I missed #WorldTessellationDay by a couple of days (June 17) but I figure it’s close enough to the time to celebrate it and also that I am posting to my blog for the Math Forum at NCTM for the first time! All’s right with the world.

I missed #WorldTessellationDay by a couple of days (June 17) but I figure it’s close enough to the time to celebrate it and also that I am posting to my blog for the Math Forum at NCTM for the first time! All’s right with the world.

To appreciate the significance of this is to know that it was my interest in tessellations, the webpages I learned to write as a result of participating in a Math Forum Summer Institute and the interest those pages received, that helped me land my Math Forum job. In celebration, here are a few of my favorite photos of tessellations:

While we were visiting our older son and his wife during our Thanksgiving holiday we noticed that their across-the-street neighbor turned his Christmas lights on Friday at dusk.

We first noticed them at around 4:00 pm as he was checking everything (quite a lot to check!) and adjusting here and there. By 5:00 pm or so it was dark enough for the lights to really stand out and sparkle. I found myself thinking:

* the effect of the holiday lights is more dramatic when it’s truly dark

* a full moon could lessen the contrast

* a timer could be quite handy because you could have it automatically set and not have to remember when the sun has fully set

And as I had these details running through my mind, I stopped myself and thought, “What does it matter?” When is timing really important? Does it matter that the holiday lights are on at dusk and they are still competing with the light in the sky? If Friday was an unusual day because he was just setting up, perhaps, all of the days after (when I wouldn’t be there to watch!) could be timed perfectly to take full advantage of the contrast of the dark sky.

I found my mind shifting to the classroom – how is timing important?

What makes for good timing? Is “perfect timing” achievable? How do we find the right time during a class period to present concepts or activities to maximize the effect? Is contrast important? Is set-up important? How do we know when we’ve hit our perfect stride? Is our (the teacher’s) perfect stride also our students’ perfect stride? Do they ever happen at the same time?

I often reflect on how a class may have seemed to me (since, after all, the teacher is a learner just as the students are — I should reflect on what happened to me during the class period). I try to reflect on how a class may have seemed to the students but how can I really know unless I ask them. Is there time? There should be.

Recently I had the opportunity to introduce the online Problems of the Week routine to four classes of sixth-graders. I prepared ahead of time by:

* setting up all of their logins

* creating a handout sheet with step-by-step instructions — individualized for each student (I used the “merge” function with MS Word and Excel — very handy!)

* selecting a problem with their teacher – we chose “A Cranberry Craving” — seemed fun since it has a Thanksgiving theme

My goals for the class period were to have the students:

* comfortable logging on to the Problems of the Week with their individual username/password

* be introduced to this system with a “step one” approach rather than a “final” approach to their problem solving

By setting up the **training** session (going over the technical aspects of the PoWs) in this way, I was attempting to influence their learning. I want them ultimately to be comfortable with the problem solving process. I want them to think of problem solving as something you do over time:

*The goal is not to be over and done. The goal is to think, express, reflect, and revise.*

Although I included that sentiment on their login instruction sheet, I didn’t dwell on it during the training. If the training is structured well, however, and the students practice each of the steps. I think with time they may understand the process I am hoping they adopt for problem solving.

What are your training routines? How do they support your teaching/your students’ learning?

In my role at the Math Forum I work with math teachers in their classrooms and from that vantage point I often view these “players” interacting with each other:

students <-> students

students <-> teachers

teachers <-> teachers

teachers <-> other professional development providers (other than me)

teachers <-> school administrators

teachers <-> district administrators

school administrators <-> district administrators

I find myself thinking of two common themes.

The first theme is from parenting — “**Do as I say!**” The TV show *Mad Men* comes to mind where the parents are drinking and smoking and it comes as a surprise to them when the young daughter tries to sneak a smoke in the bathroom. She’s just modeling the behavior of the parents, right? Is she completely to blame for an action that has been modeled by her parents?

As I think of that phrase “Do as I say” the implication is “and not as I do.” In many of the interactions that I view, the person of authority in any of the pairings is trying to improve the behavior of the other. I’m using “behavior” to include “instructional behavior” or, in other words, how the classroom is managed or functions. The classic example is when you find yourself being lectured to when the theme of the professional development is student-led instruction or something that is the opposite of lecturing!

The second theme is **valuing** — this has always been an underlying theme of my interactions with the Math Forum from my very first encounter in July, 1995. Each individual has value and the way that we acknowledge their value is to listen to them before suggesting any action or change. An example of how this works is our Noticing/Wondering activity and it turns out that it is extremely powerful!

If I pose a math context (without any question to distract us) and I ask students “What do you notice?” I am immediately valuing their input. As I listen and/or record their noticings, I am continuing to value their thoughts. And, when used well, I value and make use of those thoughts as we move forward with our mathematical thinking.

