Let me note where I think some difference in our opinions is occuring—

Chapter 2.6 Special Angles on Parallel Lines

There are some really nice patty-paper inductive explorations here, where students fill in blanks to make conjectures like “If two parallel lines are cut by a transversal, then corresponding angles are ___.”

Where deductive reasoning hits is on page 129:

You used inductive reasoning to discover all three parts of the Parallel Lines Conjecture. However, if you accept any one of them as true, you can use deductive reasoning to show that the others are true.

Solution: (then is given immediately after)

I could imagine in a class having the students stop reading and derive the thing for themselves, but the big chunk is devoted to inductive exploration, there’s no faciliation of creating the deductive proof in the text, and no followup problems where students create deductive proof independently.

Also, the “if you accept any one of them as true” bit is a little mysterious and completely hiding the momentous occasion of creating a new axiom. This is, I know, Intentional, but it is hard for me to see how the section puts equal value on deductive and inductive proof.

I’m not even saying it’s impossible to have a class with a fairly equal mix using the text (plenty of counterexamples in this very thread); but the text as written (where, let’s face it, most teachers will plow through at face value) does not seem to encourage such an approach.

This is well worth the read by anyone in this discussion thread, even if you have not seen the Discovering Geometry text. Great information about building toward formal deductive proof and how the van Hiele levels relate to student understanding of proof.

In regard to our recent conversations about the instances of proof in the DG text, I encourage you to read about the “Developing Proof” problems as well as “What’s Wrong with this Picture” exercises that are spread throughout the book, beginning in Chapter 2. For students at any level on the van Hiele scale, building toward a solid understanding of proof begins with practice putting their arguments into writing. The What’s Wrong with this picture problems are excellent spots for this development of proof to start.

I would echo @Andy’s sentiments about judging a book by it’s review section. The review sections tend to be focused on individual skills and recognition of stand alone properties (with a small handful of proof based problems worked in). A closer look at each chapter would reveal a much deeper examination and progression toward student understanding of proof. Starting with Chapter 2 (which half of is devoted to the development of inductive reasoning for problem solving), I count 15 instances of proof based, deductive argument problems and examples. In subsequent chapters, the opportunity for deductive argument and eventually formal proofs continues to ramp up.

But I think that there is also a bit of difference of opinion here about what deductive proof needs to look like. Even though you don’t seem to be a fan of the 2 column, it sounds like you’re looking for formality in the write up (format and language used). I do see the value in that, esp for a student on path for a university degree in math related fields; but since the vast majority of my students are no where near that, I don’t emphasize that until they’ve been convinced that there is even a need to justify their thinking. Convincing them takes time (months).

Second, using the review sections as an assessment of this book’s (or any book for that matter!) handling of proof is misleading. There are entire sections of a few chapters in the first half of the book *devoted to writing* proofs. Yes, the review sections don’t necessarily reflect that, but I’ve never taught a book (or evaluated–formally or informally–a book for that matter) using the chapter reviews.

As stated many times so far throughout this thread, a better assessment of how this text handles deductive reasoning can be found in the “Tracing Proof” document Serra links to above.

1) I agree with Franklin that, it seems to me, no-one is any more or less inspired for having spotted for themselves that, say, the opposite angles in a parallelogram are always equal. In fact, taking the time to spot that is arguably time wasted. A teacher can exposit that fact, and if students then either worked on proving the fact, or using it to solve angle problems, then that process can be very enjoyable – here is where I agree with Franklin, and with Daniel Willingham, we are naturally wired to enjoy problem solving, provided the problem doesn’t feel impossible.

2) I’ve used induction successfully (by my own measure!) in a way prescribed by Engelmann to teach conceptual ideas such as what a parallelogram actually is. In my case, I’ve used it for Prisms and Surds. On both occasions, if I tried to define what they were I figured I would only overload the working memory of many students – the explanation could never be sufficiently clear and simple, nor concrete, for a novice. On the other hand, by just showing lots of examples and non-examples in a carefully designed sequence, almost everyone seemed to get it, and many (not all) were able to articulate some kind of definition for themselves. I would then often have to provide them a more correct definition, however.

@Jason I’d be extremely interested in hearing your concerns with van Hiele especially as it relates to deduction and the current discussion. Please do share your thoughts when you have time to do so.

