My name is Laurel Pollard and I am the computer lab tech at Hanover Street School, a K-4 Elementary school in Lebanon, New Hampshire. Although I am not a classroom teacher, I am the go-to person for the Math Forum at my school. 

We have been using the Math Forum as math enrichment in the 3^rd and 4^th grades for a number of years, but this is only the second year we’ve included second graders. This year I work with two groups of second graders.  One is a group of four students I see only once a week when their whole class is in the computer lab.  

The other group I see twice a week.  The second graders do the weekly current PoW in the Primary and Math Fundamentals band or I use the online PoW Library to look up a problem that will hit areas of the Common Core curriculum where these students need work.  I meet with the classroom teachers to develop a growth plan using each student’s individual results from the computer-adaptive standardized assessments (NWEA Map tests) we give at Hanover Street School.

For the 2^nd graders this year I adapted an answer worksheet that I use with 3^rd-4^th graders. The second grade answer worksheet consists of a single page with sentence starters like:

• I wonder
• I noticed
• First I did
• Then I did
• Here is a number sentence I used.
• Here is a picture I drew.
• The best part was
• The hardest part was

At the beginning of the year we mainly used a hard copy of the worksheet to work the problem.  We’ve now transitioned to mostly typing the answers online and using the whiteboard for drawing the pictures.  I still prompt them with questions like, “What did you do first?” and “Did you show your number sentence?”  I continue to read the scenario to them when they first arrive in the computer lab. I try to let them do most of the typing, but when I want them to expand on an idea or explain why they did what they did more thoroughly, I will type in their own words.

I find it most difficult to balance the amount of time to allow for a given problem.  The second graders tend to be impatient once they have figured out the answer and will only be receptive to a few prompts to add more details. Some of the questions I am working on are: How important is it for them to go back and rethink their solution or other possible ways to solve problems?  When do you move on to the next problem? Like in the art process, how do you know when your painting is finished?

Most rewarding for me are the surprising learning opportunities that come up as the students work on the Math Forum problems.  I never know which of the current PoWs will spark the interest of a group of students, often promoting conversation beyond the question posed.  But when it does I try to be ready to run with it. This year I’ve seen it twice. Once was with the PoW Miracle Miranda and the Mascot, which got the 4th graders interested in the very large numbers generated by doubling. With a simple spreadsheet calculation and cut and paste they were able to watch numbers grow astronomically and across the screen.  Even I joined in the fun and looked up how to say those large numbers like (“317 octillion!”).

The second time was the Voting Time scenario.  I usually stick to the Pre-Algebra and Math Fundamental band levels for 4^th graders, but I thought one group of 4^th graders could look at this problem as they were studying graphs in their classrooms. With November elections that week, it was very timely.  We videotaped our results, which you can find on the Math Forum site. That sparked another interest– becoming famous. How cool it is that by solving a math problem you could be famous?

With the encouragement of the Math Forum team, my students have started making video scenarios of Math Forum problems.

From my perspective, The Math Forum is a truly unique opportunity for students to stretch their math brains. The only other place you find such constructed math problems is … in the real world! While we do want to “know how their brains work,” I am most excited for the students themselves to understand how their own brains work.

The “Forget the Question” Activity

To put the focus more on the process, introduce the class to the problem by removing The Question. This can be done as a whole class (which is how we might start) or in small groups. The students must analyze the situation and focus on reading and interpretation instead of coming up with The Answer.

1. Give students the text of the problem without the question (the overhead works great for this) or draw the associated picture on the board and tell them only what they need to know to understand the situation.
2. Go around the group and have each person list one thing they “notice”. Responses might be as simple as “the lines go up”, or even “there is one blue line and one red line”, or as complex as “the blue line is going up twice as fast as the red line”. Everyone can contribute something, and all the “noticings” are recorded for the group (on the class data pad or whiteboard, etc.) with minimal discussion.
3. Ask the students which items on the list they are wondering about (we often use the language of “wondering” instead of asking them what they don’t understand). For example, a student might ask, “I’m wondering how you know that the blue line is going up twice as fast as the red line.” Let the students respond to these questions. “Who would like to try to explain?” If possible or necessary, have more than one student explain each idea so that more student voices get heard.
4. At this point, we often ask students to pose a question for the situation presented. You might learn that sometimes math is pretty predictable—in my experience, kids almost always come up with a question that is a lot like the original question!
5. Pose the actual question (or choose a student question) and talk about it as a group:
   • Have students list the observations they think will be helpful in answering The Question.
   • Let some kids take a stab at answering The Question. Depending on the readiness of your students, you may do this as a whole class or have students work in pairs.

