Meaningful learning occurs when individuals connect their existing knowledge with their current understanding. Unfortunately, many teaching practices neglect the learners' subconscious knowledge, also known as their "funds of knowledge." The result is a disconnect between school and home knowledge, social injustice towards students whose…

For a semester within a transition-to-proof course, mathematics majors explored two famous conjectures: The Twin Primes Conjecture and the Collatz Conjecture. Students were scaffolded into exploring the conjectures through directed activities but were also expected to create their own methods of exploration. We documented students' experiences…

Feedback in exploratory learning systems has been depicted as an important contributor to encourage exploration. However, few studies have explored learners' interaction patterns associated with feedback and the use of external representations in exploratory learning environments. This study used Fractions Lab, an exploratory learning environment…

Proof and proving are important components of school mathematics and have multiple functions in mathematical practice. Among these functions of proof, this paper focuses on the discovery function that refers to invention of a new statement or conjecture by reflecting on or utilizing a constructed proof. Based on two cases in which eighth and ninth…

The purpose of this research was to: 1) development of learning devices based guided discovery model in improving of understanding concept and critical thinking mathematically ability of students at Islamic Junior High School; 2) describe improvement understanding concept and critical thinking mathematically ability of students at MTs by using…

An emerging consensus suggests that guided discovery, which combines discovery and instruction, is a more effective educational approach than either one in isolation. The goal of this study was to examine two specific forms of guided discovery, testing whether conceptual instruction should precede or follow exploratory problem solving. In both…

In 2001, the Chinese Ministry of Education issued Guidelines for Preschool Education (GPE) (trial version) to call on early childhood practitioners to use a child-centered and play-based approach to teaching and learning. The guidelines also include mathematics within the science domain and described its standards in a way that significantly…

This article presents an instructional approach to constructing discovery-oriented activities. The cornerstone of the approach is a systematically asked question "If a mathematical statement under consideration is plausible, but wrong anyway, how can one fix it?" or, in brief, "If not, what yes?" The approach is illustrated with examples from…

In many mathematics classes, students are asked to learn via the discovery method, in the hope that the intrinsic beauty of mathematics becomes more accessible and that making conjectures, forming hypotheses, and analyzing patterns will help them compute fluently and solve problems creatively and resourcefully (NCTM 2000). The activity discussed…

"Principles and Standards for School Mathematics" (NCTM 2000) proposes that mathematics instruction provide opportunities for students to engage in mathematical inquiry and in meaning-making through discourse. Mathematics teachers are encouraged to build on student discoveries in designing subsequent instruction. In this article, the author…

Discussed are generalizations and problem solving, unsound uses of generalizations, generalization as data, and student discovery of generalizations. (CT)

A discussion on the general equation of a line shows how students can be lead to discover mathematical properties, find how one discovery leads to another, see how different branches of mathematics can lead to a solution, and be provided with a starting point for studying special mathematical topics. (MP)

This paper describes an approach to task analysis which seeks to identify potential sources of difficulty in the self-discovery of improved procedures by students who have been taught simpler procedures. The approach considers novices' procedures in terms of the changes needed to produce an expert procedure; the knowledge required to make those…

Using mathematics as an example, the nature of the confusion between "knowledge" and "coming to know" is elucidated. The range of curriculum choices available to structuralists once the distinction is clarified is larger, and teaching a body of knowledge and teaching by discovery are compatible. (Author/MLF)