This morning I was looking through my photos and remembered that I hadn’t yet posted any of these photos of patterns that caught my eye. These are from different trips that I took last year!

What do you notice? What do you wonder?

This morning I was looking through my photos and remembered that I hadn’t yet posted any of these photos of patterns that caught my eye. These are from different trips that I took last year!

What do you notice? What do you wonder?

Yesterday I noticed that in celebration of March 14, Richard had added our Pi graphic to the Math Forum homepage and also a link to our T2T® **FAQ: Pi Day**

As the author/editor of that FAQ I thought I should take a few moments to check the links since as happens with many webpages, some go offline. I found a few to remove from the **Resources on the Web** section and I added a link to Scott Steketee’s recent **π Day 2014** blog post — great Sketchpad resources! And as I was checking the various resources I started to think about ways that I might use them if I were transported back in time to my middle school classroom (a position I left fourteen years ago).

**IDEA 1: I Notice, I Wonder™**

Before first period begins I’d set up my computer and video projector or SMARTBoard to Scott’s page. I’d scroll down to have **Jaws of π **displayed:

When the first student arrives (throughout the day) I’d ask him/her to be the “driver” and click on “Unroll and Fill Arcs” and “Separate Arcs and Roll Up” as needed. That would leave me time to ask students “What do you notice?” and call on them as they had a response. After a few minutes I would ask them “What do you wonder?” In all I might give 5 minutes for this activity. No conclusions — just food for thought!

**IDEA 2: I Notice, I Wonder™ in Pairs in a Computer Lab or Using iPads**

My last classroom was a computer lab – it is natural for me to imagine having a few students who drop in before school help me set up the computers to this webpage, add it as a “favorite” or “bookmark” so that when my first math class enters I would have the students work in pairs and have notice/wonder conversations with each other. This same idea would work well with iPads (I just checked to make sure that Scott’s resources work on an iPad — they do!)

Both Idea 1 and Idea 2 might be something I would do as part of a Pi Day celebration. As I was looking over the T2T® **FAQ: Pi Day ****it occurred to me that the resources we link to could be thought of as appropriate to be included:**

- during a one day celebration
- as an aftermath of that celebration (or possibly next year as something leading up to the celebratory day)

**ONE DAY CELEBRATION RESOURCES**

Besides the idea of having students engage with Scott’s Sketchpad resources, here are some of my “oldie but goodie” favorites linked from the FAQ:

- Pi Day Songs that Carolyn M. Morehouse shared with us (first in 2002 and then added some in 2003 and again in 2006) : http://mathforum.org/te/exchange/hosted/morehouse/
- The Derivation of (Pi) a Math Forum Web Unit that Jon Basden wrote in 2002.
- Pi Day: Making a Pi Necklace an activity developed by Diana Funke in 2000.

**AFTERMATH IDEAS**

I am assuming that it takes time and exposure for many middle school students to develop an appreciation of the significance of π. Like many math topics we can present an idea to students but until they have a chance to make the ideas something meaningful to them, they haven’t really learned. Maybe they’ve been “taught” but they’ve not chosen to “learn.” Here are some activities that I’ve been thinking about:

*AFTERMATH IDEA ONE*

Extend the initial I Notice, I Wonder™ experience of Scott’s Sketchpad activities. Have students work in pairs to explain what is really going on in the activity. How do the four compare? How are they alike? How are they different?

*AFTERMATH IDEA TWO*

Use one of the **Problems of the Week linked from the FAQ*******. For example, How Fast Is a Minute? One of the teacher resources is the Teacher Packet and I just took a screenshot of Rishi’s solution. It’s not something students who haven’t had practice problem solving and communicating will write automatically … but … it’s an example of what we want them to work toward!

*****If you are a PoW member you have access to the Problems of the Week listed on the FAQ. If you are not a member, know that you can sign up for a (free) Trial Account. For 21 days you’ll have free access to the Current Problems and you can view up to 5 problems from the Library!

* * *

Are you planning any Pi Day AfterMath ideas? Do you have any blog posts or resources pages to suggest that I add to the FAQ: Pi Day page?

My colleagues recently blogged about Noticing and Wondering in High School (Max – @maxmathforum) and Noticing and Wondering in Elementary School (Annie – @MFAnnie) and as I read both of their blogs, so much of what they write about applies to a middle school classroom. In my experience the biggest bang for your buck in using this strategy is engagement of all students! As I’ve worked in elementary classrooms the feel is a little different from middle school — the younger the students the more I feel I’m tapping into enthusiasm that hasn’t been dampened yet. As I work with fifth grade or sixth or seventh or eighth graders I often feel that there are more years of disappointment and/or disillusionment that have to be countered.

