My name is Mary Wren and I currently teach freshmen at Great Falls High School in beautiful Big Sky Country: Montana. I have used PoWs with both middle and high school students for over three years now. It has been the best “investment” of time that I have ever made.

I learned about them from a colleague at our state teacher convention. I have always tried to incorporate new ideas and technology into my classroom, so learning about the Problems of the Week (PoWs) was exciting for me. The enthusiasm was so contagious it spilled over to the students!

I gave the students a challenge: Be the first one to have your name “highlighted” on a solution posted on the Math Forum’s website. The students worked hard on their math, but our school also had a mission to improve writing. So, for me, I found the Math Forum as another way to incorporate writing into my classroom. With each student name recognized on the Math Forum’s website, it created more motivation and determination among all the students.

What happened next was truly exciting. Students were talking about the Math Forum’s PoWs OUTSIDE of the classroom. They were solving problems. And writing. And revising. And…(get this)….having fun!

By implementing the PoWs, my students collaborated and shared ideas. They were reading. They were thinking. They were writing. And doing the math. Lots of it. Students submissions were using more math vocabulary and reflecting. As for me, I pushed myself to give better feedback to students to motivate them and push their thinking. I tried to encourage collaboration and perseverance. They got lots of feedback from me, but they were even more excited when they received feedback from a volunteer mentor.

It is so exciting to see “student thinking.” One of my favorites was when there was a problem about a farmer having so many dogs and so many birds. (The problem is Feathers and Fur, available in the Problems of the Week Library as #421.) Students were given a number of heads and total number of legs and needed to figure out the total for each animal. Many students were attempting algebra and various strategies, while another student quickly drew the appropriate number of circles on his paper (to represent the heads) and then proceeded to draw “2 legs” on every head. Then, with the extra “legs” proceeded to put 2 more legs on the “animals” until he ran out of legs. And voila! He determined the number of birds and the number of dogs in a matter of minutes, using nothing more than good problem-solving skills.

Aside from seeing the students improve their mathematical thinking and problem-solving skills, I feel I have been more effective in questioning students in order to stimulate mathematical thinking. Kudos to all the individuals at the Math Forum who help teachers to become better teachers while helping students become better problem solvers!

]]>*“Since I already use this Math Forum with my upper students this course allowed me to gain the strategies and courage to introduce the program to my struggling students.”* — Stephanie Davis, Lebanon, NH

*“Its helped me think of ways to help my students learn to tear apart problems and to scaffold the problems for my struggling learners.” * — Lois Burke, Charlottesville, VA

*“The course was the catalyst for me to read and participate in the POW program and take advantage of the many resources to help teachers and students alike improve their communication skills concerning mathematical problem solving.”* — Margaret McCloskey, Rosemont PA

*“Stick with the course. The whole package is fantastic and brings a lot of it together. I’d love other teachers in my school/district to take THIS course.”* — Sarah Winne, Bar Harbor ME

**Courses begin October 2nd.**

**PoW Membership: Resources & Strategies for Effective Implementation **

- learn more about the resources provided with each of the Math Forum’s Problems of the Week (PoWs) and how they can help you enhance student competence and confidence in problem solving and communication.
- develop concepts of mathematical problem solving and communication, both your own and your students’.
- enhance your understanding of the Common Core State Standards’ Mathematical Practices and the role of PoWs in addressing them.
- learn more about assessing student work and providing effective feedback.
- expand your toolkit of strategies for managing problem solving in your classroom.
- participate in an ongoing community of teachers who are using PoWs with their students.

**Learning from Student Work: Make the Most of Your PoW Membership**

- have the opportunity to mentor student work submitted to the Math Forum from classrooms around the world.
- become comfortable as well as successful in prompting students to develop sound mathematical practices through written feedback.
- after honing this skill with a diverse range of students, you might want to enroll in “Mentor Your Own: Supporting Strong Development of Mathematical Practices” to learn how to integrate written feedback into your own classroom practice.

