Examining the Factor Structure of the Home Mathematics Environment to Delineate Its Role in Predicting Preschool Numeracy, Mathematical Language, and Spatial Skills

A growing body of evidence suggests that the ways in which parents and preschool children interact in terms of home-based mathematics activities (i.e., the home mathematics environment; HME) is related to children’s mathematics development (e.g., primarily numeracy skills and spatial skills); however, this body of evidence is mixed with some research supporting the relation and others finding null effects. Importantly, few studies have explicitly examined the factor structure of the HME and contrasted multiple hypothesized models. To develop more precise models of how the HME supports children’s mathematics development, the structure of the HME needs to be examined and linked to mathematics performance. The purpose of this study was to extend prior work by replicating the factor structure of the HME (as one general HME factor and three specific factors of direct numeracy, indirect numeracy, and spatial) and using those factors to predict direct assessments of children’s numeracy, mathematical language, and spatial skills. It was hypothesized that the general HME factor would be related to each direct assessment, the direct numeracy factor would be related to both numeracy and mathematical language, and the spatial factor would be related to spatial skills. Using a sample of 129 preschool children (M age = 4.71 years, SD = 0.55; 46.5% female), a series of confirmatory factor analyses were conducted. Results diverged somewhat from prior work as the best fitting model was a bifactor model with a general HME factor and two specific factors (one that combined direct and indirect numeracy activities and another of spatial activities) rather than three specific factors as had previously been found. Further, structural equation modeling analyses suggested that, in contrast to expectations, only the direct + indirect numeracy factor was a significant predictor of direct child assessments when accounting for age, sex, and parental education. These findings provide evidence that a bifactor model is important in understanding the structure of the HME, but only one specific factor is related to children’s outcomes. Delineating the structure of the HME, and how specific facets of the HME relate to children’s mathematics skills, provides a strong foundation for understanding and enhancing the mechanisms that support mathematics development

Kindergarten Predictors of Formal Understanding of Mathematical Equivalence in Second Grade

A longitudinal study was conducted to identify unique sources of individual differences in later understanding of the equal sign as a relational symbol of equivalence (i.e., formal understanding of mathematical equivalence). The sample included 141 children from a mid-sized city in the Midwestern United States (Mage = 6 years, 2 months in kindergarten; 88 boys, 53 girls; 71% white, 8% Hispanic or Latine, 7% Black, 3% Asian, 11% multiracial or other race/ethnicity; 42% qualified for free/reduced lunch). They were assessed on three categories of skills in kindergarten including number knowledge, relational thinking, and executive functioning. These skills were hypothesized to provide a foundation for a formal understanding of mathematical equivalence (assessed in second grade) by preventing a specific, narrow misunderstanding of the equal sign that hinders learning. Results showed that kindergarten relational thinking, particularly tasks assessing non-symbolic equivalence understanding, uniquely and positively predicted formal understanding of mathematical equivalence and negatively predicted the specific misunderstanding of the equal sign in second grade controlling for IQ, free/reduced lunch status, and gender. Exploratory analyses unpacking the categories of skills into individual tasks also indicated specific areas of kindergarten instructional focus that may help children construct understanding of mathematical equivalence in future years

Research-Based Teaching Practices for Improving Students’ Understanding ofMathematical Equivalence Have Not Made it into Elementary Classrooms

Elementary math instruction traditionally has emphasizedprocedures rather than concepts. Thus, students tend to lack astrong understanding of foundational concepts likemathematical equivalence. Cognitive scientists andmathematics educators have found small yet effective ways tomodify traditional arithmetic instruction to promote students’conceptual understanding of math equivalence. Educationalstandards also now reflect this academic research. However, itis unclear whether classroom practices have caught up withresearch and policy. In the current study, we observed teachers’practices during arithmetic instruction. The goal was todetermine if teachers are using research-based practices thatpromote understanding of math equivalence and if variation inuse of research-based practices is associated with students’growth in understanding of math equivalence across the schoolyear. Eight second and third grade classrooms (M students perclassroom = 23) were observed twice during math instruction.Students completed a math test both before and after theobservation period. Research-based practices were rarelyobserved in any classrooms, so there was not much variation inclassroom use of research-based practices to predict studentgrowth. Students improved their performance on all problemtypes tested, but performance on math equivalence problemswas significantly lower than on other problem types. Resultssuggest that policies and practices designed to improvestudents’ understanding of math equivalence may not havefiltered down to affect instructional practices in classrooms

