For a non-empty set I, the sub-defect of an
$ I\times I $
I
×
I
doubly substochastic matrix
$ A=a_{ij}_{i,j \in I}, $
A
=
a
ij
i
,
j
∈
I
,
denoted by
$ \mathrm {sd}(A), $
sd
(
A
)
,
is the smallest ...cardinal number α for which there is a set J with
$ \mathrm {card}(J) =\alpha, $
card
(
J
)
=
α
,
$ I\cap J =\emptyset, $
I
∩
J
=
∅
,
and there exists a doubly stochastic matrix
$ D=d_{ij}_{i,j\in I\cup J} $
D
=
d
ij
i
,
j
∈
I
∪
J
which contains A as a sub-matrix. In this paper, we show the set of all finite sub-defect matrices is closed under multiplication. We also show that the inequality
$ \max \{\mathrm {sd}(A), \mathrm {sd}(B)\} \leq \mathrm {sd}(AB) \leq \min \{n, \mathrm {sd}(A) + \mathrm {sd}(B)\} $
max
{
sd
(
A
)
,
sd
(
B
)
}
≤
sd
(
AB
)
≤
min
{
n
,
sd
(
A
)
+
sd
(
B
)
}
which is obtained by Lei Cao et al. remains valid for all finite sub-defect matrices.
•Set-to-number and number-to-set tasks measure related but distinct aspects of cardinal number knowledge•The Give-N and How-Many? tasks cannot be used interchangeably•The differences between tasks ...are not due to general cognitive or language factors
Children’s understanding of the cardinal numbers before entering school provides the foundation for formal mathematics learning. Two types of tasks have been primarily used to measure children’s knowledge of cardinal numbers: set-to-number and number-to-set tasks. However, there has been a continued debate as to whether the two types of tasks measure the same conceptual construct, allowing comparison and interchangeable use, or whether they measure different but related constructs. To answer this question, we analyzed the relation between task and item level performance on representative set-to-number (e.g., How-Many?) and number-to-set (Give-N) tasks in a large group of 3- to 4-year-old preschoolers (N = 204, median age = 3y 10 m). By constructing and comparing models with different latent variable structures, we found that the best-fitting model was a bi-factor model, where performance on set-to-number and number-to-set tasks is best explained by both overlapping and some distinct aspects of cardinal number knowledge. Further analyses ruled out the idea that differences between tasks were due solely to non-numerical, general cognitive or language factors. Together these results suggest that set-to-number and number-to-set tasks have some commonalities but also retain at least some significant conceptual distinctness. Based on these results, we suggest these two types of tasks should no longer be used indiscriminately to inform theory or educational assessment of numerical abilities in preschool children.
Cardinal number knowledge-understanding “two” refers to sets of two entities-is a critical piece of knowledge that predicts later mathematics achievement. Recent studies have shown that ...domain-general and domain-specific skills can influence children’s cardinal number learning. However, there has not yet been research investigating the influence of domain-specific quantifier knowledge on children’s cardinal number learning. The present study aimed to investigate the influence of domain-general and domain-specific skills on Mandarin Chinese-speaking children’s cardinal number learning after controlling for a number of family background factors. Particular interest was paid to the question whether domain-specific quantifier knowledge was associated with cardinal number development. Specifically, we investigated 2–5-year-old Mandarin Chinese-speaking children’s understanding of cardinal number words as well as their general language, intelligence, approximate number system (ANS) acuity, and knowledge of quantifiers. Children’s age, gender, parental education, and family income were also assessed and used as covariates. We found that domain-general abilities, including general language and intelligence, did not account for significant additional variance of cardinal number knowledge after controlling for the aforementioned covariates. We also found that domain-specific quantifier knowledge did not account for significant additional variance of cardinal number knowledge, whereas domain-specific ANS acuity accounted for significant additional variance of cardinal number knowledge, after controlling for the aforementioned covariates. In sum, the results suggest that domain-specific numerical skills seem to be more important for children’s development of cardinal number words than the more proximal domain-general abilities such as language abilities and intelligence. The results also highlight the significance of ANS acuity on children’s cardinal number word development.