Middle school teachers (and, of course, also high school teachers) who are trying to encourage their students to embrace the Mathematical Practices need to have patience. It isn’t easy to change from a “No Child Left Behind” test-prep routine to a student-centered approach. Using I Notice, I Wonder activities can definitely help. For several years Erin Igo (@igomath) and I have worked in her middle school classroom to have students use Noticing and Wondering and last year we worked on using those two phrases in giving feedback to students. Erin worked on giving written feedback to her students Problems of the Week work using only the two phrases,

- I notice … (and she valued one thing in their submission).
- I wonder … (and she asked one question hoping students would reflect and revise/add to their submission).

As she worked on this she quickly emailed me what happened in class each day using these three prompts:

- some gauge of student reaction to what you did (of course, from your viewpoint)
- some prediction of what students will do during your next session
- some reflection on what you predicted and what you now observed

It turned out that Erin’s quick (5 minutes tops!) reflection on what happened in class helped her work through the process. I found it interesting to read (and now I have something to look back on and refer to for this post) … but … Erin and I both agree that the time she took to write her own “teacher exit ticket” was most valuable for her.

Here are some excerpts:

Even after I wrote my summary of what happened today I sat in my seat for a minute and just took a long deep breath and told myself…it’s a process. I asked myself, what opportunity could I create for the student to engage? I know the opportunity is just time…the students need time to adjust to the newness of this in my class and I need to allow the students to go with it!

I did notice that students were including more ideas in their explanation based on my “wonderings” from previous problems.

They seem to be more comfortable with getting on (the computer) and reading my replies…now the question is are they really reading my replies?

I know it’s a process and I have to remind myself everyday but I thought maybe the students would progress a little faster. I do like what I am seeing. I want to them to interact with each other more.

I predicted to see the same behavior but I am wondering if I could change my questioning to get them more engaged in the problems and rubric. I want the students to talk more with each other about the work.

I think that student get off task because once they have finish the task that the teacher wants them to do…they truly don’t know what to do…because the teacher hasn’t told them. I think students are programed to follow directions and the moment they feel like they complete a task…they don’t know what to do with their time. Its almost like we (teachers) have programmed our students not to persevere….

Patience is definitely a virtue and is not easy to have. I have noticed that with time my students have started to use my Notices and Wonders in their new explanations…without even thinking about it now. I have to continue to tell myself that this is a new type of learning for the students that they are not use to and that it will take time to get use to…actually thinking on.

Students were engaged…but trying to finish..answering the questions instead of completely understanding method. They just wanted to be finished!

[*This is the activity Erin was using*: Ostrich Llama Count–Examining Solution Methods]

I thought they would read the method and try to figure out and understand it before answering questions…big mistake!!! Next time I would structure it into smaller chucks to make it more manageable for the student and make the task feel like it was easier to accomplish.

I love reading this! It proves the idea that you never know whats really going to happen until you try it and invariably it takes time to get it to work! (It’s as much a process as the process you’re trying to get the kids to embrace.)

Some of my previous blog posts on the I Notice, I Wonder™ theme:

And in December, 2010, Marie Hogan and I had our article Problem Solving–It Has to Begin with Noticing and Wondering published in an issue of the CMC ComMuniCator, the journal of the California Mathematics Council.

I’ve been thinking back to an EnCoMPASS pre-institute activity and a conversation I had in an online discussion forum. Here’s what happened:

The EnCoMPASS Fellows were asked to view this video clip:

4th graders explaining their Eating Grapes solution

They were also given the text of the problem the girls and their classmates were working on: Eating Grapes [Problem #4507]. The Fellows were asked to notice and wonder about what was happening.

What the Fellows weren’t aware of was that:

- I was the voice prompting the students in the video clip.
- The two students were in the 4th grade.
- I was a guest in the classroom.
- Asking the girls to come to the front of the room near the end of the class period was completely impromptu. I hadn’t known if presenting to classmates was part of their classroom culture. I had listened to them talking with each other before they came to the front and their two-party conversation seemed worth sharing with the class and getting on tape since videotaping students’ problem solving and communication had been my goal of the day at that school.

As I’ve watched the videoclip several more times and reflected I find it fascinating to think about if the two students were comfortable with what they’d written on the papers they were holding. **Did they own it yet? **

I had worked with the class for a full class period and during it I :

- was introduced to the class by their teacher – “Miss Suzanne from the Math Forum at Drexel will be teaching the class today!” (It continues to amaze me that teachers allow me to take over their classrooms for a full period. It is really a treat!)
- told them I would start by reading a story. I read them the Eating Grapes Scenario (no question).
- asked the full class “What did you hear?” and I quickly pointed to student after student to listen to their response.

*(click on each of the photos embedded here to view a larger version)*

- read the “story” again and again asked them to tell me what they had heard and this time generated an “I Notice…” list on the chalkboard. I introduced the idea of “I Wonder…” and we included that in the chalkboard list as well.

