One of my primary goals as I teach mathematics to my middle schoolers is to help them see how much fun math truly is.  The fun I describe to them is not the fun you might associate with the phrase you sometimes see from online vendors that claim they “Make Math Fun”.  Typically, this is some sort of game that is a reward for completing so many math drills.  That’s not the fun I mean.  The fun I mean is the fun a person gets from becoming totally immersed in an intellectual act, being challenged, rising to the challenge, sometimes succeeding, sometimes experiencing frustration, attempting to overcome frustration, overcoming frustration, talking to friends and more experienced math learners, and succeeding.  Now that’s fun!

(Please note – there are many online sites and math games that teach worthwhile mathematics.  Those are fun, too.)

I’ve used the PoWs as a challenge for individuals, with small groups in Math Club, and with whole classes.  I’ve settled into a routine over the past three years that I like very much.  I’ll be doing it again this year.

My middle school prides itself on each grade sending out a weekly newsletter.  Each teacher reports on classroom activities, homework and test schedules, field trips, assemblies, etc.  I add an extra section to my part of the newsletter.  It’s simply called “Problem of the Week,” and links to a Math Forum problem I’ve selected.

That’s where the fun starts.  Students are encouraged to work the problems on their own time.  They can work by themselves or with a friend (or several!). I encourage parents to get involved and work the problems with their child.  I tell students I will award a little bit of extra credit for their work but nobody really cares about that.  Many, many students jump in and work the problems.  There’s no pressure.

Sometimes a student will stay in at recess to work a problem with me.  What a great way to spend recess!  Our Language Arts teacher always works the PoW.  Students sometimes work them with her.  Parents, either with their child or without, work the problems.  The best exchange I had was around The Oracle’s Crowns [Problem #16999].  I received this email:

Hi Seth-

We are giving up on this problem, have you posted the answer anywhere?  Or could you explain it?  This one stumped us.

Jean

I replied:

Hi Jean.

I won’t give you a solution but I’ll give you a hint or two.

which I did.

Here’s the next email:

Mr. Leavitt,

Before the meeting in late January we briefly discussed with you the Oracle’s Crown problem.  We were doubting that Caleb could know with certainty which crown he had on.  I happened to have passed the problem along to a co-worker of mine who also found it interesting.  He in turn passed it along to his boss who is, I think one of the most intelligent people I know.  The two of them discussed it and also came to the conclusion that Caleb could not know with certainty what crown he had on.

Do you have access to the official answer or could you show us how you think Caleb could know?

I have shared some of this other extra credit problems at work too.  People love them.  They think it is great that you are challenging the kids (and their parents) with these problems.

Thanks

Jerome

The family — student and parents — continued working on the problem until they understood (with help) how Caleb could deduce that he had a crown made of tin.

My last email from Jerome was:

That explains it.  Thank you.

As an amusing follow-up to my note yesterday, the boss (who I said was one of the most intelligent people I know) left a voice mail message on my co-worker’s phone at 3:36 this morning.  It simply said, “Tin, tin, tin! The crown is made of tin!” He obviously must have figured it out during the night and had to let him know.

Thanks again!

What fun and what a great way to interact with students and their parents.  I love it.  And I owe all my enjoyment to the Math Forum and the great Problems of the Week.

Here is a glimpse into our Math Forum experience.

Gina DiDomenic

Hi, I’m Gina, I’m currently entering my junior year at Drexel University and am just now finishing my second co-op.

As stated above, I had the privilege of working at the Math Forum, and it has been a wonderful experience.

It had been very difficult for me to find an education-based co-op, since most education jobs are field experience-based, which you do in your later years at Drexel, but I was lucky enough to find this one.

I got to do things I love every day at work. I would solve various math problems each day (I love math!!) and try to think of methods for how to solve them. At first I found this a little challenging, because when solving a problem you are used to doing it your own way — and that’s it. However, with this I had to think outside the box. Solving problems for elementary students was probably the most difficult. Even though the problems were much easier, I had to keep in mind that they did not have all of the same schooling as me yet. For these problems I used various methods such as Drawing a Picture, or Using Manipulatives.

I had a great time at the Math Forum, and I’m definitely going to miss it. I will carry on the skills I learned here into my own classroom one day.

Casey Sneider

My name is Casey Sneider and I am a soon-to-be junior at Drexel University! My major is Secondary Mathematics Education with a minor in Mathematics and for the past four months, I have been completing my Co-op at The Math Forum.

