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This article discusses a problem-based curriculum that aims to help students understand the Multiplication Principle (MP) for counting in combinatorics. The curriculum consists of various exercises that focus on different aspects of the MP, such as the distinction between multiplication and addition, the role of order in counting, and the concept of independence. These exercises are designed to engage students and promote independent and collaborative learning. The curriculum follows the principles of variation theory and encourages full-class discussions to enhance students' conceptual understanding of the MP. [Extracted from the article]

As a parent, the author stepped into his child's class on a Friday morning to a room buzzing with activity. Parents walked around the room, coffee and bagel in hand, reading stories that their child (and others) had drafted, revised, written, and illustrated. Students eagerly shared their stories and drawings, cherishing the comments and praise from their parents (and others). Writing was a favorite subject, and "reaching a higher reading level" filled the hallways as students' goals for the year. All of this was great. The problem was the lack of anything remotely comparable in mathematics. So, the author's daughter's second-grade teacher and the author, a mathematics teacher educator, brainstormed ideas to do something similar in mathematics (at least once). They decided that this project would have to accomplish a few things: It must: (1) be a problem that students would work on, revise, and reason about for multiple days; (2) result in a student-created work at the end, that students' parents would have to explore; (3) be enjoyable; and (4) have some meaningful mathematics that related to what they already had learned or were learning. And so it--what would later be dubbed "Math Madness"--began. The project ultimately introduced students to graph theory--in particular, graph coloring. In this article, the author describes the activity and some of the thinking behind the decisions as well as some of the reactions to it.

In an undergraduate Calculus III class, we explore the effect of "flipping" the instructional delivery of content on both student performance and student perceptions. Two instructors collaborated to determine daily lecture notes, assigned the same homework problems, and gave identical exams; however, compared to a more traditional instructional approach, the flipped instructor utilized videos to communicate more procedural course content to students out-of-class, with time in-class spent on more conceptual activities and homework problems. Findings from two semesters indicate similar performance on more procedural problems and small to moderate gains for the flipped students (N = 74) over their traditional counterparts (N = 77) on more conceptual exam problems. However, student perceptions remain mixed, with flipped students reporting increased communication during class but traditional students perceiving more effective use of class time, despite the gains in performance for flipped students.

This manuscript presents findings from a study about the knowledge for and planned teaching of standard deviation. We investigate how understanding variance as an unbiased (inferential) estimator--not just a descriptive statistic for the variation (spread) in data--is related to teachers' instruction regarding standard deviation, particularly around the issue of division by n-1. In this regard, the study contributes to our understanding about how knowledge of mathematics beyond the current instructional level, what we refer to as nonlocal mathematics, becomes important for teaching. The findings indicate that acquired knowledge of nonlocal mathematics can play a role in altering teachers' planned instructional approaches in terms of student activity and cognitive demand in their instruction.

This article draws on semi-structured, task-based interviews to explore secondary teachers' (N = 7) understandings of inverse functions in relation to abstract algebra. In particular, a concept map task is used to understand the degree to which participants, having recently taken an abstract algebra course, situated inverse functions within its group structure (i.e., the set of invertible functions under composition). In addition, their particular conceptions of functions and function composition throughout the interviews were then also considered as a means to explore further their responses during the interviews. Findings indicate that only two participants showed evidence of the desirable mathematically powerful understandings from abstract algebra in relation to inverse functions, and further analysis suggests a variety of challenges in terms of developing meaningful connections, which were more related to conceptions about secondary content than to the abstract algebra content. Implications for the mathematical preparation of secondary teachers are discussed.

Future teachers often claim that advanced undergraduate courses, even those that attempt to connect to school mathematics, are not useful for their teaching. This paper proposes a new way of designing advanced undergraduate content courses for secondary teachers. The model involves beginning with an analysis of the curriculum and practices of school mathematics and its teaching, and then using those to build up to the advanced mathematics--in this case, real analysis. After developing definitions, examples, theorems, and proofs, the model then reconnects to practice, asking the teachers to translate ideas from real analysis in ways that are appropriate for teaching high school content to students. To illustrate the model, we provide and discuss two example tasks.