Without going back to look at the tapes, I actually am not sure of what I did next! I remember at some point the class decided to work on the question, How many grapes did Angela eat on Monday? And the students were working in pairs or groups of three to find an answer but to also be able to explain their thinking and how they arrived at that answer.

And at one point we talked about strategies and listed those on the board.

I also remember that as I moved around the room listening to the students talk with each other, I was particularly struck by the drawing the girls had on their paper.

I find myself thinking, what helps **students own their **own mathematical thinking and helps them be confident in their explanations of that thinking? I imagine that time and practice are critical.

At what point in the problem solving/communication process do students really identify with what they write down? What is going on when they have generated something on a piece of paper but then are asked to ”present”? If they had had a document camera (or a SMARTboard displaying a PDF of their work) would the focus have been more on their thinking as they generated the work on their paper and less on re-creating that work (with accompanying explanation) on the chalkboard?

What do you notice as your students present in class? What are the signs that they feel that they **own their work**? How are you facilitating their process or, in other words, what is working for you (and them)?

Life isn’t simple. From the time we wake up until the time we fall asleep there are always surprises each day. Just when I think I have a handle on what I’m doing for the day, something comes up and I adjust.

I wonder if we try too hard to present mathematics and, in particular, problem solving too simply to our students. Is that why we have a tendency to:

- bring closure to a problem during a class period rather than using a Take 5 Minutes approach and let time elapse between engagement with a problem?
- want to help too soon when students are struggling?
- want to confirm “yes, your answer is correct” instead of asking “why do you think that?” or some question that encourages explanation rather than right/wrong?

In October of 2011 I tried something during a workshop that I’d actually had slip back to the recesses of my mind … but … as I’ve been thinking more about helping teachers use more problem-solving activities in classrooms, it suddenly came forward again.

Here’s what a group of teachers in the workshop and I tried:

- Read-aloud
*scenario*: Eating Grapes [Problem #4507] - Look-at-the-picture
*scenario*: Measuring Melons [Problem #5144] - Look-at-the-picture-and-the-graphs
*scenario*: Filling Glasses [Problem #5104]

We spent about 10 minutes on each first noticing and wondering orally and then taking a few minutes to individually write down some things that were noticed and wondered.

[**NOTE**: Some teachers encourage students to notice/wonder individually before anything is said aloud. Because my own classroom experiences were with struggling learners and I often work in classrooms now with teachers and students who are struggling, I tend to encourage a quick oral exercise of noticing and wondering before we ever get to the point of writing. So many of the students I've worked with would give up and think they can't participate if at first I asked them to write.]

After we had noticed and wondered on those three problems (not finding answers at all particularly because the *scenarios* didn’t include any questions!) we paused to reflect on the experience.

- Was it stressful?
- Were they on overload?
- Were they considering trying it in their classrooms?

They responded no, no, yes to these questions.

Thoughts?

If you try this, I’d love to hear stories!

Yesterday as Max and I were planning a workshop we’ll be facilitating in early August a recurring thought came to me — after spending hours of time on problem solving teachers sometimes comment to us, how will I ever find time to do this with my students?

As I ponder this issue, I wonder if at the heart of it is that

- the teachers realize that the amount of time we spend on one problem is worth the time?
- there is no apparent transfer from a condensed one-day workshop to a full-year class?
- teachers’ learning experiences are different from their students’ learning experiences?

From formative assessment during the workshop and evaluations at the conclusion of our workshops, teachers indicate that the time spent is worth it for them.

While teachers would like to transfer the ideas, this seems hard to achieve. So many school “routines” get in the way.

Are teachers more likely to be in control of their own learning? Are classrooms/schools ready at this point to have students be in control of their own learning?

How can teachers find time for rich problem-solving experiences for their students?

Advantages:

- Using just 5 minutes at the end of a class period is manageable.
- Starting and stopping reinforces problem solving as a process.
- Perseverance is also reinforced.

Do you see any disadvantages?

Does it make sense that taking this approach could reinforce the idea of problem solving as a **process** and that it’s not something to rush to finish just to be over and done? How might this idea fit within your classroom routine?

Last week a fourth grade teacher in Philly invited me to work with her students. She emailed me to explain, ”We are moving into intersecting, parallel, and perpendicular lines. Would it be possible to focus your lessons Tuesday and Wednesday around this skill?”

