An international perspective of early number sense: Identifying components predictive of difficulties in early mathematics achievement

Abstract

From preschool to high school, number sense is included as an important outcome in mathematics syllabi. It is a term associated with the development of concepts and skills in mathematics and as such appears frequently in the teaching literature. However number sense is a term yet to be operationally defined. The present study used a modified Delphi procedure to determine the level of agreement amongst six early mathematics experts from Britain, Europe and the United States, with respect to a variety of components of early number sense. The results of this study are discussed in relation to a similar Australasian study and the emerging literature on early number sense assessment. This study revealed some components of number sense worthy of further investigation in relation to early mathematical achievement but also confirmed a lack of consensus on what exactly constitutes early number sense.     

Relations between mathematics achievement growth and the development of mathematics self-concept in elementary and middle grades

Creating an educational program that results in positive science, technology, engineering, and mathematics or STEM-oriented outcomes for all students is an important education objective and federal policy directive in the United States. In addition to developing strong mathematics foundations in Grades K–8 that are closely associated with successful STEM outcomes, intrapersonal skill development is also critical to academic and postsecondary success. Using nationally representative data from the Early Childhood Longitudinal Study, Kindergarten Class of 1998–99 the current study applied structural equation growth modeling to examine relationships among mathematics achievement in Grades K–1 and Grades 3–8, and mathematics self-concept development across Grades 3–8. Study results revealed that students reported declining levels of mathematics self-concept across the middle grades. Importantly, third grade reports of self-concept were associated with both mathematics achievement at kindergarten entry and third grade mathematics achievement; however, mathematics achievement at kindergarten entry was also associated with the development of mathematics self-concept in third through eighth grade. In addition, after accounting for mathematics achievement in Grades K–1, mathematics achievement growth in Grades 3–8 was directly associated with mathematics self-concept development across the same time period. Implications for future research, mathematics instruction, and curriculum development are discussed.     

Prevalence,stability, and functionality of achievement goal profiles in mathematics from third to seventh grade

This study examined the longitudinal trajectories of achievement goal profiles in mathematics from third to seventh grade in a sample of 302 German students. Latent profile analyses were conducted separately for each school year and revealed three subgroups of students with distinct goal profiles labeled high multiple goals, moderate multiple goals, and primarily mastery-oriented. Only about one third of the students held the same goal profile across all school years. The amount of students pursuing moderate multiple goals increased over time, which supplements previous findings and theorizing about an ongoing differentiation of achievement goals during early adolescence. There were remarkably few differences in educational outcomes (interest, effort, achievement) between students from distinct goal profile groups. However, high multiple goal students showed the lowest test scores in sixth and seventh grade. Moreover, if students showed low performance during one school year, they were more likely to adopt a high multiple goals profile in the following year. Results are discussed in relation to individual cognitive developments, changes in school environments, and special characteristics of educational systems.     

The structural relationship between writing attitude and writing achievement in first and third grade students

This study tested three models of the structural relationship between the writing achievement of primary grade students and their attitude towards writing (defined here as an affective disposition involving how the act of writing makes the author feel, ranging from happy to unhappy). The three models tested were: (a) writing attitude influences writing achievement in a unidirectional manner, (b) writing achievement influences writing attitude in a unidirectional manner, and (c) the effects of writing attitude and achievement are bidirectional and reciprocal. The model that best fit the data was based on the assumption that writing attitude influences writing achievement. In addition, the direct path between attitude and achievement in this model was statistically significant. Although third grade students were better writers than first grade students, there was no statistical difference in younger and older students’ attitude towards writing. In addition, girls were more positive about writing than boys, but there was no statistical difference in their writing achievement related to gender. This research extends models based on the cognitive and language processes of writing to include the role of attitude, which is an affective component of motivation.     

