In my last post, I discussed Every Minute Counts, a book that influenced me early in my career as a high school teacher in the 1980’s. It was mostly useful because of David R. Johnson’s suggestions on how to run a class discussion, and his insistence that the teacher needs to hear from every student, not just the two or three most vocal participants. I had received training in a slightly different approach to those same goals in the 1970’s when I was a “Community Teaching Fellow” (CTF) back in my graduate school days. As a CTF, I spent 40 minutes a day teaching math to 3rd graders, four times a week. On the fifth day, I spent 40 minutes observing another CTF. It was a two-way street: other CTFs attended my classes, and gave me feedback. This was a powerful approach to training a beginner: I learned not only how to run an effective discussion, but also that I should aim high and respect the intelligence of my students.
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!
PS: At the height of the math wars in the 1990’s, I was involved in a discussion with Chakerian. It ended well. Read about it here.
PPS: Chakerian, Crabill and Stein also wrote books for Algebra 1, Algebra 2, and Trigonometry. Those were not as successful as the geometry book, but I still got many good ideas from those, including the 10cm circle and a geometric approach to complex numbers.