Conference PaperPDF Available

Identification of first-grade students at risk of developing mathematical difficulties through online measures in arithmetic and pattern tasks: A study using error rates and response times

Authors:

Abstract and Figures

For researchers and practitioners, it is important to identify students at risk of developing mathematical difficulties. The aim of this pilot study was to investigate whether it is possible to identify first-grade students who are at risk of developing mathematical difficulties (RMD) through online measures in arithmetic and pattern tasks. In our study, 54 first-grade students worked on 75 tasks in twelve sets on a computer screen. We also carried out a standardized mathematics test to identify students as RMD. We then investigated if error rates and response times as online measures allow to replicate the identification of students as RMD. Using a logistic regression model, we found that the error rates and response times allow identifying students as RMD with acceptable accuracy. We also found that tasks on symbolic number comparison, completing color patterns, and enumeration of small sets were particularly informative to identify students as RMD.
Content may be subject to copyright.
Identification of first-grade students at risk of developing
mathematical difficulties through online measures in arithmetic and
pattern tasks: A study using error rates and response times
Lukas Baumanns1, Demetra Pitta-Pantazi2, Eleni Demosthenous2, Constantinos Christou2,
Achim J. Lilienthal3 and Maike Schindler1
1University of Cologne, Germany; lukas.baumanns@uni-koeln.de, maike.schindler@uni-koeln.de
2University of Cyprus, Cyprus; dpitta@ucy.ac.cy, demosthenous.eleni@ucy.ac.cy,
edchrist@ucy.ac.cy
3Örebro University, Sweden; achim.lilienthal@oru.se
For researchers and practitioners, it is important to identify students at risk of developing
mathematical difficulties. The aim of this pilot study was to investigate whether it is possible to
identify first-grade students who are at risk of developing mathematical difficulties (RMD) through
online measures in arithmetic and pattern tasks. In our study, 54 first-grade students worked on 75
tasks in twelve sets on a computer screen. We also carried out a standardized mathematics test to
identify students as RMD. We then investigated if error rates and response times as online measures
allow to replicate the identification of students as RMD. Using a logistic regression model, we found
that the error rates and response times allow identifying students as RMD with acceptable accuracy.
We also found that tasks on symbolic number comparison, completing color patterns, and
enumeration of small sets were particularly informative to identify students as RMD.
Keywords: mathematical difficulties, early identification, digital tools.
Introduction
Mathematical difficulties often begin early, before primary education and can—due to insufficient
support—cascade into severe mathematical problems (Geary, 2013; Moser Opitz, 2013; Sasanguie
et al., 2012). Longitudinal studies confirm that students who enter school with mathematical
difficulties do generally not overcome these during primary school (e.g., Viesel-Nordmeyer et al.,
2019). It is significant for teachers to be aware of such difficulties, be able to identify them, and to
provide adequate support for students. Yet, identifying student difficulties at an early age is
challenging—among other reasons, because young students often lack the ability to report their
difficulties (Klein et al., 2010). There are numerous tests for identifying students at risk of developing
mathematical difficulties at an early age (Hellstrand et al., 2020). These tests, some of which are
conducted individually, require a considerable amount of time for both conducting the tests and
evaluating results. Also, they are often not feasible to use by practitioners at school. A digital
screening offers the possibility to identify, also in practice, a large number of children who are at risk
to develop mathematical difficulties. Digital tools are particularly suitable as both the collection of
data and its evaluation can be conducted on the same digital device. The Erasmus+ project DIDUNAS
builds on this idea. It aims to develop an app, which incorporates tasks in the domain of early
arithmetic and pattern tasks, that identifies first-grade students in need of support. For this
identification, the app uses data such as error rates and response times.
This pilot study aims to investigate whether it is possible to identify first graders who are at risk of
developing mathematical difficulties through online measures (response times and error rates) of
students’ work on tasks in the domain of early arithmetic and pattern tasks.