**Differentiated Math Instruction: Using Rich Problems to Reach All Learners**

- learn how rich problems help teachers differentiate instruction for learners with diverse developmental levels and learning styles.
- discuss how problem analysis enables teachers to take better advantage of a problem’s potential.
- develop strategies for adapting problems to make them more accessible or more challenging.
- learn management strategies that allow all students to grow.
- explore ways in which assessment of problem solving can inform instructional decisions.

A full listing of all of our upcoming courses can be viewed **here.**

Look for Suzanne Alejandre, Lynn Palmer and Tracey Perzan

**Session 10: Breaking the Barrier: Risk Taking Beyond the Grade**

10:15-11:15 a.m. Room 222

*Lynn Palmer and Suzanne Alejandre*

(Grades 3-5)

My students love to memorize formulas. They don’t want to struggle or go deeper. We’ll share our classroom practices designed to help students persevere. We’ll focus on problem solving and communication and provide ways to help students embrace these practices.

**Session 63: The Math Forum @ Drexel’s Problems of the Week (PoWs)**

1:00-2:00 p.m. Room 222

*Suzanne Alejandre*

(General Interest)

How are teachers using the Math Forum’s PoWs to effectively provide feedback, encourage students’ math writing (drafting, writing, and revising), and develop students’ mathematical practices? Together we will look at the features of the online PoW environment and student work.

**Association of Teachers of Mathematics of Philadelphia and Vicinity (ATMOPAV) in Royersford, Pennsylvania **

**Annie Fetter presents**

*Using Screen-Capture Movies to Assess Quadrilateral Constructions in Sketchpad*

**Association of Mathematics Teachers of New Jersey (AMTNJ) in New Brunswick, New Jersey**

**Friday, October 24 from 12:30 – 1:30**

Join us for an hour of exciting Ignite presentations.

Bill McCallum, Distinguished Professor and Head Department of Mathematics, The University of Arizona will be emcee of the event!

The Ignite Speakers:

- Norma Boakes
- Neil Cooperman
- Phil Daro
- Annie Fetter
- Melissa Jackson
- Bob Lochel
- Max Ray
- Jim Rubillo
- Dianna Sopala
- Patsy Wang-Iverson

**Pennsylvania Council of Teachers of Mathematics (PCTM) in ****Hershey, Pennsylvania**

Look for Suzanne Alejandre, Annie Fetter, Max Ray and Steve Weimar

**November 6-7, 2014**

We will have a Math Forum booth in the Exhibit Hall and we will also have a “dedicated” room for all the Math Forum talks. [**pdf**]

**Utah Council of Teachers of Mathematics (UCTM) in Layton Utah**

**November 7-8, 2014**

Look for Annie Fetter and Max Ray Ignite talks and sessions.

**California Mathematics Council – Northern Section (CMC-North)**

**December 5-7, 2014**

**Friday Mini-conference (1:30-4:30 PM)**

*6-8: Max Ray, “Does That Make Sense in the Story?”: Launching and Exploring Rich Problems*

What does it take to get a room full of middle-school students persisting on a rich task? Lots of careful set-up and planning! We’ll explore some rich tasks using a range of representations, analyze stories and videos of teachers implementing rich tasks, and learn about effective ways to help students understand problems well enough to solve them and persevere during independent work time.

**Saturday, 8:00 AM – 9:00 AM
Suzanne Alejandre**

What’s possible when students become active doers rather than passive consumers of mathematics? In the spring of 2013, the Math Forum partnered with Christopher Columbus Charter School in Philadelphia to produce videos of some of our favorite problem-solving activities. Our videos are freely available on our website [http://mathforum.org/pps/video.html]. We’ll tour the collection and discuss ways the community can add to it! Problem prompts and problem-solving activity handouts will be provided.

Grade Band: 3-8

**Saturday, 1:30 PM – 3:00 PM
Steve Weimar**

What does it mean to get good at looking for structure or reasoning abstractly and quantitatively? We will explore problem solving activities designed to help students get good at mathematical practices and the central role of sense making in that process. This session will draw on insight’s from the Math Forum’s experience mentoring thousands of students in the Problems of the Week program.

