Proof and Consequences

A conversation was taking place over at Dan Meyer’s Blog ( about proofs, which is a topic that I find myself faced with about this time every year.

This isn’t a new conundrum for me. I’ve been working for while now trying to make this idea of proof, which, when compared to the typical form of textbook Algebra I should be an easier sell. But it just isn’t.

Here are some discussions of my previous attempts to sell it. Posts from Nov 2, 2012, Nov 16, 2012, Dec 7. 2012 are a few examples of my thoughts from around a year ago when geometry hit this place last year.

The problem I have is that the academic norms seem to prefer deductive reasoning to inductive and use of the theorem names. Those two things seem important to decide on before starting the journey of proof. If you are going to prefer deductive measurements, it rules out using measurements in proofs and it requires a lot more formal geometric language.

The problem that I see is that to rule out measurements (at least from the very beginning) and to strongly increase the formal geometric language in a way that makes deductive proofs possible from the very introduction of proofs creates… well… what Christopher Danielson is quoted as saying in Meyer’s post… “one of the most lifeless topics in all of mathematics.”

In order to breathe life into the topic, from the experience I’ve had, you need to let students engage in ways that make sense to them at first. The target to start the process is simply to get them comfortable with the idea of designing a functional persuasive argument about a mathematical situation. This requires recognizing that they need to start with a clearly stated claim (preferably something that is provable) and then start supporting it.

I find it helpful to let them pull measurements from pictures first and use those in the proof. The idea of comparing two things by length and NOT measuring them to get the length seems to a lot of kids like we are making the math difficult simply because we want it to be difficult. If they sense there is an easier way to solve a problem, then the explanation for why that method is against the rules had better be very strong, or else buy-in is going to suffer some pretty heavy causalities.

Once they get the hang of making an argument, then we can start by having discussions about what kinds of evidence are more compelling than others. This is usually where the students can figure out for themselves that each piece of information needs its own bit of mathematical support.

Next we can start deliberately exposing the students to different ways of proving similar situations. Triangle congruence seems to be a popular choice. We can have conversations about proving a rigid motion or proving pairs of sides and angles. Eventually certain kinds of explanations become more and more cumbersome. For example, using definition of congruent triangles to prove that two triangles are congruent as shown here:

Do we really need to keep going to find the three pairs of congruent angles?

Do we really need to keep going to find the three pairs of congruent angles?

Then, we can start pushing into shortcut methods. Mostly because those angles are going to be somewhat tricky to find (and why do more work than you need to… the students DEFINITELY identify with that.)

By using this method, I am trying to create what I’ve heard Meyer call “an intellectual need” for additional methods to prove this claim. (Keyword: trying… not sure how successful it is, but I’m trying.)

Then, that transitions fairly smoothly into stuff like this:

2013-10-28 08.28.32

… where we standardize the side lengths of two different triangles and see how many different triangles we can make and in what ways they are different.

Now, the tougher question is whether or not you allow the class consensus following the “Straw Triangle Activity” (which was a gem that came out of Holt Geometry, Chapter 3) to count as proof of the SSS theorem. In an academic sense, now we should “formally prove” SSS theorem. To most of the students, it’s settled. Three sides paired up means the triangles are congruent. What are we risking by avoiding the formal SSS proof? Do we risk giving the impression that straws and string are formal mathematical tools? But wait… aren’t they? What do we risk by doing the formal SSS proof? Do we risk our precious classroom energy by running them through an exercise there isn’t a lot of authentic need for right now?

Am I able to say that this is the definite recipe for breathing life into geometry proofs? Not even close. I am sure there are students who are completely uninspired by this. I can say using anecdotal evidence that engagement seems significantly and satisfyingly higher then when we used to run deductive two-column proofs at students from the very beginning.

But, we’ll have to see what the consequences are as we keep going.


What Should Professional Development Do For You?

What expectations should teachers have when they attend a PD session?

What expectations should teachers have when they attend a PD session?

I’ll acknowledge that the job of a professional development facilitator is not easy. I’ve done it twice and each time, I felt like I didn’t do a very good job. So, I’ll be the first to admit that it is a lot easier to sit and criticize than it is to do the good job.


I went to a PD session today that sold itself as a session on developing “deeper understanding of mathematics pedagogy” and “the capacity to utilize an inquiry approach to instruction.” Add to that the Michigan Association of School Boards awarded the project Michigan’s Best Education Excellence Award just this year. That’s pretty good, right?

