time-lapse can make them feel confused and behind.

Math as a Cumulative Discipline

Math know-how is cumulative, which means it works much like a stack of building blocks. You have to gain understanding in one area before you can effectively go on to “build upon” another area. Our first mathematical building blocks are established in primary school when we learn rules for addition and multiplication, and those first concepts comprise our foundation.

The next building blocks come in middle school when students first learn about formulas and operations. This information has to sink in and become “firm” before students can move on to enlarge this framework of knowledge.

The big problem starts to appear sometime between middle school and high school because students very often move on to a new grade or new subject before they’re really ready. Students who earn a “C” in middle school have absorbed and understood about half of what they should, but they move on anyway. They move on or are moved on, because

• They think a C is good enough.
• Parents don’t realize that moving on without a full understanding poses a big problem for high school and college.
• Teachers don’t have time and energy enough to ensure that every single student understands every single concept.

So students move to the next level with a really shaky foundation. The outcome of any shaky foundation is that there will be a serious limitation when it comes to building and real potential for complete failure at some point.

The lesson here? Any student who receives a C in a math class should review heavily to make sure to pick up concepts they'll need later. In fact, it is smart to hire a tutor to help you review any time you find that you've struggled in a math class!

Making Math Less Difficult

We have established a few things when it comes to math and difficulty:

• Math seems difficult because it takes time and energy.
• Many people don't experience sufficient time to "get" math lessons, and they fall behind as the teacher moves on.
• Many move on to study more complex concepts with a shaky foundation.
• We often end up with a weak structure that is doomed to collapse at some point.

Although this may sound like bad news, it is really good news. The fix is pretty easy if we’re patient enough!

No matter where you are in your math studies , you can excel if you backtrack far enough to reinforce your foundation. You must fill in the holes with a deep understanding of the basic concepts you encountered in middle school math.

• If you’re in middle school right now, do not attempt to move on until you understand pre-algebra concepts fully. Get a tutor if necessary.
• If you’re in high school and struggling with math, download a middle school math syllabus or hire a tutor. Make sure you understand every single concept and activity that is covered in middle grades.
• If you’re in college, backtrack all the way to basic math and work forward. This won’t take as long as it sounds. You can work forward through years of math in a week or two.

No matter where you start and where you struggle, you must make sure you acknowledge any weak spots in your foundation and fill the holes with practice and understanding!

• How to Tell If You Are Right-Brain Dominant
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• Innovative Ways to Teach Math
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• Biology Homework Help
• College Preparation in Middle School
• Why Learning Fractions Is Important
• Multi-Sensory Instruction in Math for Special Education

Math has always been my strength ; UA&P

Maheshkushwaha 1 / 3   Jul 26, 2013   #2 First, I could not get your topic of the essay you have written. Second, the content of the essay makes the reader make a negative remark on you for you've tried to show yourself the best throughout the essay. Frankly speaking, it's not good to flatter about self in the essays especially written to the admission officer. They seek for uniqueness not talent actually. There are millions of people who're good at Maths, thousands of people can play piano and of course all except some are active. Try to show what is the quality in you that no others have. Try to show them the real YOU. and I hope that works. You can write better.

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Using Positive Feedback in Math Classrooms

Providing math students with positive feedback can help them clarify their thinking, take risks, and apply concepts in new contexts.

If you were a “good” math student, your teacher may have filled your papers with a checkmark next to each correct answer. But handing back “perfect work” with a slew of checkmarks is a missed opportunity for math teachers. Every time we provide a student with feedback, we have a chance to send a message. We can signal what we value and the strengths we see in the student’s work, providing insight into areas for growth and further learning. In math we value clarity and logic, creative solutions, perseverance, and curiosity. Used strategically, positive feedback can reinforce these cornerstones of the discipline. Although the examples I provide in this article are geared toward middle and high school math classes, positive feedback can be used at a variety of grade levels.

Reinforcing Effective Communication

Math teachers frequently ask students to show their work. When students give us insight into their thought processes by writing enough on the page, we can provide positive and specific comments in return. Our reinforcement helps students learn what specific aspects of their work were effective at clarifying their logic.

Here are some examples:

• Your explanation here helps me see how you got from one step to the next.
• The picture you drew helped me to understand how you’re thinking about this.
• When you defined the variable before using it, I was able to follow your reasoning.
• The sentence at the end helped me to see that this is your final answer.

