In this article, we analyze the complexity of using instructional dialogues in the teaching of mathematics. We trace a 10-lesson unit on functions and their graphs taught by Magdalene Lampert to a 5th-grade classroom. We use this trace to help analyze and systematize the complexity of the classroom discourse. Analysis shows that Lampert's…

Many years ago, Arons pointed out the incomprehension science students exhibit of the basic mathematical operations multiplication and division and the need to address the problem in physics classes to assure student understanding of the physical world. McDermott et al.'s Physics by Inquiry program does address this need directly and in detail (by…

Few things capture the spirit of spring like flying a kite. Watching a kite dance and sail across a cloud spotted sky is not only a visually appealing experience it also provides a foundation for studies in science and mathematics. Put simply, a kite is an airfoil surface that flies when the forces of lift and thrust are greater than the forces of…

This article analyses the modeling approach used by one student in a business problem. It is argued that if we use previous frameworks we are not able to classify the students' approach to modeling as purely theoretical or empirical. Instead the student used a theoretical approach when constructing a real model, but abandoned it when she had to…

This article discusses the role of the commutative, associative, distributive, and identity properties of addition and multiplication in preparing children for future success in algebra. (Contains 3 figures.)

This article, in honor of the 100th anniversary of "Mathematics Teacher," touches briefly on mathematics and the history of mankind.

Confidence intervals (CIs) give information about replication, but many researchers have misconceptions about this information. One problem is that the percentage of future replication means captured by a particular CI varies markedly, depending on where in relation to the population mean that CI falls. The authors investigated the distribution of…

Innovative teachers can make writing as an integral part while teaching math to students. The teachers will have to implement the math instruction that enables students to organize and consolidate their mathematical thinking through communication, analyze and evaluate the mathematical thinking of others, and use the language of mathematics to…

Simon's (1995) development of the construct of hypothetical learning trajectory (HLT) offered a description of key aspects of planning mathematics lessons. An HLT consists of the goal for the students' learning, the mathematical tasks that will be used to promote student learning, and hypotheses about the process of the students' learning.…

If a curriculum developer's goal is to create a single linear sequence of tasks that lead to the development of some important mathematical concept, then some researchers have suggested that these sequences should follow progressions similar to stages of development that have been identified in Piaget-like research on the relevant concept(s).…

Suggests that there is continued poverty in student understanding of proof. Reports on how proof attitudes could be inculcated in students by offering them a course design that is somewhat faithful to the historical genesis of modern mathematics. Contends that such a design can enable students to discover a sense of proof for themselves. (Contains…

We present a lottery project for lower level mathematics courses that demonstrates financial decision making in a context familiar to students. Students use elementary principals of probability and financial mathematics to compare the expected values of buying Powerball lottery tickets to some basic savings plans. Then, they compare the expected…

Most serious rock climbers are familiar with a counter-intuitive fact about their sport: The force experienced by a falling climber due to the rope as it arrests his fall does not depend simply on the length of the fall, but rather on a ratio called the fall-factor. This article explains, using elementary physics and simple differential equations,…

As with natural numbers, a greatest common divisor of two Gaussian (complex) integers "a" and "b" is a Gaussian integer "d" that is a common divisor of both "a" and "b". This article explores an algorithm for such gcds that is easy to do by hand.

The authors give a rigorous treatment of the differentiability of the exponential function that uses only differentiable calculus. It can thus make "early transcendental" courses complete.