Colin Freel
Rogers City High School, Rogers City

Teaching Geometry from a different angle

As a teacher of geometry I see many students who have difficulty with spatial reasoning and following complex proofs. In order to address this problem I have written a digital geometry text. Each student is given a printed copy of this text as well as a copy on CD-ROM. The text has 60 lessons with discussion and problems. One of the benefits of this text, which addresses the difficulties stated above, is the use of embedded objects that utilize PowerPoint files, that step students slowly and visually through demanding proofs, diagrams, and problems. The goal for this project has been to “ease the pain” of the average geometry student, and I am confident this digital text does just that.

In the classroom students are introduced to a new theorem and its proof in a very visual manner. Rather than seeing the entire proof of the theorem in all of its complexity, which sometimes intimidates them, the students step progressively through the proofs as each step appears in the PowerPoint slide. Angles, segments, lines, arcs, etc. are analyzed as they appear, but not all at once. At home or in the computer lab students can review the animations step-by-step at their own pace.

Another built-in benefit of this digital text is the PowerPoint review games. These games are patterned after the game show Jeopardy. Students choose from categories of geometry problems and work the problems for review. The games provide opportunities for healthy competition coupled with relevant geometry review. We usually play these games in a classroom setting, but students tell me they like to play them at home as well.

A third benefit of this digital text is the use of images created with Mathematica (a mathematical software package). The graphics packages of Mathematica are excellent, illustrating vividly three-dimensional shapes in an understandable manner. This aspect of the text again addresses the spatial reasoning difficulties some students have.

A fourth benefit is the ability to insert images into the text of the Word lessons. This allows real-world problems relevant to students to be seen and examined. For example, in one of the lessons we worked on a problem dealing with the angular velocity of the wind generators in Mackinaw City. A photograph of the wind generators was inserted into the text. Students could see the actual size of these massive structures with people standing alongside for perspective. Of course, this is an ongoing process as new and relevant problems will constantly be found and added to the text, along with inserted pictures.

The challenges to a project of this magnitude are daunting. These challenges range from the very minor problems of typesetting (mathematical equations, drawings, etc.) to the very major problems of determining logical flow, learning Mathematica code to produce graphical images, or addressing all of the Michigan Benchmarks. Overall, the response from the students has been so positive that the effort has been worthwhile.

This is the first full year that the digital text has been used at our high school. For this reason it is difficult to offer quantitative measurements that compare past years geometry students with those taking geometry with the new text. One interesting statistic is that the number of students taking geometry this year compared with last year has increased by 17 students, from 30 in the 2004/2005 school year to 47 in the 2005/2006 school year. Though I cannot say assuredly that the rise in numbers is attributable to the new, more visual text, it certainly may factor into students’ decisions. These statistics aside, I am confident that in the coming years student performance will rise in performance-based testing.

At this point I would like to direct you to the digital text contained on the enclosed disk. By opening the Word file “Table of Contents” you can preview the contents of the book, and get a feel for how the topics are developed. Next, you may want to read through some specific lessons. You can do this by simply opening a lesson. Read through the text. As you come to an embedded PowerPoint file, double click on the icon, which will take you through a detailed explanation of a proof, problem, or theorem. These lessons illustrate some of the aforementioned items:

Lesson 21 – This lesson contains embedded links to a proof of a theorem (AIthenP.ppt), a visual of a geometric construction (ConstrucParallel.ppt), and a review game (Geoparty11-20.ppt) patterned after Jeopardy.

Lesson 41 and 42 – These lessons show some difficult examples in PowerPoint files.

Lesson 48 – Illustrates transformations as well as shows the process of their basic construction.

Lesson 51 – Shows the process of matrix multiplication.

Lesson 53 – Shows geometric solids generated with Mathematica.

Feel free to sift through any or all of the other lessons. If you have any questions concerning the lessons or if you find any errors, please contact me.