Two Reviews of Geometry: A Guided Inquiry

1. By a leading math educator, Henri Picciotto

In 1981, I moved from K-5 into high school. This transition felt almost like starting a new career. In many ways, what I had learned about pedagogy as an elementary school teacher still applied, but I needed to learn a lot more to grow into my new job. That’s when I came across Geometry: A Guided Inquiry (GGI), a way-ahead-of-its-time 1970 textbook by Chakerian, Crabill, and Stein. It was just what I needed, and had an enormous impact on me, both as a teacher and later on as a curriculum developer. In fact, of all the books I’ve seen in my 49 years in math education, this is probably the one that taught me the most, by far. In this post, I will try to explain why by listing some specifics.

1. Group work helps student learning. The front matter of the Instructors’ Edition explained the benefits of students working in collaborative groups. (Here, I have to trust my memory, because I only have a copy of the student book.) Students get a deeper understanding if they discuss the math with each other; and this setup allows the teacher to pay attention to (e.g.) eight groups rather than 32 students. The authors also argued that groups of four are optimal, because in groups of three one student is often left out, and groups of five are unwieldy. 

2. Tradition is not a good guide to the sequencing of topics. Or at least, it should not prevent a teacher or curriculum developer to consider other options. For example, starting with definitions of points, lines, rays, etc. is a terribly boring start to a course. Or, the concept of deductive reasoning need not be introduced prior to doing interesting geometry. And so on. In particular, it verges on insanity to start with formal proofs of self-evident results, such as:

Midpoint Theorem: If M is the midpoint of AB, then AM = AB/2 and MB = AB/2

There is no quicker way to convince students that math is a weird twilight zone where their teacher is an idiot, and yet they need to do what they say. (OK, I’m exaggerating, but the point of proof is to dispel doubt. The idea that obvious statements need proof is an advanced idea, completely inappropriate for the first few days of a 9th or 10th grade course.) 

3. Most students don’t learn things that they only see once. Each chapter of GGI is organized in three parts. Central is where the ideas are introduced. Review goes back over those ideas, using many interesting problems. And Projects include extensions which are not required for the book’s sequencing to work. At a certain point, I realized that forging ahead to the next Central while assigning  Review problems as homework was a way to extend student exposure to the concepts. This was the seed of my idea about lagging homework. (Read about that here.)

4. Guided inquiry provides the right balance between student discovery and direct instruction. I can’t get into that here, but in short: neither is sufficient without the other. Students cannot hear answers to questions they do not have / students cannot discover all of math. The key to a healthy combination of discovery and direct instruction is the use of worthwhile problems that are both accessible and challenging, both before and after the key results are presented explicitly.

Anchor problems and activities help to introduce big ideas. Chapter 1 of GGI starts with the “burning tent” problem: a camper who happens to be carrying an empty pail near a straight river needs to run to the river, fill the pail with water, run to the tent, and put out the fire. What is the shortest path to accomplish this? Chapter 2 starts with the question: which polygons tile the plane? These lessons are engaging and accessible, and lead to many important and interesting ideas. Give me these openers any day over “the segment addition postulate” and the like.  (I wrote about anchors in these posts: Mapping Out a Course and Sequencing.)

Problems need not be sequenced in order of increasing difficulty. This is countercultural, but effective. When students don’t know if the next problem is going to be easy or difficult, they are more likely to give it a shot. If the problems get harder and harder, many students will reach a point where they decide they can go no further. 

Practice need not be boring. For example, the book had many entertaining problems in the form “what’s wrong with these?” which showed figures which violated one or another theorem. 

Answer-getting is not the point. In fact, some answers are given right there in the margin, which allows students to check their understanding as they work. Most other answers are given at the end of each chapter.

5. Inquiry and proof are not mutually exclusive. Some geometry books prioritize inquiry at the expense of teaching proof. Others prioritize proofs at the expense of motivation and access. GGI keeps these in balance. Also, while Chakerian et al present both paragraph and two-column proofs, they make clear that the former is the standard in mathematics. If I remember correctly, in the Instructors’ Edition, they point out that students will be writing paragraphs for the rest of their lives. Two-column proofs, not so much.

Having absorbed all these ideas, I was ready to start developing curriculum myself, and to lead my department away from textbooks in our core classes. But even decades after using our own materials in geometry, we continued to use some of the problems from GGI.

I will forever be grateful to Chakerian, Crabill, and Stein. I was so lucky to have run into Geometry: A Guided Inquiry early in my high school career!

— Henri

2. By David Chandler (My review on Amazon, which explains my enthusiasm for this textbook … 5 stars, of course)

Best geometry text I’ve ever used in 35 years of teaching–A Classic

Of all the geometry texts I have used over the past 35 years, this one stands out as by far the richest, most intuitive, and most interesting. This text is unique.

Most geometry textbooks present a long list of facts about geometric figures organized in a rigid logical order, working generally from simple to more complex. Applications of these facts may or may not be made clear to the student. Geometry: A Guided Inquiry starts each chapter by posing an interesting geometric problem (puzzle), called the “Central Problem” for the chapter. Clusters of geometric facts are introduced, as needed, in the process of solving these problems. The usefulness and relevance of the new facts are therefore apparent from the moment they are first presented.

Most geometry textbooks, especially those written under the influence of the “New Math” era of the 1960s, put heavy emphasis on precise use of technical vocabulary and mathematical notation. Geometry: A Guided Inquiry emphasizes the underlying geometric and mathematical ideas and works to help the student understand them intuitively as well as logically. Overemphasis on technical vocabulary and complex notation can actually stand in the way of understanding, so the authors use simplified vocabulary and notation wherever possible.

Most geometry textbooks start each problem set with lots of routine, repetitive problems, gradually working up to an interesting problem or two at the end of the assignment. Geometry: A Guided Inquiry puts the best problems right up front! From the very beginning the student is given problems worth solving.

Most geometry textbooks read like they were written by a committee following a prescribed agenda. Most in fact are! The life is squeezed out of the narrative in the process. Geometry: A Guided Inquiry has a distinct sense of authorship. The authors are good mathematicians, good teachers, and good writers. Their joy in the pursuit of mathematics shows through their writing.

Geometry: A Guided Inquiry makes frequent use of compass, protractor and ruler activities, data tables, guess and check methods, model-building, and other techniques of intuitive exploration in preparation for general solutions. Each chapter begins with a “Central Problem” that provides the focus and motivates the discussion in that chapter. The Central section presents all the essential new material. Along the way the student is led to a solution of the Central Problem, while exploring its connections with other topics. After the Central section is a Review section, and each of the first seven chapters are followed with a short Algebra Review that stresses algebra topics related to the current work.

Next comes the best part. Each chapter has an open ended Projects section with problems that are extensions to the material in the Central section, sometimes carrying the discussion in new directions. (The Project sections include some of the most interesting material in the text!) In a classroom setting, where students work at their own pace, the quicker students would work on the Project section while the slower students finish the Central and Review sections. In a home study environment the student should read through the whole Project section and work on as many of the project problems as possible within the time frame available. Students who find the work easy, rather than going faster, you should instead take more time and go deeper!