Benefits of the “Forget the Question” activity

The goal is to get students engaged in the process of thinking mathematically and about how to solve problems. It is not about finding the solution, at least not initially. You will be able to judge the success of this activity as you listen to the buzz in the classroom and see how many more students are participating.

Posing math scenarios without questions can:

• Inspire the students who are afraid to get the wrong answer to share – if there’s no question to answer, how can they be wrong?
• Slow down the students who race to be done – if there’s no question to answer, what does “being done” even look like?
• Show students that math problems come from real people wondering, not from some mean guy in a windowless room somewhere writing worksheets!
• Engage students in solving a problem they came up with. They can’t say “I don’t get it!” if it’s their question, and that might be a little more motivated to solve it if it’s genuinely what they wonder.

How to Find “Forget the Question” Scenarios

Every Math Forum Current PoW includes a print-friendly scenario only version. You can print and distribute the “Scenario Only” PDF linked in the blue box on each PoW, or display it using a projector, document camera, etc.

We also have a blog that features a new Scenario Only (with no questions, ever!) here: http://mathforum.org/blogs/pows/

Video Scenarios

Some PoWs even have video scenarios. We first started making these for fun, and the scenarios like Charlie’s Gumballs and Val’s Values they were so fun that they inspired the 3rd and 4th graders at Hanover Street School to create their own video scenarios! Here’s their video scenario for the Baseball Cards PoW:

Do you think your students might enjoy making a video showing the story of a PoW, or a math story from their own experience? We’d love to feature it on the Math Forum! Email Max if you’d like to play along.

Spring 2013

Look for us in Denver! We’re speaking, presenting and exhibiting at both NCSM and NCTM. You might even catch us square dancing.

Here’s where you can find us:

NCSM – Booth 500
Win a PoW Prize Package or a NCSM Membership and Registration for New Orleans, 2014.

Session schedule: (get all the details)
Monday April 15th – Max and Val #182, Annie #188, Hope and Jason #189
Tuesday April 16th – Erin and Suzanne #205, Steve #244, Val and Ellen #245.
Wednesday April 17th – Suzanne, Erin and Beth #352,  then Max and Annie IGNITE it up #382

NCTM – Booth 741
Win a new iPad or a PoW Prize Package or a Current PoW Membership.

Session schedule: (get all the details)
Research precession Tuesday April 16th – Jason #110
Thursday April 18th – Hope and Jason #143
Friday April 19th – Val #482, Erin and Beth #531, Suzanne, Erin and Beth #586
Saturday April 20th – Cheryl and Diane #680

Summer 2013

July 10-12  Conference for the Advancement of Mathematics Teaching – CAMT
Look for Suzanne Alejandre and featured speaker Max Ray

July 25-28 Twitter Math Camp
This year camp comes to Drexel University! Tweet with us: #TMC13. There is still time to register if you’d like to participate.
See highlights from last years conference. Read about Max’s experience here.

Ongoing:

*Let us help you plan and present a mathematics workshop for the professional development of your teachers. We offer workshops during the school year on professional development days or as weekend sessions. If your school, district, or region is interested in having a Math Forum staff member present a workshop, please complete this form. See our sample agendas from workshops that we’ve presented.

This issue we look at the first Common Core Mathematical Practice, “Make sense of problems and persevere in solving them.” What are some ways that teachers support their students as they make sense of a Problem of the Week?

Noticing and Wondering/Forget the Question

When teachers do PoWs in class, a very popular activity for getting students started at making sense of the problem is to display, hand out, or read out loud the problem scenario, without a specific math problem to be solved. Then we ask students to share what they Notice about the scenario, and what they Wonder.

With younger students, we often read the problem aloud and ask, “What did you hear?” first. Older students might read the problem as they listen, and then write down or tell a partner what they noticed and wondered.

We also sometimes do a whole group brainstorm or a class go-round so everyone can hear the noticings and wonderings of their classmates. Making a big public list of noticings and wonderings also helps students have something to come back to when they get stuck or want to make sure their answer is right.

