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The article focuses on how teachers could communicate mathematical ideas in the classroom. It is noted that during the early stages of dynamic view including learning, graphing, substitution, using the metaphor of continuous motion and working only with continuous functions is helpful. It is asserted that it is not sufficient to communicate mathematical ideas to students although knowledge of mathematical content is necessary.

The article discusses topic related to the mathematical problem chords of circle to triangular numbers to volumes of spheres. It denotes that this kind of delving deeper is different from its other kind since arose from a telephone conversation inspired by mathematician Don Crossfield. It also presents the conversation between Hypatia and Ptolemy concerning the problem raised by Crossfield which was published in "The Oregon Mathematics Teacher." According to the article, the conversation is composing of a casual talk and confusing reference that occurred in a casual conversations.

The article discusses topic related to the importance of learning various methods in solving mathematical problem. According to the article, students such as those that can easily find a way to solve problem will become better problem solver and have a deeper understanding of mathematical concept is they can recognize and adapt various problem solution strategies. The article also discusses the Principles and Standards for School Mathematics (NCTM), define proficiency in mathematics as the ability to use knowledge flexibility.

The article discusses topics related to the proof and prealgebra curriculum. It highlights that prealgebra curriculum offers opportunities for motivating students engage in proof and for cultivating proof readiness. It explains that prealgebra curriculum presents numerous opportunities that encourage the students to justify their interference. It accordingly requires the students to clearly communicate and to reason out their solutions to mathematical problems with appropriate mathematical languages and notations.

The article discusses topic related to the importance of diagrams and representations in understanding mathematical solutions. It highlights that mathematical representations helps the students organized thinking, since they bring a visual components to abstract ideas and serves as template for computation with comprehension. On the other hand, the inevitability of representations is not accordingly a guarantee that they can be used effectively. According to the article, the focus of the mathematical representation and their connection began during the publication of "Curriculum and Evaluation Standards for Mathematics" (NCTM).

The article discusses topic related to the relationship between the slopes of perpendicular lines. It denotes the use of the Geometer's Sketchpad (GSP) to allow student discover a property of slopes. It examines the range of values the slopes might have and provides an example of result the student might generate. It also presents the laboratory sheet with instruction and notes for teachers to help students who encounter problems. According to the article, with the use of GSP, the students might be able to discover that slopes of perpendicular lines are negative reciprocal to each other.

The article presents the answers and solutions for several mathematical problems published in the previous issues of the journal. It includes the solution of the problem number one which shows degree of vertices at the circle labeled A-H. Solution of the number two problem concerning the radii of circles related to the problem number one. The solution of problem number three is also mentions related to the vertex angle of the isosceles triangle at the center of the square.

The article discusses topic related to the particulars of proofblocks. It explains that among the difficult topics in geometry includes the concept of proof. It discusses the intuitive approach of teaching proof, which is the proofblock, that makes proof accessible to students without sacrificing the rigor of traditional approaches. It denotes that with the use of ProofBlock, the student will be able to find it natural from both ends of their attempted proof. It also mentions that ProofBlock is an inherently visual approach which uses manipulative that addresses the need of visual and kinesthetic learners.

The article discusses the use of mathematics in modeling and designing picket fences. It highlights the need for students to recognize the need of using mathematics in particular, concrete context and choose the function the represents the situation mathematically. It explores the use of picket fences as mathematical models. It cites the identification of mathematical forms in contemporary styles and types of fences. It also indicates the reversion of the process in identifying mathematical models for picket fences.

The article presents a response by Kaminski and colleagues addressing a commentary by Jones regarding their research on the effects on figurative transfer of learning mathematics. Kaminski and colleagues disagree with Jones' comment regarding the similarity of the generic representations used in their research to the transfer domain than the concrete representations. They argue that their research have data supporting the success of domain transfer when the concrete representations presented in quantities. They cite that sound theory and theory-grounded empirical findings may encourage the understanding of learning and transfer. Additionally, Kaminski and colleagues expressed their gratitude to Jones on raising the concerns on the query regarding which components affect transfer.

The author reflects on the generalization given by Kaminski and colleagues in their pattern game study that sets to establish the effectivity of generic instantations in presenting mathematical concepts over concrete examples. He notes that the transfer task is biased since it is more similar to the rules involved in the generic instantation condition. He states that more studies similar to domain problems of Koedinger and colleagues must be made before making sweeping generalizations.

The article focuses to explain that student's attempt to interpret mathematical problems fosters their mathematical investigations. It employs the technique of implementing discussion based mathematical investigation in a classroom. It uses the Principles and Standards for School Mathematics, which encourages students to discuss their mathematical problems with their classmates. It presents a detailed description of the investigations, which includes assigning problems to students, discussing their solutions in large groups, nourishing their problem solving skills. It describes the progression of problem solving in students, while solving the cereal box problem, that involved arranging 45 cereal boxes. It also stresses on the teacher's role in developing students' mathematical skills.

The article discusses a mathematical problem, that asked to arrange the digits 5, 6, 7, 8, and 9 in a two-digit and a three-digit number, so that their product produces maximum result. It explains methods employed to achieve the solution. It describes an informal approach, which is based on common sense and presents four cases using this approach. Another approach explained is the formal approach, which compare all the four cases in pairs. It also addresses other problems that arise, while solving the original problem. It arranges the problem in 3 cases to analyze various probabilities that result from the problem.

The article presents methods of coordinating discourse in a classroom by using an inductive model of teaching. It discusses univocal discourse that involves dialogues and feedbacks from a teacher only, and the dialogic discourse involving exchange of ideas between teacher and students. It reports that the National Council of Teachers of Mathematics has realized that the type of discourse and its quality promotes mathematical understanding. It mentions a program that involved a mix of univocal and dialogic discourse in combination with an inductive model of teaching. It discusses that the model was developed after analyzing an eighth-grade algebra class. It also reports that the model promoted better mathematical understanding amongst students.

The article discusses Lissajous figures in mathematics. It explains that Lissajous figures are not usually included in most secondary school mathematics curricula, however, they can provide a fertile ground for students to explore real-world mathematics. The article presents a historical background of Lissajous figures, which were created by Jules Antoine Lissajous, a nineteenth century French physicist for whom the figures are named. The article also discusses the generation of Lissajous figures in a calculator.