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Mathematical modeling has taken on increasing curricular importance in the past decade due in no small measure to the Common Core State Standards in Mathematics (CCSSM) identifying modeling as one of the Standards for Mathematical Practice (SMP 4, CCSSI 2010, p. 7). Although researchers have worked on mathematical modeling (Lesh and Doerr 2003; Pollak 2003, 2012), much remains to be learned about modeling and how to teach students to use modeling appropriately, particularly at the secondary level. This article uses high school classroom episodes to address an aspect of modeling called "mathematizing," the process of translating a given situation into its mathematical model (Blum and Ferri 2009). These classroom episodes are used to illustrate and analyze: (1) the ways of thinking that help students' engagement with mathematizing; and (2) how teachers can support students in developing such ways of thinking.

The Common Core State Standards for Mathematical Practice (CCSSI 2010) elevate the ways of thinking used to create mathematical results to the same level of importance as the results themselves. These mathematical habits of mind (MHoM), or "specialized ways of approaching mathematical problems and thinking about mathematical concepts that resemble the ways employed by mathematicians," are increasingly the focus of professional development efforts for precollege mathematics teachers. Educators across the country are considering how to implement this approach to thinking about, doing, and teaching mathematics. One way is through an immersive experience in which teachers experience mathematics as learners and rely on colleagues, and themselves, to unpack mathematical ideas. Such experiences provide exposure to an approach to teaching mathematics that embeds and highlights MHoM. The Park City Mathematics Institute (PCMI) provides a high-quality, immersive experience; but as a three-week intensive residential experience, it costs about $3500 per participant, since it includes the cost of housing, tuition, and travel. To help more teachers have access to this experience at a lower cost, we designed a similar program online. In this article the authors list the key aspects of the in-person experience and the design decisions made to preserve the PCMI experience while using the strengths of an online environment. They discuss the changes that occurred as a result of doing this work online and end with takeaways and remaining questions. (A bibliography is included.)

Building coherence in the development of mathematical ideas across the grades is key to improving students' mathematical learning in the United States. Knowing the mathematical experiences, understanding, skills, and habits of mind that students bring to a grade level and what the expectations are for the following grades can help teachers bridge the transitions for students on each end. In this article, the authors discuss the implications of the habits-of-mind approach for high school curricula, using the Center for Mathematics Education (CME) Project as a source of examples. They examine general mathematical habits that are used across high school as well as specific geometric and algebraic habits. The authors found that using mathematical habits of mind as an organizer can bring genuine and often surprising coherence to a curriculum. (Contains 3 figures.)

Educational Development Center, Incorporated, organized a professional development program for elementary, middle and high school mathematics teachers, with algebraic thinking as the main theme. Integrated learning enabled teachers to assess and analyze the themes that enhance algebraic thinking across all grade levels. Describing patterns, functions and algebraic structures, and generating adequate pedagogy were identified as some of the factors fostering algebraic thinking. Problem activities that helped teachers to evaluate algebraic thinking in detail are described.

Standards of mathematical education can be improved by teachers' usage of pictorial representations of function machines to provide students with insight on the hierarchy of operations, iteration and properties of addition, multiplication. Exposure to drawings of function machines greatly aids in simplifying complexities in topics such as algebra.

The National Council of Teacher's of Mathematic's Curriculum and Evaluation Standards for School Mathematics advocates enhanced usage of modern computing technologies in bestowing students with better understanding of mathematical phenomena, thereby aiding their growth in stature as real-world problem-solvers. Empowering teachers with the nuances of modern techniques and upgrading the quality of instructional materials are the initial strides toward effective classroom usage of advanced computing technologies.

The use of computer technology in a geometry class in a high school introduced new methods of thinking about geometry. The computers allow students to construct geometric figures according to their needs, helping them to understand geometry better. They inspire classroom discussions, encourage students to form conjectures and check them with experiments, and suggest a proof. The students can also relate geometry to algebra and functions.

Interactive geometry programs can help students develop ways of thinking that underlie calculus and analysis. These help in solving optimization problems and can also be applied to the topic of regression lines in statistics.

A method for making divergent series converge is described. Proofs of the procedure are presented. (MNS)

The planned revision of the math curriculum, if implemented, has the potential to empower students to conduct research in the subject, thereby gaining valuable experience for future use in real-world problem-solving. Proposals have been mooted for revamping the content of the subject's syllabus and enhancing the standards of math teaching through better curriculum management for ushering in excellence in the study and teaching of math.

In this article the author discusses teaching discrete mathematics to his class. When trying to determine a closed-form solution to any recurrence relation, guessing is allowed sometimes, and it is imperative that the teacher explain this fact. The says that teachers do not want students simply substituting numbers into equations to solve for unknowns in a typical algebra equation; they would rather want that they use algebraic rules to get the unknown by itself and thereby solve the problem. The same is true for difference equations, the discontinuous cousins of differential equations. Long before college, students might well become curious about solving a difference equation in the course of investigating the Fibonacci sequence often introduced in middle or even elementary school mathematics classes. An equation characterizes a typical recurrence relation--the nth term of the sequence is defined by using previous terms, and the two seed numbers give a starting point. If the nth term had depended on three previous terms, there is a need of three seed numbers.

Elaborates on the concept of permutations with the aid of examples. Definition of the term, Permutations; Step by step solution for the examples along with charts to show all possible arrangements; Derivation of the formulas used in the solution for the examples.

Discusses several ideas on mathematics tackled during the High School Teachers Program of the Park City Mathematics Institute. Graphical representation of the Gaussian integers; Application of Gaussian integers in the derivation of Pythagorean numbers; Creation of algorithms for example problems and solutions in algebra.

An essay is presented on the benchmarks for the evaluation of a mathematics program in high school education. It offers information on the curricular emphases and instructional approaches which could make reasoning and sense making foundational to the content that is being taught and learned. It stresses the need for a curriculum to be based on a set of organizing principles that bring the kind of coherence teachers need to support mathematical thinking.

Merit pay for teachers is discussed. The use of student performance on standardized tests as the measure of merit and the loss of cooperation among teachers are among the reasons why the author considers merit pay a poor solution for improving instruction. (MNS)

Offers tips on teaching the factorization of quadratic trinomials. Approach used in textbooks; Method for obtaining one factor; Examples of calculations. INSET: Editors' Note.

Presents a response by the author on the article "How to Solve Any Triangle: First, Forget the Law of Sines and the Law of Cosines."

Discusses the mathematical depth that is potentially present in simple mathematical problems. Example of an expanded motion problem; Connection between the motion problem and geometry; Practices that results from mathematical maturity.