All of these comments acknowledge the time that students put into explaining their reasoning. Providing positive reinforcement when students effectively convey their thought processes helps them develop into effective communicators of mathematics.

Recognizing a Mathematician’s Craft and Choices

Math teachers constantly remind their students that “there is more than one way to solve a problem.” This is true—the art of problem-solving allows students to discover elegant paths to a solution. Math talks have become popular because of their emphasis on multiple problem-solving techniques. The comments that we make on student work can reinforce the value of mathematical thinking.

You might write something like these:

• I didn’t think of this strategy!
• This is a clever implementation of factoring.
• This step reminds me of the example we looked at when _____.
• I like how you adapted the idea from _____ to this problem.

These comments will help students see their technique within the space of many pathways to a solution. When we reference the strategy that they used and contextualize it, we help them connect their problem-solving process to the content.

Celebrating Growth and Perseverance

Math is hard! Math teachers must find ways for students to grapple with the content and engage in productive struggle. If we give students credit for their progress and perseverance, we can encourage them to push through challenges the next time they arise.

Here are some examples of how you might emphasize growth and praise perseverance:

• Great job catching the mistake here.
• I can tell this was a long and messy computation. By keeping your work organized and sticking to your plan, you persevered.
• I noticed that you had trouble with this skill in the last unit, but you have mastered it now! Great job sticking with it.

Students don’t always notice their own growth. When we can point it out to them, they see that their hard work and struggle is worth it.

Encouraging Reflection

When students perform at a high level, positive comments can push their thinking beyond the standard content. With feedback, we can inspire their curiosity and encourage deeper thinking.

Here are some reflective comments you could make:

• What about this problem helped you realize that you needed this particular strategy?
• How did you check your work?
• What strategies did you use to keep track of all the steps needed to solve this problem?
• How will you remember the connection that you made here?
• What do you think would happen if the problem were slightly different? How would you have to adapt your approach?

All of these questions send the message that learning doesn’t stop at a perfect test score. We should continue to ask questions, pursue our curiosity, and find new and novel ways to apply what we’ve learned.

I hope that these ideas inspire you to provide comments beyond the simple check mark. We should be putting just as much care and attention into the feedback we provide to high-achieving students as what we give to students who need more practice.

While some may think that math teachers have it easy when it comes to grading, we shouldn’t take the easy way out. Writing a few comments might add a minute or two to the time it takes to mark a paper, but I assure you that the gains you will see in student confidence, motivation, creativity, and understanding will be worth it. By taking the time to provide meaningful, detailed, specific, and positive feedback, we provide all of our students the opportunity to grow.

Note: I wrote this piece after reading Alex Shevrin Venet’s “ How to Give Positive Feedback on Student Writing ,” because I was struck by how many of the guidelines that she provides are applicable to the math classroom. I recommend taking a look if you haven’t already!

How to talk about weaknesses in MBA essays and interviews

“I have my flaws. I sing in the shower, sometimes I volunteer too much, occasionally, I’ll hit somebody with my car.” – Michael Scott, Regional Manager, Scranton Branch, Dunder Mifflin Paper Company.

In two delightfully short sentences, Michael Scott from The Office , with his trademark humor, summed up the contradictions and complexities involved in confronting that most confounding of all corporate questions – tell us about your weaknesses.

Talking about weaknesses in your MBA essay and interview is like walking a tightrope. Talk too less and you end up looking arrogant, like Michael Scott. Talk too much and you start sounding like an agony aunt column in an adolescent magazine, also like Michael Scott.

Is there a sweet spot somewhere in between that is just right and that isn’t Michael Scott?

Yes, there is, and we are here to tell you how to hit it.

Why do you need to talk about your weaknesses anyway?

Most MBA essays have a section that requires you to talk about your strengths and weaknesses. Alternatively, there might be a section where you are asked to describe a situation where you failed or talk about a time when you received criticism for your work and how you handled it.

The same set of questions could be posed to you in an interview as well.

Strengths are easy to talk about. You know what you’re good at and you can talk about it endlessly. Talking about weaknesses is the tricky part.

The admissions committee judges your answers to evaluate two things – how self-aware you are, and how well you bounce back from failure. The point to really understand here is that you are applying to a school. A business school, yes, but a school nonetheless. A school is a place where you go to learn things. And you only learn things you do not know yet, or which you aren’t very good at yet.

