Suzanne at the Math Forum

Display the two paragraphs.

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?

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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!

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!

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?

I’m running some online courses right now designed to help teachers make the most of the Problems of the Week and one theme that has emerged is –> students working individually vs. students working in pairs vs. students working in groups.

Some comments have been made about one student doing all the work or the social conversations that happen when pairs or groups are used. I find myself commenting that I’ve become more and more convinced that if you start your problem solving with Noticing/Wondering, those issues are minimized. And, I think this is true whether the problem solving is with a Math Forum PoW or a prompt in your curriculum or just a problem solving activity you’ve devised.

I’ve thought for some time that some behavior management issues are caused because students are not comfortable with the task. If it’s too easy, they’re bored and they act out to entertain themselves or turn to socializing because they think they’re “done.” If it’s too hard, they give up and in frustration they act out or socialize to get away from the unpleasant situation. And there are always those few students for whom the task is just right and they stay on task! Using a problem solving prompt with the question removed and doing a Noticing/Wondering activity creates a task that all students can do and moving from that (or during that activity) into pairs or groups is reasonable.

What type(s) of work do you have your students do?

• individual
• pairs
• pairs when they turn and talk
• pairs when they talk and then turn and work individually
• pairs to groups of four
• groups of four

Is there value in mixing the order? Do you mix the types of work within a class period?

At the Math Forum we’re just starting to work with a videographer at Drexel to help us capture some of the practices we use with students learning to be better problem solvers and communicators. Recently Mr. Reo and his fifth grade students welcomed us into their classroom. Wooden Legs was the Problem of the Week that I introduced to his students. In preparing for the session I had these goals in mind:

* use the Scenario as a “story”
* start with a read aloud (with no visuals either projected or on student handsouts)
* ask students “What did you hear?”
* as students respond, try my hardest not to REPEAT their words

Those were the initial goals that I think the video clip linked here may demonstrate. I had other goals as the lesson unfolded but I’ll save those if/when I have other video clips to share.

You’ll notice that I said “… starting to work …” and one thing we discovered is that the audio of the students’ voices is not up to our standards. We have plans to tape again, however, for this discussion — trying to show what it might look like if a teacher doesn’t repeat students’ responses — I think the fact that you can’t quite make out what each student is saying, is “okay” (not great but not dreadful). And, you’ll notice there are some subtitles to help with the audio of the students.

Why do I think this is an important goal?

As I mentioned in another blog entry I’ve thought a lot lately about how our students are to develop the CCSS Mathematical Practices. If students are to “Make sense of problems and persevere in solving them.” we need to help them own their own learning. When the teacher repeats the student’s response and/or restates the student’s response, the focus is back to the teacher and relieves the student of holding on and owning those thoughts.

Here’s the video clip:

Wooden Legs Video 1: What Did They Hear?

What do you notice?
What do you wonder?
Do you see any slip-ups from me?
When you have whole class sessions like this, do you find yourself repeating what students respond?
Have you tried not to repeat? If, yes, how was it different? Did it change any of your classroom dynamics?

Learning to speak a language has intrigued me since my own sons were born and then continued to be something that I observed, reflected, and questioned when we lived in Germany and then Spain when our sons were young.

When our sons were infants my husband and I audio taped their sounds. Later when they were older we listened to those tapes and we were fascinated to hear language development! When you’re so close to an infant you don’t hear as much but if you can put yourself in a situation where you can objectively listen and it’s amazing what you hear.

In Germany and Spain I worked as a conversational English instructor at Berlitz School of Languages and it was that training that convinced me that the underlying motivation for learning to speak a language is the need to communicate. The more I could create a real need for communication the more a student would try to find the word to explain their thoughts.

Have you experienced trying to communicate an idea to someone whose first language is not English? If you really need to have them understand, you phrase the ideas you’re trying to express in different ways if the first way meets a blank stare. As they ask you questions, you respond. You ask/answer/ask/answer until you have communicated what you need them to understand. It’s not one way — it’s an exchange.

Mathematics is a language. We have to establish environments in which students feel a need to communicate. How do we create the need?

When you are communicating with someone and you don’t quite understand what they mean, you ask questions.

