Scholarly article on topic 'Multi-Year Longitudinal Investigation of Children's Early Mathematics Development'

Multi-Year Longitudinal Investigation of Children's Early Mathematics Development Academic research paper on "Psychology"

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Abstract of research paper on Psychology, author of scientific article — Lília Marcelino, Óscar de Sousa, Vitor Cruz, António Lopes

Abstract According to Jordan et al. (2006), most researchers agree that number sense in young children can be defined as the ability to subtilize small quantities, to discern number patterns, to compare numerical magnitudes and estimate quantities, to count, and to perform simple number transformations. Number sense, as assessed by NSB - Number Sense Brief Screener (Jordan, Glutting et al., 2008), is a powerful predictor of later mathematics outcomes at the end of first and third grades (Jordan et al., 2010). The results of Jordan et al. (2010) are in consonance with those of other investigations, suggesting that weaknesses in number competences concerning to counting, number relationships, and basic operations underlie most mathematics learning difficulties (e.g. Gersten et al., 2005; Geary et al., 2007; Landerl et al., 2004). With this study we aim: 1) to adapt the NSB - Number Sense Brief Screener (Jordan, Glutting et al., 2008) for the Portuguese population in order to identify early learning difficulties of mathematics; 2) to analyze the predictive capacity of the battery at the end of first grade; and 3) by a longitudinal and qualitative study, to follow the mathematics learning trajectories during the first and second grade. After kindergarten and at the moment of first grade initiation we measured number sense in Portuguese public schools’ children (n=860, average 6.43 years old). At the present moment we are on the process of the multi-year longitudinal investigation of children's mathematics development. The preliminary NSB results suggest that first-grade lower performance NSB group exhibits weak counting procedures, weak number comparisons (e.g. identify numbers in a numerical sequence), and inaccurate verbal problems’ resolution. Presently we are extending the number of participants; nevertheless, the findings indicate that screening early number sense development is useful for identifying children who will face later math difficulties which support the previous studies of Jordan et al. (2006; 2007; 2008).

Academic research paper on topic "Multi-Year Longitudinal Investigation of Children's Early Mathematics Development"

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Procedía - Social and Behavioral Sciences 69 (2012) 1911 - 1920

International Conference on Education and Educational Psychology (ICEEPSY 2012)

Multi-year longitudinal investigation of children's early mathematics development

Lília Marcelino*a, Óscar de Sousaa, Vitor Cruzb, Antonio Lopesc

aCeiEF, Universidade Lusófona de Humanidades e Tecnología, 1749-024 Lisboa, Portugal bFaculdade de Motricidade Humana, Universidade Técnica de Lisboa, 1495 - 688 Cruz Quebrada, Portugal _cCEPCA, Universidade Lusófona de Humanidades e Tecnologia, 1749-024 Lisboa, Portugal_

Abstract

According to Jordan et al. (2006), most researchers agree that number sense in young children can be defined as the ability to subtilize small quantities, to discern number patterns, to compare numerical magnitudes and estimate quantities, to count, and to perform simple number transformations. Number sense, as assessed by NSB - Number Sense Brief Screener (Jordan, Glutting et al., 2008), is a powerful predictor of later mathematics outcomes at the end of first and third grades (Jordan et al., 2010). The results of Jordan et al. (2010) are in consonance with those of other investigations, suggesting that weaknesses in number competences concerning to counting, number relationships, and basic operations underlie most mathematics learning difficulties (e.g. Gersten et al., 2005; Geary et al., 2007; Landerl et al., 2004). With this study we aim: 1) to adapt the NSB - Number Sense Brief Screener (Jordan, Glutting et al., 2008) for the Portuguese population in order to identify early learning difficulties of mathematics; 2) to analyze the predictive capacity of the battery at the end of first grade; and 3) by a longitudinal and qualitative study, to follow the mathematics learning trajectories during the first and second grade. After kindergarten and at the moment of first grade initiation we measured number sense in Portuguese public schools' children (n=860, average 6.43 years old). At the present moment we are on the process of the multi-year longitudinal investigation of children's mathematics development. The preliminary NSB results suggest that first-grade lower performance NSB group exhibits weak counting procedures, weak number comparisons (e.g. identify numbers in a numerical sequence), and inaccurate verbal problems' resolution. Presently we are extending the number of participants; nevertheless, the findings indicate that screening early number sense development is useful for identifying children who will face later math difficulties which support the previous studies of Jordan et al. (2006; 2007; 2008).