So I went through the text; there’s proof earlier than I remember, but it still strikes me as pretty scant. For example, in the review section for each chapter, here’s how many problems involve making a proof:

1. 0 out of 57
2. 0 out of 26
3. 0 out of 32
4. 3 out of 37
5. 1 out of 27
6. 0 out of 33
7. 0 out of 28
8. 0 out of 47
9. 0 out of 29
10. 0 out of 28
11. 0 out of 21
12. 0 out of 28
13. 14 out of 31

Jason: I would agree with Ryan above that deduction is presented way before the final chapter. In fact, I’ve never taught the final chapter due to time constraints and (more importantly) because I’ve never taught students who were ready for this.

But if I ask my students to complete an angle chase (or play Mastermind, or complete a Sudoku puzzle) and *justify their series of steps*, that is deduction in practice. If you’re familiar with the “What’s Wrong with this Picture?” theme Serra threads throughout the text, these problems are absolutely brilliant. These little tasks opened the door to some of my struggling students. I’ve had students verbally creating beautiful deductive arguments–the same students who probably couldn’t write this up formally in whatever format they choose.

Jason: my impression is that you’re *not* considering a student’s 2 step argument in a little task like I just outlined as a deductive proof? I totally do. If I’m wrong here, then I’m missing something in my understanding of what you wrote (sorry!) Other than adding formality and more rigor, I’m not sure what the book’s final chapter does for *most* kids (I’ve rarely taught a 9th or 10th grader ready for this formality . . . maybe this is more a difference in the populations we all teach?).

Serra has said more than once above to consider the “Tracing Proof” document that goes with his text. Even if you haven’t taught (or don’t teach) from DG, this is an important document. It helped me better understand how I present deduction to students.

I’ve been teaching from the Discovering Geometry curriculum for the past 7 years. It is hands down the best curriculum I have taught from. From a pedagogical standpoint, the progression of this course is brilliant. As Mr. Serra pointed out in previous comments, the teacher must understand where his or her students fall in the van Hiele levels of geometric understanding. Most high school students are not able to understand the role of deductive proof as a systematization of geometry. Many students do not appreciate or see the need for formal deductive proof at all. Many will be convinced, for example, that triangles have an angle sum of 180 degrees based on 2-3 measured examples or the fact that their 6th grade math teacher told them so. Prove it? “Why do I need to?” they will ask. They most certainly are not ready for formal deductive proof at the beginning of a Geometry course.

That being said, I have to say that deductive proof is woven throughout the whole of Discovering Geometry. Serra’s “Developing Proof” exercises are an integral part of the process. Are they formal proofs? Of course not. But they lay the groundwork for explanation in a deductive way that is less intimidating to the average geometry student. Students working though DG are experiencing informal deductive proofs as early as Chapter 2, and the students that catch on are doing formal deductive proofs by Chapter 4. This is most certainly not a curriculum that saves deductive proof for the last chapter of the book.

@Jason, you mention that you’d like to mix both types of proofs equally. Discovering Geometry is for you then. However, you have to look at a typical lesson as more than just the steps of an investigation followed by work time on exercises. The opportunities for deductive reasoning exist in nearly every lesson starting at Lesson 2.4. Early on, they are intended for group and class discussions. Have students share arguments in groups and share on large whiteboards. Encourage debate. Challenge students with “devil’s advocate” scenarios. Help students formulate more reasoned explanations. By chapter 4, your middle to high end students will be grasping more formal deductive reasoning via flow chart proofs. By the end of chapter 6, most of your students, including the lower kids, will be able to at least reason deductively. The improvement in quality of students’ arguments over the first 6 chapters is a cool thing to see.

This has been a great discussion to follow. I appreciate the perspectives that have been shared. Having taught Geometry from other curriculum in the past, I have found that students are turned off by a more “traditional” approach to geometry. They resort to procedural steps and memorization of postulates and theorems without developing a conceptual understanding of deductive argument. I remember my high school geometry teacher who had us write reasons in our two column proofs like “Theorem 3.4.2” which I assume made his correcting job easier. I learned nothing from this method. I really feel that unless you are teaching high end honors students, you’ve got to meet them where they are at – van Hiele level 1 and then build deductive understanding from there.