Abstract
In 2017 and 2018, the Research Laboratory in Educational Opportunities (LaPOpE) at the Federal University of Rio de Janeiro conducted a study with public school children attending the beginning of compulsory schooling – kindergarten (children of age 4). Only 59.1% of the 4-year-old children were able to count four fixed objects. Therefore, a new counting test was developed and applied to some children. In this test, several aspects were considered: (i) quantity (perceptual, elementary, and complex numbers), (ii) arrangement of the objects (canonical form, randomly arranged, and in a circle), and (iii) categories of counting (loose, fixed objects, and object selection). The results showed that the way in which the objects were arranged influences the way children count as well as the quantities offered to them. This test can be used by teachers as a tool to assess the students’ counting skills and, then, to propose relevant interventions in the classroom.
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Notes
- 1.
Educational system in Brazil comprises four stages: early childhood (for children of ages 0–5, nursery for children of ages 0–3 s, and kindergarten for children of ages 4 and 5), elementary school (for children/teenagers of ages 6–14), high school (for teenagers of ages 15–17), and higher education. Until recently, attendance was mandatory from age 6 (first year of elementary education). From 2009, schooling in Brazil became mandatory from age 4 (first year of kindergarten), and, according to the National Educational Plan goals, local educational systems should have guaranteed universal attendance for children of ages 4 and 5 until 2016.
- 2.
An associated private school (in Portuguese: escola conveniada) is a private or philanthropic institution that becomes a “partner” of the municipal public-school system in order to fill any vacancies that were not attended by municipal government.
- 3.
The PIPS are tools with a high level of accuracy that measure cognitive development of children at the beginning of compulsory schooling. They were developed by researchers from Durham University and tested for over 20 years in the UK and other countries. The aim of PIPS is to create a baseline measure in this initial period of schooling. For more information on PIPS adaptation to the Brazilian context, see Bartholo et al. (2020a).
- 4.
With this statistical model, the authors were able to examine the level and growth of numerical competence (from kindergarten to the first year of elementary school) in relation to the performance and learning of mathematics from the first to the third grade.
- 5.
We have considered conceptual counting as an ability that goes beyond the simple enumeration and sequencing of number words; that is, the child understands the counting principles, the concept of counting itself. On the other hand, the procedural aspect of counting is rather related to a memory process; this does not necessarily reflect the child’s mastery of the ability for counting, which is essential for early childhood education and for the development of other mathematical skills throughout their schooling trajectory.
- 6.
The paper mentions neither the place where the study was conducted (the country) nor the age group of the evaluated children.
- 7.
In the complete sample, there were 1375 children with valid data on math performance scores in kindergarten mathematics, but due to sample attrition, at the end of the fifth grade, only 785 students remained in the study. The analysis used a subsample of 781 children who had data on math performance scores present in all measurements from kindergarten through the fifth grade.
- 8.
de Oliveira (2020, p. 31) brings an in-depth appreciation of how early mathematics tests such as “Early Childhood Assessment in Mathematics (ECAM), Child Math Assessment (CMA), Test of Early Mathematics Ability, Third Edition (TEMA-3), Measuring Early Learning Quality and Outcomes (MELQO), and Performance International Indicators in Primary Schools (PIPS)” that measure counting abilities.
- 9.
The study included three waves of data collection using the Brazilian adaptation of the PIPS instruments: at the start of the first grade of kindergarten/compulsory education (March/April 2017), at the end of the grade year of kindergarten (November/December of 2017), and at the end of the second grade of kindergarten (November/December of 2018). For more detailed information, see Bartholo et al. (2020b).
- 10.
As stated by this author, 0.5 is the minimum reliability value. Starting from 0.8, the value indicates enough reliability for decision-making in terms of the validity and internal consistency of the test; the closer the value is to 1.0, the better.
- 11.
In general, the index for people is lower than the index for the test and follows the same parameters in terms of minimum acceptable and adequate values for the analyzed instrument.
- 12.
According to information from the developer of the software Winsteps (available at https://www.winsteps.com/winman/table26_1.htm), for each of these statistics, the MNSQ is available. MNSQ is a mean-square statistic computed for all scores responses, excluding responses in extreme total scores. This is a chi-square statistic divided by its degrees of freedom. Its expectation is 1.0. Values substantially less than 1.0 indicate overfitting. Values substantially greater than 1.0 indicate underfitting. Values greater than 2.0 mean that the off-variable noise is greater than the useful information; such values undermine the measurement and thus should be monitored. Values greater than 1.5 imply a noticeable off-variable noise and neither promote nor degrade the measurement. Values between 0.5 and 1.5 signal a good performance of the item for the measurement. While values below 0.5 suggest overly predictable measurements, there is no differentiation between them, which could lead to a “ground effect.” That is, every individual (or almost every individual) can correctly answer the item. In other words, the item does not establish any kind of differentiation.
- 13.
This would be the case, for example, if the child, after counting eight bottle caps (1, 2, 3, 4, 5, 6, 7, 8) and being asked to place eight caps in the examiner’s hand, placed only the eighth cap counted in their hand.
- 14.
The results observed individually demonstrated that of the 13 children who got it right, 11 were in kindergarten, and the other 2 were in nursery.
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Acknowledgments
This study was financially supported by Rio de Janeiro State Research Support Foundation (Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro/Faperj) and Coordenação de Aperfeiçoamento de Pessoal de Ensino Superior (CAPES) of the Brazilian Ministry of Education.
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de Oliveira, A.L.R., Yokoyama, L.A., Koslinski, M.C. (2021). 1, 2, 3... Let’s Count: The Development of Counting at the Beginning of Compulsory Schooling. In: Spinillo, A.G., Lautert, S.L., Borba, R.E.d.S.R. (eds) Mathematical Reasoning of Children and Adults. Springer, Cham. https://doi.org/10.1007/978-3-030-69657-3_6
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