You can find a print-friendly Scenario Only version of each Current PoW in the blue box with all the teacher resources, to make it easy to share the PoW without telling students the specific problem they’ll be solving. We like the Scenario Only version because not having a specific problem:

• helps keep some kids from blurting out an answer right away, and helps them stay focused on understanding
• helps students who are scared of getting wrong answers… anything they can notice or wonder is valued, and there’s no question to get wrong
• gives the students a chance to come up with a problem to solve, via their wonderings, which helps them be motivated to actually solve it
• shifts the focus from getting the answer and being done to coming up with as many math ideas as possible

To learn more about I Notice, I Wonder brainstorming, you can visit the Understand the Problem strategy page, read this story by Annie Fetter [PDF] about the first time she did I Notice, I Wonder with students, and check out these articles about noticing and wondering in the ComMuniCator:  http://mathforum.org/pow/teacher/articles.html.

Change the Representation

Annie tells a story about doing some math with her niece, Olivia, who was in second grade at the time. Olivia was a very good reader, and Annie figured it was important for her to know that being good at reading can really help you do math. (Olivia’s parents are an anthropologist and a fine silver jewelry maker, so Annie feels like she always needs to make her nieces and nephew think about math!)

Annie asked her to think about this story: “A man has three bunches of roses. Each bunch has six roses.”

Olivia immediately said, “He’s got 18 roses.” Annie told Olivia that a lot of students think the man in the story has nine roses — to which Olivia exclaimed, “WHAT?!? Why would they think that?” Annie and Olivia talked about how she was good at understanding stories, and how that helped her also be good at math. Olivia thought that was pretty cool — she wouldn’t be tricked into adding the three and the six if she stopped to think about what was happening in the story. The conversation helped Olivia think about how she could use her strengths to connect to math, and it helped Annie think about the importance of supporting students to understand the story behind a math problem.

When we work with students who are stuck, our first question is often, “What’s going on in the story?” or “What is this story about?” That gives us a window into what the students do and don’t understand about the context.

Annie also wondered if Olivia, who likes to draw (though not as much as she likes to read), was able to visualize stories in her head as she read them. We’ve started to help students make visualizing and illustrating a part of their sense-making routine too. Especially for those discouraged students who leave their sense-making ability at the door of math class, drawing a picture can be a great way into a math story. Many of the same students who would answer “9″ for the bunches of roses problem would be able to draw an accurate picture of 18 roses if they are just asked to “draw a picture illustrate this story.”

If your students are having trouble connecting to the action in story problems, seeing what the problem is about, or connecting the problem to their own experience, you might try:

• Asking stuck students, “How would you illustrate this problem?”
• Dividing into groups of no more than 4 and each group coming up with a skit to act out the action in the problem.
• Inviting small groups to draw a picture to help them solve the problem, and then sharing out each picture before going on to solving the problem. Talk about what makes the pictures similar and different, and which would be most useful, and why.

Share and Compare

Lots of PoW submitters write and tell us, “When I first solved the problem I was thinking this, but then when we talked about it as a class, I realized I was wrong.” Sometimes it takes outside input to realize hidden assumptions!

For example, I just wrote the commentary to accompany student solutions to the PreAlgPoW “Totolospi.” It was a tricky probability problem and students had a range of answers. Several submitters wrote to say, “At first I thought there were only four possible outcomes, but after we talked as a class I realized there were eight.”

A great moment to have those class discussions is after students have had a chance to read the problem for themselves and start doing some initial work. Questions come up like, “What do they mean by ‘all the possible tosses?’” or “Wait, how many possibilities are there for each die?” or even “What’s a cane die?” Pausing the group early on for a “distributed summary” — a chance for students to quickly report on their current thoughts and questions — can help problem-solvers get on the same page before they commit to a lengthy process based on a wrong assumption.

Good questions to ask for a distributed summary early in problem solving are:

• Are there any questions that have come up about the story?
• Is there anything in your thinking that you’re not too sure about?
• How are you understanding the story?
• What’s one idea that you’re trying right now?
• Is there anything you’ve ruled out? An answer that’s definitely too high or too low?
• What’s your best estimate/hypothesis about the answer right now? How could you test that?

All of these strategies help students engage in sense-making, and avoid doing random calculations without thinking.

… And Persevere in Solving Them

The other key component of Mathematical Practice #1 is perseverance. How can we support students to persevere when they see math as too hard, too frustrating, or even too boring?

