The goal of this design-based research study was the creation and evaluation of a mini-unit intended to foster perceptually grounded understandings of the concept of slope in middle-school students. Central to this unit was an innovative device designed to create a productive pedagogical space between student intuition for steepness and formal…

Although there is no consensus in regard to a unique meaning for abstraction, there is a recognition of the existence of several theories of abstraction, and that the ability to abstract is imperative to learning and doing meaningful mathematics. The theory of "reducing abstraction" maps the abstract nature of mathematics to the nature…

The authors believe teaching for understanding begins with the development of a few essential orientations. Teachers must have an appreciation for student centered mathematics teaching, valuing an approach that builds on student thinking. In addition, teachers must appreciate the complexity of students' mathematical thinking and ideas. Once these…

This study reports on our steps toward achieving a conceptually coherent and empirically validated learning progression for carbon cycling in socio-ecological systems. It describes an iterative process of designing and analyzing assessment and interview data from students in upper elementary through high school. The product of our development…

THE GENERAL PROBLEM UNDER INVESTIGATION CONCERNED TWO PROCESSES OF CONCEPT FORMATION - THE DEDUCTIVE REASONING PROCESS AND THE ASSOCIATIVE PROCESS - EMPLOYED BY CHILDREN OF ELEMENTARY-SCHOOL AGE. IT WAS ASSUMED THAT BOTH OF THESE PROCESSES ARE FOUND IN VARYING DEGREES AMONG CHILDREN OF DIFFERENT AGES AND MENTAL ABILITIES, DEPENDING UPON THE…

THIS EFFORT WAS AN INVESTIGATION OF THE PROCESSES WHICH OCCUR WHEN CHILDREN ARE SET TO FORM CONCEPTS AND, ALTERNATELY, THOSE WHICH OCCUR WHEN CHILDREN ARE EXPOSED TO MATERIAL AND NO DELIBERATE EFFORT IS MADE TOWARD CONCEPT FORMATION. THE EXPERIMENTAL PLAN WAS AN INDEPENDENT GROUP DESIGN IN WHICH THE MAJOR VARIABLES WERE INTENTIONALITY OF SET…

A REVIEW OF THEORY AND EMPIRICAL RESEARCH ON "THE LEARNING BY DISCOVERY HYPOTHESIS," AND RELATED LITERATURE ON CONCEPT-LEARNING AND PROBLEM-SOLVING, WAS PREPARED FOR A CONFERENCE IN NEW YORK CITY, 1965. RECENT CURRICULAR ACTIVITIES IN THE SUBJECT AREA WERE DESCRIBED. ATTEMPTS WERE MADE TO IDENTIFY INDEPENDENT AND DEPENDENT VARIABLES AND…