This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by teachers' knowledge of abstract algebra are developed. In each of the four content areas (arithmetic properties, inverses, structure of sets, and solving equations), descriptions and examples of the transformational influence on teaching these topics are used to depict and support ways that study of more advanced mathematics can influence teachers' practice. Implications for the mathematical preparation and professional development of teachers are considered.

The work that mathematics teachers do is frequently mathematical in nature and different from other professions. Understanding and describing common ways that teachers draw upon their content knowledge in the practice of teaching is important. Building on the descriptions by McCrory et al. ("Journal for Research in Mathematics Education" 43(5) 584-615, 2012) of two primary ways teachers use their content knowledge--decompressing and trimming--this paper differentiates between micro- and macro-levels of mathematical complexity as a way to extend and further conceptualize the moves that teachers make at a more nuanced grain size. This paper explains and portrays their counterparts, micro-level trimming and macro-level decompressing, which we discuss as concealing and foreshadowing complexity, respectively. Support for their use in teachers' work is provided through specific examples. Implications for the mathematical preparation and professional development of teachers are discussed.

The purpose of this study was to investigate teachers' subject matter knowledge relevant to the teaching of informal line of best fit. Task-based interviews were conducted with nineteen pre-service and in-service mathematics teachers. The results include descriptions and categorizations of teachers' conceptions, criteria for placement, accuracy of placement, and interpretation of the informal line of best fit. Implications regarding teacher preparation for the teaching of this topic, including current status and recommendations for future preparation, are discussed.

Finding and designing tasks that allow for students to make connections among mathematical ideas is important for mathematics educators. One such task, which affords students the opportunity to make connections and engage with significant mathematical ideas through a variety of problem-solving approaches, is described in this article. Three solutions to the task that draw on statistical, probabilistic (and stochastic), and algebraic methods are described. This task has been used with both secondary school students, mainly juniors and seniors, and in-service mathematics teachers during extended 90- to 120-minute blocks. Although serving slightly different purposes with different groups of students, the task supports introducing, developing, and reviewing some important mathematical ideas, with connections between at least some of the methods presented here being discussed.

The recent trend to unite mathematically related disciplines (science, technology, engineering, and mathematics) under the broader umbrella of STEM education has advantages. In this new educational context of integration, however, STEM teachers need to be able to distinguish between sufficient proof and reasoning across different disciplines, particularly between the status of inductive and deductive modes of reasoning in mathematics. Through a specific set of mathematical conjectures, researchers explored differences between mathematics (n = 24) and science (n = 23) teachers' reasoning schemes, as well as the confidence they had in their justifications. Results from the study indicate differences between the two groups in terms of their levels of mathematical proof, as well as correlational trends that inform their confidence across these levels. Implications particularly for teacher training and preparation within the context of an integrated STEM education model are discussed.

Today, the Common Core State Standards for Mathematics (CCSSI 2010) expect students in as early as eighth grade to be knowledgeable about irrational numbers. Yet a common tendency in classrooms and on standardized tests is to avoid rational and irrational solutions to problems in favor of integer solutions, which are easier for students to comprehend and check. Common irrational numbers as well as their applications in the world are frequently approached from their traditional developments. This article provides an exploration of how the irrational number "e" emerges from various solution strategies to an unconventional problem and an example of how problem posing can lead to interesting mathematics. A bibliography is included.

Facilitating the transition of STEM teachers into the teaching profession represents an important challenge in teacher education. We argue that it is those aspects of excellent teaching that beginning teachers believe to be important that may be the central foci for teacher preparation. In the context of the nationally replicated UTeach program, we explore how beginning UTeacher's beliefs about important instructional approaches (Study 1) relate to observed classroom practices (Study 2). UTeachers valued classroom practices such as: designing engaging, inquiry-based activities in real-world contexts; responding flexibly to student needs by modifying instruction, differentiation, and using questioning strategies; and effectively communicating content. However, UTeachers showed limited mastery of some practices--they were successful at designing engaging activities but struggled with in-the-moment teaching behaviors such as higher-level questioning and modifications. We also found that while UTeachers reported valuing their content knowledge, they sometimes had difficulty effectively translating their content expertise into a K-12 teaching context.

