Students often look for real-life situations where they can apply the concepts they learn. A project of measuring the speed of moving cars demonstrates that they learn communication skills, teamwork skills, and develop patience specially when work in a group with a common purpose.

Two ideas of mathematics that the students discuss here, demonstrate how sometimes a teacher can learn something new from the students. The examples given use geoboard and show relationships of the two ideas with the Pick's theorem.

The challenges in developing discursive classrooms in high-poverty, rural schools motivate less verbal students to communicate. Students tend to resist mathematics education reforms and classroom discourses.

Under a revised assessment system called EMRF, a four-tier rubric is used to evaluate students' work and is beneficial to the teachers as well as the students. It is carefully designed and its descriptors are available to the students also.

Mathematics students should understand what proof is and its necessity and role in mathematics. They must make their own conjectures, which make them feel that they can "do mathematics" and enjoy the sense of accomplishment too.

Counting can be done using a linear, exponential method or by using a technique incorporating a recursive process which gives a visual analysis of population data. Population estimates are based on assumptions about change brought about by immigration, emigration, deaths and births.

Mathematical contests should be decentralized to allow access to non competitive activities so as to enable students to participate and succeed in Mathematics. These activities should be compellingly interesting so that students enjoy learning about proofs and abstract structures rather than just solving problems.

The greatest benefit of including leap year in the calculation is not to increase precision, but to show students that a problem can be solved without such presumption. A birthday problem is analyzed showing that calculating a leap-year birthday probability is not a frivolous computation.

The factors responsible for the shortage of middle and secondary mathematics teachers in the U.S. are discussed. Seven strategies are suggested to reach out and recruit more mathematics teachers in the future.

Illustrations on how algebra students can detect patterns and form conjectures are presented. In constructing algebraic proofs, students can see the crucial connections between whole-number operations and algebra and can proclaim, "Yes, it's always true."

Teachers often wrongly presume that all students would interpret a three-dimensional image presented in a two-dimensional manner in the same way. They need to be specific as to how questions are stated especially in real-life situations, where alternative interpretations are possible.

While helping high school students develop statistical thinking, teachers need to engage them in all phases of investigative cycle. Students should master some of the nonmathematical elements of the cycle involved in identifying a problem, creating a plan of attack and gathering necessary data.

The preproblem pondering strategy of "anticipate the answer" involves attempts to anticipate the form of the answer and the answer's relationship to the conditions of the problem. It draws on the skills of recognition, identification, interpretation and builds confidence.

In a course on proofs, a number of problems deal with identities for Fibonacci numbers. Some general strategies with examples are used to help discover, prove, and generalize these identities.

While exploring the subject of geometric proofs, boolean logic operators AND and OR can be used to allow students to visualize their true-or-false patterns. An activity in the form of constructing electrical circuits is illustrated to explain the concept.