Grade Band: 9-12

**Saturday, 1:30 PM – 3:00 PM
Max Ray**

“Ursula is Undecided,” a problem from the world of discrete mathematics, provides a challenge for students of all ages. We’ll attempt to solve this low floor, high ceiling task, and learn strategies for facilitating students as they make sense of the story and answer their own questions about it. We’ll generate facilitation questions that help students create simpler versions of the problem and reflection questions that help students add “Solve a Simpler Problem” to their strategy toolkit.

Grade Band: General Interest

**Saturday, 3:30 PM – 5:00 PM
Annie Fetter**

Description of Presentation: The practices of Noticing and Wondering can help all students generate mathematical ideas and make connections between them. Noticing and Wondering pave the way for the development of other problem solving strategies and support a classroom culture that gives every student a way to contribute mathematically and treats math as a creative process.

Grade Band: 3-8

**Saturday, 7:30 – 9:00 PM
Ignite hosted by the Math Forum and CMC**

Dan Meyer will be emcee of the event! And as we go to press today, our speakers include Michael Fenton, Annie Fetter, Arjan Khalsa, Max Ray, Brian Shay, Elizabeth Statmore, and four others!

Here’s a quick survey to help you think about how *you* interpret MP4. Does “Model with mathematics” mean:

- Students should use a lot of manipulatives?
- Students should solve problems set in real-world contexts?
- Teachers should demonstrate how to solve problems?
- Students should physically construct miniature versions of objects/situations in the problems?
- Students should write equations or number sentences to solve problems?
- None of the above?
- All of the above?

I would say that none of these to me really captures the *thinking* that students need to do to model with mathematics. It is certainly true that mathematical modeling comes from “real-world” contexts, and that often equations and number sentences are part of the modeling process, though (especially for younger children), physical objects or virtual manipulatives can also be part of the modeling process. However, none of these to me really capture what modeling *is*.

The Common Core State Standards document describes modeling with a nifty diagram:

But even this diagram hides a little bit of what’s going on. For example, consider this problem by John Golden, posted on Fawn Nguyen’s VisualPatterns.org:

.

In order for students to find the perimeter of a shape made of 43 hexagons joined in a long line, we hope they will formulate an equation or some other way to quickly calculate the perimeter of any number of hexagons, plug several examples into their formula, interpret their results as perimeter, and confirm them against examples that they draw and count by hand. They’ll then adjust their formula if needed, and when they have a formula or equation that always seems to work, they’ll calculate the perimeter of the shape made of 43 hexagons, and report out.

Is that mathematical modeling? It definitely draws on a lot of modeling skills and thinking, such as determining what quantities to pay attention to, figuring out how quantities are related, writing that in a compact, algorithmic or algebraic way, and verifying the abstracted formula to examples one can draw and check.

Contrast that with this Math Forum Problem of the Week: Teddy Bears’ Banquet [#4651]

Ursinus Hotel is one of the world’s few hotels just for bears. The tables in its banquet room are regular hexagons with room for one seat along each side. In other words, one table standing alone seats six bears.To make more room for dancing at the Teddy Bears’ wedding banquet, the staff arranges the tables in a long row along one side of the room. When they connect two tables together, here’s how the seating looks:

- How many guests can sit at 10 tables?
- How many guests can sit at 25 tables?
- How many guests can sit at 100 tables?
Explain how you found your answers and how you know you are right. Describe any patterns that helped you.

Note: Here is a link to virtual pattern blocks that might help you solve the problem: http://nlvm.usu.edu/en/nav/frames_asid_170_g_2_t_2.html

Extra 1: Use either words or numbers and symbols to write a rule for calculating the number of bears that can sit at any given number of tables.

Extra2: How many tables would it take, arranged in one straight row, to seat 120 bears?

When we ask students what they notice and wonder about this story, they often wonder things like, “Why do they need so many tables?” or “How many bears are coming?” or “What are the other hotels for bears?” or “Will they need extra tables for the food or for unexpected guests?” The context of the problem means that students have wonderings about things that are realistic concerns when it comes to actual event planning… but which aren’t relevant to the math of analyzing patterns.

Now contrast both of these examples with giving students the task of figuring out how many tables they will need to request to host an actual event in their school’s gym or cafeteria or library.

