Rudimentary stages of the mathematical thinking and proficiency

Abstract

A national-level dataset (n = 7770) at grade 1 of primary school is re-analyzed to study preconditions in proficiency in mathematical concepts, operations and mathematical abstractions and thinking. The focus is on those pupils whose preconditions are so low that they are below the first measurable level of proficiency in the common framework with reference to mathematics (CFM). At the beginning of school, these pupils may not be familiar with, e.g., the concepts of numbers 1–10, they may not be aware of the consecutive nature of numbers, and they have no or very limited understanding of the basic concepts of length, mass, volume, and time. A somewhat surprising finding is that the key factor explaining the absolute low proficiency in mathematics appeared to be a low proficiency in listening comprehension. This variable alone explains 41% of the probability of belonging to the group of pupils who are not able to show proficiency enough to reach the lowest level in any of the criteria. It is understandable that, if language skills are underdeveloped in general, a child is not expected to master the specific mathematical vocabulary either and, hence, the low score in a test of preconceptions in mathematics too. Other variables predicting the absolute low level or preconditions of mathematics are the decision on intensified or special support, status of Finnish or Swedish as second language, and negative attitudes toward mathematics.

Attenuation-Corrected Estimators of Reliability

“…Two hypothetical numerical examples illustrate how attenuation correction (10) affects reliability when applied in different estimators: Case 1 represents tests with extreme difficulty level causing radical attenuation in reliability; this case is comparable with the real-life case by Metsämuuronen and Ukkola (2019) discussed above. Case 2 represents tests with incremental difficulty levels in items comparable with the traditions in the achievement testing.…”

“…Because the estimates of correlation by PMC between items and the test score are always attenuated and deflated, this causes attenuation in the estimates of reliability when using the traditional estimators such as coefficient alpha, theta, omega, and maximal reliability. Examples show that, in the extreme cases, the estimates by alpha may be radically deflated to the extent of 0.40–0.50 units of reliability and, in real life-settings, even more than 0.60 units of reliability (see Gadermann et al, 2012 ; Metsämuuronen & Ukkola, 2019 ). As a specific solution for the attenuation in the reliability, a new kind of attenuation correction is proposed to replace PMC in the formulas: the attenuation-corrected PMC ( R AC ) as the proportion of observed correlation of the maximal possible correlation with the given variables.…”

“…Metsämuuronen (2021a, 2022a, 2022b), specifically, discuss using the formula (1) as a basis of estimating MEC-corrected alpha where the error-causing ρgX is replaced by a totally different estimator that would be less affected by MEC. Some options of this kind of coefficients have been suggested in earlier works: R PC (see Gadermann et al, 2012; Metsämuuronen, 2022a; Zumbo et al, 2007), G and dimension-corrected G ( G 2 ; Metsämuuronen, 2021a, 2022a) as well as D and dimension-corrected D ( D 2 ; Metsämuuronen, 2020b, 2021a, 2022a; Metsämuuronen & Ukkola, 2019). Of these, in simulations, R PC , G, and G 2 appear to be MEC-free in many conditions of MEC affecting obviously in PMC (Metsämuuronen, 2021a, 2022b).…”

“…The first example comes from Metsämuuronen and Ukkola (2019) . They used a very easy sub-test of preconditions in mother language at the beginning of the first year in school as a part of a National level assessment of learning outcomes ( n = 7770); 72% of the test takers got the highest possible score in 8-item, 11-point screening test for basic skills of understanding auditive instructions.…”

“…The obvious reason for the remarkably higher estimate by αD and αORD than by the traditional α and ρMAX is that PMC and factor loadings are severely affected by MEC in items with extreme difficulty level while D and R PC are less affected by MEC in the binary case (see Metsämuuronen, 2021a, 2022b). In the case of Metsämuuronen and Ukkola (2019), the magnitude of estimate would have been even higher if R PC or Goodman–Kruskal gamma ( G ; Goodman & Kruskal, 1954) was used instead of D in the formula because the estimates by D are usually more conservative than those by G (see a re-analysis of the dataset in Metsämuuronen, 2022a). Some exceptions of the conservativeness of D are discussed by, for example, Metsämuuronen (2021b), and, in the case of binary items, the difference between the estimates by R PC and G is nominal (Metsämuuronen, 2021a).…”