At the Math Forum, our focus in perseverance is on supporting students to revise their submissions to the PoWs — we’ve found over the years that that’s when most of the learning happens, and since our only contact with most PoW members is through mentoring student submissions and reading their revisions, well, that’s what we’ve gotten good at.

Here are some principles that work well at encouraging students to persevere and revise:

• Value what the student has done so far — there is nothing more encouraging than feeling heard! We always find at least one thing to say to show that we recognize the thinking the student has done and appreciate it.
• Remember to praise the things the student did well — there was a study once with two teams of bowlers who watched video of their bowling and got feedback. One team got all feedback about what they could do better (i.e. what they’d messed up and needed to change). The other team got all feedback about what they were already doing well. Guess which team improved most? The team that got only specific positive feedback. It’s counter-intuitive but true!
• Stick to one or two steps you want the student to take next. We’ve found that students only respond to at most two requests, and so if you don’t choose your top two, your students will choose for you… Sticking to one or two requests means you can prioritize!
• Baby steps! Trying to make every student an expert in one revision isn’t going to work and is frustrating. What’s one thing each student could do to make a little bit more progress?
• Consider what this student would find interesting? For example,
  • instead of “How did you know the answer was eight?” (kind of a boring question) we might ask “Some students said the answer was four because RFF was the same as FRF (order doesn’t matter). How would you explain why you’re right and they’re wrong?” (fun because you get to argue!).
  • instead of “Can you write a rule or equation?” we might ask “What would happen if there were 100 of them? 1000? 1,000,000? How could you figure it out?” (more fun because big numbers are fun and because the student has a reason to write a rule, they aren’t just doing it ’cause we told them too).

If you’d like to read more about helping students persevere, you might enjoy this article by Suzanne entitled “Supporting Sense-Making and Perseverance,” or the article by Math Forum Teacher Associate Marie Hogan entitled “Pairing Like Students for Persevering in Problem Solving.”

We hope that some of the strategies for sense-making help students get started, and that through feedback and multiple revisions we can help them persevere on the PoWs as well!

I think financial education offers teachers a nice way to connect mathematics to a real world context. It is certainly not the only meaningful real world context, but there are a handful of nice ways to offer students mathematically sound examples, at many levels, of financial decision options that have them doing great math, enjoying doing it and talking about it. One of the neat things about financial decisions, in particular personal financial decisions, is that people can have good reasons for making different and sound decisions – like, you and I might choose different cell phone plans – and that makes for a rich discussion.

What can teachers do to help parents continue the financial lessons at home?

I think it can be as simple as reminding parents to include their children in any conversations that they feel comfortable discussing with them – small or large. For example, groceries, cell phone plans, renting vs. owning, how mortgages work, opening bank accounts with/for children and talking to them about how all the fees work, explaining how checking accounts work and showing children how and when they pay their bills, and helping children set up savings goals and keeping track of how much they are saving and how close they are to their goals.

Given teachers’ busy schedules and curriculum demands what are some ways to incorporate financial examples and problem solving into their classes?

There are a bunch of neat projects to be done – following a set of stocks and graphing them, exploring loans and interest rates, creating business plans. Lately, I’ve been thinking a neat way to do a little financial education everyday would be to do a series of Do-Now activities to start class that were financially related and could generate discussions about math and money related decisions. For example:

• 20% or $25 off – which is better?
• Put up two companies’ cell phone plans and ask students to talk about them.
• Clip a part of a mortgage agreement or a credit card statement (perhaps the part that says how long it will take to pay it off at the minimum rate) and post that for students to talk about and ask questions about.
• At your job you make $X a week and you have to pay $Y for the bus to get to your job. You want to save for a new phone that costs $Z dollars, write an expression that represents how long it will take to save.
• Compare the meal deals at a local fast food restaurant to the individual prices of each item.
• Ask students to play with an online loan repayment calculator for a loan of a particular size varying the interest rate and tell you what they’ve learned or graph what they’ve learned.

  After a while, I would challenge students to bring in Do-Now financially related activities that they’ve encountered in their lives – they could bring in ads, letters, stories, or  photos taken on their cell phones.

What’s the best/hardest financial lesson you learned growing up?

Hmm, that’s a tough one. One good lesson I’ve learned is about setting up automatic deposits to savings accounts. A wise person told me that if I set up a deposit to my savings account on every payday, I wouldn’t miss the money and I would be able to save consistently. I’ve been doing that now for many years and I always have a bit of money for when I want to do something special or need it for something unexpected.

