Reasoning and Sense Making in Geometry

I've finally caught my breath after returning from a fantastic 3-day Institute in Orlando about motivating students to learn mathematics through Reasoning and Sense Making (the link goes to a page with handouts from the many presenters.) What a wonderful experience being together with about 700 other math teachers from all over the country (including Puerto Rico and the Virgin Islands - but only one other person from my neck-of-the-woods) who are also concerned about the state of math instruction these days.

Funny thing is, there has been concern about the way mathematics has been taught in this country since way back in 1833, when a guy name Colburn wrote about using the Pelozzi method for teaching arithmetic. He complained that arithmetic was all drill and memorization, not reasoning. Sound familiar? The NCTM president Mike Shaughnessy went through a long list of early quotes, including 1923, 1935 and more recently since the '80s. I hope he'll upload his PowerPoint, because I'd love to have those quotes!

I mostly followed the sessions about Geometry, since I couldn't figure out last winter how to get students interested in doing proofs, which is what I think is the fun part of geometry. It turns out that kids are turned off by having to prove the obvious, when we ask them to prove things like vertical angles as congruent.

Michael Battista, in his presentation, The Role of Proof in Geometry, said that proof in Geometry is a caricature, since we are teaching the form of proof, rather than the content. We start too low to teach the method of proof at a point where it just doesn't make any sense to prove things. Kids experience that as busy work, which just doesn't make sense! We should start where it takes some thinking, reasoning, struggling, to figure out how to get from the given to what is to be proved. When the students have figured it out and can explain it - that is when they will be open to learning how to do formal proofs. The proof is just a final written justification of the work they have already done. No mathematician would start writing the proof before having reasoned his way to a solution. We shouldn't expect our students to do so either. Proof is a personal sense making, he said, where we go from saying what is "true" to "why" it is true. We are explaining our reasoning to others. Students move through 5 levels of geometric understanding, he said (known as the van Hiele levels:)

1. A visual, holistic examination of the shape
2. A description of the parts and their relationships
3. The interrelating properties (like vertical angles and the properties of parallel lines)
4. Conceptual proofs (explaining verbally)
5. Formal proof.

We have been trying to get students at level 2 to write formal proofs. They need to work with geometric shapes a while to get to level 4. At that point we can introduce formal proofs.

Battista has written a lot about using the software Geometer's Sketchpad to help students reason about geometry so that it makes sense, and has contributed to several books published by the NCTM. The man behind Sketchpad, Michael Serra, honored us with a wonderful collection of Investigations in Geometry from his textbook Discovering Geometry. We had a nice break in all the talk working in groups to figure out a variety of geometric problems. I worked with Origamics problems, developed by Kazuo Haga. (I just ordered the book. What a fun way to work with geometric reasoning!)

Jeffery Wanko provided a fun session, Developing Proof Readiness with New Logic Puzzles. He uploaded his materials to the Institute website, so you can enjoy them, too: Presentation (PDF)and Handout (PDF).We started with puzzles and their solutions, which we studied to figure out the rules. Then we did a big one together (this was a whole roomful of Sudoku addicts, of course,) and worked individually and in pairs to solve some smaller ones. He provide several pages of puzzles, which gave me something to do besides reading on the long plane trip back home! He recommended the Japanese puzzle magazine, Nikoli.com. Students can become ready for writing formal proofs through talking about puzzles like these with each other, getting to at least levels 3 and 4 listed above.

All that confirmed my previous experience that geometry is fun. I hope I can inspire my students the same way!

I will try to find time to write more about my experiences at the Institute in another blog. Besides swimming every day,of course, the most valuable part of the Institute was the many discussions with teachers from all over the country with many different school experiences.

Memory and Sense Making

On Wednesday I am off to Orlando to participate in the National Council of Teachers of Mathematics Summer Institute for High School Teachers on Reasoning and Sense Making. I am looking forward to being with a group of teachers who really want their students to understand mathematics. Too often during teacher training I ran across teachers who were more of the "drill and kill" school.

With my experience with myself, my own children (now successful adults,) and the children and young people I have taught, kids don't learn because you force them to memorize something or give them drills to do whatever time and again until it sinks in. Kids learn because they are curious about something and want to find out about it. If they have a reason to learn something that means something to them (and I doubt "to get into college" or "because it's in the standards" are reason enough for most students,) they will want to learn it, and will dig into a topic until it is theirs. They might even ask someone for the answer - or help to find the answer.

I read a short article yesterday about some research that implies that people don't remember as well as they used to because now they can just Google stuff to get answers they don't have to remember. Evidently some people were tested on how well they remembered things (probably a list of unrelated facts) and some were given the opportunity to enter them on a computer. That last group, of course, forgot them immediately. But that doesn't prove the thesis that we remember differently now. The author of the article pointed out that Socrates was just as worried that the new-fangled techniques of writing would ruin people's ability to memorize things - which is probably true, of course. I write things down so that I can go on to investigate other things. In a sense, the written word is an extension of our long-term memory.

