Geometric Structures by Douglas A. Aichele and John Wolfe

From MathWiki

This will be a report on the textbook entitled, “Geometric Structures: An Inquiry-Based Approach for Prospective Elementary and Middle School Teachers” by Douglas B. Aichele and John Wolfe. (2008)



Professor Whiteley lent me the textbook for further research into the applicable activities provided within the text for prospective teacher candidates. I will discuss the format and accessibility of the text to teachers. I will also write about the connections between the topics covered in this learning resource and the Big Ideas, Concepts and Procedures recently being discussed in this course.


Format and Accessibility

The textbook is definitely written with teachers in mind! It is organized into five different parts, each of which is categorized into approximately 2-6 chapters and subsequent sub-chapters.


Part 1 is called “Paper Folding”

Part 2 is called “Geoboards and Dot Paper”

Part 3 is called “Straightedge and Compass”

Part 4 is called “Computer Constructions and Explorations”

Part 5 is called “Mira (Reflecta) and Tracing Paper”


As one can quickly tell, the parts are based on certain manipulatives, which are very commonly found in a geometry classroom. The authors describe the reason for this divide:

Our intention is to use the manipulative to emphasize the importance of the appropriate use of physical models. Our experience has been that many times physical models can provide a meaningful context for individual thinking and class discussions. We believe that the quality of these discussions is vital to the process of making sense of geometry. (Aichele, Wolfe 2008)


The grade level for this textbook is meant for elementary and middle school teachers, but the activities could easily be modified in order to apply to the lower high school grades (9 and 10). For example, once a teacher understood the concept in the chapter, they could easily take one of the concepts and develop a challenging problem to be solved by their students.


The textbook is quite useful because every section is very concise and easy to understand. That is not to say that the mathematical terminology and truths are striped from the lessons. It is my understanding that the lessons are meant to be read/taught to teacher candidates in their “mathematics for teaching” course in the Faculty of Education (or a similar such course). Then once the teacher candidate understands the concept, they can practice their understanding with the questions provided in each section. Furthermore, a teacher could also use this textbook as an excellent teacher’s aid in order to develop practice problems and lessons for their students.


Let me give an example, to better support this description:


Activity 16.12 – Marking Symmetries on Wallpaper Designs


After giving a solid description of the types of symmetries which can be found in the wallpaper designs provided (reflectional, rotational, translational and glide-reflectional) the activity/exercise is written in such a way that the text can very easily be photocopied and completed directly on the page.


This makes it very easy for the teacher to:


(a) learn and practice the material before teaching it

(b) have a lesson/description of the concepts

(c) be able to give their students the photocopies activities directly from the textbook


One more very convenient and practical way the text is organized is through small photo icons. Each sub-chapter, whether it is a lesson, activity or exercise is labeled with an icon describing what type of learning will occur. For example, the sunshine symbolizes “daily activity” sheets that “provide an experiential basis for understanding the geometric concepts and relationships presented.” (Aichele, Wolfe 2008)


Connection to Big Ideas

The textbook covers many of the big ideas discussed throughout this course. Here is a list of the chapters within each Part of the text that will help to give some context to this section of my report:


Part 1 – Paper Folding

Ch. 0 Warm-up Activities Ch. 1 Polygons and Angle Relationships Ch. 2 Quadrilaterals and Their Definitions Ch. 3 Constructing by Paper Folding Ch. 4 Explorations in Three-dimensional Geometry


Part 2 – Geoboards and Dot Paper

Ch. 5 Area Ch. 6 Explorations with Geoboard Areas Ch. 7 Similarity and Slope Ch. 8 Pythagorean Theorem and Perimeter Ch. 9 Geometry of Circles


Part 3 – Straightedge and Compass

Ch. 10 Straightedge and Compass Constructions Ch. 11 Congruence Conditions and Reasoning from Definitions to Properties


Part 4 – Computer Constructions and Explorations

Ch. 12 Computer Constructions Ch. 13 Computer Explorations


Part 5 – Mira (Reflecta) and Tracing Paper

Ch. 14 Mira Constructions Ch. 15 Symmetry Ch. 16 The Four Symmetries Ch. 17 Symmetries of Mandalas Ch. 18 Symmetries of Borders Ch. 19 Escher-Style Tessellations


It is clear that it is not enough to simply connect the titles of the chapters with the big ideas being discussed in this course. Which is why I will be continually referring and modifying this section of my report as we continue to unpack and understand the big ideas in regard to teaching geometry.


Here are a few examples of some connections that I have already made throughout the text to the big ideas:


Table of contents

1. (1 BI) understanding isometries and congruence


“This includes compositions of isometries (reflections, rotations, translations) A related small idea would be connecting paper folding and mirror reflections in the plane.”

Within the textbook there are many activities which use paper folding in order to explore angles and the properties of 2-D figures. The authors of the text also use the activities with the miras in order to help teach the concepts of the altitude of a triangle and the location of lines of symmetry in the 2-D plane.


(7 BI) Geometric Measures


“Area, volume (and length) as additive for cuts in to several pieces”

In chaper 4 there are some connections to this big idea. For example, when discussing the volume of a pyramid, the authors write, “The formula for finding the volume of a pyramid is simple, but not very intuitive…Two other activities (“What Does Volume Really Mean?” and “Volume of a pyramid”) explore the validity of the volume formula” (Aichele, Wolfe 2008) The activities are very interesting as one involves looking at the net of a 3-D object and calculating its volume, and the other is an activity of physically finding the volume of the different parts of a 3-D object.


(10 P) Manipulatives / virtual manipulatives

“Process/Technology around hands on materials and the use of virtual representations in both 2-D and 3-D.”


As was discussed earlier, the text is purposely divided into parts pertaining to the use of manipulatives within the geometry classroom.


Final Thoughts

One of the most interesting aspects of this textbook is that at the end of every chapter there exists a sub-chapter called “How Do I Know if I Understand?”


It is an interesting title given to a sub-section which provides a list of “Basic Relationships, or Big Ideas” giving Big Ideas specific to this chapter as well as Big Ideas for geometry in general. There is also a list of “Basic Types of Problems” specific to each chapter.

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