New Podcast: Why I love this picture

So, I get the pleasure of supporting a different school community than I have been these past couple of years. As a part of that work, I have decided to start a short video podcast. (The Geogebra post was the first episode.)
Occasionally, that means I’ll be borrowing “thegeometryteacher” content and podcasting new life into it. This is one of those examples. I originally posted this last fall…


Here’s the picture…


I took this picture this morning in Lansing, MI during some wonderful small group math talk. There is one device, an iPad, with an Osmo setup attached to it.

So, here’s why I love this picture.

There’s tech and…

… manipulatives and whiteboard markers and collaboration. Tech fits among the variety of tools available. It’s not the best tool unless it best supports the learning. And sometimes other tools work better. And in this case, the students were being led into learning with all the different tools.

The activity is built around the social nature of learning. 

The kids are clearly sharing their answers with each other and the teacher… there is a constant back-and-forth, sharing ideas and discussing them. They were seeing each other ideas, but…

Their strategies aren’t all the same. 

One girl is using an array. One girl is using groups of three. One girl wasn’t quite sure what to do (so it was a good thing she could see the other two girls’ work.)

The teacher’s hands are off. 

The students are doing the reaching, arranging, manipulating. Remember, the one that does the work will be the one that does the learning.

That’s why I love this picture. it captures so many wonderful things about the right kind of teaching and learning.

A toolbox that is truly full

When I think of a math activity that really flexes it’s muscles, I’m reminded of the activities that could reasonably be solved multiple different ways with no method being preferable on the surface. These are tricky to create (or to find and steal). They are also somewhat taxing on your students (especially if they aren’t used to this kind of problem-solving). It requires a more intense, higher-order level of thinking.

photo credit: Jo Fothergill - Used under Creative Commons

photo credit: Jo Fothergill – Used under Creative Commons

Many schools are reaching the stage where students are carrying around smart devices. Increasingly schools are issuing them (we are up to 4 districts in our country that are now 1:1). Also, in many districts, students can be trusted to bring smart phones in with them. With all of these devices available, it seems like we could integrate a new set of tools into our tool box for consideration. We should design activities that allow the students some control over what mathematical techniques they choose to employ. But increasingly, it’s making more sense to also allow the students some control over what tech tools they are making use of.

That is a nerve-racking idea for some, especially since as soon as students start dabbling in technology that the teacher is unfamiliar with, they become their own tech support. There is a very real (and perfectly understandable) anxiety over students using technology pieces that the teachers aren’t familiar with. But we could flip that on it’s head.

First, we could model some of the tech pieces that we are familiar with.

Consider an activity where in the students use a Google Form to poll their classmates, then enter the data into a Desmos sheet to do the analysis. The formative assessment of the analysis could be done on Socrative or Google Forms.

I mean, this is a standard math task: Gather some data, represent it visually, analyze, and produce a product to submit to your teacher. What makes this different it two-fold: First, some of the more annoying parts are going to be relieved by the technology (namely recording the data, plotting the points, and drawing/calculating the best fit line… those are also the parts that create barriers to our students with special needs). Second, you are giving the students meaningful experience using their technology for something that makes their school work easier and more productive. (Imagine that… both easier and more productive… both…)

One of the goals of an activity like this is the students gaining an appreciation of the roles of each of those different technology pieces, in the same way as we give them specific prescribed practice with the math skills to gain comfort. But it should stay there in either case. At some point, the students need to build in a working understanding of each of the tools in their tool box – math, tech, or otherwise. (I was even vulnerable to whining… the right kind… used at the right time. Proper tool for the proper job.)

Then, when we unleash our students to solve a problem by any means necessary, with a proper foundation underneath them, we run a much lower risk of them choosing something completely off-the-wall. Students might replace our technology with choices of their own, but if they know that we have a standby that will work, then often the replacement is something they find more useful… and it might be something we’ve never seen before… and they might be able to teach us how to use it.

And never underestimate the power of allowing a student to be the expert in the room once in a while.

Full disclosure: The data that I plugged into the Google form came from here. Many thanks to who is named as the owner on the site.

Reflecting on the Common Core…

photo credit: flickr user "Irargerich" - Used under Creative Commons

photo credit: flickr user “Irargerich” – Used under Creative Commons

A lot has been said about the Common Core State Standards in the last year. Some of it has been by me. Some has been by guys like Glenn Beck who is not a big fanMost (if not all) states have some sort of a “Stop Common Core” group. There is even a #stopcommoncore hashtag on Twitter that turns up quite a few results (although some use that hashtag as a means of highlighting objections in the arguments of CCSS opponents.)

The pub isn’t all negative. Some groups, The NEA among them, have come out in favor. Phil Valentine has some good things to say in support.