My co-workers have become my mentors over the past few months, as they have taken me to observe in classrooms, taught me how to come up with valuable methods for teaching children math, and talked to me about what I should anticipate to experience when I become a future teacher one day. Learning about the Problems of the Week and being able to help Suzanne create packets for the problems has been extremely helpful in teaching me how to apply these writing and thinking skills to making teacher lesson plans. Making video scenarios* with Gina all over Philadelphia has been not only fun, but has taught me how there are more ways to teach than by just lecturing in front of a class.

I’ve also been able to meet with many teachers from all over the country, primarily through The Math Forum’s EnCoMPASS Project. They talked to me about how connecting with teachers through social media is such a valuable source to a young teacher who is just starting out in the classroom for the first time.

I’ve gained so many skills and learned many new things about the teaching world through my co-op, so I know that The Math Forum will always have a special place in my heart and in my classroom one day.

* to view the video scenarios – YouTube or Vimeo

October is a very very busy month for us!

Northwest Mathematics Conference in Bellevue Washington

Max presents: Getting Good at Problem Solving Through Sense Making

The first CCSS Practice is “Make sense of problems and persevere in solving them.” But how do students become sense-makers? What can teachers do so that students are making sense of problems for themselves? And what do teachers do that gets in the way? In this hands-on workshop we’ll use different activities to make sense of some interesting math problems.

Suzanne presents: Unsilence Students’ Voices

Picture a classroom. A teacher presents a problem and initiates a discussion. Some students look attentive but are quiet. A few students have hands raised, posed to talk, but at least one or two students seem disengaged. Every classroom has silenced voices. Learn activities to increase CCSSM practices (especially #1 and #3).

NCTM Regional Conference in Baltimore Maryland

Look for us at Booth 112 – Suzanne, Annie, Max, Steve and Erin Igo will all be there, be sure to stop by and say hello!

Suzanne and Erin present: Moving Beyond the Right Answer: Developing Students’ Math Communication Skills

The Math Forum’s rubric emphasizes a combination of good problem solving and strong mathematical communication. We score in six areas, including interpretation, strategy, accuracy, completeness, clarity, and reflection. We’ll share stories from online and classroom exchanges of our efforts to help students develop mathematical communication skills.

Steve presents: Notice and Wonder: Beyond Engagement to Reasoning

The notice-and-wonder approach to problem solving has gained popularity for overcoming student anxiety and for connecting to student thinking. We will look at the activities and developmental progression that show the approach’s value for expert problem solvers and facilitate the connection to strategic planning and higher-order thinking.

Annie presents: “Noticing and Wondering” as a Vehicle to Understanding the Problem

The activity Noticing and Wondering can help all students generate mathematical ideas and make connections among them. It paves the way for developing other problem-solving strategies, supports a classroom culture that gives every student ways contribute mathematically, and promotes the practice of sense making and perseverance.

ATMOPAV Mathematics & Technology Conference in Horsham, Pennsylvania

Annie presents two sessions:

Grades K-5:  Sense Making?  Aren’t We Already Doing That in Literacy?

The very first Mathematical Practice, “make sense of problems”, includes ideas that have long been foci of literacy instruction.  Yet when “math” start, everyone leaves those good habits behind.

Grades 6-12:  Ever Wonder What They’d Notice (If Only Someone Would Ask)?

Sense-making is an important part of math, and a powerful way for teachers to learn what students understand about a problem or situation.  How can “noticing” support this mathematical practice?

December 2013

CMC North Asilomar 2013 in Asilomar California

Annie Fetter presents two sessions:

Strategic Uses of Technology to Promote Conceptual Understanding

Sense Making? Aren’t We Already Doing That in Literacy?

Steve Weimar presents:

Notice & Wonder: Engage in Formative Assessment of Mathematical Thinking

Suzanne Alejandre presents:

Moving Beyond the Right Answer

Max Ray presents:

Becoming Better Reasoners: Supporting Students to Develop as Problem-Solvers

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.

I’ll start with a story that helped me think about what “concrete” and “abstract” mean when solving math problems with students:

The Story
I was teaching 5th grade kids about area and perimeter using this scenario: you have 36 meters of fencing and want to build a rectangular frog pen using all of it. What are some different pens you could make? If each frog needs 1 square meter of space to flourish, how many frogs can your pen designs hold? Which design holds the most?

One traditional model of teaching suggests that what’s hard for students when solving word problems is getting rid of the fluff and decoding the underlying abstract mathematics hidden in the context, and that if the teacher can restate the problem in mathematical language, it will support the students to solve successfully. Here’s what I observed when we used that model:

The next period we tried an alternate model, in which the context was used to elicit the students’ concrete ideas, and the concrete ideas were valued. We helped the students organize their ideas and look for patterns. In short, we avoided abstraction that the students didn’t suggest, and instead supported organization, pattern recognition, and referring back to the concrete.

