Encourage students to become independent investigators through the use of open-ended, hands-on math activities that require students to pose questions and tackle problems by applying the scientific method to explorations in geometry.

In his presentation to the Harvard Mathematics Club on combinatorial geometry Solomon Golomb (professor of mathematics at MIT) asked the members, “How many different ways can you combine five squares?” That day in 1953 they found that there were only twelve answers to the question. During that talk Golomb gave those twelve shapes the name pentominoes: penta (five), ominoes (from ½ of a domino).

*Modified from Polyominoes, Solomon W. Golomb, Princeton University Press, revised edition, 1994, page xi.*

Sets of pentominoes are now used as manipulatives in math classes around the world.

Although the mathematical content of this course is focused on explorations of area and volume, and related problems of perimeter and surface area; there is another important aspect here that is often missing from the traditional arithmetical/mathematical curriculum… building an authentic community of mathematicians.

Just as the Harvard Mathematics Club explored an open-ended math problem with Solomon Golomb your students will be exploring that same problem as well as many other open-ended problems that have no obvious answer. As they do so, your students will be behaving as mathematicians, and in the process, acquiring valuable problem solving skills.

As you and your students explore these open-ended investigations they will likely discover new solutions that are not included in the solution sets provided in the content of this course. I invite you to send them to me and I will include them in future revisions with footnote credit to your students.

The best way to learn mathematics is to behave like a mathematician. Students need to tackle open-ended explorations where the solutions are not obvious. Mathematics is a branch of science! Math students need to apply scientific method to solving open-ended math problems. They need to collect data using manipulatives, post the data for others to evaluate, look for patterns in the data and organize it, checking for missing solutions and complete the solution set, make generalizations about that solution set, and use those generalizations to make extrapolations leading to new questions related to, and building, on the previous investigations.

- As students are involved in this method of thinking and behaving they are mathematicians!
- The students “own” the ideas and concepts because they “lived” them! This is authentic learning! These experiences will alter their cognitive domains!

This course is most appropriate for teachers of fourth grade through middle school. (The first section of explorations of area may be appropriate for second and third grade students.) This method of teaching requires the instructor to take the risk of “diving into” an area of mathematics that they may not be familiar with. They need to be willing to ask questions they don’t know the answers to, and are willing to find the solutions along with the students.

This course may not be appropriate for teachers who prefer to be the “sage on the stage” rather than the “guide on the side.” (To tell the students the “right” answer is to rob them of the joy of discovering it for themselves.)

“Exploring with Squares and Cubes” is an opportunity for you to turn your students loose on a series of engaging, open-ended math problems that build valuable problem solving skills, creating mathematicians in the process.

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