Author Pam Harris argues that teaching real math--math that is free of distortions--will reach more students more effectively and result in deeper understanding and longer retention. This book is about teaching undistorted math using the kinds of mental reasoning that mathematicians do. Memorization tricks and algorithms meant to make math…

The decomposition of numbers when solving subtraction tasks is regarded as more powerful than counting-based strategies. Still, many students fail to solve subtraction tasks despite using decomposition. To shed light upon this issue, we take a variation theoretical perspective (Marton, 2015) seeing learning as a function of discerning critical…

The objective of this study was to evaluate and characterize the argumentation patterns used by seventh-grade students in the context of algebraic addition and subtraction problems. A qualitative case study was conducted using the Claim-Evidence-Reasoning (CER) model (McNeill & Krajcik, 2008) and argument quality framework. To describe the…

Computation competence (CC) in simple addition and subtraction using non-counting (NC) strategies is an important learning objective in Grade 1 mathematics but many children, especially low achievers in mathematics, struggle to acquire these skills. To provide these students with the support they need, it is important to have valid and reliable…

In numerical cognition research, the operational momentum (OM) phenomenon (tendency to overestimate the results of addition and/or binding addition to the right side and underestimating subtraction and/or binding it to the left side) can help illuminate the most basic representations and processes of mental arithmetic and their development. This…

This study, which used a multiple baseline across participants' design, examines the effectiveness of a modified schema-based instructional approach to improve the mathematical word problem solving performance of three students with mild intellectual disability, two of whom had an autism spectrum disorder. Following the intervention, the three…

Several working memory processes have been hypothesized to influence different arithmetic operations. Working memory has been compartmentalized into a number of different sub-processes, such as phonological memory and visuospatial memory that are believed to have unique contributions to the performance of two distinct arithmetic operations:…

Fluency in mathematics is more than adeptly using basic facts or implementing algorithms. It is not about speed or recall. Real fluency is about choosing strategies that are efficient, flexible, lead to accurate solutions, and are appropriate for the given situation. Developing fluency is also a matter of equity and access for all learners. The…

CBM for mathematics assesses growth in accuracy and fluency of basic math skills using content from a student's curriculum. CBM for mathematics can include single-skill measures (SSM), skill-based measures (SBM), and general-outcome measures (GOM). Past research on growth rates in CBM for mathematics has focused on GOMs and has relied on…

Central elements of adaptive expertise in arithmetic problem solving are flexibility, using multiple strategies, and adaptivity, selecting the optimal strategy. Research shows that the strategies children actually use do not fully reflect the strategies they know: there is hidden potential. In the current study a sample of 147 third graders from…

Mathematical writing is one way for primary students to communicate their mathematical thinking. Research in the field of writing has shown that to become an effective teacher of writing, preservice teachers must have experience engaging in the kinds of writing given to their students. The study reported in this paper explored how 27 preservice…

To live independently, it is critical that students with disabilities maintain the basic mathematical skills they have acquired so they may apply these skills in daily life. To support maintenance of mathematical skills among students with developmental disabilities, the researchers used a multiple probe across participants design to examine the…

This chapter focuses on the development of concepts that children draw on as they work toward understanding negative numbers. Framed from a conceptual change lens, I discuss different interpretations children have of minus signs, numerical order, numerical values, and addition and subtraction operations and how children draw on these varied…

Successful math word-problem solving is difficult for some students. Getting the right answer involves students engaging correctly in an amalgamation of actions. Schema instruction is an instructional approach aimed at supporting students in identifying the underlying math problem structure to yield an appropriate solution plan. The focus of this…

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…