For many of my students, Real Analysis I is the first, and only, analysis course they will ever take, and these students tend to be overwhelmed by epsilon-delta proofs. To help them I reordered Real Analysis I to start with an "Analysis Boot Camp" in the first 2 weeks of class, which focuses on working with inequalities, absolute value,…

The importance of reasoning quantitatively is reflected in both mathematics education research and mathematical standards for K-12 students. In this article, we detail how a quantitative reasoning framework can be used to help differentiate two quantities we have found students often struggle with: arc length and the measure of a central angle. We…

Of all the arguments directed against presenting mathematical topics within a cultural and historical context, the most serious ones are those that we will refer to as the time and the content arguments. In this paper, after briefly describing these stances, we will endeavor to evaluate and refute their rationales using a social constructivist…

Mathematical argumentation, justification, and proof are practices at the heart of mathematics. Yet mathematics teachers generally operate without clear definitions of these practices making it difficult to communicate expectations, decide what to accept or expect from students at different grade levels, and distinguish these activities from…

Using student mathematical thinking during instruction is valued by the mathematics education community, yet practices surrounding such use remain difficult for teachers to enact well, particularly in the moment during whole-class instruction. Teachers' orientations--their beliefs, values, and preferences--influence their actions, so one important…

In this study, based on the analysis of a teaching experiment with middle school students, we propose a framework for describing meanings of a point represented on a plane in terms of multiplicative objects in the context of graphing. We classify those meanings as representing: (1) non-multiplicative objects; (2) quantitative multiplicative…

Along with the rapid development of technology, videorecording teachers' lessons for professional development or educational research have become commonplace. Although the benefits of the use of videorecording has been well documented, few studies have attended to teachers' affective experiences in relation to videorecording. In this study, we…

Mathematical proof is a logically formed argument based on students' thinking process. A mathematical proof is a formal process which needs the ability of analytical thinking to solve. However, researchers still find students who complete the mathematical proof process through intuitive thinking. Students who have studied mathematical proof in the…

In educational research, the cognitive activity of problem posing is recognized as an important component of mathematics teaching and learning. Compared to the prevailing educational paradigm of problem solving, problem posing features less commonly in classroom instruction. During the past 20 years, numerous studies examining the use of problem…

The research literature on middle grades mathematics teachers' mathematical literacy indicates that mathematical problem solving and problem posing are essential skills for mathematics instruction, and that teachers must be skilled in mathematical problem solving and mathematical problem posing. However, it is not well known how these two skills…

After being introduced to "ambitious" teaching methods during teacher education, only some beginning teachers attempt to enact those methods, as others gravitate toward the conventional practices that overwhelm K-12 settings. To learn more about why, I conducted a multiple-case study of four beginning secondary mathematics teachers who…

An increasing number of high school students are enrolling in statistics courses. Hence, an increasing number of teachers are needed to plan and implement these courses. Yet, many teachers report feeling unprepared to teach statistics. When planning and implementing these courses, teachers draw upon their knowledge and beliefs when making…

While teacher evaluation policies have been central to efforts to enhance teaching quality over the past decade, little is known about how teachers change their instructional practices in response to such policies. To address this question, this paper drew on classroom observation and survey data to examine how early career teachers' (ECTs')…

Justification, an essential mathematical practice, is well-known for its role in promoting rigor and developing mathematical understanding. Equally powerful is its role as a discursive practice that provides students with access to mathematical thinking and reasoning (the mathematics behind the answers) and promotes student agency with respect to…

Sickle cell disease is a very common disease throughout the world. We used it to contextualize probability for upper elementary school students. Using Punnett squares, activities guide students' evaluation of inheritance using theoretical probability. Then, students perform trials using tools to compare theoretical and experimental probabilities.