The morning session began with a 75-minute talk about the history of the program, the grant, and the logistics of continuing education credits. At this point, the energy was completely out of the room.

… and it never came back.

The afternoon was a bit better largely because they broke us up into groups of teachers and gave us a fairly rich task that included about an hour’s worth of discussion choosing an engaging problem to try with our classrooms and discussing how to pitch the problem, anticipating what methods the students might use to engage the problem, and the possible struggle points.

This isn’t to say that there wasn’t potential for some fantastic discussions. The facilitators brought some wonderful starters to the table. “What are some common traits among mathematically powerful students?” and “What makes a rich mathematical task?” are great hooks as long as the fish are biting. Unfortunately, by the time we got there, no one was in the mood to discuss anything.

But, it would be wrong to sat that I came away with nothing. I came away with this: Teachers are an overstressed and undercompensated group of people. Their time is incredibly valuable. So, when they step out of the classroom, the time needs to be effective, efficient, and potent.

But I’ve seen some pretty effective professional development models. I am thinking of the examples put forth Dan Meyer (@ddmeyer) and the MathTwitterBlogoSphere (#MTBoS). I got to be a part of one of Dan’s “Perplexity Sessions” this summer and the #MTBoS and I have a longstanding relationship that includes some excellent professional growth. So, why are these PD’s more effective than the one that I attended today? Well to start with they…

1. Start with a bang! Potential buy-in is never higher than at 8:00 am when everyone is still an open book to what may or may not happen over the next few hours. Plus, everyone is nicely caffeinated and a little energized from the change of scenery. If you waste this energy, you won’t get it back. #MTBoS usually hooks people in because educators come looking for help solving a specific problem. The first experience is an effective one. Dan received a bit of an intro from the hosts, but once he got the floor, he wasted no time getting the entire group started DOING something so that he could…

2. Model effective instruction! Dan did this all the way through his session (which lasted 6 hours). The #MTBoS nails this one too. Many, many members of this group are willing to open their classrooms up to the masses through their blogs. I love the question: “What effective teacher moves did you see me do just now?” In order to do that, you mustn’t have the group looking back at your introductory lecture that lasted longer than the class periods that any of those teachers teach.

3. Jam pack the session: Fill it end-to-end with rich tasks one-after-another. I was floored at the amount of ground covered in the Perplexity Session, #MTBoS handles their business in this respect as well. You can go from blog to blog, twitter feed to twitter feed for days on end. You should at least be able to put enough rich tasks in front of the group that keeps them actively engaged practically the entire time so that you can…

4. Draw as much as you can from the group. This one sort of goes back to modeling effective instruction, but that having been said, no one knows the classrooms of the group members like the group members, so build into your individual segments repeated think-pair-shares where the central ideas are being developed, expressed and generalized by the group members. After all, it’s their classes that this material has to go back to. Developing these ideas can help the facilitator, too, because you should…

5. Do a tight, concise debriefing session at the end of each segment that not only ties together the main objective of that segment, but relates it strongly to the rest of the overall session. This needs to address a couple of main points: Why will this help my students, why will this be worth the effort to implement. Dan actually gave the group members a chance to share their “secret skepticisms” to draw out the roadblocks and address them.

It is to these 5 ideas that I will commit. If I should ever facilitate a PD session, I will do my very best to meet each of these five expectations. If you should be there, please hold me to them, because it seems like you should be able to expect your PD to do those things for you.

Glimmers of Twitter Hope

A tool is just that. Tools are inherently valueless. They aren’t good. They aren’t bad. They are tools. They are perfectly meaningless and useless separated from a user. This is true of classical tools as well as much more recently-invented tools. Pencils, hammers, pillows, frying pans, coffee mugs are all just tools that could be used to do productive things or destructive things.

Twitter is an example of a tool that I have found incredibly productive. The #MTBoS has helped to define and demonstrate Twitter as a tool for professional development, peer review of the goofy things that rattle around my head, and a warehouse of effective resources for the effective instruction of mathematics. However, it goes without saying that those are not the only uses that Twitter has. In fact, the general student population seems fairly inexperienced with Twitter as a academic networking tool. Their use of Twitter has been largely social, which isn’t necessarily bad, but reflects a missed opportunity to take advantage of a resource that could be helpful and supportive.