The third task differs from the others in really important ways. The first way is that students have to figure out what is important to know or find out. It may be that there is plenty of space and so the hard part will be figuring out how many people are likely to come. Students will spend a lot of time figuring out the guest count, and once they’re confident, they’ll divide the people amongst tables, round up a little, and voila! But maybe the library is small and the tables are big and so the students will need to measure the dimensions of the library and the tables and use manipulatives and calculations to figure out how many tables can fit and the best way to arrange them. Once they know the maximum seating capacity, they can build a guest list. Or it may be that cost is the prohibiting factor because tables have to be rented, and so they might figure out the most number of people they can seat if the number of tables is limited. In the Teddy Bears’ Banquet and Visual Patterns problems, students have to make decisions about the relative importance of facts based on what the story or image does — or does not — mention.

For example, no measurements were given in the Visual Patterns problem, so students had to think of the perimeter in terms of units or segments, or make the deliberate choice to introduce inches or centimeters arbitrarily. In the Teddy Bears’ Banquet problem, how to arrange the tables was given, and how many were needed or available was vague, to allow students to focus on the function relationship.

The second way is that students have to deal with a lot more than mathematics to be successful, even though mathematics will be part of the success. Students have to consider things like, “how many people will want to come to our event?” and “how can we predict how many people will show up vs. how many RSVPed?” and “how close together will people be comfortable sitting?” In the Teddy Bears’ Banquet problem, those concerns were either already addressed or purposefully left out of the story, and in the Visual Patterns problem, those concerns aren’t part of the situation at all.

The third way is that the “right answer” to how many tables are needed will depend on factors other than logical or mathematical reasoning. The students could plan their event to a T, and then have terrible rain the night of the event, and end up with a lot of extra seats. The analysis of how well their model fit the actual situation will have to take those factors into account. In the Teddy Bears’ Banquet and Visual Patterns problem, the right answer can be confirmed with a drawing (so it’s not just a “take my word for it” situation), but it is not affected by whether any teddy bears got food poisoning the night before or whether any of the segments didn’t want to be placed next to other segments. There’s no unexpected variability to account for.

So what does this mean for trying to make MP4 part of your classroom? I DON’T think it means get rid of visual pattern problems, other non-contextual problems, or traditional word problems. What makes modeling so special and different from other kinds of reasoning is precisely that the mathematics used is complex, messy, subject to interpretation, and not always confirmed by the actual outcome. I would not use the question of, “How many tables will we need for our event?” for the same teaching purposes that I would use the Visual Patterns or Teddy Bears’ Banquet problem. I couldn’t anticipate that the mathematical issues I was hoping for would come up or turn out to be most useful.

Instead, I would use “How many tables will we need for our event?” to teach the specific skill of modeling. I would NOT use the Teddy Bears’ Banquet problem or the Visual Patterns problem to help students work on MP4. I think students are only working on MP4 when they are asking themselves, “What’s important?” and “What do I need to know/research/measure to get started?” and “What are the things I could represent with numbers/calculations? What can’t I represent with numbers/calculations?” and “How could those calculations be affected by non-mathematical things?” and “How do I account for all the things I left out of my model?” and “If my prediction doesn’t match what happens, does that mean I need to change my model or is this variability expected?”

However, while I was teaching any of these three tasks, I’d be on the lookout for my students’ abilities to:

- notice quantities
- make sense of how doing calculations on quantities results in new quantities
- interpret the results of their calculation in the context of the situation
- use multiple representations to aid them in understanding a situation
- coordinate multiple representations, making sure their words match their drawings and their drawings match their expressions/number sentences
- choose increasingly abstract, numerical representations, such as moving from diagrams to tables, or tables to expressions
- check their results against the situation, what seems reasonable, simpler examples, and others’ thinking
- articulate their assumptions, and how different assumptions could change the results
- explain and justify their thinking in terms of the original context

All of the above skills, that cut across many types of problem solving, are foundational to modeling. Modeling is like doing all the thinking which goes into word problems, but with the added complexity of not ignoring all of the context that we have so often been trained to ignore when trying to get the one right answer based on one set of (sometimes unrealistic) assumptions.