Linking university mathematics classroom environments to student achievement: The mediation of mathematics beliefs

This study examined the relationships between university mathematics classroom environments, mathematics beliefs, and achievement in the context of mainland China. As a type of affect in mathematics learning, mathematics beliefs including mathematics conceptions and efficacy were examined as mediators. Based on analysis of a sample of 1443 undergraduate students from a university in northern China, the study not only found that the mathematics classroom environment was associated with both affective and cognitive mathematics outcomes, but also confirmed the mediating role of mathematics beliefs. Specifically, a desirable classroom environment was related to students’ cohesive conceptions, enhanced self-efficacy, and higher mathematics achievement. In contrast, students’ perceived difficulty of learning mathematics was connected to their fragmented conceptions, reduced self-efficacy and lower mathematics achievement. Implications of the study are discussed.     

Relationship between teacher knowledge of concepts and connections, teaching practice, and student achievement in middle grades mathematics

The mixed method sequential nested study examines whether and how the cognitive type of teachers’ content knowledge is associated with student achievement, and correlated with teaching practice. In the context of this study, the cognitive type refers to the kind of teacher content knowledge and thinking processes required to accomplish a task successfully, in terms of knowledge of facts and procedures (Type 1), knowledge of concepts and connections (Type 2), and/or knowledge of models and generalizations (Type 3). A sample of 102 middle school mathematics teachers (grades 6–8) was assigned to the study from a number of school districts in an urban area in the Southwestern US. Teachers were tested using the Teacher Content Knowledge Survey. Student level data of about 2,400 middle grades students’ standardized test passing rates including percentage of students meeting the state standards by objectives were collected. The type of teachers’ content knowledge was assessed and tested for association with student achievement on the state-mandated standardized test using multivariate methods including tests for variance and independence. The nested research consisted of three phases. Substudy-1 focused on quantitative analysis of the association between cognitive type of teacher content knowledge and student achievement. Substudy-2 aimed at the correlation between cognitive type of teacher content knowledge and teaching practice. Finally, substudy-3 was a case study on examining middle grades mathematics teachers’ knowledge and understanding of fraction division.     

Classroom instructional quality,exposure to mathematics instruction and mathematics achievement in fifth grade

This study examined the quality of teacher–child interactions and exposure to mathematics instruction as predictors of 5th grade student’s mathematics achievement. The sample was a subset of the children involved in the NICHD–SECC longitudinal study (N = 657). Results indicate that, even after controlling for student demographic characteristics, more exposure to mathematics instruction was related to increased fifth grade mathematics achievement for both calculations and applied problems assessments, but there was no main effect for improved instructional quality. Findings also indicate that, in classrooms where lower instructional quality was observed, greater exposure to mathematics instruction predicted improved mathematics achievement. Findings are discussed in terms of differing aspects of mathematics instruction and the possible compensatory role of exposure to instruction in classrooms of lower quality.     

Kindergarten reasoning and achievement in grades K-3

Performance of kindergarten children on reasoning (Piaget tasks of logical thinking), visual-motor integration, and verbal development was related to achievement scores in kindergarten, second grade, and third grade. Subjects were 52 children in the kindergarten classes of a middle-class, suburban/rural school. Reasoning and visual-motor integration were related to achievement on the Metropolitan Achievement Test at the end of kindergarten. For the 43 children remaining in second grade, there was a relationship between reasoning in kindergarten and achievement on the Reading and Math Concepts sections of the Iowa Test of Basic Skills. For the 38 children remaining in third grade, kindergarten reasoning was related to total and Vocabulary scores of the California Achievement Test. Kindergarten visual-motor integration and verbal development were not related to achievement test performance at the end of the second and third grade for these subjects. Early intervention programs that encourage the development of thinking and match the child's reasoning to his academic tasks may stimulate both reasoning and achievement.     

Building number sense with Number Worlds: a mathematics program for young children

Number sense is easy to recognize but difficult to define and hence, to teach. In this article, number sense is defined in terms of the knowledge known to underlie it, as identified in cognitive developmental theory and research. A preK-2 mathematics program, called Number Worlds, that was specifically developed to teach this knowledge is described and five instructional principles that lie at the heart of the program, and that are grounded in recent theories of learning and cognitive development, are used to illustrate several of its features. Evidence of the program’s effectiveness in teaching number sense and in enhancing the mathematics learning and achievement of at-risk children is briefly presented.     

