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This article was submitted to Educational Psychology, a section of the journal Frontiers in Psychology

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Early numerical competencies (ENC) (counting, number relations, and basic arithmetic operations) have a central position in the initial learning of mathematics, and their assessment is useful for predicting later mathematics achievement. Using a regression model, this study aims to analyze the correlational and predictive evidence between ENC and mathematics achievement in first grade Portuguese children (

Difficulties in mathematics are pervasive and can have lifelong consequences (

Identifying these basic skills has been a mathematical cognition research concern for the past two decades, and some key findings relevant to later achievement in mathematics were pointed out by

The present study attempts to contribute to this research which aims to analyze the correlation and predictive evidence between early numerical competencies (ENC) (which are pointed out as one of the foundations of mathematical competence) and mathematics achievement at the end of first grade. We expect to enhance the transcultural evidence and consistency of international studies that analyzed and verified the predictive relation between these two variables in different contexts, in this case, in the Portuguese setting.

Our study may not only contribute to the international literature, as mentioned above, but may also give theoretical support to national literature, due to the fact that the National Council of Education (

In this paper, we refer to the ENC as a set of symbolic numerical abilities (also known as symbolic number sense) received from cultural and learning inputs, which may depend on the development and the integration of multiple basic cognitive abilities (

Although there is no one definition of ENC, several researchers agree that this set of symbolic competencies in the 4- to 6-year-old range refers to: (a) counting in a small set of objects; (b) number identification; (c) making relations about numbers (e.g., 4 is closer to 3 than to 6) and their magnitudes (e.g., 5 is more than 3); and (d) basic arithmetic operations, which means transforming sets of numbers by adding or taking away items (e.g., 3 and 2 makes 5, and taking away 2 from 5 is 3) (

To represent larger sets precisely, children need to learn how to count (

Identifying numbers implies the ability to identify or recognize a number symbol (e.g., 13) or number symbols combined to represent any number (e.g., 128) presented as a visual stimulus. To answer the question “What number is this?” children must learn number-words using long-term memory.

Comparing numbers on the most basic level implies that children look at two numbers (e.g., 4 and 9) and answer the question, “Which is bigger?”(9) or “Which is smaller?”(4). Preschool students, when presented with two non-symbolic sets for comparison, often do not count when comparing the two sets. Typically, students rely on visual (i.e., non-symbolic) inspection (

Adequate counting, comparing, and symbol knowledge skills are necessary to carry out most addition and subtraction problems presented to students in early elementary school (

Some studies indicate that ENC, especially with respect to basic arithmetic operations, are closely tied to the mathematics curriculum in elementary school. For instance, the results of the

These findings suggested that ENC is a crucial predictor of early mathematics achievement. Children who started first grade with advanced ENC will consequently progress faster in arithmetic and generally in symbolic numerical tasks in the first grade (

The present study aims to understand to role of ENC in mathematics achievement in Portuguese first school children. The main purpose is to analyze the correlation and predictive evidence between ENC and mathematics achievement at the end of first grade. The following research questions were considered:

What is the correlation between ENC and mathematics achievement at the end of first grade?;

Do ENC at the point of school entry predict mathematics achievement at the end of first grade?;

What is the relation between low, moderate, and high numerical competencies and mathematics achievement? In this research question, two hypotheses were tested: (a) children who started first grade with low numerical competencies remained low mathematics achievement at the end of first grade; (b) children who started with moderate numerical competencies, finished the first grade with moderate to higher gains on mathematics achievement; (c) children who started with high numerical competencies, finished the first grade with high mathematics achievement.

The NSB – Number Sense Brief Screener (

Children were split into the low, middle, and high grounds based on the normative values of NSB. Children were assigned to the low numerical competence group (A) when they performed at or below the 25th percentile; average numerical competence (B) and high numerical competence (C) children’s groups were defined as performing on the 25–75th range, and above the 75th percentile, respectively.

The participants were children who, after kindergarten, attended the first grade (first year of formal schooling in Portugal) in four elementary public schools. The average age was 6.37 (

The public schools are located in a residential area of western Lisbon with different background characteristics concerning the level of education and socioeconomic level of the local population. The southern part has a family-unit urbanization with houses, where the level of education and the socioeconomic level is medium-high. The northern part has buildings (median five stories) with a medium to low socioeconomic level.