]]>My main job at the Math Forum is that of a Software Engineer – meaning that I build, maintain, and support software. Currently, my main focus is the **EnCoMPASS **software and so I think of myself as an EnCoMPaSS software engineer.

*Tell us about your experiences at the EnCoMPASS Summer Institute?*

The **EnCoMPaSS Summer Institute** aka ESI 2014, was a great experience for me – especially as one of the newer employees of the Math Forum. It was the first time that I got to meet with our external end-users and observe them doing their work using the software. For a software engineer, there is no greater pleasure than this.

While watching the teachers work, and listening to their discussions, I found myself by turns fascinated and enlightened by teachers’ concerns. Having been a student for my whole life, I found that the other side of the coin had its own troubles – and was surprised to learn that many of their troubles were relatable… I guess as a student, one never really sees more than one side of a teacher, because your grades are the primary reason for most, if not all communication between you. It was of course also interesting to see them use the software in unexpected ways.

*What do you think is exciting about the EnCoMPASS project for teachers? For computer programmers?*

Programmers and I would say engineers in general, are simple people. We love to build cool things that solve problems; We love to see our inventions solving those problems; and we love to see the innovation and creativity that is often spurred on just by the existence of such a tool.

So for me, as a programmer, EnCoMPASS is exciting for all these reasons:

1. It is cool technologically speaking (uses some of the latest tools in software development)

2. It is being built to address a real problem: That teachers rarely have the time or space to deal with their students as effectively as they would like to.

3. It will allow teachers to do what they do best, in even more effective and impactful ways.

I think #3 should excite teachers too.

**I hear you’re a poet. Is there a poem you think our newsletter readers might enjoy?**

I wrote a poem recently which was inspired by my experience so far at the Math Forum. It is called “Problem Solvers.”

“Time! More Time!” the teachers cry

As six by ten seconds fly by

“I started in the afternoon

How did it get this late, so soon?”

“Why!? Why!?” principals sigh

As two in three students fall shy

“I know we did, all that we could

Why is it not being understood?”

“How!? How!?” the district asks

Can we solve for [tool] in [task]?

“What can we use across the map

That gets our grades up to the cap?”

“Whoa! Whoa!” the gurus say

Let’s look at this a different way

“A problem underlies these facts

Let’s analyze..

and do the math”

I would like the Math Forum to be thought of as the “gurus” in this piece… and EnComPASS to be the tool that helps to address all the problems outlined therein.

Want to know more, check out his website.

]]>My primary responsibilities involved software quality assurance, and drafting teacher packets for the Problems of the Week. I had a lot of fun crawling around in our EnCoMPASS software hunting for bugs and characterizing improvements for current and future features.

When I wasn’t working on technical-engineery stuff, I helped to write teacher packets for the PoWs with Val and Max. This was an interesting challenge, especially considering my skills are more suited to calculations and planning. While tedious, I quite enjoyed looking through student solutions and comparing them to how I would solve the problem. Keep an eye out for Trig/CalcPoW and AlgPoW Teacher Packets with “Daniel” at the bottom!

Beyond the daily tasks assigned to me, I was also employed as event staff for the EnCoMPASS Summer Institute and the 2014 Philadelphia Engineering & Math Challenge. I’ve also personally taken on the mantle and stethoscope of @AskDrMath on twitter, and tweeted cool math videos and interesting exchanges in our archives.

Now that all is said and done, I’m proud to say that I’ve worked at the Math Forum and with everyone on the team. I look forward to becoming the official Math Forum birthday singer. So long, and thanks for all the donuts!

Follow @AskDrMath for twice weekly tweets about Math and how to do it.

Follow @Hecatonkheries for puns and musings from an engineering student.