Once we established that when frog farmers say “pen” they mean fenced-in-space-for-keeping-animals-safe, not ink-based-tool-for-writing, there was enough going on in the context that the students had some ideas about how to draw different pens, check if they fit the farmer’s specifications, and try to fit the frogs into the pens.

The Common Core Math Practice encourages students to decontextualize and contextualize. Sometimes that means that as teachers, we need to get out of the way and stop helping students decontextualize. Instead we can support them to:

• determine if their thinking makes sense in the story
• organize their thinking to help them see patterns
• ask students to make conjectures (guess) and check their thinking based on the context
• encourage multiple representations for the same problem

When we let students be in charge of decontextualizing and making use of contexts, sometimes they surprise us!

PS — If you want to be surprised like I was at how often we do all the contextualizing and decontextualizing for students, watch Annie Fetter blow you away with 5 minutes jammed with examples of teachers “helping” students by thinking for them!

It’s hard to pick just one story but since I’ve been watching the videos that we took last May at Christopher Columbus Charter School I was reminded of one of my favorite PoWs to present to learners of all ages. It’s called Eating Grapes – you can find it using the problem number 4507. It’s a Math Fundamentals PoW but I also wrote A Cranberry Craving [Problem #3284] as a PreAlgPoW version — same problem with a slightly different story. In both cases the kids often wonder why Angela (or Carissa) eat so many grapes (or so many cranberries)!

This problem is perfect to use just the Scenario and tell the kids, “I’m going to read you a story!” And then once you’ve read the scenario aloud, ask the students, “What did you hear?” I love having ALL students involved from the outset and this technique has never failed me.

Another reason this problem is one of my favorites is because of the variety of methods that can be used to solve it. We’ve included six different methods in the Teacher Packet [PDF] and there could easily be more. One of my favorite solutions was presented by two students in the fourth grade at Christopher Columbus Charter School. I included the video clip in this blog entry: Owning It

If you could get teachers to take away one important aspect of the PoWs, what would it be?

I am happy when teachers focus their students more on the “process” of problem solving rather than just being quickly over and done with a PoW. Using the Scenario [pdf] link is how I think of starting the process. Having students go online to submit just their I Notice, I Wonder ™ responses is a great second step. Having students talk about the problem and then submit a draft of their solution online is a third step. And the process can continue with teacher online feedback and reflection/revision. An ultimate goal is to have students return to their online work later in the year to view their electronic portfolio of problem solving!

Notice and Wonder™ is a frequent theme in both your blog and the free scenarios. How important is this to problem solving?

It’s very important because it evens the playing field for all students. Students who normally would say “I don’t get it” and put their heads down or act out and disrupt the classroom, don’t do that – instead they’re engaged. Students (not as many but there’s always at least one or two) who race to find the answer and then they’re done, can’t do that — there’s no question yet! So, using I Notice, I Wonder™ has the effect of engaging all students in the problem solving process.

What did you have the most fun working on as part of your job at the MF in the last 2 months?

I’d have to say that the EnCoMPASS Summer Institute was the most fun. When we host an institute I always have memories of the first Math Forum Institute I attended as a participant in 1995. I enjoy helping the participants feel comfortable and to help them make the most out of the experience.

What are you looking forward to most this upcoming year?

September 28 is the date that I’m most looking forward to because it’s the date that our book, Powerful Problem Solving: Activities for Sense-Making with the Mathematical Practices should be in our hands! Max Ray has written such a friendly and accessible book that I think will give teachers a real sense of how to make the most of the PoWs with their students. I’m going to be first in line to get my copy autographed by Max!

There are many steps a teacher has to take when going over certain problems with their students. First of all, the teacher needs to make sure their student understands the problem. How can you help your student understand the given problem when mentoring them? One teacher named Barbara states that in order to guarantee your student will be able to solve/understand the problem is to have previously taught them various ways of solving problems. Making sure they are aware of different methods that they can try out, so in case they get stuck on one, they can try a different approach. Knowing different strategies allows students to talk to one another and compare what each other did, Barbara says. Students comparing each others’ methods will then most likely result in viable arguments and criticism between students. These are not bad things! These arguments can open students’ eyes to new and exciting methods.

Another strategy you can use while mentoring your students is to help them to read abstractly. Make sure they make sense of these quantities and the relationships they hold within the problem. Have the students relate them to everyday experiences and manipulate the representing symbols to understand them better. Using diagrams, graphs, flow charts, and manipulatives with students is also very helpful. Visualizing the problem with your student helps them get a better understanding of it.

Finally once the problem is done, have your student reflect what they did. Ask them their thoughts on the problem and how if they could, what they would do differently.

More on helpful strategies can be found on the Elementary Math Practices blog.

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!