Related work
Early identification of students with mathematical difficulties
A variety of reliable and valid standardized tests are commonly used in identifying mathematics
difficulties. To develop these tests, researchers focus on identifying variables that appear to be good
predictors. Some tests assess speed and accuracy with which students can identify object sets (e.g.,
Geary et al., 2009). Early numeracy skills, such as quantity comparison, number identification, and
counting, appear to have much predictive power for mathematical difficulties (Gersten et al., 2005).
Most of these tests include tasks on: Object counting, number comparison, sequencing, connecting
numbers to quantities, number recognition, and counting back and forth. Fewer tests include number
calculations such as additions and subtractions with symbols or word problems. Even fewer tests
include patterning tasks such as copying or extending a pattern or identifying the pattern unit.
However, a growing number of researchers (e.g., Verschaffel, et al., 2017) argue that that for
investigating mathematical abilities of young students, patterning tasks also need to be considered.
Students’ number sense and awareness of structure and patterns contribute substantially to students’
later success in mathematics (Wijns et al., 2019). Pittalis et al. (2018) in a longitudinal study with
first-grade students suggested that that the growth rate of algebraic arithmetic has a direct effect on
the growth rate of conventional arithmetic, and subsequently the growth rate of conventional
arithmetic predicts the growth rate of elementary arithmetic.
Error rates and response times as measures to identify mathematics difficulties
Several studies investigated differences in error rates and response times of students with and without
mathematical difficulties. Some of these studies addressed students’ enumeration ability and
indicated that students with mathematical difficulties have longer response times within the subitizing
range as compared to students without such difficulties (Moeller et al., 2009; Schindler et al., 2020;
Schleifer & Landerl, 2011). Other studies addressed students’ comparison of symbolic and non-
symbolic numbers and found no significant differences in error rates and response times between
children with or without mathematical difficulties (Mussolin et al., 2010). However, the slope of
response times was significantly steeper for students with mathematical difficulties. Other studies
found that response times and error rates for non-symbolic number line estimation are significant
predictors of mathematical achievements (Sasanguie et al., 2012). These studies indicate that error
rates and response times for suitable tasks can be a predictor of mathematical difficulties. Thus, we
intend to investigate: To what extent can error rates and response times of early arithmetic and
pattern tasks be used to identify students that may be at risk of developing mathematical difficulties?
Methods
Participants
The study was conducted with 54 first-grade students (age: M = 7.38; SD = 0.55) from two primary
schools in Germany. In the German federal state that the study took place in, a social index classifies
schools into levels from 1 to 9 which is based on factors such as child and youth poverty, family
language, and special educational needs of students. Index 1 represents the most favorable conditions.
The two participating schools had an index of 7 and 6 which means they tended to have a higher
number of students in need of support. Of the students, 34.6% had German as their mother tongue.
Procedure and tasks
For the study we used two tests: (1) the standardized ZAREKI-K test to identify students at risk of
mathematical difficulties and (2) a self-developed computer screen test with twelve sets of arithmetic
and pattern tasks (75 tasks in total).
Standardized mathematics test: ZAREKI-K is a standardized test for identifying children at the
transition from kindergarten to primary school of being at risk of developing mathematical difficulties
(von Aster et al., 2009). The test battery is constructed as an individual procedure and consists of 18
subtests. For the present study, an adaptation of ZAREKI-K was used, which requires only six
subtests: (a) Counting up to 30, (b) Numbers that precede or follow, (c) Word problems, (d) Visual
calculation, (e) Number conservation, and (f) Writing numbers. This adaptation has been shown to
yield excellent prediction rates for identifying students at risk of developing mathematical difficulties
(Walter, 2020). The students took an average of 14.6 minutes to complete the ZAREKI-K.
Early arithmetic and pattern tasks: The students worked on twelve sets presented on a computer
screen (Fig. 1). Every set had an example task to get the students acquainted with it. For sets (1), (4),
and (5), students were to determine the number of objects. For set (2), students were asked to
determine a number on a number line. Set (3) asked for the number behind the sun. Set (6) asked how
many dots needed to be added or subtracted to make it equal to the number shown on the right. In set
(7), the largest number was to be determined. In set (8), the number of persons’ legs behind the wall
was to be determined. In set (9), students were to determine the number of bricks of the tower behind
the white blob. In set (10), the result of an addition/subtraction problem was to be determined. In set
(11), the students were to compare quantities. For set (12), a color pattern was to be completed.