For more Financial Education in the Math Classroom, visit http://mathforum.org/fe/

After covering every whiteboard in the room with their thinking, they boiled down what they were looking for into two big categories, each of which had three sub-categories:

Problem Solving

• Interpretation
• Strategy
• Accuracy

Communication

• Completeness
• Clarity
• Reflection

After determining those rows, Suzanne and Annie turned their attention to the ratings, or columns.  Since Suzanne values problem solving as a process to improve upon, she felt that communicating where students stood on a spectrum of mastery would be better than numerical scores, so they came up with the levels of progress: Novice, Apprentice, Practitioner, Expert. Also, not having numerical scores helped each rubric category stand separately. Rather than averaging the scores together to tell students “you got a 15 out of 24 on the problem” the rubric communicates “You’re still a novice at writing your steps completely and I couldn’t tell if your strategy was sound, so you’re a strategy apprentice. But you’re a practitioner at interpreting problems, calculating accurately, and writing clearly. You’re a novice at reflection.”

Each current FunPoW, PreAlgPoW, AlgPoW, and GeoPoW comes with a problem-specific rubric that the teacher can use to understand more about what it takes to be a practitioner on this PoW (and which teachers sometimes share with students after they solve the problem). In addition, teachers and students can view generic Math Forum rubrics that use student-friendly language to explain each rubric category. We offer generic rubrics for every service, from Primary up through Trig/Calc!

Many PoW teachers report that they start with one category at a time. Some focus on Problem-Solving, starting with Interpretation or Strategy, while others just want the students to get used to writing their thoughts, and so focus at first on Completeness. A simple way to achieve that focus when mentoring online is to click the check-box for “Hide the Rubric from Students” then simply type their score on the focal category in the mentoring response. Teachers can also check “Choose Not to Score this Submission” and leave the rubric blank.

Rubrics are a great way to communicate a lot of information about student progress very compactly. We hope the PoW rubric is useful to you, and that you enjoyed learning a little more about the history and philosophy of the rubric!

By Fawn Nguyen

I love teaching problem solving for a very selfish reason: I always learn something new from it. I learn from struggling with the problem, I learn deeper mathematics, I learn from my students’ different solutions and their non-solutions, I learn from other teachers, I learn that I have soooooo much more to learn.

I use three types of PSs (problem solving activities)— weekly, in-class, and group.

Weekly PS

The first time that I introduce this, the process roughly takes on this form:

1. I pass out the PS. (To save paper and photocopying, student just gets a strip of paper that has the problem on it.)
2. I call on one student to read the problem aloud. I then ask everyone to read the problem again quietly on their own. If it’s a particularly lengthy one, I ask them to read it again.
3. Questions I tend to ask, “What are you being asked to find out?” “What information do you already know?” “What are your immediate thoughts about this problem?” “Have you seen a similar problem before?” “Do you have ideas on how to start the problem?” “Wanna give me a guess on what the answer might be?”
4. I tell students that since this is the first time they do a PS in my class, I’ll walk them through the process of writing up a PS. I walk them through Polya’s four steps:
   • Understanding the Problem
   • Devising a Plan
   • Carrying out the Plan
   • Looking Back
5. I blah-blah-blah about the importance of writing in mathematics, and that I don’t ever want to hear any whining, especially of this sort “but this is not an English class…,” because help-me-God if I do hear it.
6. I emphasize that all write-ups must be on notebook or grid paper.
7. Students have one week to turn in the PS. They get a new PS each Monday, it’s due the following Monday. This is the only assignment that I do not accept late.

That’s roughly my introduction. Then I help them begin the problem in class. This should take up to a full period. Before I dismiss them, I hope for this exchange:

Me: When is this PS due?
Ss: Next Monday!
Me: When next Monday?
Ss: At the beginning of class!
Me: What if you don’t turn it in next Monday?
Ss: That’s too bad!