This first step of valuing could go a long way in working with teachers. Instead of imposing the next round of professional development “on” them, I wonder what might happen if we were to pose a classroom situation and ask them what they notice. It might take a little extra preparation but it would provide the valuing that is so needed. Teachers, just like their students, are not blank slates!

Think of a professional development session you have recently experienced. Was your initial state of mind valued? How successful was the experience for you?

This morning I attended a meeting at the School District of Philadelphia. The presenter included in his remarks references to “making learning fun.” I wondered how he might respond if I asked him to tell me more about what he means by “fun.” For students to have fun, does that imply entertainment? Does it imply games or video games or other media that we think will capture their attention? Do we connect games to fun? Might something be fun if we’re feeling successful at doing it? If we “get” something and we’re smiling and talking and engaged, does that show that we’re having fun? Can we be having fun if we look serious? How do we define “fun” and, in particular, how do we define “fun” in an educational setting?

I find myself thinking of math software that is designed for students to answer a certain number of questions and then they are rewarded with a short time of some sort of “fun” game. I wonder if they think the fun part rests the student’s mind or is such a reward that the student will work hard to be able to have fun. I also wonder at what point the student is more engaged — the question/answer part or the fun game part.

I’ve known teachers who feel that if only they could entertain their students (like someone who’ve they’ve watched at a conference or, perhaps, a colleague at their school) then learning would be fun and their students would do well.

I have developed a different idea of what “fun” is in a classroom. For me learning is fun when students are given access to the subject. Students who are able to make sense of what they’re being asked to do, make it be their own learning, have facilitators to help them engage, are encouraged to persist and re-engage — I’ve seen those students have smiles on their faces and excitement as they learn. There is a lot of fun to be had when a student feels they’re in control of their own learning.

I would claim that Max and Steve are having fun here even though they’re not smiling!

What is “fun” in your classroom?

We were walking across the street entering “Kristian’s Park” (a name we had made up because that’s where we had met Kristian) and Lee suddenly asked us:

*When did the world turn color?*

I remember being surprised by the question but I also remember that my husband and I took our 5 year old’s question seriously. Since we didn’t quite know what prompted his question, we asked him some questions to get an idea of what his frame of reference was. I don’t really remember what we asked him but I do know that he provided this additional information:

*The photos in our albums are black-and-white but the photos now are in color. When did the world change?*

Wow! Now we had something to work with!

Lee, my husband, and I were reminiscing the other day and this story came up. We all remembered that we had a good conversation and Lee, in particular, remembers that we valued his questions and responded accordingly. Lee and I compared the “picture” we each took of that moment in time! As we each described what we remembered of the moment and where we were, it was uncanny how much they matched. We agreed that we are very visual and it seemed quite appropriate that Lee would have thought of that interesting question.

I told him that it reminded me of one of the practices at the Math Forum. We encourage teachers and ourselves to value what a student is thinking. If we’re not sure where the statement or question is coming from, we don’t discount it but instead we ask some questions. Valuing each individual is key.

Take a look at this **sample** of our Problem Solving and Communication Activity Series!

In early April I posted thoughts about thinking of students as lumps of clay versus sprouting seeds. This morning I started thinking of clay vs. seeds again but this time in conjunction with the CCSS Standards for Mathematical Practice. The first practice is “Make sense of problems and persevere in solving them.”

When I approach this as a teacher who is nurturing students and creating a healthy environment for each of them to grow, I am hopeful. I’m not in a hurry. I allow the students some space. I encourage them to take accountability. I encourage them to talk with each other to share ideas. There is no rush to be over and done but instead our goal is to think, reflect, discuss, and revise. We might work on a problem a little bit at a time over a month. We might let a problem fade into the background and re-engage with the ideas a week or two later. There is no rush because we want to make sense of things and we want to persevere.

Can a teacher who believes that students are lumps of clay that need molding have a different reaction to “Make sense of problems and persevere in solving them.”? I think they probably can. My guess is that they might approach that phrase to mean that the students listen quietly as the teacher explains how the problem makes sense. And, the students need to persevere and finish by the end of the class period! Yikes, that vision is quite the opposite of what I would want my students to experience.

As I read the paragraph of explanation under that first stated practice, the tone points toward the student having control of their own learning. I just can’t ignore statements that include:

… students start by explaining …

… make conjectures …

… consider …

… try …

… monitor and evaluate …

… change course if necessary …

… help conceptualize …

… continually ask themselves, “Does this make sense?”

For many classrooms this requires quite a shift in the environment. If we even follow this first practice knowing that there are seven others to consider, how does that change things? I’d love to think that we’ll move away from lumps of inanimate clay and more to living, breathing, and individual beings who are trying to make sense of their worlds.

Lee Alejandre making sense of the chairs on Swarthmore’s campus. The photos I took of him on that day inspired the Problem of the Week that we named Lee’s Lawn Chair!