© 2012 The Authors. Published by Elsevier Ltd.

Selection andpeer-reviewunder responsibility of Dr. Zafer Bekirogullari of Cognitive - Counselling, Research & Conference Services C-crcs.

Keywords: Number sense; Learning difficulties; Mathematics; Early assessment, adaptation

* Corresponding author. Instituto da Educagao, Universidade Lusófona de Humanidades e Tecnologia, Campo Grande 376,1749-024 Lisboa, Portugal. Tel.: +353-21-7515500; fax: +353-21-7577006. E-mail address: lilia.marcelino@gmail.com

1877-0428 © 2012 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of Dr. Zafer Bekirogullari of Cognitive - Counselling, Research & Conference

Services C-crcs.

doi:10.1016/j.sbspro.2012.12.145

1. Introduction

Due to the fact that the number is a fundamental parameter for understanding the world around us, the poor performance in mathematics can have serious consequences for educational and professional performances (Jordan, Glutting et al, 2008). Poor numeracy (e.g. difficulties to deal competently with numbers, tables and graphs), imposes difficulties for functioning in all areas of life (Parsons & Bynner, 2005). The longitudinal studies "Does numeracy matters?" (Bynner & Parsons, 1997; Parsons & Bynner, 2005) shows that people with poor numeracy have a tendency to leave full-time education at the earliest opportunity. Moreover, most of their jobs are low skilled and poorly paid and offered few chances of training or promotion. At the age of 30 men and women with poor numeracy are more than twice as likely to be unemployed as those with competent numeracy (Parsons & Bynner, 2005). So, logical-mathematical reasoning is much needed in day-to-day operations as in success courses related to science and technology at higher level (Jordan, Kaplan, Ramineni & Locuniak, 2009).

Furthermore, according to the National Association for the Education of Young Children (NAEYC) and the National Council of Teachers of Mathematics (NCTM)(2002) to succeed in school and in life, young children need a strong foundation in mathematics. Mathematics helps children make sense of their world outside of school and helps them construct a solid foundation for success in school, not only in math courses but also in science, social studies, and other subjects. Once out of school, as an adult, they need a broad range of basic mathematical understanding to take informed decisions in their jobs, households, communities, and civic lives.

While considering the impact of mathematics throughout an individual's entire life, mathematics education is rapidly becoming a top priority among U.S. policy because proficiency in mathematics is essential to success, especially when U.S. children's mathematical proficiency is far below that of many other industrialized nations, and the mathematics gap is widest for children living in poverty and those who are members of ethnic, cultural, and linguistic minority groups (NAEYC & NCTM, 2002; National Mathematics Advisory Panel, 2008).

According to the Programme for International Student Assessment (PISA) in 2009, Portugal and U.S. are below (487 points) the OECD average (496 points) on the mathematics scale. Comparing to other countries Portugal and U.S are in 37th ranking position. In 2011 the Portuguese national evaluation on mathematics at the end of 4th grade points to a success rate of 79.1%. It means that 20.9% of Portuguese children are in risk of failure on mathematics. Although in the last four years there is a downward trend in "mathematics national results" at the end of 4th and 6th grade early mathematics education is not yet a national top priority policy.

The importance of adapting the NSB to the Portuguese population is supported by their predictability on mathematics achievement, and in a practical perspective, by the Portuguese scenario in mathematics. The present study also provides additional cultural evidence of the potential importance of number sense screening for early identification of mathematics difficulties in a country other than the United States.

