Laura Bofferding

Middle and High School Students' Conceptions of Climate Change Mitigation and Adaptation Strategies.

Both scientists and policy-makers emphasize the importance of education for influencing pro-environmental behavior and minimizing the effects of climate change on biological and physical systems. Education has the potential to impact students’ system knowledge – their understanding of the variables that affect the climate system – and action knowledge – their understanding of behaviors that can impact the system. Research on climate change education has largely focused on system and action knowledge that address mitigation while overlooking equally necessary adaptive responses. This study used a pre/post-test format to identify aspects of middle and high school students’ climate system knowledge and action knowledge of both mitigation of and adaptation to climate change. Results indicate that adolescents currently conflate climate change mitigation strategies with unrelated environmental problems far less than in previous surveys. However, students demonstrated limited understanding of adaptive responses to climate change. After engaging in an instructional unit on climate change, students expressed stronger system and action knowledge, but significant misconceptions remained that conflated mitigation of and adaptation to climate change.

Comparing the Effectiveness of Virtual and Concrete Manipulatives to Teach Algebra to Secondary Students With Learning Disabilities

A sizable body of literature exists studying various technologies and pedagogical practices for teaching secondary mathematics curriculum to students with a learning disability in mathematics. However, with the growing footprint of computer-based technologies in today’s classrooms, some areas of study, such as the use of virtual manipulatives, lack sufficient exploration. Although concrete manipulatives were studied for many decades for students with a learning disability and are considered a best practice, the research base for virtual manipulatives is notably less. With a specific focus on algebraic instruction, this study sought to compare the benefits of both forms of manipulatives to assist secondary students with a learning disability in mathematics to solve single-variable linear equations using a single-subject alternating treatment design. Over the course of 30 sessions of intervention, three students exhibited over 90% average accuracy solving problems using both virtual and concrete manipulatives, while the concrete manipulative earned higher scores for two of the three students.

Learning to "See" Less than Nothing: Putting Perceptual Skills to Work for Learning Numerical Structure.

How can childrens natural perceptuo-motor skills be harnessed for teaching and learning mathematical structure? We address this question in the case of the integers. Existing research suggests that adult mental representations of integers recruit perceptuo-motor functionalities involving symmetry. Building on these findings, we designed a hands-on curriculum that emphasizes symmetry to teach integer concepts to fourth graders. Compared to two control conditions, children who went through the experimental curriculum showed evidence of incorporating symmetry into their mental representations of integers and performed higher on problems beyond the scope of instruction, including negative fractions and algebra-readiness problems. Gains did not come at the expense of basic integer computation skill. This study has direct practical implications, as current integers curricula generally omit symmetry. The research demonstrates an approach to designing instruction that involves identifying perceptuo-motor functionalities underlying numerical cognition and creating learning activities to recruit them.

Nuances of Prospective Teachers’ Interpretations of Integer Word Problems

This chapter identifies the ways in which 15 prospective teachers engage the strands of mathematical proficiency as they solve word problems involving integer addition and subtraction. The prospective teachers, through think-aloud interviews, demonstrated a strong focus on solving problems using procedures, which some did not explain and others explained in detail. Number line representations were popular ways to illustrate solution methods, especially to highlight distances to and from zero. Further, some problems elicited a variety of strategies, while others mainly elicited procedures. The collective think-aloud data reveal strong, interconnected strands that could help individuals reflect on procedural versus conceptual knowledge and how best to explain and make connections among the ideas involved in the problems.

Challenges of Promoting Conceptual Change with Instructional Contexts

This chapter focuses on the interaction of two first graders as they attempt to make sense of a particular instructional context for learning negative numbers. The context is one where they move an elevator to a building’s floors above and below ground in order to model integer addition and subtraction problems. In particular, the focus of the activity was to discover that solving problems such as 4 – 1 and 1 – 4 will result in different answers. The two students misinterpret each other, model the problems in multiple ways with the elevator (and to varying extents), but also work cooperatively at times as they complete the activity. Their efforts highlight the difficulties they encounter when working with ready-made contexts and in obtaining solutions that do not fit their prior experiences. The results present a brief view of the conceptual change process and support a stronger focus on connecting to students’ prior thinking when introducing new instructional contexts.

Conclusion: Reflecting on the Landscape: Concluding Remarks

The conclusion contains a response to the chapters and commentary in this book describing the thinking, models, and metaphors for integer addition and subtraction. This response includes three main sections: establishing landmarks, valuing emergent thinking, and critiquing integer instructional models. First, we further discuss the need to establish landmarks, or use clearly defined language (e.g., order, magnitude, strategies), in our work with integers. Second, we suggest that valuing emergent thinking within the research on thinking and learning of integer operations is important and entails less focus on correct strategies and places more value on the development of integer understanding. And, last, we critique the consistent rhetoric of “meaningful” for both contexts and instructional models by highlighting that what is meaningful to children may not be meaningful to teachers and researchers (and vice versa). We end the conclusion by posing questions for future research in the realm of thinking and learning within integer addition and subtraction.

Playing with Integer Concepts: A Quest for Structure

How children play around with new numerical concepts can provide important information about the structure and patterns they notice in number systems. In this chapter, we report on data from 243 second graders who were asked to fill in missing numbers on a number path (encouraging them to play around with numbers less than zero) and to solve integer arithmetic problems (encouraging them to play around with the concepts of addition and subtraction involving negative integers). When playing around with the number paths, students made patterns, continued the number sequence in interesting ways, and used invented notation. When playing around with the operations, they interpreted negative signs as subtraction signs or added negative signs to their answers. Their play with the number path often connected to their play with operations, revealing that although some students were attuned to the pattern in the order of numbers and operations as movement in a particular direction, others focused more on numerical values and operations as changes in amount. The various ways in which children played with integers provide insight into their conceptual change process and can provide guidance for ways teachers could help students build on their logic.

Improving the Numerical Knowledge of Children with Autism Spectrum Disorder: The Benefits of Linear Board Games

A lack of numerical knowledge early on can impede a childs academic development. In past research, playing linear board games improved childrens understanding of numerical relations, which the authors theorised could extend to children with autism spectrum disorder. For this pilot study, 10 children played a board game where they moved tokens across coloured tiles displaying the numbers 1–10. During gameplay, students in the experimental group focused on the numbers on the tiles, while the control group focused on the colours. The assessment consisted of a number line estimation task, where students placed numbers ranging from 1 to 9 on a blank number line. Results showed children in the experimental group experienced statistically significant improvement in their understanding of numerical relationships of numbers on the number line estimation task. Findings indicate extended experiences with linear board games can support the numerical development of children with autism spectrum disorder.