Visit our webpages to learn more about:

EnCOMPASS and The Summer Institute

2014 Philadelphia Engineering & Math Challenge

*How I found POWs, used them, lost them and eventually came back to them!*

I actually found the Math Forum POW’s many, many years ago (we won’t talk about how long). When I first started using them I taught mostly Geometry and I really loved finding non-traditional problems for my kids to try. I started by working a few in class, in small groups. I was trying to get my kids to take some risks and to start diving in to problems that weren’t quite as “direct” as the ones often offered in textbooks. I was looking to stretch their thinking and increase their confidence. Over time my kids become great at taking those risks and really did jump in whenever I gave them a POW. They competed with one another as to who could come up with the most unique solution and whose solution might be the most elegant – and what that meant. As a student myself, I always did the problem the “hard” way — I was the brute force mathematician — so it was always fun to see how they would work the problem. The ideas they would come up with were amazing!

Then came state testing and suddenly, I didn’t feel like I had time for the PoWs. Had to prepare for those state tests. My kids did well but the ones who didn’t do well were struggling. Why? What weren’t they getting? After looking at test questions and working with and talking to kids for a while, it seemed that the kids who were struggling just didn’t seem to be able to handle any problem that didn’t look like the five examples we had done in class. Anything new threw them for a loop. They just didn’t know how to attack it. Enter the PoW’s. I decided that the only way my kids could build their problem solving muscles was to do exactly that – problem-solve!

The love affair starts again. Now I make a habit of including PoW’s as often as possible. We take time to notice and wonder and I’ve noticed (no pun intended) that noticing and wondering has made the transition over to the regular ‘ol math lessons on the typical stuff too!

*“Ms. Burke, I noticed that when you multiplied (x + 3)(x – 3) the middle canceled out. I wonder if that always happens when multiplying two binomials when one is positive and one is negative?” *

*“I noticed that when you have one root at 4 + i then you have another at 4 – i. I wonder why that is?” *

Now that being said…. It didn’t happen overnight. The first one went “ok” – and that’s about as enthusiastic as I can be about it. I was still getting used to teaching with the PoWs; they were still getting used to learning with them. The noticing and wondering we did in class was crummy. For example, when we did The Function Challenge (#628), they noticed that there were five functions; and they noticed that two of them were quadratics — and that was it. Come on… really… that’s it? Nothing about the lines? And I was at a loss as to how to get them to think more deeply. The solutions weren’t much better – they lacked detail and were often just an answer, even though I had given them a rubric and gone over it in class. Again… really…Yuck… what to do next?

Enter the importance of feedback and revision! I started having my kids submit their answers to me in a Google doc. I told them that I would give them feedback, provided they worked on the PoW early – before it was due (this was a mistake, by the way; I’ll explain shortly). I provided lots of comments in their documents, pushing them to think more deeply and explain more thoroughly. It was hard to come up with comments that didn’t give them answers. I decided, though, that this might be a good place to model for them some noticing and wondering.

*“I noticed that you looked at the graphs and which was higher. Good thinking! I wonder if you thought to compare the compositions? What function is created by B(D(x))? What about D(B(x))? Which is larger algebraically or graphically? “*

The students who worked on it ahead of the due date got lots of feedback. Be prepared: giving quality feedback takes time but it is so worth it! The kids who took the time, read the feedback and even asked questions back did much better than the ones who left it until the last minute. Lesson learned….

Enter next PoW and the importance of the scenario vs. the actual problem we did Don’t be Square #736. The first time I had used the scenario but I wasn’t as prepared as I should have been. This time I was ready! We put up EVERY single thing they noticed – EVERYTHING! We put up EVERY single thing they wondered – EVERYTHING! Crazy stuff went up there! It was fun! Then we started analyzing our thinking. What seemed important? What did we think the question might be? This bugged them at first – no question was there….

*“I can’t do this … there isn’t a question? Why are you putting up a problem that’s not a problem?”*

They also hated that I wouldn’t give any value one way or another to their responses. I just asked them to repeat for the class and the class decided what they thought we should think about and how one idea could connect to another. It was probably one of the best classes I’ve had in a long time! They got it.

Their responses were so much better too! Everyone had to submit a rough draft this time. No exceptions. Two due dates: one for the rough draft that I gave them feedback on, and one for the final. This way I got to comment on ALL of them. No one got to wait until the last minute. It was a beautiful sight to behold. Even parents got into it! One parent helped his daughter with the assignment rough draft and actually thought they needed Calculus. She took great pleasure in going home and explaining it to him “the easy way!”

