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Tremendous variation in elementary school children’s mathematical achievement can partly be traced back to differences in early domain-specific quantitative competencies. While previous research mainly focused on numerical magnitude representation and counting, we tested the long-term effects of relational quantitative reasoning. Before children (N = 51) entered school (i.e. at age 5-6), we assessed this competence with a test that required no knowledge about Arabic numerals. Two and a half years later, when children were in third grade of elementary school, we gauged mathematical achievement, general reasoning ability, and reading skills. A multiple regression analysis with mathematical achievement as outcome variable revealed a small but unique impact of children’s relational quantitative reasoning in kindergarten on their later mathematical achievement after controlling for general reasoning and reading abilities. Thus, a considerable amount of individual differences in mathematics achievement in elementary school results from differences in early relational quantity understanding that emerge before systematic instruction starts.

Mathematical competencies are pivotal for educational and professional success of individuals and for societal functioning and welfare (

Researchers have amassed evidence that the human brain is hard-wired for processing quantitative information. For example, preferential looking studies in infants indicate that already few months old babies can discriminate between sets with different numbers of elements (

In several longitudinal studies starting at kindergarten age, researchers have identified particular numerical precursors of later mathematical achievement. These precursors comprise quantity to number words linkage (

Although the strong impact of symbolic magnitude representation on later mathematical achievement is uncontested, it is not the only domain-specific precursor. Another important source of mathematical achievement is the spontaneous representation of order. Children as young as 11 months (but not as young as 9 months) can discriminate between an increasing and decreasing row of quantities independent of the magnitude of sets in preferential looking tasks (

The concept of order lays the foundation for relational quantitative reasoning or seriation, as labeled in Piagetian tradition. Seriation skills help to integrate features across quantities and thereby allow drawing inferences based on this integration. In this sense, 6, for instance, is not just understood as a term for naming a set of 6 items, but is also represented as a predecessor of 7, a successor of 5, or the double of 3. Researchers typically assess this concept by so-called non-symbolic seriation tasks: children have to rearrange a disordered collection of sticks differing in length to construct an ordered series. While even 2- to 3-year-olds master this task when presented with 1 up to 3 sticks, many 5-year-olds acted by trial and error when presented with six sticks (

Besides the domain-specific abilities, also domain-general reasoning abilities contribute to mathematics learning (

Apparently, an individual’s domain-general reasoning abilities guide the acquisition of knowledge in terms of quality and speed, which will subsequently impact further learning. Nevertheless, several studies with elementary (e.g.,

One might object that most of the studies have controlled for domain-general reasoning abilities in kindergarten children (e.g.,

Consequently, the unique predictive validity of early numerical precursors on later mathematical achievement could have been overestimated. This might have been the case because most of the studies control for domain-general reasoning ability with tests like intelligence tests, but their relatively low stability and reliability in assessing young children’s performance may be problematic. As a consequence, the impact of domain-general reasoning abilities on mathematics achievement may have been underestimated in studies which assessed domain-general reasoning abilities in preschool or kindergarten. Furthermore, in addition to domain-general reasoning abilities, several researchers have shown that reading comprehension explains variance in mathematical achievement because it is related to conceptual understanding and application of mathematics (e.g.,

Despite the large number of studies on early precursors of mathematical achievement, research has been biased in three ways. First, a major focus has been on children at risk and deficiencies in the lower performance range (

In a longitudinal study, we focused on assessing the predictive value of the concept of order, that is, relational reasoning about quantities and seriation skills in kindergarten children growing up under normal conditions (i.e. no special focus on children at risk). Will relational reasoning about quantities still explain unique variance in third grade mathematical achievement after simultaneously controlling for intelligence and reading ability? If this is the case, it would strengthen the domain-specific view on learning mathematics.

Addressing this research question requires an assessment of relational reasoning about quantities that indicates individual differences among normally developing 5- to 6-year-olds. We decided to not include number symbols like Arabic digits, because they were not systematically taught in the kindergartens from which we recruited participants (even though most German 6-year-olds can be expected to know the Arabic digits in the range from 1 to 6,

Fifty-one children (21 girls) participated from kindergartens located in a metropolitan middle-class area. Children were initially tested in kindergarten a few months before entering formal education (_{age} = 6.5 years, _{age} = 8.9 years).

To assess kindergarten children’s relational reasoning about quantities, we administered the Quantity Sequence Test (QST,

However, each trial will only be scored when the child selects the correct three cards at the first try. For example, Trial 1 (see

Two example trials (A represents Trial 1 and B Trial 4 of

Trial | Positions Presented |
Positions to be Completed |
Cards to Choose From for Positions D-F | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

A | B | C | D | E | F | |||||||

Practice | 1 | 2 | 3 | 3 | 6 | 0 | 4 | 9 | 5 | |||

1 | 2 | 3 | 4 | 8 | 7 | 9 | 5 | 6 | 10 | |||

2 | 1 | 0 | 2 | 0 | 4 | 9 | 0 | 3 | 1 | |||

3 | 0 | 1 | 0 | 3 | 0 | 9 | 8 | 4 | 2 | |||

4 | 0 | 2 | 0 | 5 | 4 | 3 | 0 | 2 | 10 | |||

5 | 4 | 0 | 3 | 5 | 2 | 0 | 10 | 0 | 6 | |||

6 | 6 | 5 | 4 | 1 | 6 | 3 | 0 | 2 | 8 | |||

7 | 10 | 9 | 8 | 7 | 4 | 2 | 6 | 1 | 5 | |||

8 | 0 | 2 | 4 | 7 | 10 | 6 | 2 | 0 | 8 | |||

9 | 10 | 8 | 6 | 3 | 7 | 8 | 4 | 0 | 2 |

We measured mathematical achievement in third grade with a 36-item test (henceforth referred to as MAT). Items belong to two categories: (1) word problems (16 items), and (2) arithmetic multiplication, addition, and subtraction problems (20 items). We scored each item with one point if answered correctly, and with zero if answered incorrectly.

