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Noticing and Wondering in Middle School

by Suzanne Alejandre
September 15th, 2013

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:

day 4
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!
day 6
I did notice that students were including more ideas in their explanation based on my “wonderings” from previous problems.
day 9
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.
day 10
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.
day 11
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.
day 15
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.
Suzanne’s response to Erin on day 15
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.)
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.
Categories CCSS Mathematical Practices, problem solving
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Owning It

by Suzanne Alejandre
August 4th, 2013

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)?
Categories CCSS Mathematical Practices, Videos, problem solving
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Mix and Match

by Suzanne Alejandre
July 27th, 2013

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:

  1. Read-aloud scenario: Eating Grapes [Problem #4507]
  2. Look-at-the-picture scenario: Measuring Melons [Problem #5144]
  3. 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!

Categories CCSS Mathematical Practices, problem solving
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Finding Time

by Suzanne Alejandre
July 24th, 2013

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?
Worth the Time
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.
Transfer
While teachers would like to transfer the ideas, this seems hard to achieve. So many school “routines” get in the way.
Learning Experiences
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?
 
Idea: If students are encouraged to take charge of their own learning, our job is not to “lead” them through problem solving but instead to create environments that encourage them to embrace the process. Try the “At the End of the Period: Take 5 Minutes” approach.
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?
Categories CCSS Mathematical Practices, problem solving
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A Tessellation or Not?

by Suzanne Alejandre
May 5th, 2013

… that is the question.

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”!

Categories problem solving
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Learning from Watching

by Suzanne Alejandre
April 5th, 2013

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!

Categories CCSS Mathematical Practices, Videos, problem solving
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PoW Teacher Packet Idea

by Suzanne Alejandre
October 4th, 2012
Baking Blackberries is a Pre-Algebra Problem of the Week that just went into “preview” in our Current Problems today. As I was preparing things (writing the Teacher Packet with CCSS Standards alignment, possible solutions, student solutions, etc. and the other resources that are linked from the “blue box”) I came across Annie’s “real-life” solution that she had included in her comments when the problem first ran in 2007. I love how she described what she really did in the kitchen to decide which pan to use! I included it as one of the solutions in the Packet, Method 2: Annie’s “Hand” Estimation.
This morning I was thinking about how teachers might use that resource. I know that I would want to have them see/read Annie’s solution. Do you share it after students have worked the problem probably using straight-forward math including formulas for area? Do you share it before?
And then it suddenly occurred to me — sometimes (particularly as you’re developing students’ problem solving habits) why not use the “Our Solutions” section of the Teacher Packet and ask them to make sense of the solution we present? Here is how I thought I would use Annie’s “Hand” Estimation:

Display the two paragraphs.

  • Ask “What do you notice?”
  • (my guess is that many students don’t have cooking experience and there could be a lot of wonderings at the same time)
  • Interesting would be to see if students could then craft a math problem out of what Annie did.
  • Interesting would be to show students the problem and have conversations about how they might do it.

Another idea instead of displaying the two paragraphs would be to say, “I’m going to read you a story.” Read the first paragraph and ask “What did you hear?” Read the second paragraph and ask “What did you hear?” And then, perhaps display the two paragraphs to do a “Noticing/Wondering” activity.

If you try this idea of using either student solutions or our solutions from the Packet, please leave a comment to tell your story. What did you notice as your worked with your students? What did you wonder?

Some “Baking Blackberries” links in case you are interested:
Categories problem solving
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Tips on Managing Mentoring

by Suzanne Alejandre
February 25th, 2012

As a middle school teacher I know that it’s difficult to make time to individually connect with each of your students since you may be dealing with 130 to 180 students (depending on how many classes and how many in each class). Elementary teachers usually don’t have the volume of students that middle or high school teachers have but because mathematics is usually just one of the subjects they are responsible for delivering to their students, their time is similarly precious when considering adding yet another task to their never-ending list of things to do.

Often I ask teachers who think that using the Math Forum’s online feedback/mentoring functionality, what writing their students are already doing. For example,

* do you have students keep journals? How often do you collect them? How often do you comment on them?
* do you have students write responses to problem solving prompts on paper? as classwork? as homework? as projects? How often do you collect them? How often do you comment on them?
* do you have students reflect on feedback and revise?

Another thing I ask teachers who are contemplating this is, how organized are your students? If they start writing in your class on one day, do they have the paper with them the next day? Do you keep their papers in folders and they stay in the classroom? Do they keep their papers in their own notebooks?

The reason that I ask these questions is that it’s possible that using an online system just might save time in the long run.

My main tip, however, is in how you provide feedback. I recommend that teachers make only two comments per student following the format:

I notice ….
I wonder ….

The “I notice” statement notes one thing that you value in the student’s solution. In other words, a sentence of praise. The “I wonder” statement is a question with the intention that as a result the student will reflect on their draft, revisit it and add more. Along with this, I recommend that teachers check these two boxes in our system so that they bypass using the full rubric:

    Choose not to score this submission.
    Hide the scoring grid from students.

