I used to think the same way. I have to cover x, y, and z by the end of the year and I wouldn’t have enough time to do the Math Forum Problems of the Week or problem solving or (fill in the blank of your favorite thing we don’t get to). This year I decided that I was going to do the Algebra Problems of the Week (PoW) with my Algebra 1 classes. On the first day of the new AlgPoW cycle, I project the scenario for my students. Our current routine is that I read them the scenario, students list their noticings and wonderings as I read it to them and for a moment or so afterwards. They then get about 2-3 minutes to share their noticings and wonderings with a partner. I then ask each group to share one notice, which I compile. Even if all of their noticings are up there, they have to tell me which notice they had and I add an asterisk to indicate that more than one group had it. After each group has had a chance to contribute, I ask for any additional noticings, which I add to the list. We repeat this for their wonderings. This whole process takes about 15 minutes total.

My students have made incredible strides just in their noticings and wonderings over the course of the first 12 weeks or so of school. The quality of their noticings and wonderings is far superior to where they were at the beginning of the year. Now they are looking more for the mathematics and their wonderings are more mathematical than they were at the beginning of the year.

Do we solve every problem in class? No. What will happen next is I will give them the Problem of the Week that has the question. They are expected to work on it outside of class. Right now, I am working with them to focus on attempting and revising a solution. When we have work time in class, they can be working on the PoW. I have some students who diligently use their class time to work on the PoW so that they can bounce ideas off of other students or ask me for some direction. Other students will work on submitting it online. Out of my approximately 70 Algebra 1 students, about 20 of them consistently submit at least one draft to a PoW. This is probably the biggest area I am struggling with right now. I would like this number to be higher. Since it is my first year doing this, I am just kind of going with the flow right now.

What is my goal with giving my students the PoW? I want them to be exposed to mathematical situations. I want them to be able to find the pertinent mathematics in a problem situation and be able to use it to solve a problem. My hope is that by the end of the year, my students are more confident in attempting these types of problems because from what I have seen on the PARCC exam, these are skills they will need to be successful on them. But most importantly, they are skills that they will need to be successful in solving any type of problem in life. Will they encounter quadratic equations in their daily lives? Probably not. But will they encounter problems? Yes. Being a good problem solver is an important life skill. If I don’t cover all of the material in my course, yes, they’ll be lacking a little bit when heading to the next course. But if they are good problem solvers, they’ll be able to figure it out and apply what mathematics they do know to the situation. The mathematics will come. Meanwhile, I will keep plugging away at helping my students be better problem solvers. I know it is time well-spent. I can see the improvement in my own students in just over 12 weeks (we are doing our 7th Noticing and Wondering / Problem of the Week).

So, Teacher, wherever you are, give it a try. And I don’t just mean give it one try. One is not enough. When we began, this is where my students started (you will find it at the beginning of the post). This is where my students were after 2 PoWs. This is where my students were after 4 PoWs. And now you see where my students are today (scroll further down in the post). 15 minutes of class time to do Noticing and Wondering every 2 weeks plus another 10 or so minutes of class time to pass out the problem and give some further direction (such as sharing all of the noticings and wonderings or having students look back at their own lists) isn’t a lot of time.

You don’t need to give up a whole day of instruction. A little bit at a time will help your students out. If you have a PoW membership, you can find PoWs that will work with what you are teaching right now, which will let you work with two things at once – reinforcing your content and teaching problem-solving skills. It took me three years to finally get there and I don’t regret doing it at all with my students. Just try it. I don’t think you’ll regret it.

This was originally posted by Lisa on her blog: An “Old Math Dog” Learning New Tricks.

]]>Classroom teacher and EnCoMPASS participant Andrew Stadel is excited about using CueThink. He recently posted on his blog: **“ I was giddy exploring this app for the first time; seeing how well it could support students through the problem-solving process, seeing the functionality for feedback, and having a teacher dashboard. …Think of the potential.”**

Do your students have access to iPads? Are you working with them to make sense of problems, talk mathematically with their peers, and persevere through the problem-solving process?

The best way to tap into the fun is to take a video tour on their website at cuethink.com and download the application. Already have a registered CueThink account? Take the next step and have your students Notice and Wonder using the CueThink platform.