Last night, I got to see this on full display. I ran my first (EVER!) Twitter review with my geometry students. I invited the students on for an hour and tried to lead a Twitter discussion. It’s the first time I’ve ever done that and it was the first experience many of the students had had with that type of reviewing structure.

In general, the discussions were teacher-to-student, which has it’s value. I’d ask a question, they’d respond. They’d ask a question, I’d respond. And despite my best efforts, I had difficulty fostering those meaningful student-to-student interactions to push this chat to the next level of group learning.

But then this happened (I apologize for the grainy photos. I’m still working on learning some photo editing software… that blurring out trick would be killer! Anyway… ):
It started with a question that I tossed out in response to a student’s statement of confusion:

Twitter Exchange 1

Then that student-to-student started happening…

Twitter Exchange 2 Twitter Exchange 3

Notice what happened: The student got some advice as to where to find the definitions directly from another student. Then UH-OH! The geometry binder got left at school. But this is the 21st Century! That is TOTALLY not a problem, because…

Twitter Exchange 4 Twitter Exchange 5

She went on to send her both sides of handout G.CO.4 as well. This is a powerful moment. It is a moment that sets a precedent. Marissa just found a resource that she can trust. Twitter is beginning to be redefined in her mind as a potential academic tool.

This is a fantastic example of what is possible. This is what we in the #MTBoS have been doing for a while now. We are used to networking, leaning on trustworthy sources, and asking for help (in 140 characters or less). Most students aren’t used to using twitter in this way. Students need to be taught how to use their networks for more than simply social needs. If the academic world is becoming more digital, then let’s take the digital world and make it academic.

Last night’s review session gave me a glimmer of twitter hope that this social media machine that is being driven by the young people in our classes can be as effective a learning tool for them as it has been for me.

The Real Value of the #MTBoS

Our friends over at Explore The MathTwitterBlogosphere have prepared a variety of opportunities for us math teachers to become better acquainted with this community that exists with one goal in mind. Take math teachers in and make them better. What distinguishes the #MTBoS as it has come to be known from other grass roots movements is that this movement is completely inclusive. We WANT you to join us because we all have something to share and something to learn.

As part of the exploration, Sam Shah (@samjshah) has asked us to consider an interesting question: What happens in my classroom that makes it distinctly mine?

Within my district, my practices and classroom policies are a little off the mainstream. My students get boatloads of exploration and practice activities. That isn’t unusual, except that more than half of the work I assign I neither collect, nor assign points to. Oh, and I try to avoid homework at all costs. I try to put students in an ability to use mathematics in ways that seem authentic. As a geometry teacher, this means that I use proof and algebra probably more sparingly than most, opting instead for visual explorations when appropriate. Also, I have an aversion to textbooks.

I believe in authenticity above all things. There is a belief that most students won’t do work that doesn’t have points attached. I have found the opposite to be true. I have noticed that students are more willing to explore and practice when we are honest about what it’s for. The same with homework. Homework doesn’t tend to be a very meaningful learning activity. So, I don’t use it very much.

We follow the same idea with Algebra. We use algebra… when it makes sense in the context to do so. We graph… when it makes sense in the context to do so. But transformational geometry doesn’t require these things. In fact, there are many different ways to explore the strategic transforming of shapes and figures. From an understanding of transformations, you can move toward congruence and similarity discussions. Similarity leads to proportional reasoning and then trig naturally follows. In my opinion, this is  fairly simple way to look at geometry course, but it has lead me in directions that are, as I said before, a little off the mainstream. It forced us to ditch our textbook, which was heavily, heavily algebraic. We had to replace it with our own handouts which were homemade or found on the #MTBoS. That opened up a ton of conversations about what good math looks like and what it will take to lead our students to it.

This is where the real value of the #MTBoS has shown off. The #MTBoS means never having to be alone, regardless of your thoughts or ideas. There’s someone who has thought that thought and discussed the possible implications on a blog somewhere. There’s someone who has tried an activity and tweeted about how it went. There’s someone who has a link to a handout and an e-mail address so that you can ask questions about how it got delivered. If you try it and blog about it, there’s people who will read your post and leave comments. It’s a form of individualized PD that is difficult to find elsewhere.