To put it simply, the admissions committee wants to hear about your weaknesses so that you enter school and learn to improve upon them.

It follows that what the admissions committee wants to see in your essay or hear from you in the interview are two things – an awareness of what your weaknesses are, and a capacity or a will to improve upon them provided the right guidance and environment.

Knowing this is the key to crafting a great response to the adcom’s questions.

So how do you put this knowledge into actionable items?

Below is a 7-point checklist on what to do and what not to do when talking about your weaknesses in MBA essays and interviews.

• Don’t take it personally
• Be original
• Don’t hide your weaknesses
• Don’t try to turn a positive into a negative
• Don’t dwell on your weaknesses too much
• Avoid freudian slips, red flags, alarm bells
• Take it easy

Let’s try to understand what each one means.

How to Talk About Weaknesses in MBA Essays and Interviews

1. don’t take it personally.

Your MBA essay is not your personal diary and your interviewer is not your therapist. While it’s a good habit to keep a diary and try therapy for self-improvement , your MBA essay/interview is not the right place for these.

When the adcom asks you to talk about your weaknesses, what they have in mind are your weaknesses or failures in a professional setting. They aren’t very interested in your character flaws or your life struggles unless they have a direct bearing on your professional performance.

So what you are expected to describe here are situations at your workplace, or examples from your academics where you struggled.  

2. Be Original

This one’s a little obvious but it bears repeating. Only talk about weaknesses that you really have. Do not try to mention a weakness simply because it sounds cool, or because you heard someone else talk about it and get through.

Remember how in school, when we were asked our hobbies, half the class would say listening to music and playing cricket?

And then there would be one wiseguy ( it was almost always a guy for some reason) who, refusing to be dragged down to the level of average Joes and plain Janes possessing mundane hobbies, would come up with a hobby no one had ever heard of, simply to sound cool. Like lepidopterology.

Don’t be that guy in your MBA essay.

It is perfectly fine to have perfectly ordinary hobbies like listening to music and playing cricket. It is also perfectly fine to have outrageously unheard of hobbies like lepidopterology. Just make sure that whatever you say, you mean it. If you talk about a certain weakness, you need to make sure that it is something you’ve lived through.

Oh, and a lepidopterologist, by the way, is a butterfly collector.

Read more on the best extracurricular activity for college admissions .

3. Don’t Hide Your Weaknesses

Remember how Tyrion Lannister from Game of Thrones advised wearing your weaknesses like armor so no one could use them against you?

He had a point.

Everyone has weaknesses. If you try to hide yours, the admissions committee will see right through it.

Instead, be honest and talk about your weaknesses openly. You can even use them as an opportunity to show how you’ve grown and changed over the years.

For example, let’s say you have a weakness in math. You can talk about how you used to struggle with math in school and how you had to work hard to improve your skills. You can talk about how you’re now much better at math and how you’ve even been able to help other people improve their own math skills.

This shows that you’re honest about your weaknesses and that you’ve taken steps to improve upon them.  

4. Don’t Try To Turn a Positive Into a Negative

This is a classic mistake, related closely to the previous point that a lot of candidates make. It involves describing as a weakness something that is not usually considered a flaw.

For instance, when you mention things like being a perfectionist, or being too kind, or working too hard, what you are in effect doing, is dressing up desirable qualities to pass them off as weaknesses.

This feels disingenuous, like trying to stick colorful feathers on a chicken to pass it off for a peacock.  Your interviewer can easily turn it around into a sticky situation and trap you.

Imagine the following exchange:

Interviewer: What is your biggest weakness? You: I think I am too kind. Interviewer: I don’t think kindness is a weakness. I think it is a strength. *awkward silence*

To avoid this trap, instead of saying you’re too kind, you need to phrase it to convey that you lack the firmness to deal with people or that you’re not very good with people skills.

In other words, call a spade a spade.

5. Don’t Dwell on Your Weaknesses Too Much

This follows directly as a counter to the previous point. When talking about your weaknesses, don’t overdo them. You need to keep a balance.

For example, let’s say you’re asked about a time when you failed. You don’t want to spend the whole interview talking about that one time you failed.

Instead, you want to focus on what you learned from that experience and how it made you a better person. This shows that you’re able to learn from your mistakes and that you’re not afraid to fail.

What you want to emphasize here is that you’re aware of your weaknesses and that you’re taking steps to improve upon them.