Promoting Children’s Relational Understanding of Equivalence

Deep understanding of mathematical equivalence is criticalfor later mathematical understandings. However, researchstudies and national test results have repeatedly demonstratedthat many students fail to develop adequate understanding ofequivalence. Recent work from McNeil and colleaguesproposes that this failure is partly due to the format oftraditional instruction and practice with highly similarproblems. Specifically, the change-resistance account(McNeil & Alibali, 2005) proposes that students struggle withequivalence because they have developed overgeneralized“rules” that affect how they process and approach mathproblems, (e.g., the operators are always on the left side, theequal sign means to “do something” or “give the answer”)and fail to see equations having two separate sides that arebeing related to one another. Extensive practice withproblems in a similar format (e.g., those that present allarithmetic operations on the left side of the equal sign)encourages students to develop ineffective mental models ofproblem types. We replicate and extend prior work that bringscognitive science research to the classroom. Our findingsindicate that applying research-based design principles toarithmetic practice improves student understanding ofmathematical equivalence enough to support transfer to novelproblem types

The Roles of Mathematical Language and Emergent Literacy Skills in the Longitudinal Prediction of Specific Early Numeracy Skills

Mathematical language (i.e., content-specific language used in mathematics) and emergent literacy skills predict children’s broad numeracy development. However, little work has examined if these domains predict development of individual numeracy skills (e.g., cardinality, number order). Thus, the aim of the present study was to examine longitudinal relations among mathematical language, emergent literacy skills, and specific early numeracy skills. Participants included 114 preschool children, ages 3.12 to 5.26 years (M = 4.17 years, SD = 0.59). Specifically, this study examined if mathematical language and three emergent literacy skills (print knowledge, phonological awareness, and general vocabulary) in the fall of preschool predicted 12 individual early numeracy skills in the spring, controlling for age, sex, rapid automatized naming, parent education, and autoregressors. Results indicated that mathematical language predicted development of most of the early numeracy skills (e.g., set comparison, numeral comparison, numeral identification), but findings for emergent literacy skills were not robust. Among the three emergent literacy skills, only print knowledge was a significant predictor of development in some specific numeracy skills, including verbal counting, number order, and story problems. Results highlight the important role of mathematical language in children’s numeracy development and provide the foundation for future work in designing interventions to improve early numeracy skills

Perceptual and Number Effects on Students’ Initial Solution Strategies in an Interactive Online Mathematics Game

This study investigated the effects of 1) proximal grouping of numbers, 2) problem-solving goals to make 100, and 3) prior knowledge on students’ initial solution strategies in an interactive online mathematics game. In this game, students transformed an initial expression into a perceptually different but mathematically equivalent goal state. We recorded students’ solution strategies and focused on the productivity of their first steps—whether their initial action led them closer to the goal. We analyzed log data within the game from 227 middle-school students solving four addition problems and four multiplication problems consisting of a total of 1,816 problem-level data points. Logistic regression modeling showed that students were more likely to use productive initial solution strategies to solve addition and multiplication problems when 1) proximity supported number grouping, 2) 100 was the problem-solving goal, and 3) students had higher prior knowledge in mathematics. Furthermore, when problem-solving goals were non-100s, students with lower prior knowledge were less likely to use productive initial solution strategies than students with higher prior knowledge. The findings of the study demonstrated that perceptual and number features influenced students’ initial solution strategies, and the effect of number features on initial solution strategies varied by students’ prior knowledge. Results yield important implications for designing instructional activities that support mathematics learning and problem-solving

Next Directions in Measurement of the Home Mathematics Environment

This paper synthesizes findings from an international virtual conference, funded by the National Science Foundation (NSF), focused on the home mathematics environment (HME). In light of inconsistencies and gaps in research investigating relations between the HME and children’s outcomes, the purpose of the conference was to discuss actionable steps and considerations for future work. The conference was composed of international researchers with a wide range of expertise and backgrounds. Presentations and discussions during the conference centered broadly on the need to better operationalize and measure the HME as a construct – focusing on issues related to child, family, and community factors, country and cultural factors, and the cognitive and affective characteristics of caregivers and children. Results of the conference and a subsequent writing workshop include a synthesis of core questions and key considerations for the field of research on the HME. Findings highlight the need for the field at large to use multi-method measurement approaches to capture nuances in the HME, and to do so with increased international and interdisciplinary collaboration, open science practices, and communication among scholars

Next directions in measurement of the home mathematics environment: an international and interdisciplinary perspective

This article synthesizes findings from an international virtual conference, funded by the National Science Foundation (NSF), focused on the home mathematics environment (HME). In light of inconsistencies and gaps in research investigating relations between the HME and children’s outcomes, the purpose of the conference was to discuss actionable steps and considerations for future work. The conference was composed of international researchers with a wide range of expertise and backgrounds. Presentations and discussions during the conference centered broadly on the need to better operationalize and measure the HME as a construct – focusing on issues related to child, family, and community factors, country and cultural factors, and the cognitive and affective characteristics of caregivers and children. Results of the conference and a subsequent writing workshop include a synthesis of core questions and key considerations for the field of research on the HME. Findings highlight the need for the field at large to use multi-method measurement approaches to capture nuances in the HME, and to do so with increased international and interdisciplinary collaboration, open science practices, and communication among scholars