Algebraic structures are a necessary aspect of algebraic thinking for K-12 students and teachers. An approach for introducing the algebraic structure of groups and fields through the arithmetic properties required for solving simple equations is summarized; the collective (not individual) importance of these axioms as a foundation for algebraic thinking is discussed by way of an example of an abstract group. Results of this approach in a mathematics for teachers course are presented, with K-12 teachers (n = 12) having increased their content knowledge both about arithmetic properties and abstract algebra, as well as having their teaching beliefs and intended practices influenced. Implications for classroom teaching about a vertical slice of the K-12 curriculum related to algebraic structure are examined.

With the adoption of the "Common Core State Standards for Mathematics" (CCSSM), the process of mathematical modeling has been given increased attention in mathematics education. This article reports on a study intended to inform the implementation of modeling in classroom contexts by examining students' interactions with the process of creating mathematical models and analyzing the types of models so generated. Results indicate students' tendencies to oversimplify or overcomplicate the modeling process. Implications of the study are discussed, especially for understanding which aspects of the modeling cycle might be most helpful for teachers to focus on in order to develop students' modeling abilities.

Last year the United States unveiled the Common Core State Standards (CCSS) in English and Mathematics for grades K-12. In particular, the authors included two possible sequences of 8-12 mathematics courses that would fulfill the standards. Most notably, the courses titled "3a" and "3b" in these two sequences have become gatekeepers to Pre-Calculus (and consequentially Calculus). Taking "3b" would not prepare students to take Pre-Calculus, but at that juncture students would be prepared for a variety of other possibilities among mathematics courses--Statistics, Finance, Modeling, Linear Algebra, Discrete Mathematics, and Computer Science (those in "3a" would have access to all of these and Pre-Calculus). Employing Harvey & Knight's analytic framework on educational quality, this article compares who has access to taking various high school mathematics courses in three countries: the U.S., Finland, and Singapore. Using the framework as a lens to discuss various statistics and different measures of quality, the new CCSS offer a relatively wide variety of courses for high school students, aiming to make the mathematics classes required useful to the students who take them, while simultaneously keeping options open for higher level mathematics.

As the national impetus to advance education continues to look toward improving the teachers sent into the classroom, more research needs to be done to in the area of mathematics teacher education. A number of recent studies have summarized, synthesized, and developed useful information about what should be viewed as important in mathematics teacher training. This paper looks at some of these claims from current research and uses the UTeach program from the University of Texas at Austin as a means of elaborating on how some theoretical issues might be addressed in practice. A number of implications are proposed in response to the discussion, one in particular addressing a gap in current research about understanding when beginning mathematics teachers learn the attributes helpful for good mathematics instruction.

This paper discusses one approach for teaching proof in a real analysis course that makes use of a contemporary dynamic technology. The dynamically reconstructed proof visualizations discussed in this paper are intended both to make abstract ideas in real analysis more visually dynamic and also to provide an environment for students to interact with the particular givens and arguments in a claim in order to foster insight and comprehension about its proof. [ABSTRACT FROM AUTHOR]

Combination problems are a cornerstone of combinatorics courses, but little research has been done examining the ways that students perceive and differentiate among different combination problems. In this article, we investigate how mathematics education students (n = 18) in a discrete mathematics course view two categorically different combination problems (Category I and II combination problems). In particular, we look at how participants conceptualized each problem's sets of outcomes, counting processes, and formulas, while also exploring the means by which students justified their relationships. Review of the data collected suggests that students tend to be less consistent and have more trouble utilizing and justifying combinations with a collection of ordered indistinguishable objects (Category II) than they do with a collection of unordered distinguishable objects (Category I). Based on these findings, we provide recommendations for the teaching and learning of combinations in combinatorics education. [ABSTRACT FROM AUTHOR]