6. Avoid Freudian Slips, Red Flags, Alarm Bells

This one follows directly from the previous point. When you talk about your weaknesses, make sure you do not reveal more than is necessary.

An example?

Suppose you say something like you have trouble being fully functional at work until you have had your fourth cup of coffee. Or You say that you have trouble waking up most mornings and reporting to work. Or You say that you prepare all your PowerPoint presentations after 7 PM with a sundowner in hand.

All these are behavioral traits that can be viewed as being symptomatic of deeper mental or physical health issues. They are best kept away from an MBA essay or interview.

7. Take It Easy

Finally, don’t spend too much time figuring out the perfect weakness.We know this is a kind of a long and somewhat intimidating list, but once you sit down with a pen and paper to think things through, it will all come together naturally.

Remember, that everyone has weaknesses and that you’re not alone in this. Give it a good thought, but it shoudn’t sound too stressed.  

Industry insider tips to answer the weakness question in Interviews

Some business schools do not specifically ask about strengths and weaknesses in their MBA essays. So how does the admissions committee judge these aspects?

“We practice holistic admissions and evaluate candidates across a wide array of behavioral-based evidence in their application packages,” says Erin O’Brien, assistant dean and chief enrollment and marketing officer in the University at Buffalo School of Management.

“We choose to explore a candidate’s strengths and weaknesses as part of our interview process, so that we can probe their responses in a deeper and more thoughtful way than limiting it to a written personal statement,” she adds.

We took the opportunity to dig a little deeper, and Erin had some interesting insights, advice and tips to share.

MCB: What advice do you have to applicants who are apprehensive about revealing their vulnerabilities during the interview?

Erin: Let’s face it, interviews for any graduate management program can be scary. While it’s a delicate balance, it’s also OK to reveal vulnerabilities during an interview – personally, I think it shows authenticity. So many interviews sound the same, and sometimes vulnerabilities can help positively differentiate you from the rest of the candidates. They have the power to show honesty, self-awareness, and a focus on continuous improvement – traits we strongly desire in our students. They can also show the human side of a candidate, beyond quantifiable academic performance statistics.

But, it’s best to be planned and directed when sharing vulnerabilities. As you prep for your interview, ask yourself how will sharing my weakness also provide evidence of something more positive – can I use it to frame resilience or persistence, skill or competency growth? Can I weave it into my motivation for applying to my graduate management program? And, make sure you include how you’ve worked to overcome this weakness or vulnerability, even if you haven’t quite solved it yet. Show how you have made the effort to improve. If you have actual results or behavior-based evidence of success, highlight them.

I think it is important to have guard rails, though, in this type of response. There are potential pitfalls if you choose to reveal a weakness or vulnerability that may be baseline required skills for entrance into the program. You’ll want to avoid those types of responses. Also, as with every answer in a graduate management admissions interview, don’t make anything up – be truthful and honest.

MCB: What are some of the top traits your team looks for in candidates that can help them get an admit in spite of weak areas like a low GMAT score or low GPA?

Erin: I tell candidates all the time, no one thing can rule you in or out. Just because you show up with a great standardized test score or outstanding undergraduate GPA, you still need to bring a portfolio of both cognitive and non-cognitive skills with you into your application. We want to see evidence of strong work experience, be it professional post-baccalaureate, volunteer or internship. We want to see behavioral-based evidence of skills like leadership, teamwork, communication, resilience, creativity, etc. We want to see positive progression, motivation and the ability to self-manage and adapt. These are all equally or sometimes even more important than GPA or test score.

Look at your application as a whole, an entire portfolio of elements. Draw a line between what may be a plus or a minus. If you have a minus, like a lower GPA or a lower test score, make sure you have a plus above the line that far outweighs any negative impact.

For example, if you have a lower GPA, but your undergraduate career was more than five years ago and you now have outstanding work experience showing lots of leadership potential, that’s going to minimize the negative impact of that GPA in our admissions decision-making. If you volunteered in a global experience, leading a team of others in a social impact project, that may far outweigh any negative impact from test scores.

Here’s an insider pro tip: you’re far more interesting than your GPA or test score. Tell us all about it. Chances are it will improve your potential of admission.

MCB: What is your advice to those who have career related issues including frequent career switching, career gaps or lay-offs?

Erin: My advice to applicants with resumes that may have gaps or frequent switches is really a call-back to my previous responses above: you are far more interesting than any single application data point.