1.1. Number sense

In order to improve the foundations for learning advanced mathematics, it is necessary to develop the basis of number competencies. It has been consensual among researchers interested in the cognitive development of children and in early identification and intervention of the mathematics difficulties, that many mathematics difficulties in elementary school can be traced to weaknesses in understanding the meaning of numbers and number relationships, which means that number sense has a central position in the initial learning of mathematics (Gersten, Jordan & Flojo, 2005).

According to Jordan, Glutting et al. (2008), number competencies or number sense in the 3 to 6 years old range, refers to ability to apprehend the value of small quantities immediately without counting, counting items in a set of at least five numbers with the knowledge that the final count word indicates how many are in the set; make judgments about numbers (e.g. 4 is closer to 3 than to 6) and their magnitudes (e.g. 5 is more than 3); and transforming sets of numbers by adding or taking away items (e.g. 3 and 2 makes 5, and take away 2 from 5 is 3).

Number sense allows children to make connections among mathematical relationships, principles and procedures (Gersten et al., 2005). Children gradually learn that numbers in the counting sequence have larger quantities than earlier numbers and that numbers have magnitudes. Children use these skills eventually to construct a linear representation of numerical magnitudes, which allows them to learn place value, to know the number decimal system and perform mental calculation (Jordan, Glutting et al., 2008). This kind of competencies allows children to

learn advanced mathematics. If not worked out, mathematic difficulties will be cumulative and worsen with time. Therefore, a good knowledge of the concept of number is an important precursor for future success in mathematics (Griffin & Case, 1997; Gersten & Chard, 1999). Before failure occurs, early screening at the kindergarten and first grade levels can identify children in need of educational support or intervention (Jordan, Kaplan, Locuniak, & Ramineni, 2007).

1.2. Predictability of number sense

A body of research primary from psychologists who are interested in the early identification and interventions in mathematics focus on the concept of number sense and his implications in mathematics proficiency (Dehaene, 1997; Greeno, 1991; Okamoto & Case, 1996 in Gersten et al., 2005). Using this framework, Baker, Gersten, Flojo et al. (2002) explored the predictive validity of a set of measures that assess student's number sense to predict subsequent performance in arithmetic. Nowadays, there are consensuses that number sense predicts math outcomes (Gersten & Chard, 1999; Griffin & Case, 1997; Gersten et al., 2005; Jordan et al., 2006, 2007, 2008, 2010). Other studies suggest that weaknesses in number competences concerning to counting, number relationships, and basic operations underlie most mathematics learning difficulties (e.g. Gersten et al., 2005; Geary et al., 2007; Landerl et al., 2004).

The findings of the Jordan and colleagues (2009), demonstrate the importance of kindergarten number competence for setting children's development learning trajectories in primary school mathematics. The children who start first grade with good number competencies are likely to advance their mathematics learning, whereas children with weak number competencies. Therefore, the early number competencies are important to later mathematics functioning.

Jordan and colleagues (2007) found that the development and performance of the number sense in kindergarten children explain 66% of the performance of mathematics at the beginning of first grade, and assessing the number sense identifies approximately 52% of children with difficulties in fluency in numerical calculations (Locuniak & Jordan, 2008). In a recent study, Jordan, Glutting & Ramineni (2010) found evidence that the effect of the number sense, even at an early stage of the kindergarten, maintains a strong prediction in mathematic achievement until at least the end of the 3rd grade. Furthermore, Duncan and collaborators (2007) suggests that the relationship between the number sense and mathematics performance continues throughout schooling.

1.3. Measuring number sense

Jordan, Kaplan, Olah & Locuniak (2006) in a longitudinal study, focusing on the development of math skills in children at risk, inserted into the Children Math Project at the University of Delaware, developed a Number Sense Battery (NSB) to be applied in kindergarten and in first years of schooling, aged between 4 and 8 years.