Among the demographic characteristics of the three working samples, Group C (high numerical competence) has more boys than girls, and the parents’ education is higher when compared to Group A (low numerical competence).

The demographic information of the overall sample and the three working sample is portrayed in

Demographic information for participants.

NSB | Group A | Group B | Group C | |
---|---|---|---|---|

Male | 57.70 | 53.70 | 53.80 | 71.40 |

Female | 42.30 | 46.30 | 46.20 | 28.60 |

Primary education | 6.80 | 10.30 | 5.00 | 5.30 |

Secondary education | 52.50 | 61.90 | 54.60 | 42.90 |

Higher education | 40.70 | 27.80 | 40.40 | 51.80 |

The NSB is a shortened version (

The NSB raw score means for the Portuguese population is 22.87 points (

The composite achievement score in NSB (NSB overall) was the combined raw scores for seven subtests assessing the following ENC:

After the child finished counting a set of five stars, the examiner asked the following question: “How many stars were on the paper you just saw?” Counting sequence also included counting to 30.

To assess counting principles, children were asked to recognize correct, incorrect counts (e.g., counting the first object twice), and correct unusual counts (e.g., counting from right to left or counting the yellow dots first and the blue dots afterward).

Children were asked to name a visually presented number (e.g., 13) with the question “What number is this?”

Children were asked to make numerical magnitude judgments in three different ways: (a) given a number (e.g., 7), children were asked what number comes after the given number and what number comes two numbers after the given number; (b) given two numbers, children were asked to indicate which of two numbers was bigger or smaller; (c) and given three number (e.g., 6, 2, and 5), each placed on the point of an equilateral triangle, children also were asked to identify which number was closer to the target number.

On non-verbal calculation (presence of objects but without verbal stimuli) the examiner showed a set of chips, covered them, and then performed the addition or subtraction transformation (by removing or adding chips). Children were then asked to indicate how many chips were then under the cover.

On story problems (objects referents with verbal stimuli), children were orally phrased to three addition and two subtraction story problems and asked to solve them (e.g., Jose has 3 cookies. Sarah gives him 2 more cookies. How many cookies does Jose have now?).

On number combination (no object referents with verbal stimuli), children were asked to solve four addition and two subtraction computations (e.g., How much is 3 and 2?).

Achievement in mathematics was assessed using a formal Portuguese school evaluation of mathematics. The MSE was applied by teachers to measure mathematics achievement at the end of first grade in the classroom for 1 h and a half. The measure is based on 25 items. Results were normalized to a total of 100%. The items included four subtests using the quantities 1 up to 99.

The composite achievement score (MSE overall) was the combined raw scores for subtests assessing counting and arithmetic operations, place value, applied problems and basic geometry.

The psychometric analysis showed that MSE is reliable with an internal-consistency reliability of 0.87 (

Children were asked to solve exercises concerning object counting to 10, counting by fives up to 30, counting money up to 10 (e.g., 1€ + 2€ + 5€), ordering numbers up to 99, adding and subtracting facts up to 99 (e.g., 15+25 = ___, 50-40 = ___) and missing values with addition facts (e.g., 80 = 40+___).

The tasks were related to units and tens identification with abacus and Cuisenaire rods models. Children also had to identify predecessor and successor numbers up to 99 (e.g., __ 49 __), and to identify several numbers on a number line (with interval from 10 to 90) partially filled with numbers.

Children were asked to solve three addition and subtraction word problems up to 50. Another word problem concerned the interpretation and counting data in a 2×2 table.

The tasks were related to geometry (e.g., to identify basic geometric figures), time (e.g., to identify the days of the week) and spatial sense (e.g., to design a symmetry figure).

This study was carried out in accordance with the recommendations of the National Direction for Education with written informed consent to contact the national groups of schools. After one national group of schools with four primary public schools located in Lisbon had given permission to apply the present study, the sample were collected with the parents’ written informed consent, with a notification that any elements of research will be covered by the anonymity of the participants.

Children were assessed individually with the NSB at the point of school entry by examiners who were fully trained in the testing procedures. The examiners were graduate students in Psychology and Education Sciences. Based on the NSB initial application protocol (

To determine the association between ENC (assessed by NSB raw scores) and mathematics achievement (assessed by MSE raw scores), bivariate correlations were analyzed and are presented in

Correlations between early numerical competencies and dependent variables.