Next Monday comes around…

True to my promise, I ask the kids to pass forward their PS write-ups. As I quickly leaf through the pile of papers in my hand, I can expect (and you should too) to witness the following:

• Their papers: ripped holes, ripped corners, half-sheet, unlined, spilled cappuccino.
• Their write-ups: red inked, work written sideways and upside down (a complete mess), Dad’s handwriting, four papers are identical (including the non-legible and nonsensical steps), only two papers include the “looking back” step, Mom’s writing at the top of paper asking for extra time.
• Their solutions: all the numbers in the problem got added, all the numbers got multiplied, oh look, this student performed all four operations on the numbers, $850,000 for the bicycle, Victor is 48 years old (while Victor’s father is 36), the building is 756,411 feet tall.
• Their reasons for not having it done: I don’t get it, I forgot it at home, my Dad accidentally threw it out, my sister who’s in calculus couldn’t even do it, my uncle who’s an engineer couldn’t figure it out, I don’t think there’s an answer for it, I was absent when you gave it out, remember? I’m-sorry-I-forgot-to-do-it-but-I-love-your-dress-Mrs-Nguyen!

I take a deep breath. Mentally embrace these precious children. Remind myself that I’ll be with them all year. Worse, they have to be with me all year.

So at this time I pass out a PS scoring rubric and carefully go over it. I’ve always used a 6-point rubric but I really want to change it to 4-point. [Ed note: For an example of a 4-point rubric, you might like to download a PDF of the rubric we use]

I give them a new PS for the coming week and only do the first three steps — reading the problem and checking for understanding — as I outlined above. I remind them that I offer PS help at lunch time.

In-Class PS
I don’t grade these. Because I encourage kids to do math with their family and anyone with a pulse, it’s nice to learn once in a while (about one per quarter) what they can do completely on their own. One class period.

Group PS
I don’t grade these either. Of course this is my favorite type of PS because I get to watch the kids do the math, ask them questions, and listen to their discussions. Before getting into their groups (I almost always assign kids in groups randomly), students have about 10 minutes of quiet individual time to work on the problem. I do make a conscious effort to follow the 5 Practices [Ed note: Fawn is referring to 5 Practices for Orchestrating Mathematical Discussions by Margaret S. Smith and Mary Kay Stein ] whenever kids work in groups.

I genuinely hope that you’ll incorporate problem solving in some fashion, if you haven’t already, into your curriculum. If you don’t think you have time because you feel you have to cover x-y-and-z content standards, then please make the time. Learning math is a social endeavor, a really fun one, please provide students with lots of opportunities to think critically and struggle productively. I think it’s a beautiful thing when we can develop a classroom culture of doing mathematics so contagious that it spreads beyond school boundaries.

You can follow Fawn at http://fawnnguyen.com/

Annie: I can still remember when Suzanne and I sat in this room for the entire day and asked ourselves “What do we value in students’ problem-solving work?” We decided that both problem solving and communication should be valued and, in the end, we had three sections for each for a total of six, including interpretation, strategy, accuracy, completeness, clarity, and reflection.

Suzanne: And I remember when we first started writing the problem-specific rubrics, we talked a lot about “double dipping.” A problem solver might be scored as a “practitioner” in Completeness or Clarity even if the problem had been scored as a “novice” in Interpretation or Strategy.

Both Annie and Suzanne have given workshops on using rubrics, including:

Assessing Problem Solving and Writing With Constructed Response Problems
Constructed response problems give unique insight into students’ mathematical understanding. A rubric that addresses problem solving and communication also provides structure for assessing student work and driving instruction.

Moving Beyond the Right Answer: Developing Students’ Math Communication Skills
The Math Forum’s rubric emphasizes a combination of problem solving and strong mathematical communication. We’ll share stories from online and classroom exchanges illustrating how we help students develop these skills.

During another Friday meeting someone noticed that we refer to our rubric as a  “scoring” rubric and not a “grading” rubric. Suzanne explained that when a Math Forum staff member views (or listens) to student thinking, our focus is on what to value. We’re not looking for ideas to mark wrong, but instead, for ideas that are good starting points of mathematical thinking. We score students’ problem solving and communication efforts to give an indication of where they are in the process but our ultimate goal is to provide just the right amount of feedback to encourage the student to reflect, revise, and continue in their problem solving process.

After these and other conversations during our Friday meetings we have been shifting our focus from the more limited thought of producing rubrics to a more encompassing topic of formative assessment. What environments associated with the Problems of the Week might make it easier for teachers to provide feedback on specific problems? How might they share what they’re doing, and build up a stable of likely comments and feedback to students that everyone can use to encourage their students to persevere and improve their mathematical thinking?

Scoring rubrics and the role they play in formative assessment, problem solving, and communication are becoming increasingly important to the development and strengthening of our students’ mathematical practices as the Common Core State Standards are implemented. And so we are also interested in how an online process of creating, sharing, discussing, and using formative assessments might help all of us have a professional community through which we feel supported and that supports our professional development.