I suggest this abbreviated method for several reasons, including

* it doesn’t take very long per student
* it reinforces problem solving as a process
* … but … most importantly, the student’s thinking and problem solving remain in THEIR possession and is not transferred to the teacher

Recently I’ve realized that when a teacher repeats everything a student says or when they give detailed feedback, in some way they are taking over the student’s thinking. If the student is to embrace the Mathematical Practices of …

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.

… they have to continue to own their work. They have to reflect and revise!

Thoughts?

What does this really have to do with my blog post? Nothing! I just love the photo. This is a sea dragon that I saw at the Monterey Bay Aquarium. (Click on the small photo to view a larger version.) I just love dragons!

Categories problem solving
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Who Gives Your Students Feedback?

by Suzanne Alejandre
February 11th, 2012

Sometimes when I describe how the Math Forum’s Problems of the Week service works a teacher and/or their administrator get very interested when I mention the online feedback options. Imagine reading or hearing a version of this description:

Students are encouraged to submit solutions explaining how they arrived at their answer, as the beginning of a process designed to develop their communication and mathematical thinking skills. These solutions may be mentored by volunteer or paid mentors, or by their own teacher. The Math Forum offers an instructional rubric for scoring student work and detailed instructions on giving helpful feedback to students. The mentoring process promotes reflective, thoughtful problem solving.

[Note: in a previous blog post I talked about slowly introducing our rubric.]

Often what a teacher/administrator hears is the time-saving idea of having others (the Math Forum’s volunteer or paid mentors) give feedback to the students!

Consider these possibilities:

* student submits online, receives no feedback
* student submits online, receives feedback from someone besides the teacher, the teacher doesn’t have time to look online (it’s an activity only between the student and the mentor)
* student submits online, receives feedback from someone besides the teacher, the teacher reads the online exchanges

With the first and second possibilities the teacher is saving time because the online problem solving interaction is something they’re having a student complete either alone or with interaction from others. With the third possibility mentioned it might take more time for the teacher to read the exchanges than to be the one involved in the first place. My guess is that even with the best intentions, a teacher who planned to read the online exchanges might not be able to keep up with that idea.

In conversations with teachers/administrators inevitably my next question is, What does saving time mean? Is one of your goals of having your students work on the Problems of the Week to encourage them to practice  ”making sense of problems and persevere in solving them“?  How good are the students currently with this practice? Are they developed enough so that they don’t need their teacher’s help to build the practice? If that’s the case, I can buy the argument that having students work on their own and possibly receive mentoring from someone else could save the teacher time. But, I think it’s more likely that students need a great deal of scaffolding to embrace this practice. If the teacher gives feedback to their own students and coordinates that with what they’re doing in the classroom, in the long run, that will save the most time.

Stay tuned … in my next blog post I’ll share some ideas I’ve used that can help you manage your time with these activities!

Categories problem solving
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Introducing Problem Solving Rubrics

by Suzanne Alejandre
February 5th, 2012

This morning I was working in a section of my online course [PoW Membership: Resources & Strategies for Effective Implementation] and one of the posts reminded me of how we introduce our problem solving rubric to students. Particularly if students are just starting the process of problem solving and communication, it’s important not to overwhelm them (or yourself!).

Whether you’re using the Math Forum’s Problems of the Week or other problem-solving prompts from your curriculum or other sources, these ideas might be helpful. For reference, the rubrics I’m referring to are (freely accessible) from this page: Teaching with the Problems of the Week

Just scroll down until you reach The Rubrics section on that page and you’ll find links to PDFs for Primary, Math Fundamentals (elementary), Pre-Algebra, Algebra, and Geometry.

This is the order I might use to unveil each of the six sections of the rubric:

1. Interpretation
At the Math Forum we think that our Noticing/Wondering activity takes care of this quite nicely! Students are learning and practicing the first half of the CCSS Mathematical Practice #1 (Make sense of problems…)

2. Completeness
Even though I might introduce this second, it’s not something students will be able to do well if they’re new to problem solving. It takes time and a lot of reinforcement to have students develop the second half of Mathematical Practice #1 (… and persevere in solving them.)

3. Strategy
This then can be emphasized in conjunction with introducing some various strategies (hopefully your entire school is on board and students will have been introduced to strategies starting in kindergarten…but…maybe not!). We have summarized a nice set of strategies on our Problem Solving Activities page (linked from the left sidebar on most PoW pages).

4. Clarity
I like to explain this idea to students by saying … write your solution so that a classmate can follow what you did. For some reason emphasizing their classmate instead of their teacher is more motivating!

5. Accuracy
You may wonder why I would recommend this so far down the list and that’s because a thorough problem solving process most likely will result in accurate problem solving. I like to de-emphasize “getting quickly to the right answer” and instead emphasize the process … but, of course, bottom line is to get to the correct answer!

6. Reflection
We save the best for last! It’s tough to get kids to reflect on their process but it is very, very valuable.

Have you introduced rubrics to your students? What have been your successes? What have been your challenges?

Categories problem solving
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