Students could skip a task if they found it difficult by saying “next”. There are no identical tasks
between these sets and ZAREKI-K. However, both include cardinal and ordinal aspects of numbers.
Additionally, set (12) is a pattern task from early algebra, which is not included in ZAREKI-K.
Students answered by tapping on the computer screen. The answers to each task were given with a
single tap on the screen. For sets (1)–(10), a number field with the buttons labeled with 1 to 20 was
shown at the bottom of the screen. Set (11) had a yellow, a blue, and an equal (“=”) button for
answering. Set (12) had a yellow, a blue, and a red button for answering. It took students an average
of 20.8 minutes to complete these tasks (including all instructions, explanations, and trial tasks).
Figure 1: Example tasks of the twelve sets
(1) Enumeration (2) Number line (3) Sun
(4) 10-field (5) Objects
(6) Difference
(7) Biggest number (9) Towers (10) Calculations(8) Hidden legs
(11) Quantity comparision
(12) Dot patterns
Measures
We use the following data sets:
(1) Identification of mathematical difficulties at risk: ZAREKI-K identifies whether a student
is at risk (RMD) or not at risk (¬RMD) of developing mathematical difficulties. We used an
Excel spreadsheet provided by Walter (2020) for entering the students’ individual points
achieved in each subtest, which then calculated the risk of students to develop mathematical
difficulties. Of the 54 students, ZAREKI-K identified 18 as RMD and 36 as ¬RMD.
(2.a) Error rates: Mean error rates were calculated for all 75 tasks in total as well as each of the
twelve sets separately. We considered tasks, where the students answered wrongly or did not
answer at all, as being not solved correctly. We considered all tasks, which were not being
solved correctly, as error.
(2.b) Response times: For each task, the time from when the stimulus was first shown to when the
student typed the response on the computer screen was measured. Only response times of
correctly answered tasks were taken into account, since tasks that were not understood by the
students sometimes were quickly skipped and since students partially rashly guessed wrong
answers. Mean response times were calculated over all tasks that were answered correctly as
well as for each of the twelve sets separately.
Statistical analysis
We followed the guidelines of logistic regression analysis and reporting by Peng et al. (2002).
Logistic regression was performed using SPSS 27 in order to calculate a probability value between 0
and 1 for each student using mean error rate and mean response time over all tasks. For different cut-
off values p between 0 and 1, students are identified as RMD or ¬RMD. A ROC (receiver operating
characteristic) curve was then plotted, which indicates the sensitivity (true positive rate) and
specificity (true negative rate) for all cut-off values p as an indicator of the overall classification
accuracy. The area under this curve (AUC) is a measure of the classification quality.
Next, we ask which of the twelve mean error rates or the twelve mean response times are most
informative for identifying students as RMD, according to ZAREKI-K. We thus carried out a
backwards selection, subsequently for both mean error rate and mean response time for all twelve
task sets in early arithmetic and patterns using our multiple-logistic regression model.
Results
We conducted a t-test to compare mean differences of error rates and response times between students
identified as RMD and students identified as ¬RMD. Using the Shapiro-Wilk test, the normal
distribution of mean error rates (W(54) = .974, p > .05) and mean response times (W(54) = .967,
p > .05) was checked. Using the Levene test, the homogeneity of variances of mean error rates
(p > .05) and mean response times (p > .05) were checked. Thus, variance homogeneity exists
between the groups. With regard to the mean error rate, the 18 students identified as RMD had a
significantly higher mean error rate (M = .29, SD = .12) as compared to the 36 students identified as
¬RMD (M = .19, SD = .09; t(52) = –3.48, p < .05). With an effect size of r = .43, this is a medium
effect. With regard to the mean response time, the 18 students identified as RMD did not have a
significantly higher mean response time (M = 7.04s, SD = 1.63s) as compared to the 36 students
identified as ¬RMD (M = 6.65s, SD = 1.29s; t(52) = –.957, p = .34, r = .13).