We used word problems from a set of empirically validated word problems for elementary school students developed for the Munich Longitudinal Study (

We assessed reading comprehension (RC) with a standardized test - Salzburger Lese-Screening (

We used the Culture-Fair Test for Intelligence (CFT 20,

At the first measurement point, we administered only the QST. A trained experimenter tested the children individually in a quiet room of their kindergarten. The mean testing time was 19 minutes (

Test | Minimum | Maximum | ||
---|---|---|---|---|

QST (max. 9) | 5.33 [4.81, 5.86] | 1.86 | 1 | 8 |

MAT (max. 36) | 19.45 [17.31, 21.60] | 7.63 | 4 | 33 |

RC (max. 70) | 39.25 [36.49, 42.02] | 9.85 | 20 | 69 |

CFT (max. 46) | 28.83 [27.42, 30.31] | 5.13 | 13 | 38 |

First, we computed Pearson correlation coefficients between the test scores (see

Variable | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1. QST | - | |||

2. MAT | .54 [.31, .71]* | - | ||

3. RC | .37 [.11, .57]* | .38 [.12, .59]* | - | |

4. CFT | .51 [.27, .69]* | .42 [.16, .62]* | .04 [-.24, .32] | - |

*

To examine whether the QST-performance in kindergarten explains unique variance in third grade mathematics achievement, we computed a hierarchical regression analysis (see ^{2}^{2}^{2}

Step | Variable | Unstandardized B [95% CI] | β | Δ^{2} |
Total ^{2} |
||
---|---|---|---|---|---|---|---|

1 | .305 [.11, .50] | ||||||

CFT | .017 [.007, .027] | .401 | 3.329 | .002 | |||

RC | .008 [.003, .013] | .362 | 3.006 | .004 | |||

2 | .370 [.18, .56] | ||||||

CFT | .010 [-.001, .021] | .241 | 1.762 | .085 | |||

RC | .005 [.000, .011] | .251 | 1.984 | .053 | |||

QST | .037 [.003, .070] | .324 | 2.211 | .032 | .065 |

In the introduction, we emphasized that previous research on domain-specific numerical precursors of later mathematical achievement is limited in three ways. First, the majority of studies have focused on children at risk and deficiencies in the lower performance range. Second, measures of the relational quantitative reasoning (i.e. the concept of order and seriation skills) have not been considered in the same way as tests on symbolic magnitude representation and counting. Third, measures of domain-general reasoning abilities have often been assessed together with the precursors in longitudinal studies. These early measures suffer from low reliability which might lead to an underestimation of the impact of domain-general abilities. In a longitudinal study, we examined the relation between relational quantitative reasoning assessed in kindergarten and mathematical performance in the third grade of elementary school in normally developing children.

Our results provide evidence that relational quantitative reasoning (as assessed with the quantity sequence test, QST;

In contrast to the widely used number comparison tasks which address only the questions of “What is more [or less]?”, the QST used in the present research required a deeper understanding of ordinality, namely, understanding the sequence of quantities which refers to the question “What comes before or after?”

One could object that a more direct test of the role of symbol processing in understanding the concept of order would require the inclusion of a symbolic version of the QST. However, as there was no compulsory kindergarten curriculum in Germany at the time of data collection, not all children could be expected to be familiar with digits (i.e., Arabic numerals) required in a symbolic version. Nevertheless, future studies should include several other measures of mathematical precursor skills to allow assessment of the unique contributions of the different skills. Our main aim was to show that early relational quantitative reasoning does indeed predict later mathematical achievement.

Another criticism might be that we did not control for social background or general reasoning abilities, which might account for differences in the QST-performance. It is beyond doubt that a low-stimulating home environment affects mathematical development and that children from low-income families fall short of their potential (e.g.,

Different from several other studies, we presented the test of general reasoning abilities together with the outcome variable, not together with the predictor. As reliability and stability of measures of intelligence have been shown to markedly increase after children enter school, our kindergarten measure of domain-specific quantitative abilities had a stronger competitor compared to many other studies. Even if domain-general reasoning abilities are measured together with the outcome variables, as we did, their predictive value became smaller when we included quantitative relational reasoning as a domain-specific predictor in the regression model. Although our study can thus be considered a rigorous test of the impact of a domain-specific precursor, an additional measure of general reasoning in abilities in kindergarten would have strengthened the explanatory power. Incorporating such a measure would have enabled us to empirically replicate our claim about the increasing stability of such measures within our study: Nevertheless, the unique proportion of variance in third grade mathematical achievement explained by informally acquired seriation skills indicates that relational quantitative reasoning is a sound domain-specific source of mathematical competencies.

The present findings stimulate questions for future research concerning early education: to what extent can pedagogical interventions in kindergarten improve a child’s understanding of the concept of order? Would such an improvement facilitate children’s acquisition of mathematical concepts later in school? How should interventions be designed in order to provide effective learning? What are the educational implications and effects of relational reasoning about quantities compared to those of other domain-specific precursors of mathematical skills (e.g., counting ability, quantity to number word linkage, and numeracy-specific attention)? By demonstrating that early relational quantitative reasoning is an important domain-specific precursor for later elementary school mathematics achievement our study complements other findings on precursor skills of later mathematical achievement (e.g.,

The authors have no funding to report.

The authors declare that no competing interests exist.

We would like to thank Uta Gühne for help in obtaining the data.