The Battery (NSB), with acceptable psychometric qualities (coefficient alpha between 0.82-0.89), comprises the following subtests: Counting skills and Counting principles, which looks at knowledge of the counting sequence, the ability to enumerate sets, number identification and understanding of counting principles (Geary, Hoard, & Hamson, 1999; Gelman & Gallistel, 1978); Number knowledge, which involves the ability to compare quantities, such as which of two numbers is larger or smaller (Griffin, 2004); Nonverbal calculation, or the ability to perform simple addition and subtraction transformations with objects without verbal stimuli (Hughes, 1986; Huttenlocher, Jordan, & Levine,1994; Klein & Bisanz, 2000); Story problems, which assesses the ability to solve simple word problems where objects were referred to but not presented (e.g., "Jill has two pennies. Jim gives her one more penny. How many pennies does Jill have now?") (Ginsburg & Russell, 1981; Levine, Jordan, & Huttenlocher, 1992); and Number combinations, which involves verbally presented calculations with no object referents (e.g., "How much is 2 and 1?").

2. Method

2.1. Participants

Participants are children who complete the kindergarten and initiate the first grade in Portuguese public schools. In this study we have 3 working samples regarding the study purposes: adaptation of NSB for the Portuguese

population (Sample 1), predictive analysis of the NSB in mathematics achievement (Sample 2) and a multi-year longitudinal investigation of children's mathematics development during the first and second grade (Sample 3). Background characteristics of children in each sample are presented in Table 1.

In the national preliminary sample (Sample 1), the children were from different regions of Portugal: North (n=249), Centre (n=68), Lisbon and Vale do Tejo (n=123), South (n=118) and Azores (n=302), leaving a final sample of 860 children, with an average age of 6.5 years between 5 and 10 years old, in which 63,4% came from low-income families.

In the second sample, children were drawn from a public school in the district of Lisbon, average age of 6.4, 71% of the children were boys, 25,6% of children were low-income, 41% high-income families, and half of the sample have parents with secondary education.

The last sample was withdrawn from the same public school in the district of Lisbon with half of the participants belonging to high-income families with higher-education.

Table 1

Demographic information for participants concerning to each sample

Variable Sample 1 Sample 2 Sample 3

(n=860) (n=118) (n=30)

Gender

Male 53,7% 71% 53,3%

Female 46,1% 52% 46,7%

Income

Low Income 63,4% 25,6% 28,3%

Middle Income 17,4% 33,8% 20%

High Income 19,2% 40,8% 51,7%

Mean Age 6,47 6,43 6,00

Kindergarten Frequency 89,7% 97,6% 100%

Parent's Education

Primary Education 37,5% 11,45% 18,85%

Secondary Education 45,05% 49,05% 32,85%

Higher Education 17,45% 39,5% 48,3%

Mean NSB 22 23 22

Mean Math Evaluation - 70,19% -

2.2. Procedure

2.2.1. Sample 1

The NSB were given to children individually in their schools by one of several trained graduate students in psychology, education science and psycho-pedagogy. It was administered at the beginning of first grade (in October) after the permission of the school's Headmaster.

In each region, groups of children were selected according to the number of public national schools in order to have, at least, a sample with at least 1200 participants. By this moment we already have 860 participants from the first collected data. Data collection was designed for two consecutive years by logistical issues, which means we are still in process of collecting data. The northern regions and Lisbon e Vale do Tejo regions will have different coefficients: in the case of North, it will be duplicated and in Lisbon e Vale do Tejo tripled, in order to be in conformation to the geographical distribution of the overall population.

2.2.2. Sample 2

Children were assessed individually on the NSB at the beginning of first grade. Children's later mathematics achievement is assessed on the Math Summative Evaluation at the end of first grade by teachers in the classroom.

They all attended to the same group public school with four primary schools in one single area of Lisbon. The group of schools was selected because they served children with heterogeneous demographic characteristics.

2.2.3. Sample 3

In the third sample, from the Sample 2, participants were drawn randomly into three achievement groups. The differentiation between low, middle, and high achievement in NSB allowed us to split the third working sample. After that we collect the student's mathematics achievement in 6 times (Time 1 - December; Time 2 - February; Time 3 - Mars; Time 4 - April; Time 5 - May; Time 6 - June) based on the results of mathematics school formal evaluations. At the present moment we are on the process of this multi-year longitudinal investigation to follow the mathematics learning trajectories during the second grade. This is the reason why we are not able at this moment to present any preliminary results.