The Likelihood ratio test indicates that the logistic regression model is significantly more effective
than the null model (constant only) (χ²(2) = 11.67, p < .05). Goodness-of-fit was assessed using the
Hosmer-Lemeshow test, indicating a fit of the logistic model (χ²(8) = 5.15, p > .05). Wald test
indicates that mean error rate of the 75 tasks is a significant classifier of RMD (χ²(1) = 8.53, p < .05).
The mean response time is not a significant classifier in this regard (χ²(1) = .987, p > .05).
The logistic model calculates a probability value between 0 and 1 for each student based on the error
rates and response times. The cut-off value p then defines at which probability value a student is
identified to have RMD or ¬RMD based on the logistic regression model. Choosing cut-off values of
p thus means to trade off sensitivity (true positive rate) and specificity (true negative rate) as they
change diametrically. Table 1 shows the sensitivity and specificity for different cut-off values p. The
total accuracy is computed as the number of all correctly identified results in relation to all results.
Table 1: Sensitivity, specificity, and total accuracy of the model for different cut-off values p
Cut-off value p
.05
.1
.15
.2
.25
.3
.307
.35
.4
.5
.6
Sensitivity (%)
100.0
100.0
88.9
83.3
72.2
72.2
72.2
61.1
61.1
55.6
33.3
Specificity (%)
5.6
13.9
25.0
44.4
61.1
69.4
72.2
77.8
80.6
91.7
94.4
Total accuracy (%)
37.0
42.6
46.3
57.4
64.8
70.4
72.2
72.2
74.1
79.6
74.1
For the identification of students at risk of developing mathematical difficulties at an early age, a high
sensitivity is often desirable even at the expense of a decreased specificity. A high sensitivity would
ensure that only a few students with mathematical difficulties at risk are missed. However, this has
the consequence that the specificity decreases and students with mathematical difficulties at risk are
not detected. For the cut-off value p = .307, a reasonably high sensitivity of 72.2% is achieved at a
still high specificity of 72.2%. Table 2 displays the classification of the students identified to be RMD
and to be ¬RMD through the participants’ error rates and response times compared to the students
identified as RMD and ¬RMD from the standardized ZAREKI-K for cut-off value p = .307.
Table 2: Classification tablea
Identification
Percentage correct
¬RMD
RMD
ZAREKI-K
¬RMD
26
10
72.2%
RMD
5
13
72.2%
Overall percentage
72.2%
acut-off value p = .307
The ROC curve (see Figure 2) is the generalization of a single classification table (see Table 2). Each
point of the ROC curve indicates sensitivity and (1–specificity) for a given cut-off value p. The drawn
diagonal would be expected if the classification was purely random. A measure of the classification
quality of the model is the area under the ROC curve (AUC). Following Hosmer et al. (2013), the
classification accuracy can be considered “acceptable” with an AUC = .761.
Figure 2: ROC curve of the general model (left; AUC = .761; ellipse marks cut-off value p = .307)
and the reduced model (right; AUC = .841; ellipse marks cut-off value p = .351)
To identify those sets whose error rates and/or response times are particularly good for the
identification of students at the risk of developing mathematical difficulties, logistic regression was
performed through backwards selection. This backwards selection is first done with all twelve mean
error rates of the sets. Step by step, the twelve mean error rates are removed from the model, starting
with the one that has the lowest significance for predicting the ZAREKI-K outcome. All variables
that are significant to replicate the classification based on the ZAREKI-K outcome at the p < .1 level
remain included according to the Wald test. At the same time, the Likelihood ratio statistic is used to
check whether the model would improve by adding another variable. After eleven steps, the mean
error rates of sets 7 (symbolic number comparison) and 12 (completing color patterns) (see Figure 1)
remained. For the response times, the mean values of set 9 (completing growing number patterns)
could not be included, since the number of incorrect answers were too high for which response times
were not considered. After eleven steps, only the response time of set 1 (enumeration of small sets)
remained. Applying logistics regression onto these three variables identified through backwards
selection, Likelihood ratio test indicates that the logistic regression model is significantly more
effective than the null model (constant only) (χ²(3) = 22.99, p < .05). Goodness-of-fit was assessed
using the Hosmer-Lemeshow test, indicating a good model fit; χ²(8) = 3.58, p > .89. Furthermore,
Wald test indicates that the mean error rate of set (12) (χ²(1) = 6.88, p < .05) and the mean response
time of set (1) (χ²(1) = 6.91, p < .05) are significant classifiers of developing mathematical
difficulties. The mean error rate of set (7) is not a significant classifier in this regard
(χ²(1) = 3.256, p > .05), but since p < .1 it remained in the model. Following Hosmer et al. (2013),
the classification accuracy can be considered “excellent” with AUC = .841. With a cut-off value of p
= .351, this model has a higher specificity of 80.6% compared to the previous model at a sensitivity
of 72.2%. The total accuracy of this model is 77.8%.