MSE | CO | Pv | Ap | FS | |
---|---|---|---|---|---|

NSB | 0.57ˆ** | 0.50ˆ** | 0.47ˆ** | 0.39ˆ** | 0.50ˆ** |

Counting | 0.10 | -0.03 | 0.07 | 0.08 | 0.19ˆ* |

Counting principles | 0.18 | 0.18 | 0.10 | 0.12 | 0.13 |

Number identification | 0.42ˆ** | 0.36ˆ** | 0.34ˆ** | 0.32ˆ** | 0.28ˆ** |

Number comparisons | 0.37ˆ** | 0.35ˆ** | 0.32ˆ** | 0.24ˆ** | 0.34ˆ** |

Non-verbal calculation | 0.20ˆ* | 0.18ˆ** | 0.12 | 0.12 | 0.22ˆ* |

Story problems | 0.50ˆ** | 0.46ˆ** | 0.41ˆ** | 0.34ˆ** | 0.43ˆ** |

Number combinations | 0.49ˆ** | 0.39ˆ** | 0.42ˆ** | 0.34ˆ** | 0.47ˆ** |

^{∗}p

^{∗∗}

Regarding the NSB subtests and MSE overall, number identification (

The main purpose of the study was to determine the contribution of the NSB in predicting mathematics achievement at the end of first grade. To accomplish this goal, students’ scores on the NSB were regressed on the outcomes of ASM overall. As we are conducting a simple regression analysis on two variables which, on a first approach, are linearly dependent, the major assumptions concerns normality and homoscedasticity. In this regard, normality P–P plots and Q–Q plots of the standardized residuals were conducted. Shapiro–Wilk test of normality of the standardized residuals reveals a significance of 0.043 which, ^{2} (1) = 3.247 with a significance of 0.072.

Variance explained and regression coefficients by early numerical competencies (assessed by NSB and subtests) in Mathematics Achievement (assessed by MSE).

R^{2} |
B | Beta | |||
---|---|---|---|---|---|

NSB | 0.33ˆ** | 2.16ˆ** | 0.57 | 7.53 | 0.00 |

Counting | 0.01 | 5.59 | 0.10 | 1.06 | 0.29 |

Counting principles | 0.03 | 5.73 | 0.18 | 1.96 | 0.05 |

Number identification | 0.17ˆ** | 6.99ˆ** | 0.42 | 4.94 | 0.01 |

Number comparisons | 0.14ˆ* | 5.17ˆ* | 0.37 | 4.32 | 0.02 |

Non-verbal calculation | 0.04ˆ* | 5.77ˆ* | 0.20 | 2.14 | 0.04 |

Story problems | 0.25ˆ** | 6.47ˆ** | 0.50 | 6.17 | 0.00 |

Number combinations | 0.24ˆ** | 4.87ˆ** | 0.49 | 6.01 | 0.00 |

^{∗}

^{∗∗}

Independent regressions of the different subareas of NSB vs. MSE were performed to recognize their relative importance on the prediction. Story problems (^{2} = 0.25) and number combinations, (^{2} = 0.24) accounted moderately for about 25% of the explained variance in MSE achievement. A stepwise regression analysis with all subareas of NSB showed that story problems by themselves accounted for 24.7% of the variance in mathematics achievement in first grade, and together with number identification accounted for 31%. Individual subareas of the NSB, such as counting and counting principles are not statistically significant in predicting mathematics achievement (

The differences between NSB outcomes groups (Group A, B, and C; low, moderate, high numerical competencies) allow us to test if children who started first grade with low numerical competencies remained low mathematics achievement at the end of first grade; and children who started with high numerical competencies finished the first grade with high mathematics achievement. Differences between NSB outcome groups were observed when mathematics achievements were measured at the end of first grade.

As data do not fit a normal distribution, a Kruskal–Wallis test was used to measure if differences between NSB groups’ means were statistically significant. In this regard, we found significant group differences (A, B, and C) in mathematics achievement scores at the end of first grade, χ^{2} (2) = 28.34,

Mathematics achievement means score overall and subareas by NSB outcome groups.