As members of the PoW Community, you may have been using both the general rubrics that we provide and also the problem-specific rubrics linked from many of our problems. You may have stories to share about how you and your students use the Math Forum’s PoW rubrics or other rubrics that are part of your curriculum.

Some of the questions we have been asking ourselves include:

• Are our PoW rubrics used by teachers? How?
• Do teachers prefer to create their own rubrics to assess their students’ work? Are our rubrics helpful starting points?
• If teachers are assessing student work using their own rubrics, what is their process and how are they using them?
• Are teachers more focused on “scoring” or “grading” their students?
• Are teachers involving students in the process? How?
• What formative assessment activities are offered at schools and/or districts?
• Are rubrics helping teachers to differentiate instruction? What successes have they experienced? What challenges have they encountered?

We are planning two initial grant-funded activities that will involve teacher participation. As we add more details, the information on this page will be updated:

http://mathforum.org/encompass/TeacherParticipation

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On December 14th we participated in Math Night here at Drexel University for the Powel School. Math Night brings parents and students together to play math games, learn about math programs, enjoy some pizza and even win some great raffle prizes.

We continue to offer PD and support local schools* in our area. We work at Colonial School District in Delaware often, having visited William Penn High School and McCullough Middle School many times so far this school year. In Philadelphia we’re often found at Powel School, Arise Academy, and the Universal Company Schools including, Audenreid, UICS, Daroff, Vare and Bluford.

Max and Suzanne will be at the upcoming T^3 International Conference in Philadelphia (our hometown) March 8-10, 2013.

*Let us help you plan and present a mathematics workshop for the professional development of your teachers. We offer workshops during the school year on professional development days or as weekend sessions. If your school, district, or region is interested in having a Math Forum staff member present a workshop, please complete this form. See our sample agendas that we’ve presented.

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.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

We focus on the practices for a few reasons. One is because in our work to support students in problem solving, through the PoWs, our Problem Solving and Communication Activity Series, and our Professional Development, we are very focused on students’ process. We want to help teachers and students think about how students are thinking and doing math, not just what they are doing. Whether you call it reasoning, sense-making, problem solving, mathematical habits of mind, or mathematical practices, it’s important to us and we’ve been focused since 1992 on helping students get better at it.

We also focus on the practices because they’ve gotten the short end of the stick, historically. The math Content Standards in the new Common Core documents are 93 pages. The Practices document is 3 pages. The Content Standards are broken out by grade level, and we can see concepts develop throughout students’ thirteen year journey through math education. The Practices are the same for all grade levels. We don’t yet know how we see the practices develop over thirteen years.

In looking at the practices, one thing that stands out about so many of them is that they require that students actively do mathematics. In order to get better at persevering, students need to do problems in which they get stuck. In order to construct arguments and critique others, students need to get into situations that are worth arguing about — situations in which they don’t know the answer or how to “mathematize” the situation best. In order to use tools strategically, students need to be in situations they’ve never encountered before, so they are forced to think hard about which of their many tools to use.

As teachers teaching with the PoWs, you clearly recognize the value of giving students challenging problems to work on and asking them to explain their reasoning and justify their thinking. You value letting students struggle productively, experience being stuck and then getting unstuck. Every time I read a PoW submission in which a student says, “I didn’t know what to do but then I had an aha! moment when….” I know that student is getting better at Mathematical Practice 1. They are learning to make sense of problems and persevere in solving them.

When I picture a classroom that is actively working at getting better at the Mathematical Practices, I picture one in which students encounter challenging mathematical situations, and teachers and students have lots of conversations about how they think about the situations, what’s working, and what they want to get better at. An observer might overhear questions like:

• What did you do when you weren’t sure how to get started?
• What does that [variable, number, etc.] mean in the story?
• When you said _________, can you tell me more about what you were thinking?
• How can you illustrate the relationship between those two things?
• What did you notice in the problem that made you think to use ________?
• Can you tell me what your answer means to the people in the story?
• What made this problem so hard?
• How did you decide to organize your work? What patterns did you see?

Did you notice there were eight questions above? One for each mathematical practice! They are the sort of questions we love to see students addressing in their PoW submissions, because we love having a glimpse of students’ reasoning, sense-making, problem solving, habits of mind, or, yes, mathematical practices.