* I’m not sure I understand.
* How does that work?
* Why did you use “(insert a word)” – can you tell me what that means?
* Why did you say that?
* Can you tell me more?

I wonder how they might be communicating?

What happens when you think of your students as communicators?

It shouldn’t come as a surprise to me that students have difficulty writing explanations of their thinking. A recent experience strongly reminded me that this is indeed the case. Here’s what happened:

1. I presented a Problem of the Week to a 5th grade class of students using the Math Forum’s Noticing and Wondering activity from our Understanding the Problem strategy.

2. I read the Wooden Legs scenario to them and asked, “What did you hear?” They responded with a variety of noticings.

3. I read it again and they confirmed some of their noticings and added more. I was pleased with their responses — all was moving along well.

4. We moved to the “What are you wondering” part and although they didn’t generate a question that we might want to switch in for the original problem question, the conversations still added to having all of the students understand what was happening in the problem.

5. Next the class was given the full problem (not just the Scenario) and manipulatives and they worked in groups to solve. They were all actively talking about the math. They were definitely engaged. Annie and I both asked groups Why? How? and Tell Me More? questions and we were encouraged by the students’ responses. We saw students in each group making notes in their math journals.

6. The next step was that we showed some students how to submit to the PoWs online.

It was what I saw later online that reinforced the fact that having students write what they did and why they did it … or … just write their answer and show how that answer works is difficult! I responded to each of the fifth graders who submitted online and when their teacher asked me if I had anything to suggest to him or his students, I told him:

It’s perfectly normal but it seems that none of the students are comprehending what I’ve written to them. It could be because:

* they don’t understand what I’ve written
* they don’t “stick on” my message long enough to read it and so they don’t comprehend (VERY normal!)
* they might read it but by the time they get to their submission they’ve forgotten what I said (VERY normal!)

…. so …. if there is a way that you can have them “talk” about what I’ve written to them, that might help. Here are some possibilities:

* ask a student if they mind having the class look at their solution and my response together — in other words, maybe the class can read everything, think about it and then that one student will submit a revision (with their suggestions)

* have students work in pairs on just one of the student’s submissions — once they read, talk about, and revise one student’s submission then they do the other student’s

* have a student log in, read my message, go to their group, report what I wrote and talk about what it might mean. Once the student has talked about it then they return to revise.

The bottom line is that you have to create steps to get students into the process and then you have to create scaffolding for them so that they have some success to build on. Students who do not have strong literacy skills require a lot of scaffolding and patience but the results are worth it!

In the work that I’m doing at the Math Forum I’m often in middle school or upper elementary classrooms and I have the Mathematical Practices on my mind. Also recently I’ve been at New York’s state mathematics conference (AMTNYS) and Pennsylvania’s (PCTM) and one of California’s (CMC-North) and the presentations and conversations have centered around the Common Core and, in particular, the Standards for Mathematical Practice. It’s occurred to me that one small detail is easily lost –> the goal is for the students to develop these practices.

What does it mean to have students “Make sense of problems and persevere in solving them.”?

Both parts of that practice require quite a shift from the practices that have become popular in classrooms feeling pressure from NCLB and the standardized tests that have been used to measure students’ success.

In classrooms where the teacher shows how to do the problem and then the students practice what was shown to them, it is the teacher who is making sense of the problem and not the students.

In classrooms where the focus is on the student making sense of problems, we should hear phrases like:

How do you know?
Can you tell me more?
What did you do?
Does that make sense?
Why do that?
Why did she say that?

If we only have to focus on one person’s practice (our own as the teacher) we have a much easier to control job than if we have to focus on each students’ practice! This change requires a major shift in our classroom environment.

Similarly, the second half of the practice “… and persevere in solving them.” also needs to shift from the teacher demanding that students persevere (or suffer the consequences) to where the students have a “practice” of persevering because they are involved in problem solving as a process.

Students who
- engage in a problem over time,
- talk about their ideas,
- use a variety of representations,
- write their ideas and receive feedback,
- reflect on their ideas and revise
… and more …
are persevering!

Establishing the expectation, believing in the students and helping them learn the routines to complete the process of problem solving is the role of the teacher. Those teachers will be helping their students develop  ”The Standards for Mathematical Practice.”