Last week as I was drafting the Teacher Packet for “I Get a Kick Out of Soccer” I was looking through all of the student submissions that we received in 1999 and I suddenly started seeing solution threads from my former students at Frisbie Middle School: Mark, Jessica, Shambria, Ciara, Alicia, Erica, Sharlene, Marvin, Leah, Octavius, Reina, Chanelle, Conseulo, Akira, Regnica, Norma, Bryant, Kathy, Robert, Keturah, Laura, Luis, Faviola, Rickisha, Xuyen, and Shirley. There were 26 of the 31 students I remember having in that math class, the last year I was at Frisbie. Their ages at that time were 12 or 13 and so I realized that now they are 23 or 24 years old. Wow!

The two students on the left are Jessica and Mark. That’s me in the middle photo and on the right Norma and Regnica are sitting together at one of the computers. In the photo below Consuelo is looking toward me. My classroom at Frisbie was a computer lab. We had twenty LC575s each connected to the Internet via ethernet. With 31 students some of them worked in pairs when we used the computers. It was always a challenge to find space for groups to work but moving keyboards out of the way or finding a spot on the carpeted floor became normal for my math students!

Another thing that happened last week was a conversation where we were talking about the order that you have students work on problem solving. We identified the possibilities of having students:

* work **individually**

* turn and **talk**

* turn and **work**

I really like the distinction between “turn and talk” and “turn and work.” And I can see that with some tasks, students would benefit from doing these three in a different order.

What are some tasks you have students work on when they “talk” before they “work” or vice versa? When do you have students work individually before working in pairs or in a group? Do you have students work individually after starting to work in pairs or in a group? Are there other possibilities to add to the list of three I’ve noted here?

Although I’m not preparing to start up my own classroom tomorrow, my thoughts still wander to what I “would do” and that brought me to reminiscing about **The Quiet Game**. Since being introduced to it in the early 90′s I’ve seen it in various settings and this morning when I googled “cooperative learning squares” I found other names for this game and the most common appears to be **broken squares.**

At the end of the game the completed puzzle pieces look like this:

If this interests you my full instructions are here:

http://mathforum.org/alejandre/quiet.game.html

And this URL results in the download of a MS Word document with these game pieces but also others — it’s great!

http://web.stanford.edu/class/ed284/csb/Broken/BC&Stext.doc

With the game/activity or any like it I think it’s important to have the culminating conversation. I asked my students to consider “offer and receive” vs “grab and take” — how are they alike? how are they different?

It takes time and effort to introduce and establish a classroom environment with these characteristics:

* during large group discussions students

- take turns

- explain their own thinking

- listen to other’s thinking

- paraphrase others

- respecting differences of opinion

- justify their own reasoning

- revise their original conjectures

* during transitions from one activity to another students

- watch for signals from the teacher

- listen to directions

- pay attention to the amount of time and pace themselves

- follow classroom routines including know the designated place for handing in their work

- move about the room as directed and then as expected

- accept consequences when disciplined

I tried not to assume that students would know how to behave in my classroom. Playing the Quiet Game was one way to introduce some of these expectations. Each time students reacted differently but it gave me an idea of where they were and what I might need to provide to help them develop a sense of community in my class. Most important for me was the culminating discussion. What was the “real” point of the exercise?

What do you do to build a community atmosphere in your classroom?

In July, Craig and I presented to the PCMI SSTP (Park City Mathematics Institute Secondary School Teachers Program) during one of the afternoon sessions. I brought him a Math Forum t-shirt (limited edition!) so that we could wear matching outfits.

Craig recently wrote me:

I wore the “What do you notice? What do you wonder?” t-shirt you gave me to school today. Several students in the hallway were intrigued by the fractal dragon (I told them to ask their Geometry teacher about it). The students in my Calculus class noticed the noticing/wondering wording on the shirt and asked me “if I had the shirt specially made.” They thought I came up with the noticing/wondering strategy! I answered them that no, neither did I have the shirt custom-made, nor was I the author of the noticing/wondering… that both shirt and idea came from the Math Forum. I took it as a very high compliment that they would attribute noticing and wondering to me!

Craig’s comment reminded me that we have an activity that we give to teachers who want to show students how to generate the fractal dragon part of our logo.

If you try this activity with your students, let us know how it went! I’m also curious to know if any of them have heard of the book* Jurassic Park* — it’s fun to look at how they used the fractal on the chapter pages. I just found it on Amazon.com and if you use the “Look Inside” feature, view the First Pages, and scroll some you’ll see what I mean.

If you’re wondering about Michael Crichton’s naming convention, you might find Cynthia Lanius’s page interesting: Is It Really The First Iteration? Check out the links in the left sidebar on her page, too. There’s a fun Java applet to try.

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