Discussion
The results of our study should be viewed and interpreted against the backdrop of the following
limitations: Logistic regression requires sufficiently many training samples, i.e., RMD and ¬RMD
cases. Having only 18 students identified as RMD and 36 students identified as ¬RMD limits the
certainty of the learned logistics regression model. A larger sample could provide further certainty.
In addition, we optimized the classification threshold for the logistic regression model on a single
data set and did not evaluate the classification accuracy on an independent test set. In practice, the
classification threshold needs to be learned on a training set, which would likely decrease the
classification accuracy on independent data.
This pilot study addressed the question to what extent error rates and response times of correctly
solved tasks as online measures in early arithmetic and pattern tasks can identify students that may
be at risk of developing mathematical difficulties (RMD). Using logistics regression, we found that
the mean error rate across all 75 tasks is a strong classifier of RMD, whereas the mean response time
was a weaker classifier. Combining error rates and response times in our study yielded an acceptable
discrimination of the model of AUC = .761. Furthermore, we investigated to what extent the error
rates and response times of twelve sets can be used separately to identify students’ RMD. We found
that the error rates of two sets (symbolic number comparison, completing color patterns) and the
response time of one set (enumeration of small sets) appeared to be particularly informative. The
influence of the set about complementing color patterns is especially noteworthy since the
standardized mathematics test focused on early arithmetic, not patterns. Combining error rates of
symbolic number comparison and completing color patterns with response times of small set
enumeration yielded a discrimination of the model of AUC = .841.
Since the DIDUNAS project aims to develop an app that identifies first-grade students in need of
support, what can we learn from these results regarding the app development? In our pilot study, we
found that it is possible to use online measurements of certain tasks or a set of them (here: symbolic
number comparison, completing color patterns, enumeration of small sets) to achieve a reasonably
reliable identification of students at risk of developing mathematical difficulties in grade 1. This is
promising for future developments. In the future, we will build on these results for developing the
DIDUNAS app, which then can be used by teachers around the world to help identify students with
risks of mathematical difficulties early on. As the app will require less effort in the conduction and
evaluation, compared with some standardized mathematics tests, the app will enable a resource-
saving use for teachers. The study presented in this paper is an important first step in that direction.
Acknowledgement
This project has received funding by the Erasmus+ grant program of the European Union under grant
agreement No 2020-1-DE03-KA201-077597.
References
Geary, D. C. (2013). Early foundations for mathematics learning and their relations to learning
disabilities. Current Directions in Psychological Science, 22, 23–27.
https://doi.org/10.1177/0963721412469398.
Geary, D. C., Bailey, D. H., & Hoard, M. K. (2009). Predicting mathematical achievement and
mathematical learning disability with a simple screening tool. Journal of Psychoeducational
Assessment, 27(3), 265–279. https://doi.org/10.1177/0734282908330592.
Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students
with mathematics difficulties. Journal of Learning Disabilities, 38(4), 293–304.
https://doi.org/10.1177/00222194050380040301.
Hosmer, D. W., Lemeshow, S., & Sturdivan, R. (2013). Applied Logistic Regression (3rd ed.). Wiley.
Hellstrand, H., Korhonen, J., Räsänen, P., Linnanmäki, K., & Aunio, P. (2020). Reliability and
validity evidence of the early numeracy test for identifying children at risk for mathematical
learning difficulties. International Journal of Educational Research, 102, 101580.
https://doi.org/10.1016/j.ijer.2020.101580.