On Monday I was still thinking about possibilities. I was walking in South Philly and suddenly thought of taking some photos with my iPhone. Here are the photos I took that day:

I combined them from photos that I took in New York City some time ago and posted here:

http://mathforum.org/blogs/suzanne/2012/07/

I made a PDF with one photo per page, sent it to the teacher Monday late afternoon so that she could project it on Tuesday during class. Here’s how the lesson went:

After greeting the students who I’ve not seen for awhile (standardized testing prep consumed their school/teacher for quite some time), I wrote these phrases on the board:

parallel lines

perpendicular lines

intersecting lines

I asked them to talk to me — I asked them, ”What do these phrases mean to you?” It quickly became quite clear that they knew a lot! After some time listening to them I told them that I took some photos on Monday. I further explained that I had been thinking about today’s visit and had “parallel” “perpendicular” and “intersecting” on my mind. We looked at the photos together (projected on the SMARTboard by their teacher). The photo that we really started talking about was this one:

What do you see?

In case you’d like a copy of the PDF I created and sent to the teacher, I put it here so that it’s accessible:

In the mid-90′s when I was working on my Tessellation Tutorial pages (which, by the way, helped me land my Math Forum job!) I had an online conversation with Michael South. I still have never met Michael in person but I remember feeling honored that he would take the time to contact me and engage in a conversation about whether some of what I was presenting as being a tessellation really was or was not. I captured part of our conversation on this webpage:

http://mathforum.org/sum95/suzanne/m.south.html

The ideas Michael and I discussed have stayed in the back of my mind and surface whenever I see something that is a repeating pattern. I often wonder what the unit is. Is the frog wallpaper in the ladies’ restroom in the Osteria Marco restaurant in Denver, CO a tessellation? Is it really made up of squares and the frogs are the decoration within each square? Or does it, at least, give that illusion? Whether it’s a tessellation (in the stricter mathematical sense) or not, it definitely caught my eye — the color, the fun frogs and the pattern. Annie had to go look for herself and she took the photo above after I showed her my photos and requested one of the “unit”!

Almost two years passed between Annie, Max, and Steve’s Ignite! debuts in Indianapolis at NCTM in April, 2011, before I made my debut. I watched their preparation, anxiety and performances.

I was learning.

In December, 2011, I watched the CMC-North Ignite! talks in Asilomar. I continued to watch, listen, and learn. In April, 2012 Annie, Max and Steve again performed at Ignite! in Philadelphia at NCSM.

I was still learning.

On October 29, 2012 I received an email from Karen Cowe and she wrote,

“You knew that one of these days I’d come knocking.” … “This will be the last Ignite! for me, so it would be great to finally get you up there!”

I decided this was my opportunity to use what I had been learning from watching. One way to cope with the pressure was not to tell anyone at the Math Forum what I was planning to do!

On Saturday, December 1, mission accomplished!

The next day I emailed:

“My Ignite! talk was successful according to several accounts. I was in good company. I was #5 out of 9 [Avery, Jennifer, Harold, Bill, me, Lew, Ruth, Scott, and Mike]. There were about 400 in Merrill Hall where it’s held in Asilomar. Even the balcony seating was full. The good news is that I didn’t even think about that. I can’t really say it was fun but I told Karen Cowe I was honored that she asked me and satisfied that I managed to do it without getting too stressed. As **Ruth Parker** said to me, she can’t remember putting that much prep time into something that only lasts 5 minutes! I agreed!”

“Well, we’ll see what the video looks like first since I have absolutely no memory now of what I said! It really is an amazing experience. You’re sitting there watching the four that are presenting in front of you and each of their 5 minutes “feels” like a real 5 minutes (or maybe even longer). Then it’s your turn and the fourth speaker comes over, hands over the mikes, you get them clipped on, you walk over to the spot, and suddenly you go into time warp and it all speeds up so quickly — it’s really, really weird — it all seemed over in about 5 seconds.”

Now that I have proof that I actually did it: Suzanne Alejandre at CMC-North Ignite I know that I really belong to the **Math Forum Ignite! Club**.

And, as often happens, I am thinking of connections between my experience of watching and learning and how that might play out in a mathematics classroom. There are students who may take time before being ready to perform. Are they watching? Are they learning? When they’re ready, will they perform? I believe there are and they definitely will. And, as I talk about in my own performance, if we create classroom environments to help **unsilence** their** voices**, there is even more of a chance that they will perform!

When I received an invitation in October from Karen Cowe to be part of the Ignite! presentations at CMC-North 2012, I was both honored and terrified! I had seen my Math Forum colleagues prepare and present at:

NCTM 2011 – Indianapolis

NCSM 2012 – Philadelphia

I wanted to be like them and give an Ignite! talk but I didn’t want to feel the pressure … so … I kept it a secret. It really happened, though, and I have the poster to prove it:

A PDF version of my **slides** are here and now (about 4 months later) we are able to view the video from Key Curriculum’s YouTube channel. **Suzanne Alejandre at CMC-North Ignite** Fun!

And recently, I made this page to keep track of the Math Forum Ignite Talks:

© Drexel University 2014. All Rights Reserved

The Math Forum is a research and educational enterprise of the Drexel University School of Education.