2.3. Materials

2.3.1. NSB - Number sense brief screener

The NSB is a shortened version of a longer number battery given to children in order to identify early difficulties in mathematics (Jordan, Glutting, et al., 2008). The NSB has 33 items. The items assess counting knowledge and principles, number recognition, number comparisons, nonverbal calculation, story problems and number combinations. The measure is reliable, at the beginning of first grade presents a coefficient alpha of .84 (Jordan, Glutting, et al., 2008).

2.3.2. Mathematic achievement

Math achievement was assessed with a formal Portuguese school evaluation of mathematics at the end of first grade. The Math Summative Evaluation (MSE) has 25 items. The tasks includes: counting & operations (counting, ascending and descending order, and written calculation); number decimal system (e.g. to identify the units and tens); applied problem solving (addition and subtraction verbal problems) and spatial & forms (e.g. pattern representation or geometric figures identification). The measure is reliable with an internal-consistency reliability of .86. Subtests coefficients alpha, inter-items mean correlations, and item-total mean correlations are presented in Table 2.

Table 2

Subtests coefficients alphas, mean inter-item (I-I), and item-total (I-T) correlations

Measure Cronbach Mean Mean

alpha I-I I-T

Math Summative .860 2.94 .710

Counting and Operations .729 .253 .439

Number System .764 .399 .540

Applied Problem Solving .804 .596 .666

Spatial and Forms .669 .223 .380

3. Results

3.1 NSB adaptation for the Portuguese population

The preliminary results of the present study, which includes the NSB adaptation for the Portuguese population at the beginning of first grade (presently with n=860) shows that the NSB is also reliable, with a coefficient alpha of .89. Until this moment, the NSB mean raw score for the Portuguese population is 22 points. The preliminaries

normative values by age level are presented in Table 3. The normative values allow us to identify the relationship of each individual NSB score to his level age group (5 to 10 years old) in percentiles.

Table 3

Portuguese normative values by age level

Age level

NSB Score 5 6 7 8 9 10

Percentiles

95 31 24 30 31 26 24 27

75 27 19 26 27 22 21 27

50 22 15 22 23 18 21 21

25 17 13 17 18 13 19 19

10 14 12 14 14 11 12 19

5 12 11 11 12 11 11 19

N 860 11 466 347 22 7 3

Mean 22,00 16,40 21,50 22,34 17,86 19,86 22,33

DP 6,57 4,00 5,98 6,12 5,71 3,76 4,16

In order to answer to the question "In which situation are the Portuguese children regarding early number competencies at the beginning of first grade?" we calculate the entire NSB success rate for each percentile presented in Table 4. As we can see in this Table, almost one quarter of children is below the percentile 25, another one quarter shows very good early number competencies (percentile > 75) and between these two percentiles there are half of the children with median number sense competencies.

Table 4

NSB success national rate by percentiles

Percentile N Success rate

>95 24 2,79 %

>75 194 22,56 %

>50 227 26,40 %

>25 212 24,65 %

>10 119 13,84 %

>5 43 5,00%

<5 41 4,77%

National success rate by NSB subtests are presented in Table 5. Concerning to counting skills, they do not have any problems to count a small set of five dots and to count at least until 10. The results shows weak ability to recognize numbers, only 6,4% recognize the number 13, 37, 82 e 124. Nevertheless, the Portuguese children in kindergarten only learn to recognize numbers until 10, maximum 20. Just one third of the children know the counting procedures. Related to number knowledge, only 23,3% of children could make judgments about numbers (e.g. 4 is closer to 3 than to 6) and their magnitudes (e.g. 5 is more than 3).

Regarding number operations, 52,3% of children can solve "nonverbal" problems with object representations but they have limited success with addition and subtraction story problems (23,4%) and number combinations (18,1%).