Klein, P., Adi-Japha, E., & Hakak-Benizri, S. (2010). Mathematical thinking of kindergarten boys
and girls. Educational Studies in Mathematics, 73, 233–246. https://doi.org/10.1007/s10649-009-
9216-y.
Moser Opitz, E. (2013). Rechenschwäche/Dyskalkulie. Theoretische Klärungen und empirische
Studien an betroffenen Schülerinnen und Schülern (2nd ed.). Haupt.
Moeller, K., Neuburger, S., Kaufmann, L., Landerl, K., & Nuerk, H.-C. (2009). Basic number
processing deficits in developmental dyscalculia: Evidence from eye tracking. Cognitive
Development, 24(4), 371–386. https://doi.org/10.1016/j.cogdev.2009.09.007.
Mussolin, C., Mejias, S., & Noël, M.-P. (2010). Symbolic and nonsymbolic number comparison in
children with and without dyscalculia. Cognition, 115, 10–25.
https://doi.org/10.1016/j.cognition.2009.10.006.
Peng, C.-Y. J., Lee, K., & Ingersoll, G. M. (2002). Introduction to logistic regression analysis and
reporting. The Journal of Educational Research, 96(1), 3–14.
https://doi.org/10.1080/00220670209598786.
Pittalis, M., Pitta-Pantazi, D., & Christou, C. (2018). A longitudinal study revisiting the notion of
early number sense: algrebraic arithmetic a catalyst for number sense development. Mathematical
Thinking and Learning, 20(3), 222–247. https://doi.org/10.1080/10986065.2018.1474533.
Sasanguie, D., Van den Bussche, E., & Reynvoet, B. (2012). Predictors for mathematics
achievement? Evidence from a longitudinal study. Mind, Brain, and Education, 6(3), 119–128.
https://doi.org/10.1111/j.1751-228X.2012.01147.x.
Schindler, M., Schovenberg, V., & Schabmann, A. (2020). Enumeration processes of children with
mathematical difficulties: An explorative eye-tracking study on subitizing, groupitizing, counting,
and pattern recognition. Learning Disabilities: A Contemporary Journal, 18(2), 192–211.
Schleifer, P., & Landerl, K. (2011). Subitizing and counting in typical and atypical development.
Developmental Science, 14(2), 280–291. https://doi.org/10.1111/j.1467-7687.2010.00976.x.
Verschaffel, L., Torbeyns, J., & De Smedt, B. (2017). Young childen’s early mathematical
competencies: Analysis and stimulation. In T. Dooley, & G. Gueudet (Eds.) Proceedings of the
Tenth Congress of the European Society for Research in Mathematics Education. ERME.
Viesel-Nordmeyer, N., Schurig, M., Bos, W., & Ritterfeld, U. (2019). Effects of pre-school
mathematical disparities on the development of mathematical and verbal skills in primary
school children. Learning Disabilities, 17, 149–164.
von Aster, M. G., Bzufka, M. W., & Horn, R. (2009). ZAREKI-K. Neuropsychologische Testbatterie
für Zahlenverarbeitung und Rechnen bei Kindern – Kindergartenversion. Pearson.
Walter, J. (2020). Ein Screening-Verfahren zur Prognose von Rechenschwierigkeiten in der
Grundschule. Zeitschrift für Heilpädagogik, 71, 238–253.
Wijns, N., Torbeyns, J., Bakker, M., De Smedt, B., & Verschaffel, L. (2019). Four-year olds’
understanding of repeating and growing patterns and its association with early numerical ability.
Early Childhood Research Quarterly, 49, 152–163. https://doi.org/10.1016/j.ecresq.2019.06.004.