Need support? sheela@cuethink.com and norma@cuethink.com are just an email away.

Help us by taking our brief survey.

]]>“**Why 2 is greater than 4: A proof by induction**” by Max Ray at the Key Curriculum Press Ignite event at Bartini’s during the 2011 NCTM conference in Indianapolis.

“**Ever Wonder What They’d Notice? (if only someone would ask)**” by Annie Fetter at the Key Curriculum Press Ignite event at Bartini’s during the 2011 NCTM conference in Indianapolis.

“**Unsilence Students’ Voices**” by Suzanne Alejandre at the Key Curriculum Ignite event at the 2012 CMC-North conference in Asilomar. Read the full paper here.

If you’ve never seen an Ignite talk before, prepare for a high-energy, fast-moving talk! In an Ignite talk, the speaker has exactly 5 minutes, and 20 PowerPoint slides, which automatically advance every 15 seconds… whether they’re ready or not!

Here’s Annie’s take on “Use appropriate tools strategically”:

If you liked this Ignite talk, check our AMTNJ 2014 Ignite Talks page, where we’ll be continuing to unveil one new Ignite talk a day from our event at the Association of Math Teachers of New Jersey 2014 meeting.

]]>- design creative solutions to engineering challenges
- present written and oral responses to open-ended mathematics problems, and
- compete head-to-head in a math quiz bowl

Our Engineering Challenges have highlighted engineering, hospitality, food science, and sound engineering programs. Judges from different departments at Drexel have lent their expertise and given the visiting students an opportunity to talk to Drexel graduate and undergraduate students about STEM.

In a survey given to participating students at the end of our **first year**, they told us what was best about the competition for them:

- “I got to learn new things and communicate as a team member and gain new skills,”
- “I had the chance to be a leader and work collaboratively with my peers and it was really fun.”
- “I like that we all worked together during the teamwork challenges.”
- “We got a chance to build teamwork skills and learned that it is okay to lose.”
- “Engineering challenge allows us to use outside and creative knowledge as well!”
- “I had learned something that I wouldn’t have learned for years.”
- “Gave me insight about how to approach different types of math problems.”
- “I liked that we worked together and that we can make a mistake without being criticized about our mistakes.”

If you are interested in tackling challenges like these with students in your classes, on your math teams, or in your math clubs, please visit **our website**. There you’ll find all of the Engineering Solution Challenges and Quiz Bowl questions (and answers!), for this year and last year. Just look for the **Documents** link in the top navigation area.

A typical day as a programmer, at least at the Math Forum, is you come in, check your email to make sure nothing is on fire, then attend to what you have been working on. This is usually a bug that someone has spotted or a new feature request from a user. Also, at least a few times a day, issues pop up, whether it’s with customers or staff. These issues range from simple questions, to generating reports, to software features that are not working as expected.

*We hear that you recently made some pretty big changes to the PoWs. Which one(s) are you most excited about?*

There is not necessarily one particular change that I am most excited about, since a lot of areas have been updated. But if I had to choose one, it would be the store front. It might not have been, at least coding wise, a huge structural change, but I’m happy that the store now has a modern look and feel. I’m also glad to finally implement some user requests in regards to threads and queues.

*Mobile Apps are huge right now. What opportunities does the CueThink partnership open up for the PoWs?*

I think the CueThink partnership opens up additional avenues for both students and educators to experience the Problems of the Week. I have noticed more and more that children spend more time on tablets than a traditional desktop. So I believe it’s vital that the PoWs are available on this emerging platform.

*How would you describe the working atmosphere at the Forum and the people around you? Don’t worry, we won’t censor you!*

The Math Forum and the people who work here are friendly and try to keep things in perspective. If you are having a tough time outside of work, you can be sure to find cards, well wishes, or help from co-workers. This is a nice change of pace for anyone coming from a seemingly anonymous corporate environment.

*What is the one thing that most people would never guess about you?*

I’m not sure if people would never guess it, since it falls in line with the programmer stereotype, but I used to be a professional video game player back in college. I travelled the country for tournaments and was sponsored by an NBA basketball player and a popular energy drink. I also am one credit short of a Russian language minor and try to practice it when I can with my Russian-speaking sister-in-law and niece.

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.

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