My students were thinking! Hallelujah! Loads of things improved: those in-class “noticings and wonderings” I mentioned earlier; their explanations to each other — and their willingness to to get up and explain in front of the whole class.They were taking risks and enjoying getting that right answer and really understanding how they got that answer. They didn’t balk at a problem that they didn’t get right away. They dove right in! Success! I didn’t feel like I was just teaching math but teaching how to solve problems – whether they were math or not. Even when the kids hadn’t seen that type of problem before, they still felt they could try it . Any problem seemed doable — even those on the state tests.

To give kids confidence … the ability to persist … the ability to communicate — that’s huge!!

Follow @lbburke or http://geekymathteacher.com/

]]>- communicating ideas and listening to the reflections of others
- estimating and reasoning to see the “big picture” of a problem
- organizing information to promote problem solving
- using modeling and representations to visualize abstract concepts
- reflecting on, revising, justifying, and extending the work.

*Powerful Problem Solving* shows what’s possible when students become active doers rather than passive consumers of mathematics. We argue that the process of sense-making truly begins when we create questioning, curious classrooms full of students’ own thoughts and ideas. By asking “What do you notice? What do you wonder?” we give students opportunities to see problems in big-picture ways, and discover multiple strategies for tackling a problem. Self-confidence, reflective skills, and engagement soar, and students discover that the goal is not to be “over and done,” but to realize the many different ways to approach problems.

The book is closely aligned with our Math Forum PoWs and Problem-Solving and Communication Activity Series. Whether you’ve been using the PoWs for years or are just starting and wondering how to help your students become better problem solvers, we think you’ll be able to find useful activities in this book.

Plus, the book has a companion website, with activities, classroom videos, and PoW alignments. The resources on the site are freely available to all PoW members and book owners. Please take some time to visit http://mathforum.org/pps/ and check out *Powerful Problem Solving*.

The very first Common Core Standard for Mathematical Practice, telling students to “make sense of problems,” includes many ideas that have long been emphasized in literacy instruction. Yet when “math” starts, both teachers and students often leave those good habits behind. We’ll look at examples of this phenomenon and explore how to translate literacy routines into good mathematical practices.

- Describing vs. Deducing: Characterizing Teachers’ Analysis of Student Work

- *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.

In all seriousness, productively critiquing the reasoning of others truly is hard work. It’s even harder to do in a way that is accountable and doesn’t leave others feeling put down or attacked. But having productive arguments in math class can be a very engaging activity, and one that helps students get into other practices, such as making sense of problems (so they can successfully convince others of their ideas), using appropriate tools strategically (as they defend their decision to solve a problem in a certain way), and modeling with mathematics (as they argue over the best way to simplify a messy quantitative situation).

Does the idea of starting with Practice 3 feel scary to you? It does to me. How would I open up my classroom to debate and argument from the get-go? What if my students are very novice at sense-making and persevering? What if they don’t know how to critique without being mean? What if someone says something mathematically incorrect and no one knows and they learn something wrong? These are all valid concerns! Maybe some of these stories will help us think about them …

**Peanut Butter Jelly Time!**

It was the very first week of school in a 9th grade class of students who were specifically identified as needing a double-dose of math every day. Their teacher was working on establishing classroom norms, especially around communication. She decided to use the classic activity, “Making a Peanut Butter and Jelly Sandwich,” in which students write directions for making a PB&J sandwich and then an alien tries to follow those directions. Hilarity and messiness ensue!

I was watching and videotaping the lesson and noticed something really profound happening. The students started out with completely reasonable steps for another normal person to follow: open the jars, spread equal amounts of peanut butter and jelly on the bread, put them together, eat. If you’ve ever seen a PB&J before, and if you’ve ever used a knife before, you could totally follow those directions.

But then the teacher started to follow the directions under the persona of Bob the Alien. She did crazy things like smush the full (open) jars of peanut butter and jelly directly onto the unwrapped loaf of bread. She used her hands to smear peanut butter all over the unwrapped loaf. She put enormous globs of peanut butter and jelly on the bread and then put the slices together, bread side in and gooey side out. The students started calling out

“Bob doesn’t have any common sense!”

