Teaching Math Dynamically

by Dr. Evan Maletsky

How would your students describe the mathematics that you teach, and how would they describe the way you teach it? Many students view mathematics as a static, fixed subject, routinely taught in a predictable, repetitive way. Yet, one of the really big ideas, both in mathematics itself and in the way we teach it, is the notion of change. As teachers, we need to focus and build on the dynamics of action and change in the mathematics classroom. Here is a fresh, new approach to some old ideas about practice and problem solving in the teaching of mathematics.

Interactive Transparencies

The new Interactive Algebra Transparencies offer a wide variety of practice problems across a broad content area. They are designed to facilitate a dynamic interaction between student and teacher. This is accomplished through a unique feature of the transparencies themselves: each transparency has a wheel that offers many different settings?from basic to more complex exercises, centered around a specific topic?simply by rotating the wheel. These settings can be changed instantly on the same transparency.

Practice and review begin with choices at levels that assure initial student success. Through interactive discussion, you build on this success as you make step-by-step changes in the structure of the exercises presented, creating progressively more challenging problems. Students often fail with more complex exercises because they never fully understand how these problems relate to simpler, more basic ones. In your hands, with a single transparency, just by choosing different settings on the wheels, you can show this connection by showing how individual problems can change. Don?t underestimate the importance of students visually seeing and processing the dynamics of these changes and connections. These are the very things you will want them to do themselves when they meet similar challenging problems on quizzes and tests.

In the new Interactive Problem-Solving Transparencies, you find a valuable blend of situations and questions that range across the mathematics curriculum, all designed to facilitate a dynamic interaction between student and teacher. Problem solving differs from routine practice and drill. It requires careful reading and thoughtful planning. With these transparencies, students see how simple problems can be changed, progressively, into more and more challenging situations. They also see how changing a word or two in a problem can change completely the approach and strategy needed to solve it. For example, consider the rearrangement of the words in these two problems:

50% of 36 is what number?
36 is 50% of what number?

Not only do students need to solve problems in mathematics, but just as important, they need to see how changing those problems often require changing the methods of solution.

Each problem-solving transparency contains a whole family of related problems. All choices of settings are visible at all times to the teacher, but only the chosen ones are projected for the students to see. When the choices are changed, students see them change in a very dynamic, visible way, and the teacher can verbally interact with the students as the changes are being made. This type of experience is critical in the mastering of problem-solving skills, since solutions to unfamiliar problems often come to light by changing them to related, more comfortable problems.

Change in Mathematics

The notion of change is at the heart of mathematics. In algebra, change is built into the concept and use of variables. Not only do expressions change in value as the variables change in value, they also change as the values of their coefficients and added constants change. By having a ready supply of closely related exercises at your fingertips on a single transparency, you can focus on change as you interact with your students in discussing these effects. Furthermore, the exercises are designed to encourage mental manipulation, so you can often assess student understanding on an immediate basis. With the simple, unique feature of having wheels that you can rotate, you can create on a single transparency, at your own level and speed, an entire family of related, increasingly challenging exercises for discussion and solution.

We are currently going through a stage in education in which teacher-directed instruction is pitted against student-centered instruction. There are strong arguments and feelings on both sides, but a good, appropriate blend of the two approaches most likely is best. Again, the importance of variation and change comes into play. Clearly, there need to be times when there is active interplay between students and teachers.

These transparency sets are called interactive because they offer an excellent and unique opportunity for both students and teachers to share in the instruction and learning process within a given classroom situation. Used interactively, with a strong emphasis on the importance of change, they make valuable and powerful new tools for teaching mathematics in your school.

Help your students see algebra and problem solving through the eyes of change. It will give them a new view of mathematics.

Dr. Evan Maletsky is Professor Emeritus at Montclair State University.

Dynamic Math Activity Pages

Using Interactive Algebra Transparencies and Interactive Problem-Solving Transparencies, teachers can create an entire family of related, increasingly challenging exercises from a single transparency. Download these sample teacher notes, worksheets, and answer pages from the book accompanying each set of transparencies.

	Interactive Problem-Solving Transparencies sample pages
	Interactive Algebra Transparencies sample pages

Dynamic Math Internet Links

Computer programs that serve up exercises at the touch of a key appear to model the notion of change in math, but many fire off random exercises rather than building knowledge and skill from one exercise to the next. Here are a few sites that seem to be heading in the right direction. Let us know what you think!

	Coolmath.com: Coolmath Algebra
	Interactive Algebra Equation Buster
	Mathforum.org: Technology Problems of the Week

March Newsletter Theme: Respect

Next month Class Ideas tackles a theme that's crucial to building core values in the classroom: respect for oneself and others.

We'll have an informative article on this core topic, links to Internet resources, and sample activity pages (not to mention a great subscribers-only special) on Didax's new Respect series for students in grades 1?6.

If you'd like to take a more purposeful approach to values instruction in your classroom, this issue is for you!