It is possible that both sides are probably overstating the impact that the CCSS will have. That being said, I will admit that I have some opinions on the CCSS. This year is our first year introducing a new geometry curriculum that we designed around the CCSS. I’ve written a few pieces before this one that have chronicled my journey through a CCSS-aligned geometry class. For example, I’ve documented that the CCSS places a greater emphasis on the use of specific vocabulary that I was used to in the past. I have also discussed (both here AND here) that the CCSS has present the idea of mathematical proof in a different light that I have found to be much more engaging to the students.

As I read the different articles that are being written, it seems like the beliefs about the inherent goodness or badness of CCSS has a lot to do with how you view the most beneficial actions of the teacher and the student in the process of learning. It’s about labels. Proponents call it “creativity” or “open-ended”. Opponents call it “wishy-washy” or “fuzzy”.

I suspect they are seeing and describing the same thing and disagreeing on whether or not those things are good or bad.

To illustrate this point further, a “Stop Common Core” website in Oregon posted a condemned CCSS math lesson because the students “must come to consensus on whether or not the answer is correct” and “convince others of their opinion on the matter.” The piece ends with “What do opinions and consensus have to do with math?”

The authors of this website are objecting to a teaching style. They are objecting to the value of a student’s opinion in the process of learning mathematics. Fair enough, but that was an argument long before the CCSS came around. I can remember heated discussions during my undergrad courses about the role of student opinion and discussion. (My personal favorite was the discussion as to when, if ever, 1/2 + 1/2 = 2/4 is actually a correct answer. One of my classmates rather vehemently ended his desire to be a math teacher that day.)

The CCSS have become a lightning rod for a ton of simmering arguments that haven’t been settled and aren’t new.

Consensus-building and opinions in mathematics vs. the authority of the instructor and the textbook. Classical literature vs. technical reading. The CCSS have woken up a lot of frustrations that are leading to some high-level decisions such as the Michigan State House of Representatives submitting a budget that blocks the Department of Education’s spending on the CCSS.

It is a little strange thinking that I am making a statement in a fairly-heated national debate every time I give my students some geometry to explore, but it seems like I do.

And I am prepared to make that statement more explicitly as I continue this reflection.

Let’s Keep This Simple

Several years ago, for about 18 months, a wave of professional development came through our school that focused on Kagan Cooperative Learning Structures. We got trainings, and books, and flip cards filled with structures with cute little names on them.

And it all seemed very… I don’t know… complicated.

Now, Cooperative Learning is a fantastic model, especially if the contexts are rich enough to make the interactions necessary. However, the Kagan model has changed the game a little bit, and I’m not sure if it is for the better. It seems to me that there are some problems with looking at cooperative learning in this way.

To illustrate, I’ll use an example from my class.

This morning, I knew three things:

1. I needed a good, quick, formative assessment of what my classes knew about right triangle trigonometry and what they didn’t.

2. To get that, I needed to engage them in a variety of simple problem-solving activities.

3. On a Monday morning like this, the last week before spring break, the Monday after the first weekend of March Madness, the first Monday in a while without snow on the ground, the week of the first sporting events of the spring season here in Michigan…

They were NOT going to engage sitting in their seats doing a quiz.

Sitting still wasn't going to work out today.

Sitting still wasn’t going to work out today.

So, I took the different types of situations I wanted, created nine different problems, printed them on sheets of paper and taped them around my room.

The I grouped the students and sent them around the room to solve one problem every 2-4 minutes and then report back what they were confident with, so-so with, and confused about.

I still needed to assess them.

I still needed to assess them.


According to Kagan, I used a Modified Gallery Walk with a possible Rally Coach with a likely Carousel Feedback. It was effective because it followed the PIES framework of Positive Interdependence, Individual Accountability, Equal Participation and Simultaneous Interaction (Kagan, 2009).


So, I taped the problems to the wall one question and a time and sent them around the room.

So, I taped the problems to the wall one question and a time and sent them around the room.


I don’t want to sound like I am making fun of Kagan. I’m not. A ton of R & D went into creating the program, the vocab, and the resources. But, at least in this case, we’ve taken a fairly simple principle of knowing what your class needs and being flexible and WILDLY over-complicated it with a ton of gimmicky-sounding vocabulary.

But teachers need to have a variety activities to use in order to be flexible, right? Without programs like Kagan, where do they go to get them?

How about the expert down the hall?

We as a teaching culture have lost the value of the classroom observation. In the schools I’ve been in, teachers hardly ever get a chance to see each other teach. It could be that this is what has caused the need for books, seminars, and flip-cards. We aren’t letting our teachers share. Seeing what the masters of the craft do when there are 25 real, live students in their room is a whole different experience… a powerful experience… an experience that we are leaving on the table.

I have been mentored by three different teachers. I was NEVER directed by my principal to observe them teach. I have mentored two different teachers. I had to ask special permission, and make all the arrangements in order to observe them or to have them observe me. It shouldn’t be this way.

When we over-complicated things, they become confusing and overwhelming. We’ve been forced into this by not letting our novice teachers watch the master teachers at work. It seems reasonable to assume that the example set by the expert next door will spread good practice a lot farther than the 400-page book that never gets read.