This commentary reviews each of the three content chapters in the integers section and offers questions to promote further discussion. In addition to the themes raised in the three chapters, I introduce the role of formal mathematical structure in generalizing systems of number, from natural numbers to integers, and analogously, from real numbers…

This paper will provide several examples of science and mathematics integration: navigation/map-reading, ecology/ecosystems/population growth, and chemistry/molecular structures. This paper underscores integrating STEM subjects with problem-based learning with technology such as video/computer simulations/programming/coding and the dynamic free…

In this Research Commentary, the author explores what is meant by "teaching for understanding" and delves into these questions: How does teaching for understanding interact with the backgrounds of the students who experience it or the attributes of the contexts in which they learn? Which empirical findings are context dependent, and…

Improving student learning is the primary goal of every teacher of algebra. Teachers seek strategies to help all students learn important algebra content and develop mathematical practices. The new Institute of Education Sciences[IES] practice guide, "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students"…

In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence--namely, in Toulmin's sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument…

The Organization for Economic Cooperation and Development [OECD] is a global policy organization that includes the United States and about half of the Western Europe countries. It administers international comparison tests, called Programme for International Student Assessment (PISA), for 15 year-old students in Mathematics and other subjects. I…

This article presents the authors' response to Asha K. Jitendra's Response to Xin and Zhang ("The Journal of Educational Research," 2009, Vol. 102, No. 6). The claim of "factual errors, including inappropriate quotes, misrepresentation of information, and inadequate scholarship in Xin and Zhang's text" are not justified. In this rejoinder, the…

Math education in the United States remains resistant to systemic change, and our country pays the price. Stemhagen's article "Democracy and School Math" further confirms this trend. Despite repeated calls for reform, decades of research on how people learn, millions of dollars invested in teacher professional development, and years of…

In this article, the author explores what Assessing Pupils' Progress (APP) is about. He contends that the predilection for testing is a catastrophe as far as the teaching and learning of mathematics is concerned; it is an outcome of the drive for collecting so-called "data" on pupils. What those people, who should know better, either choose to…

President Bush appointed a 19-member National Mathematics Advisory Panel (NMP) in 2006 to create a blueprint for advancing mathematics education in the United States. After reviewing over 16,000 research publications and reports and receiving testimonials from 110 individuals, the panel made its report public in March 2008. The NMP report provides…

It is argued that algebra is a gateway to a rich high school curriculum and later success at a four-year college. Therefore, everyone should take algebra and take it as early as possible. In this article, the author contends that it is a fallacious argument that infers causation from correlation. He believes that the kids who take algebra early…

The author responds to the recent work of Kaminski, Sloutsky, and Heckler (2008) and advances two major concerns about their research and its applicability to learning mathematics: a confounding variable that arises from the mathematical differences between the generic examples and concrete examples poses a threat to the construct validity of the…

In "Elementary School Mathematics Priorities," Wilson (2009 [this issue]) presents a list of five core concepts that students should master in elementary school so that they can succeed in algebra. As researchers in mathematics education, the authors enthusiastically endorse Wilson's recommendations. Learning algebra is key to further study of…

Flexible knowledge, knowing multiple approaches for solving problems, is a hallmark of expertise in mathematics. Frequently, the author writes, students memorize only one method of solving a certain kind of problem, without understanding what they are doing, why a given strategy works, and whether there are alternative solution methods. Comparison…

After experiencing a Developing Mathematical Ideas (DMI) class on the construction of algebraic concepts surrounding zero and negative numbers, the author conducted an interview with a first grader to determine the youngster's existing level of understanding about these topics. Uncovering young students' existing understanding can provide focus…