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
This article investigates how students with mathematical difficulties (MD) differ from typically developing (TD) students in enumeration processes of small sets of objects (of 1 up to 9 dots). We present a study with 20 fifth-grade students of which ten were found to have MD in initial diagnostics. The students were supposed to exactly enumerate sets of dots, i.e., to say how many dots they saw. This took place in three conditions: (a) in random arrangements in the subitizing range (1–4 dots), (b) in random arrangements in the counting range (5–9 dots), and (c) in canonical (dice-like) arrangements (1–9 dots). We used eye tracking (ET) to analyze student enumeration processes derived from ET video data. Whereas we did not find significant group differences in students’ error rates, we found differences in response times with longer response times for students in MD in the canonical arrangement condition. Further, we found significant group differences in students’ enumeration processes in all three conditions (subitizing range, counting range, canonical): Students with MD tended to count all dots more often whereas TD students used more advantageous enumeration processes such as simultaneous enumeration or enumeration of groups of dots more often. Our results support the assumption of qualitatively different enumeration processes between students with and without MD.
Article
Full-text available
This study investigated reliability and validity evidence regarding the Early Numeracy test (EN-test) in a sample of 1139 Swedish-speaking children (587 girls) in kindergarten (n = 361), first grade (n = 321), and second grade (n = 457). Structural validity evidence was established through confirmatory factor analysis (CFA), which showed that a four-factor model fit the data significantly better than a one-factor or two-factor model. The known-group and cross-cultural validity were established through multigroup CFAs, finding that the four-factor model fit the gender, age and language groups equally well. Internal consistency for the test and sub-skills varied from good to excellent. The EN-test can be considered as an appropriate assessment to identify children at risk for mathematical learning difficulties.
Article
Full-text available
Achtung: Dieser Text entspricht nicht vollständig dem in der Zeitschrift veröffentlichten Artikel Der zitierfähige Originalartikel ist veröffentlicht in © Zeitschrift für Heilpädagogik, 71, 238-253 2 Zusammenfassung Im vorliegenden Beitrag wird vor dem Hintergrund der Vorarbeiten von Walter (2016a,b) ein zur Einschulung verwendbares Screening-Verfahren zur Prognose einer Rechenschwäche zum Ende der 1. Klasse weiterentwickelt. Auf Basis einer Stichprobe von N = 396 Erstklässlern wird gezeigt, dass es ausreicht, sechs anstelle der 18 in ZAREKI-K verwendeten Untertests für eine sehr effektive Prognose (AUC = .89) von Rechenschwierigkeiten heranzuziehen. Ein zusätzlicher prognostisch relevanter Effekt der nichtsprachlichen Intelligenz (CFT 1-R) wurde nicht gefunden. Am Ende dieses Artikels wird ein Auswertungstool vorgestellt, das mithilfe des hier beschriebenen Prognose-Modells individuelle Klassifikationen ermöglicht. Dieses steht zum praktischen Gebrauch im Netz frei zum Download zur Verfügung. Abstract On the basis of the preliminary work of Walter (2016a,b) a screening procedure regarding the prediction of arithmetic weakness in the primary school age (first class) is presented. By a sample of N = 396 children, a set of six out of 18 predictor subscales from ZAREKI-K was found which delivers a very effective prediction (AUC = .89) without any significant loss of predictive information in comparison with the full version. There was no further classificatory strength when introducing non-verbal intelligence (CFT 1-R) into the prediction model. Finally, a risk calculator is presented which allows individual classification.
Article
Full-text available
Children’s quantitative competencies upon entry into school can have lifelong consequences. Children who start behind generally stay behind, and mathematical skills at school completion influence employment prospects and wages in adulthood. I review the current debate over whether early quantitative learning is supported by (a) an inherent system for representing approximate magnitudes, (b) an attentional-control system that enables explicit processing of quantitative symbols, such as Arabic numerals, or (c) the logical problem-solving abilities that facilitate learning of the relations among numerals. Studies of children with mathematical learning disabilities and difficulties have suggested that each of these competencies may be involved, but to different degrees and at different points in the learning process. Clarifying how and when these competencies facilitate early quantitative learning and developing interventions to address their impact on children have the potential to yield substantial benefits for individuals and for society.