Feeling number: grounding number sense in a sense of quantity

Drawing on results from psychology and from cultural and linguistic studies, we argue for an increased focus on developing quantity sense in school mathematics. We explore the notion of “feeling number”, a phrase that we offer in a twofold sense—resisting tendencies to feel numb-er (more numb) by developing a feeling for numbers and the quantities they represent. First, we distinguish between quantity sense and the relatively vague notion of number sense. Second, we consider the human capacity for quantity sense and place that in the context of related cultural issues, including verbal and symbolic representations of number. Third and more pragmatically, we offer teaching strategies that seem helpful in the development of quantity sense coupled with number sense. Finally, we argue that there is a moral imperative to connect number sense with such a quantity sense that allows students to feel the weight of numbers. It is important that learners develop a feeling for number, which includes a sense of what numbers are and what they can do.     

突出高职特点提高数学教学质量的几点尝试

简要介绍了高职数学教学过程中的几点作法:在数学教学中突出情感教育;从实际出发,切实打好数学基础;理论联系实际,学以致用;采取多种考试模式;培养学生的自学能力.     

Making sense of and with mathematics: the interface between academic mathematics and mathematics in practice

This article summarises what we can learn from research into workplace practice and vocational preparation to inform the design of mathematics education curricula for learners in general education undertaking compulsory schooling. Key findings about workplace practices are identified and explicated through the report of a case study from research in which researchers, students, teachers and workers explored workplace and mathematical practices together. Further to this, issues of learning and personal development are considered and explored from a point of view that sees learning as practice (doing) and identity development (becoming). This leads to a proposal for principles that provide a strategic vision for curriculum design. A potential approach to tactical design that facilitates curriculum structuring is illustrated in the particular instance of understanding developing measures as a modelling activity. Overall, the exercise, whilst providing some insight into possible ways forward in curriculum development, also suggests areas that require further research and development.     

Everyday common sense and school mathematics

In recent years, proposals have been made to build mathematics teaching more closely on children's informal knowledge, i.e. on concepts and strategies formed in natural, everday settings. The article focuses on questions concerning this trend, more specifically on issues concerning how and in what sense children benefit from everyday common sense knowledge when they learn mathematics at school. Three empirical examples, collected within a field-based research project at the intermediate children, when using their everyday life experience, sometimes fail to make mathematical sense of a given task. Two factors which complicate the creation of links between common sense and mathematics are discussed: one that deals with the problem of using everyday life experience as a basis for abstracting mathematical ideas, and one which concerns the pupil's abilities to interpret an assigned task in a mathematically relevant way.     

Differential contribution of personality to the explanation of variance in achievement in the first language and mathematics at high school level

This study assessed the differential contribution of personality to the explanation of variance in achievement in the mother tongue and mathematics at the high school level. The results indicated that personality, as measured by the HSPQ, aided the explication of variance in achievement in the mother tongue significantly, but had little influence on mathematics. Personality contributed as much to the explanation of variance in the mother tongue as did the composite cognitive variable. The influence of personality on achievement in mathematics was shown to be negligible. Some practical implications for the teaching of the first language are drawn.     

The role of goal orientations for adolescent mathematics achievement

This study examines the association between personal goal orientations and mathematics achievement within the trichotomous goal framework. Participants comprised approximately 2000 seventh and eighth grade White, Hispanic, and Vietnamese students in a low-income urban school district in California. Regression analysis with classroom fixed effects minimized biases arising from non-random assignment of teachers and students to schools and classrooms. While all three achievement goal orientations were correlated with mathematics achievement, only a mastery goal orientation consistently predicted achievement when a full set of prior achievement and demographic controls were included. Performance-approach and performance-avoidance goal orientations did not predict achievement in the full model.