Suzanne at the Math Forum

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.”

Both on October 22 at Germantown Academy in Fort Washington, PA and most recently on October 27 in Rochester, NY, I presented sessions about problem solving and the CCSS Mathematical Practices. In both venues we agreed that one of the more difficult challenges is to slow things down so that students have opportunities to:

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

Neither of those practices can be done well in a rushed atmosphere where the goal is to check off the “skills” that a student has mastered. Recent years have had this “quicker is better” tone. Teachers have been encouraged to cover everything that might possibly be on the standardized test. They have had long checklists of what the students must learn. Now, with the Common Core there is a return to a focus on process but how do we help make that happen? A change in classroom culture takes time and effort. Approaching problem solving as a process over time is one idea that might help.

I provided some ideas to focus on the process of problem solving and the communication that accompanies it:

* start a problem by reading it as a “story” and then ask students “What did you hear?”
* don’t plan to finish a problem in one class period — work on parts over time
* use our Noticing/Wondering activity with students working in pairs or groups
* use technology to approach a problem with virtual manipulatives when you’ve first introduced it with concrete manipulatives or vice versa
* start a problem at the end of the period, wait to re-engage until a day or two, and continue to do a little each day
* encourage students to read their solution drafts out loud to a partner and then discuss what the listener might still be wondering
* take time to give feedback to each student — it could be as short as “I notice … (one thing).” and “I wonder … (one thing).” If the teacher values the process and provides feedback, it models to the student that their initial draft is worth reflection and revision.

ATMOPAV Mathematics & Technology Conference, Fall 2011
Suzanne Alejandre, The Math Forum @ Drexel University, Philadelphia, PA

Saturday, October 22, 2011
Session 3: 1:15 to 2:15 pm
Middle School (grades 6-8)

The goal of the Math Forum’s problem-solving process is not to be over and done. It is to think, express, reflect, and revise. Leave with problems to try with students.

I. Introductions

II. The Math Forum’s Problem Solving Process — let’s experience a short version together!

III. Debrief that experience

• what did you notice?
• what are you wondering?
• CCSS Mathematical Practices
  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.

IV. More

Measuring Melons (photos)

Sports Weigh-in (photos on blog post with student comments)

V. Resources:

• Problem Solving and Communication Activity Series [pdf]
• Max’s Blog: Mastering the Common Core Mathematical Practices [webpage]
• Problem Solving–It Has to Begin with Noticing and Wondering [pdf]
• The Math Forum @ Drexel: Free Online Resources [pdf]
• Making Effective Use of Math Forum Resources [webpage]
• Samples of the Problems of the Week [webpage with PDFs links]
• Technology Problems of the Week [webpage]

VI. Links to other Math Forum ATMOPAV talks:

Annie Fetter: Using Technology to Increase Conceptual Understanding in Algebra and Geometry

Steve Weimar and Max Ray: The Mathematical Practices and Understanding Key Concepts: The Case of Fractions

Annie Fetter and Valerie Klein: Online Tools for Building Crucial Elementary Math Concepts

Max Ray: I Tweet Therefore I Learn [link coming soon]