Article
Full-text available
The purpose of this article is to provide researchers, editors, and readers with a set of guidelines for what to expect in an article using logistic regression techniques. Tables, figures, and charts that should be included to comprehensively assess the results and assumptions to be verified are discussed. This article demonstrates the preferred pattern for the application of logistic methods with an illustration of logistic regression applied to a data set in testing a research hypothesis. Recommendations are also offered for appropriate reporting formats of logistic regression results and the minimum observation-to-predictor ratio. The authors evaluated the use and interpretation of logistic regression presented in 8 articles published in The Journal of Educational Research between 1990 and 2000. They found that all 8 studies met or exceeded recommended criteria.
Article
Based on longitudinal data (n = 338) from the German National Educational Panel Study (NEPS) competence development (mathematics and language) of three groups with different mathematical preconditions (+/-1 SD) were compared between age 4 to 10. Groups were composed by first measurement of mathematical achievement in pre-school age (5/6 years). First, further mathematical development of the groups was investigated up to grade 4 of primary school. Second, group differences in Grammar and vocabulary between pre-school and primary school children were explored. Results show a consistent development of mathematical competence in all three groups from pre-school age until 4th grade. Children with low mathematical achievement measured in pre-school age were not able to overcome these deficits during primary school. In contrast, vocabulary development of the three groups varies over time: Within the group with low mathematical achievement measured in pre-school age their similarly weak vocabulary skills caught up with the vocabulary achievements of the other two groups during the following school years. For grammar, the small group-related differences in pre-school become even more pronounced in grade 1. The results contribute to a better understanding of the complex dynamic interrelationship between the development of linguistical and mathematical skills in K-4 and are discussed with respect to the importance of early mathematical promotion.
Article
In this study, we aimed to address two gaps in research on early mathematical patterning, namely the lack of attention (1) to growing patterns and (2) to the association between different aspects of patterning and numerical ability. Participants were 400 four-year olds from a wide range of socioeconomic backgrounds. Children's patterning and numerical ability were assessed by means of individual tasks. The patterning tasks assessed their performance on three patterning activities (i.e., extending, translating, and identifying the pattern's structure) for two types of patterns (i.e., repeating and growing). The numerical measure included a set of eight well-known numerical tasks. We additionally controlled for individual differences in spatial ability and visuospatial working memory. Results indicated an effect of both activity and patterning type on children's patterning performance, as well as an interaction between both. Furthermore, children's performance on four out of six patterning tasks uniquely contributed to their numerical ability above age, spatial ability, and visuospatial working memory. These findings support the importance of specific pattern types and patterning activities in the early stage of children's mathematical development and give directions for further educational practices.
Article
The aim of this study was to propose a new conceptualization of early number sense. Six-year-old students’ (n = 204) number sense was tracked from the beginning of Grade 1 through the beginning of Grade 2. Data analysis suggested that elementary arithmetic, conventional arithmetic, and algebraic arithmetic contributed to the latent construct early number sense, and the invariance of the model over time was validated empirically. Algebraic arithmetic represents the dimension of early number sense that moves beyond conventional arithmetic and encompasses an abstract understanding of the relations between numbers. A parallel process growth model showed that the three components of number sense adopt a linear growth rate. A structural model showed that the growth rate of the algebraic arithmetic component has a direct effect on the growth rate of conventional arithmetic, and subsequently the growth rate of conventional arithmetic predicts the growth rate of elementary arithmetic.
Article
Numerical processing has been extensively studied by examining the performance on basic number processing tasks, such as number priming, number comparison, and number line estimation. These tasks assess the innate “number sense,” which is assumed to be the breeding ground for later mathematics development. Indeed, several studies have associated children's performance in these tasks with individual differences in mathematical achievement. To date, however, most of these studies have cross-sectional designs. Moreover, the few longitudinal studies either use complex tasks (e.g., story problems) or investigate only one of these basic number processing tasks at a time. In this study, we examine the association between the performance of children on several basic number processing tasks and their individual math achievement scores on a curriculum-based test measured 1 year later. Regression analyses showed that most of the variance in children's math achievement was predicted by nonsymbolic number line estimation performance (i.e., estimating large quantities of dots) and, to a lesser extent, the speed of comparing symbolic numbers. This knowledge about the predictive value of the performance of 5- to 7-year-olds on these markers of number processing can help with the early identification of at-risk children. In addition, this information can guide appropriate educational interventions.