Prior studies indicate that children vary widely in their mathematical knowledge by the time they enter preschool and that this variation predicts levels of achievement in elementary school. In a ...longitudinal study of a diverse sample of 44 preschool children, we examined the extent to which their understanding of the cardinal meanings of the number words (e.g., knowing that the word "four" refers to sets with 4 items) is predicted by the "number talk" they hear from their primary caregiver in the early home environment. Results from 5 visits showed substantial variation in parents' number talk to children between the ages of 14 and 30 months. Moreover, this variation predicted children's knowledge of the cardinal meanings of number words at 46 months, even when socioeconomic status and other measures of parent and child talk were controlled. These findings suggest that encouraging parents to talk about number with their toddlers, and providing them with effective ways to do so, may positively impact children's school achievement.
•Longitudinal predictors of ANS performance in preschoolers were assessed.•Predictors of T2 ANS scores were T1 ANS, cardinality, and response inhibition.•Results suggest that the ANS task may measure ...more than core numerical skills.•Domain-specific and domain-general skills are related to performance on this task.
A large body of work has developed over the last decade examining the relation between the approximate number system (ANS) and mathematical performance across a wide range of ages, but particularly for preschool-age children. Largely, the evidence is mixed and suggests that a small relation exists that is dependent on a number of child-related or measurement-related factors. In contrast, little work has focused on understanding the stability and predictors of the ANS. These issues were examined by assessing 113 preschool children in the fall and spring of the preschool year on mathematical and cognitive assessments. Mixed-effect regressions indicated fall ANS performance was the strong predictor of spring ANS performance, suggesting moderate stability of this variable during preschool. However, cardinality and response inhibition were also significant predictors, and school-level variance was high. These findings indicate that the ANS may not be as foundational for mathematics development as previously suggested.
This study compared 2- to 4-year-olds who understand how counting works (
cardinal-principle-knowers) to those who do not (
subset-knowers), in order to better characterize the knowledge itself. New ...results are that (1) Many children answer the question “how many” with the last word used in counting, despite not understanding how counting works; (2) Only children who have mastered the cardinal principle, or are just short of doing so, understand that adding objects to a set means moving forward in the numeral list whereas subtracting objects mean going backward; and finally (3) Only cardinal-principle-knowers understand that adding exactly 1 object to a set means moving forward exactly 1 word in the list, whereas subset-knowers do not understand the unit of change.
Selective versions of θ-density Babinkostova, L.; Pansera, B.A.; Scheepers, M.
Topology and its applications,
05/2019, Letnik:
258
Journal Article
Recenzirano
Odprti dostop
In 3 the authors initiate the study of selective versions of the notion of θ-separability in non-regular spaces. In this paper we continue this investigation by establishing connections between the ...familiar cardinal numbers arising in the set theory of the real line, and game-theoretic assertions regarding θ-separability.
We provide a general criterion for Fraenkel–Mostowski models of
ZFA
(i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ...ordered” (
LW
), and look at six models for
ZFA
which satisfy this criterion (and thus
LW
is true in these models) and “every Dedekind finite set is finite” (
DF
=
F
) is true, and also consider various forms of choice for well-ordered families of well orderable sets in these models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets (
MC
ℵ
0
ℵ
0
) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets (
AC
fin
WO
) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of
AC
fin
WO
which is unknown. Models 4 and 5 are variations of Model 3. In Model 4
AC
fin
WO
is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which
2
m
=
m
for every infinite cardinal number
m
. We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false.
This paper deals with two notions: a polarized partition relations
α
β
→
γ
η
δ
λ
and product of generalized strong sequences. Strong sequences were introduced by Efimov in 1965 as a useful tool for ...proving famous theorems in dyadic spaces, i.e. continuous images of Cantor cube. In this paper we introduce the notion of product of generalized strong sequences and give the pure combinatorial proof that
α
β
→
γ
η
δ
λ
is a consequence of the existence of product of generalized strong sequences.
Brojevi kao vrsta riječi svojstveni su gramatičkomu opisu kajkavskoga narječja prema jezičnoj uporabi u pojedinim mjesnim govorima. U radu se analiziraju temeljna morfološka obilježja brojeva na ...odabranom korpusu objavljenih rječnika kajkavskih govora prema natuknicama, gramatičkim odrednicama vrste riječi, potvrđenim morfološkim oblicima, značenjima i navedenim oprimjerenjima značenja u rječničkim člancima. Predmet su analize glavni broj jedan i redni broj drugi u kontekstu brojeva i pridjeva.
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