Let’s start with lay-offs. Lay-offs happen – look at what’s happening in the tech world now. What I would want to know, as your interviewer, is, how did you pivot? This is a great opportunity to show resilience and creativity.

For gaps or career-switching, what is the story behind the gap or switch pattern? Was it intentional, e.g., did you purposefully make lateral moves to gain a wide variety of experience? Was it a condition of the industry in which you operate, e.g., were you in start-ups? What did you do with the gap time? How did you make it constructive and progressive? Was the switch or gap unintentional, e.g., is it a potential weakness?

Be truthful and honest…this may be an opportunity to showcase your adaptability and motivation for pursuing a graduate management degree, to hurdle the career plateau on which you found yourself switching job to job without advancement.

In the end, relative to all three questions and responses above, many applicants think business schools are looking for a homogenized portrait of traits and characteristics in the admissions process. Unfortunately, I think that’s a condition of being in a publicly-ranked market where applicants are trying to make themselves look like the “ideal” candidate.

However, that couldn’t be further from the truth! We are building a balanced cohort of different thoughts, perspectives and experiences for our programs. How boring would business school be if it were filled with only those who had the same profile – high test scores, similar experiences, the same type of work backgrounds.

How much more interesting will it be if you are in a cohort teamed with people who have vastly different opinions, points of view and backgrounds from yours, where you can learn new things from each other? Personally, I’d much rather be in that class.

Talking about your weaknesses is a complex task that requires self-awareness to understand and nuance to express. It is for this reason that MBA essays and interviews want to hear it from you.

Even if you haven’t been asked to talk about your weaknesses in the MBA essay, it’s good to have a few examples of your weaknesses ready to go, so that you’re not caught off guard in the interview.

By being honest and prepared, you’ll be able to talk about your weaknesses in a way that will show the admissions committee that you’re aware of them and that you’re taking steps to improve upon them.

You also want to make sure that you don’t dwell on your weaknesses too much. Instead, focus on what you’ve done to improve upon them.

A word of caution: Often, this question on weaknesses works in conjunction with other topics in the application essays or interviews. This is where it gets trickier. So make sure you understand the implications of what you’re saying, to avoid conflicting with your other answers.

MBA Crystal Ball has top admission consultants to help you answer this and other questions in your MBA application in the most effective manner. Drop us an email, if you need professional help: info [at] mbacrystalball [dot] com   Also read: – How to write great MBA essays – Common mistakes to avoid in MBA application essays – How to answer questions on the long term and short term goals – Many more top MBA essay tips – Best admissions consultant for ISB for winning ISB essay tips

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Essay on My Weakness

Students are often asked to write an essay on My Weakness in their schools and colleges. And if you’re also looking for the same, we have created 100-word, 250-word, and 500-word essays on the topic.

Let’s take a look…

100 Words Essay on My Weakness

Understanding my weakness.

Every person has strengths and weaknesses. One of my weaknesses is impatience. I often want things to happen quickly and can get frustrated when they don’t.

Impatience in Everyday Life

This impatience affects my daily life. I rush through tasks, which can lead to mistakes. It also affects my relationships, as I can be quick to react.

Working on My Weakness

Despite this, I am learning to manage my impatience. I practice mindfulness, which helps me stay calm. I am not perfect, but every day, I strive to be patient.

250 Words Essay on My Weakness

Introduction.

Everyone possesses a unique blend of strengths and weaknesses, shaping their character and defining their individuality. My journey of self-awareness and growth has led me to recognize one of my significant weaknesses – procrastination.

Understanding Procrastination

Procrastination, the act of delaying or postponing tasks, is a common human behavior. It is often misconstrued as laziness, but it is more complex. It’s a struggle with self-control, where immediate gratification takes precedence over long-term goals. As a college student, this weakness has often led me to squander valuable time, affecting my academic performance and causing unnecessary stress.

Impact of Procrastination

The impact of procrastination extends beyond academics. It has hindered my personal growth and the development of essential life skills. The habit of putting off tasks has, at times, led to missed opportunities and prevented me from reaching my full potential. It has also affected my self-esteem, creating a vicious cycle of delay, guilt, and stress.

Overcoming Procrastination

Recognizing procrastination as a weakness was the first step towards overcoming it. I have started implementing strategies like time management, setting realistic goals, and breaking tasks into manageable parts. I also practice mindfulness to stay focused and avoid distractions.

