The concept of multiple representations of functions and the ability to make translations among representations are important topics in secondary school mathematics curricula (Moschkovich, Schoenfeld, & Arcavi, 1993; NCTM, 2000). Research related to students in this domain is fruitful, while research related to teachers is underdeveloped. This…

Earlier research has found that it is useful to distinguish situations in which students construct external representations on their own from situations in which they are expected to interpret already provided external representations. One type of representations that is particularly important for teaching mechanics is the free-body diagram. In…

It has been argued that for novice students to acquire a full understanding of scientific texts, they also need to pursue a recurrent construction of their comprehension of scientific concepts. The course dossier method has students examine concepts in multiple passes: (a) through reflective writing on text before it is considered in the…

For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…

I advance a theoretically and empirically-grounded case for designing for and learning from failure, and instantiate it in a learning design called Productive Failure (PF). I describe the key mechanisms and the design principles of PF. The PF learning design comprises a generation and exploration phase followed by a consolidation and knowledge…

This study reports teachers' insights and challenges after one year of adopting a curricular material designed to move students through carefully engineered, small steps and encourage learners through success and accessible challenges. The analysis of interviews showed that teachers 'followed' the material in different ways, not necessary in-line…

The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) define what children should understand and be able to do in K-grade 12. This document also includes a description of key mathematical processes and proficiencies, the Standards for Mathematical Practice (SMPs), which provide an important overview for the kind of robust thinking…

A story problem about borrowing money, presented in this article, may be represented with positive or negative numbers and thought about in different ways. The authors describe ideas related to integers (both positive and negative) and how students used them in relation to a story problem, and how they related these ideas to equations.

Over the course of her teaching career, this author learned to create environments in which both the teacher and learners embrace understanding. She introduced new concepts with a general question or word problem and encouraged students to find solutions with a strategy of their choice. By using this instructional method, she allowed her students…

When students begin work with fraction division in fifth grade or sixth grade, they bring with them experiences from whole-number division. Many students think that a division problem should lead to a quotient that is smaller than the dividend. It is also common for students to believe that the dividend should be larger than the divisor. Many, if…

Recently, over 40 states in the United States adopted the Common Core State Standards for Mathematics (CCSSM) which include standards for content and eight standards for mathematical practices. The purpose of this study was to better understand the nature of young children's mathematical practices through an exploratory examination of the…

Abstract thinking is generally highly correlated with problem-solving ability which is predictive of better adaptive functioning. Measures of conceptual reasoning, an ecologically-valid laboratory measure of problem-solving, and a report measure of adaptive functioning in the natural environment, were administered to children and adults with and…

Mathematical learning and teaching are increasingly seen as a multimodal experience involved in cultural and social semiotic registers and means, and as such social-cultural semiotic analysis is expected to shed light on learning and teaching processes occurring in the mathematics classroom. In this research, three social-cultural semiotic…

The present study investigated the underlying reasons for difficulties faced by students when they applied the concept of inertia across varying contexts. The participants of the study included five high school students. Data obtained from interviews were interpreted from the perspectives of the coordination class and epistemological framing…

Researchers believe that the way that students talk, specifically the language that they use, can offer a window into their reasoning processes. Yet the connection between what students are saying and what they are actually thinking can be ambiguous. We present the results of an exploratory interview study with 10 participants, designed to…