During my teacher ed classes I came upon several references comparing the brain to a computer. You know, data comes into short-term memory, but it has to be connected to other information to be transferred to long-term memory. If we just give students facts, or formulas, or steps to solve problems, they may remember them long enough for the unit test, but if they don't have a way to connect those data with something else - something that makes sense to them, and they want to know about - that data we tried to stuff into their heads probably won't be around for the final, or state exams - or life.

I remember a newspaper opinion piece written by a teacher years ago in Denmark, who claimed that a teacher's job is not to fill in the holes in students' brains, but to create the holes in the brains, so that students would go around looking for what they could put into them. Learning, he said, is making holes, not filling them in. Those holes are what students create while they are making sense of their world. And the holes will never get filled. They will be dug deeper, with lots of side channels that connect up with other holes.

This was illustrated beautifully in a very moving film we saw on Saturday, Buck, which is about a guy who spends 9 months out of the year telling people how to train their horses (not break them) at clinics all around the country. Buck likes to say he's not helping people with horse-trouble, he's helping horses with people-trouble.

I kept thinking that he was talking about classroom "management," where teachers are figuring out how to train their students and need help with "student-trouble" while in reality, it's the students (who have to be there, just like the horses had no choice in the matter) who have "teacher-trouble." The movie was about the best movie on education I have seen. I kept wishing I had a notebook, so I could write down all his words of wisdom. So I bought the book that became the movie The Faraway Horses, in hopes that some of those bits of wisdom are stored there.

One of the most telling episodes in the movie was a woman who told about how Buck had changed the way she trained her horse for dressage. Evidently in the bad old days, horses were trained to get into various unnatural positions by harnessing them with torture instruments (there were examples shown in the film.) Finally the horse gave in and did as required to avoid the pain and humiliation of the harness. But the woman had participated in a sheep-herding clinic with Buck, and discovered that all those unusual positions came naturally to a horse when he was using them to herd sheep. The horse found a connection where he needed to be in that position. And then during dressage, he easily moved in the position (probably fondly remembering the weekend herding sheep.)

Are our students being difficult because they don't want to be harnessed to a school desk when it doesn't make sense to them to be there? Are we trying to break them rather than helping them make sense of what we think they should know?

At the NCTM institute, we have each selected a different area to concentrate in, which for me will be Geometry, which I think was my favorite math subject in high school. I taught some Geometry this past year, taking over from another teacher. It was very difficult teaching students to do the proofs of geometry, which is what I liked best, and which is what geometry is all about. I hope that the Institute will help me see how to present geometry so it makes sense to them. Of course it's easy enough to make sense when you're talking about things that can be represented physically, like area and volume, circles and cylinders. But the abstract high-order thinking of proofs seems to have been distracted by low-level memorization of theorems.

I expect to be a better teacher after the Institute - but it is only one of many ways I am trying to make sense of my job as a teacher.

Addendum

While reading this afternoon I happened upon a note that is so pertinent to this, that I am quoting it here:

When reviewing radioactivity for this book, I was reminded that too often in science resources, authors explain what happens without really explaining why it happens. If you can only describe occurrences,then you really don't understand what's going on, and you end up only memorizing what happens. If you can figure out a mechanism for the occurrences, though, then you can build a lasting understanding of what's going on. Even though scientists often can only describe what happens when they first encounter a phenomenon, the ultimate goal is a mechanism for the phenomenon and the resultant understanding. You can compare this to mathematics, in which there are rules to follow. Only when you understand the reasoning behind the rules do you understand math.

William C. Robertson, in More Chemistry Basics, p 109 (my italics)

Thankful that I am finally teaching again!

This is a very good way to start a job. Work one week, then get the next one off!
I'm using the week to get a better picture of my students, make new seating charts so that the weaker students aren't all sitting at the same table, and racking my brain for ways to teach my geometry students how to think "proofs," which only a few are really ready for. Having a non-math-credentialed sub for 2 weeks was not very helpful there.
But I've talked with my colleagues and have some of my own tricks up my sleeve to get their minds more focuses. Among other things, I transferred my Geometer's Sketchpad registration to my school computer, so I can at least demonstrate parallel lines and the like dynamically. I'm considering letting them work with it table by table, since we only have the one computer in the classroom. I also found some great books very inexpensively on the Key Press website. I'm looking forward to trying out Mathercize warm-ups,  which are intended to develop students' reasoning and observation skills.

Cajon Pass from the summit

We are supposed to be teaching through projects as well at our charter school. We math teachers only need to provide 2 projects a year. I'm figuring on a project involving slope in my geometry classes. I drive up through the Cajon Pass every day to and from school, which goes up to over 4000 feet on the 15 Freeway. I figured we could divide the pass up into small sections, and use a topographical map to find the y difference, (while the x-distance in the map distance (measured with a string?) Then they can each make a graph of their section, and string them along the whole wall, along with pictures from Google Earth or their own.
So back to grading papers and making new seating charts!
Happy Thanksgiving!