NSB | Group A | Group B | Group C | |
---|---|---|---|---|

NSB | 22.42 (05.83) | 15.35 (02.93) | 23.80 (02.35) | 29.60 (01.63) |

MSE | 70.39 (21.39) | 57.49 (23.01) | 71.66 (19.53) | 83.63 (10.55) |

Counting and operations | 16.86 (05.84) | 14.05 (05.71) | 16.96 (05.22) | 21.16 (03.31) |

Place value | 14.51 (04.54) | 12.20 (05.23) | 14.92 (03.98) | 17.34 (02.04) |

Applied problems | 18.84 (10.00) | 14.79 (10.87) | 19.18 (09.36) | 24.82 (06.01) |

Spatial and forms | 21.20 (06.30) | 17.58 (06.63) | 21.96 (05.92) | 25.23 (02.61) |

Early numerical competencies are considered a foundational domain-specific cognitive factor in the development of mathematical competence, allowing students to make connections with mathematical relationships, principles and procedures. Doing so, students can learn with success advanced mathematics (

The main purpose of this study was to better understand the predictive relationship between early numerical competence (or number sense) and mathematics achievement in Portuguese students. Specifically, we attempted to predict achievement in mathematics at the end of first grade by measuring early number competencies at the point of entering school (i.e., prior of formal education). We used two measures: NSB – Number Sense Brief Screener and MSE – Math Summative Evaluation in 123 children from an urban public-school setting.

The results indicated that numerical competencies (as assessed by NSB) had a moderate predictability for the performance of mathematics at the end of first year in Portuguese children.

Our findings also show a positive, significant and moderate correlation between numerical competencies (as assessed by NSB) at the beginning of first grade, and the performance of mathematics at the end of first grade. The results indicated that basic arithmetic operations had the highest correlation with mathematics achievement when compared to other individual subareas of NSB.

Consistent with these findings,

As stated by

Our results do not indicate that counting skills by themselves underlie mathematical difficulties. Noticeably, there is almost no difference in object counting tasks in the three different NSB achievement groups when compared to other mathematics contents tasks.

Related to the third research question, the results indicated that children who started first grade with low numerical competencies remained low mathematics achievement at the end of first grade; and children who started with moderate and high numerical competencies, finished the first grade with moderate and high mathematics achievement, respectively. The significant group differences found allow us to assume that ENC are important for setting learning trajectories in mathematics (

With the methodology herein presented (short-term longitudinal study) we can predict, in a reasonable way, that numeracy indicators as well as number identification, story problems and number combinations measured at the point of school entry, predict later mathematical performance, specifically at the end of first grade. As a final note, with this study we cannot state that the explained variance of mathematics achievement is a consequence of a single variable – the ENC -, but based on other works its importance may be assumed. For instance,

As the absence of control variables is one of the limitations of the present study, future studies should compare NSB with other ENC screening tools to measure concurrent analysis, with the purpose of producing a coherent and predictive model between ENC and mathematics achievement.

From a conceptual point of view, these findings support the studies of

As mathematical thinking invades the daily activities of a young child, and poor mathematics achievement has been shown to be a major influence during school-age years, this work can contribute to identify Portuguese children who are at risk for failure in mathematics. Another implication for educational practice draws attention to the importance of an ENC screening tool for use in schools, clinics and other educational settings, with the purpose of helping children build numerical competencies as early as possible, giving them the background they need to achieve in mathematics during the first years of schooling.

We declare that this study was carried out in accordance with the recommendations of the committee ethic of the MIME – Monotorização de Inquéritos em Meio Escolar, Direção-Geral de Educação, Ministério da Educação [MIME – Monitoring of School-based Inquiries, General-Direction of Education, Ministry of Education] with written informed consent to be used for all students of first-grade public schools and with anonymous reporting procedures. We also declare that the parents or legal guardian’s of all subjects gave written informed consent to carry out the current research in accordance with the Declaration of Helsinki. Finally, we state that the protocol was reviewed and approved by the Direção Geral da Educação.

Conceptualization: LM and ÓdS. Methodology: LM. Formal analyses: LM and AL. Writing-review and editing: LM, ÓdS, and AL.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The reviewer HSP and handling Editor declared their shared affiliation, and the handling Editor states that the process nevertheless met the standards of a fair and objective review.

We wish to thank all participating children who made the study possible, and their parents. We also extend our thanks to the first-grade elementary school teachers for their assistance with the Mathematics Evaluation Assessment. Moreover, we thank the director of the group of schools, Ana Mafalda Manita, for allowing the application of this short-term longitudinal study in the elementary schools.

^{a}série - N.° 59 – 25 de Março de 2015. Recomendação n.° 2/2015.