Table 5

NSB success national rate by subtests

NSB subtests Success rate

Counting 92. ,5%

Counting Principles 33. ,1%

Number Recognition 6. ,4%

Number Knowledge 23, ,3%

Nonverbal Calculation 52. ,3%

Story Problems 23. ,4%

Number Combinations 18. 1%

3.2. Number sense and mathematics achievement

Mean raw scores and standard deviations on Number Sense Brief and Math Summative Evaluation measures are presented in Table 6.

Table 6

Raw scores means (M) and standard deviations (SD) for the measures

Measures M SD

Number Sense Brief 22,59 6,00

Math Summative 71,15 21,10

Counting & Operations 17,1 5,41

Number Decimal System 14,73 4,34

Applied Problem Solving 19,28 9,85

Spatial & Forms 21,75 6,00

Bivariate correlations between the NSB raw scores and control variables are presented in Table 7. All of the correlations were positive, moderate and statistically significant (p<.01).

Table 7

Correlations between NSB and mathematics achievement

Variable Number Sense Brief

Math Summative .585

Counting & Operations .541

Number Decimal System .526

Applied Problem Solving .436

Spatial & Forms .541

The NSB accounted for 34% of the variance in mathematics achievement (Table 8). Results showed that the NSB is statistically significant to the prediction in mathematic achievement at the end of first grade (p<.01).

Table 8

Regression coefficients and variance explained by BNS in mathematic achievement

Variable R square B p t value p value

Math Summative .342 2.049 .585 7.705 .000

4. Discussion

The preliminary results show that Number Sense Brief Screener can be a reliable measure, with a coefficient alpha .89, which supports the studies of Jordan and colleagues (2006; 2007; 2008). In those studies, the NSB shows acceptable psychometric qualities (coefficient alpha between 0.82-0.89). In terms of normative values from the NSB, the raw score mean is 22 to level age 6 (which coincides with percentile 50).

This preliminary NSB results suggest that children exhibits weak counting procedures, weak number recognition (e.g. identify number "82"), weak number knowledge (e.g. comparing magnitudes), and inaccurate verbal problems' resolution and number combinations. Being the fluency in basic calculation an important tool for solving most math problems, our findings identify some children with weak basic addition and subtraction who may not be likely to achieve better fluency by second grade as Locuniak and Jordan (2008) proposed in their study. Nevertheless, is important to mention that around half of children can solve "nonverbal" problems with object representations even though they have limited success with the presence of verbal stimuli in number operations. This fact is in consonance with the study of Levine and colleagues (1992).

If we consider the value - below percentile 25 - the barrier value for children in risk of failure, our findings identify around a quarter of children with weak number competencies, and perhaps in need of educational support or intervention as suggested by Jordan and colleagues (2007) because of the relationship between number competencies and children's developmental learning trajectories in primary schools (Jordan et al., 2009). These children are likely to be at a disadvantage and may never catch to children who leave the kindergarten and started the first grade with good number competencies (Jordan et al., 2009).

In terms of mathematics achievement, the number sense, as assessed by NSB, is predictive of later mathematics achievement outcomes. Its predictability contributes 34% of the variance in first grade. It means that development and performance of the number sense at the beginning of first grade explain 34% of the performance of mathematics at the end of first grade. The relationship between earlier number sense and all mathematics achievement subareas are moderate and significant. The results are in keeping with Jordan and colleagues (2007; 2010; Locuniak & Jordan, 2008) studies.

This work can contribute for identifying Portuguese children to build number sense early (by their early number sense identification, assessed by NSB) to subsequently give them the background they need to achieve in math during the school years. From a conceptual point of view, these findings can give good indications to support the studies of Jordan and colleagues related to the NSB as a reliable early screening and identification of mathematic difficulties. However, at the present the total variance is not yet a value comparable to other studies as we have not yet achieved a consistent amount of participants. This will be done during next academic year, as will be presented the results of the third current study purpose regarding the longitudinal prospective investigation for following the mathematics learning trajectories until the second grade.

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