EDUO 9552: Exploring with Squares and Cubes (Area and Volume)

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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.

From the Instructor

If you are not currently teaching students, there is a non-teaching format for this course.  Please contact the instructor, Ron Kremer for details.

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.

Course Overview

Through the hands-on activities in this course students become independent investigators. They learn how to pose questions and tackle problems the way mathematicians and scientists do. Since mathematics is a branch of science, mathematicians also use the scientific method. In addition to the mathematical content there is another important aspect here… building an authentic community of mathematicians!

Course Objectives

The teacher will:

  • Collect and prepare materials needed for the interactive activities (experiments, model building, and art projects).
  • Conduct brainstorming sessions with the students, creating a playful, “risk-taking” atmosphere for exploratory mathematics.
  • Provide opportunities for students to engage in concrete, open-ended, hands-on explorations.
  • Guide students in the use of “scientific method” (observing, collecting and organizing data, looking for patterns, using those patterns to make predictions and extrapolations).

Course Relation to CCS or other Professional Standards

This course aligns to the standards for: K-12 California’s Common Core Content Standards for Mathematics:

• Grade Three – Geometric Measurement (Area)
5. Recognize area as an attribute of plane figures and understand concepts of area measurement.
Geometric Measurement (Perimeter)
8. Solve… mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths…

Grade Four – Operations and Algebraic Thinking
Generate and Analyze Patterns
5. Generate a number or shape pattern that follows a given rule

Grade Five – Measurement and Data
Geometric Measurement
3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement


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Ron Kremer

B.A., M.A. in Elementary Education, Elementary mathematics and science emphasis

40 years: General elementary grade teacher, school district staff trainer, and staff developer for California Science Implementation Network.

Author and guest speaker at state level conferences for mathematics. Recognized for exemplary work in the field of education by the Johns Hopkins Institute for the Academic Advancement of Youth.

My goal is to convince teachers of the importance of "discovery learning" using open-ended, hands-on explorations.

I like working on projects around the home with my family. I sing tenor in our church choir and volunteer tutor at Shriners Children's Hospital."