In conclusion, while procrastination remains a significant weakness, acknowledging it has opened avenues for self-improvement. It has taught me that weaknesses are not permanent obstacles but challenges that can be overcome with determination and the right approach.

500 Words Essay on My Weakness

Every individual possesses a unique mix of strengths and weaknesses. They shape our character, influence our actions, and guide our decisions. Acknowledging and understanding our weaknesses is not a sign of defeat but a step towards self-improvement. In this essay, I will share my personal journey of recognizing and addressing my primary weakness: perfectionism.

Understanding Perfectionism

Perfectionism is often misinterpreted as a strength. Striving for excellence and setting high standards is commendable, but when these standards are unattainable and one’s self-worth becomes dependent on achieving them, it becomes a debilitating weakness. As a perfectionist, I have often found myself in this trap, paralyzed by the fear of failure and criticism.

The Impact of Perfectionism

The impact of perfectionism is multifaceted. Firstly, it hampers productivity. The constant desire to perfect every detail often leads to procrastination and delays. Secondly, it fosters a fear of failure. The dread of making mistakes inhibits risk-taking, a crucial aspect of learning and growth. Lastly, it can lead to mental health issues like anxiety and depression, as the constant pressure to meet unrealistic standards can be overwhelming.

Recognizing the Issue

The first step in overcoming any weakness is acknowledging its existence. I realized my perfectionism was a problem when I noticed its detrimental effects on my mental health and productivity. I was constantly stressed, my work was always late, and I was never satisfied with my achievements. This realization was a wake-up call that prompted me to seek change.

Addressing Perfectionism

Addressing perfectionism requires a shift in mindset. I had to learn to distinguish between healthy striving for excellence and unhealthy perfectionism. This involved setting realistic goals and understanding that failure and mistakes are part of the learning process. I also had to learn to separate my self-worth from my achievements.

Seeking Help

Seeking professional help was a significant step in overcoming my perfectionism. Therapy provided a safe space to explore the root causes of my perfectionism and equipped me with coping strategies. It also helped me understand that my value as a person is not dependent on being perfect.

Perfectionism, my primary weakness, has been a challenging journey of self-discovery and growth. Recognizing and addressing it has not only improved my productivity and mental health but also enriched my understanding of myself. It has taught me that it’s okay not to be perfect and that our weaknesses, once recognized and addressed, can become stepping stones to personal growth and self-improvement.

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Home — Essay Samples — Life — Personal Strengths — Exploring Personal Strengths and Weaknesses for Self-Improvement

Exploring Personal Strengths and Weaknesses for Self-improvement

• Categories: Personal Strengths Struggle Weakness

About this sample

Words: 865 |

Published: May 14, 2021

Words: 865 | Pages: 2 | 5 min read

Table of contents

My strengths, my weaknesses, works cited, fostering organization as a strength, the strength of observation, confronting the weakness of confidence, resisting change: a challenge to overcome, striving for academic excellence despite intelligence challenges.

• Buckingham, M., & Clifton, D. O. (2001). Now, discover your strengths. Simon and Schuster.
• Goleman, D. (1995). Emotional intelligence: Why it can matter more than IQ. Bantam Books.
• Grant, A. M., & Dweck, C. S. (2003). Clarifying achievement goals and their impact. Journal of Personality and Social Psychology, 85(3), 541-553.
• Linley, P. A., Willars, J., & Biswas-Diener, R. (2010). The strengths book: Be confident, be successful, and enjoy better relationships by recognizing where you're strong. CAPP Press.
• Lopez, S. J., & Snyder, C. R. (Eds.). (2009). Oxford handbook of positive psychology. Oxford University Press.
• Marsh, H. W., & Yeung, A. S. (1997). Causal effects of academic self-concept on academic achievement : Structural equation models of longitudinal data. Journal of Educational Psychology, 89(1), 41-54.
• Neff, K. D. (2011). Self-compassion: Stop beating yourself up and leave insecurity behind. HarperCollins.
• Peterson, C., & Seligman, M. E. P. (2004). Character strengths and virtues: A handbook and classification. Oxford University Press.
• Rothwell, W. J. (2015). In mixed company: Communicating in small groups and teams (9th ed.). Cengage Learning.
• Stajkovic, A. D., & Luthans, F. (1998). Self-efficacy and work-related performance: A meta-analysis. Psychological Bulletin, 124(2), 240-261.

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