“Bob is crazy!”

“We’re going to have to be so specific!”

The teacher agreed, “Bob is an alien after all. He doesn’t know anything about peanut butter and jelly sandwiches.” So the students got back to work. Orally, their directions got much more precise and specific as they thought about what to change. In writing, they didn’t get nearly as much on paper as they said out loud (pretty typical!), but even their written directions improved.

What I took away from that is that students really do have a sense of audience and purpose when they argue. If they give you an incomplete argument, they aren’t necessarily being lazy or sloppy. They’re giving the (minimal) amount of information they think you’ll need to understand their ideas. With feedback about how their audience really understands their thinking, even math-phobic 9th graders see the value in revising and re-explaining their ideas.

Having an audience, someone to test their ideas against and gauge the reaction, really helped the 9th graders rise to a level of specificity and clarity they hadn’t reached before. The “critique” of their arguments wasn’t someone saying, “that’s not clear; fix it.” It was someone acting out what they thought the students meant — albeit from a very silly perspective! Students can practice the art of constructing arguments, getting feedback, and revising by writing for an audience and then having their audience read their work and say, “Here’s what I think you’re saying…. Is that what you meant?” It’s simple, motivating, not too threatening, and helps students communicate more effectively!

**Lessons from the Little Ones**

And how do students learn to be helpful audiences? This year I get the joy of spending time in Kindergarten and 1st grade math lessons, as students participate in routines like Number Talks and independent Math Centers for the first time. It’s fascinating to watch what the little ones need to learn, and what they already know how to do… and I see the same needs and knowledge in their 5th – 8th grade schoolmates upstairs!

In a Kindergarten Number Talk, the students are practicing some really important habits by talking about them all the time! Before we do a Number Talk, we discuss what it looks like and sounds like when you listen to a friend. We talk about how we stay quiet, track them with our eyes, keep our feet still, and use hand signals to show when we have a “brain match” or a question. Then we do a Number Talk, which involves the teacher putting up a visual prompt and calling on several students to tell what they see or how they think about a certain question about the visual. The Kindergarten students aren’t really sure yet how to talk about what their brains do, and so their teacher really has to work to ask questions to help them reveal their explaining.

But it’s working! The students are learning to say things like,

“I saw 2 red beads and 3 blue so I counted 3, 4, 5 and I knew there were 5.”

“I counted 1, 2, 3, 4, 5.”

“I just saw 5 and I knew it was 5.”

“I know 2 + 3 is 5.”

The students are also learning to recognize if they thought the same thing as another student. And slowly, slowly they are learning to put their “That’s not right!” reflexes into a question. So when a student says they saw 4 beads when others saw 5, instead of shouting out, the students show they have a question. Their teacher has given them some question ideas like “Could you check that again?” or “Are you sure it’s 4?” and the students are learning to use those questions instead of calling out, laughing at a friend, or saying “NO!” when they hear something they don’t agree with.

**Some Ideas to Try**

If you want to explore Math Practice 3 with your students, here are some activities you might try:

- Take a look at the Teacher Packet from a recent or upcoming PoW. Copy some of the novice and apprentice student solutions from the packet and share them with your students. Ask your students, “What do you notice about this work?” and “What’s one question you could ask this student to help them revise their thinking?”
- Invite students to share their written work on a PoW with the class. Make copies of the work from your volunteers and give students 2 colors of post-it notes. Put classmates in small groups to read the work and write “I notice…” positive statements on one color of post-it note, and “I wonder…” curiosity statements on the other color. Collect up all the feedback and give it to your brave volunteers.
- Try some Math Talks with your student. PoW user and Powerful Ideas columnist Fawn Nguyen has a great blog including a description of Math Talks that is applicable to students of many ages.
- Start some arguments! There’s a list of mathematical argument starters that ranges from upper elementary to upper grades of high school just getting started here: http://matharguments180.blogspot.com/

Enjoy!

Max

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