This research explores the interplay between students' understandings of proportional and functional relationships. Approximately 90 students participated in an early algebra intervention in Grades 3- 5. Before the intervention and after each year of the intervention, we evaluated their understandings of proportional and functional relationships.…

This paper addresses one of the foundational components of beginning interference, namely variation, with 5 classes of Year 4 students undertaking a measurement activity using scaled instruments in two contexts: all students measuring one person's arm span and recording the values obtained, and each student having his/her own arm span measured and…

In this paper, we describe our learning progressions approach to early algebra research that involves the coordination of a curricular framework, an instructional sequence, written assessments, and levels of sophistication describing the development of students' thinking. We focus in particular on what we have learning through this approach about…

The context of this paper is studies worldwide on influence of phonological factors in language development of children. Such studies reveal the significance of Phonological Awareness in development language skills: including, predictive value of phonological short-term memory for reading skills in Grade 1. This paper throws light on factors in…

Students explored variation and expectation in a probability activity at the end of the first year of a 3-year longitudinal study across grades 4-6. The activity involved experiments in tossing coins both manually and with simulation using the graphing software, "TinkerPlots." Initial responses indicated that the students were aware of…

"Make It Count" was a large scale, government-funded, project aimed at improving the mathematics learning of Indigenous students. NAPLAN Numeracy test results were used as one measure of the effect of the program. In this paper we report on the performance on these tests of Indigenous students in schools involved in the project. Group…

This paper describes a retrospective analysis of data collected during a 4-year longitudinal study on children's thinking about measurement through a teaching experiment methodology. It focuses on results from individual interviews with two students on volume measurement. Data analysis was guided by Sarama and Clements' (2009) learning trajectory…

This paper reports on the mathematical modelling of four classes of 4th-grade children as they worked on a modelling problem involving the selection of an Australian swimming team for the 2004 Olympics. The problem was implemented during the second year of the children's participation in a 3-year longitudinal program of modelling experiences…

Increasing numbers of mathematics educators, policy makers, and researchers believe that algebra should become part of the elementary education curriculum. Such endorsements require careful research. This paper presents the general results of a longitudinal classroom investigation of children's thinking and representations over two and a half…

We explore how fourth grade (9 to 10 year olds) students can come to understand and use the syntactic rules of algebra on the basis of their understanding about how quantities are interrelated. Our classroom data comes from a longitudinal study with students who participated in weekly Early Algebra activities from grades 2 through 4. We describe…

This paper is a report on the results of a 3-year teaching experiment conducted in Ithaca, New York, introducing students to the concepts of multiplication, division, and ratio as a trio, and to ratio and proportion in a project-based curriculum with heterogeneous grouping. Fractions were introduced as a subset of ratio and proportion. The paper…

A longitudinal study of growth in reading ability employed two new indicators of print exposure that use a checklist-with-foils logic and that have very brief administration times. Subjects, 52 boys and 46 girls from 4 different fourth- and fifth-grade classes in a religiously-affiliated private school, completed the Title Recognition Test (TRT)…

A longitudinal study of students' developing understanding of decimal notation has been conducted by testing over 3000 students in Grades 4 to 10 up to 7 times. A pencil-and-paper test based on a carefully designed set of decimal comparison items enabled students' responses to be classified into 11 codes and tracked over time. The paper reports on…

Through timely questioning, teachers can estimate a child's understanding of mathematical ideas. Hence, the type of question asked by the teacher must be connected to the student's present thinking about a solution. This paper illustrates and analyzes how timely questioning on the part of teachers helped third, fourth, and fifth grade students…

A study investigated how knowledge of a word's meaning, phonology, and orthography influences who will become a poor reader and who will develop adequate skill. Subjects (38 students from a small Ontario, Canada, city) were selected in grade 2 for their poor reading ability and followed until they reached grade 4. Conditions were devised in which…