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/
]]>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.
]]>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:
Enjoy!
Max
]]>The Geometry Forum began in 1992, just before the World Wide Web made the Internet accessible to a much wider audience. At that time what was your main goal for the Forum and how much has that grown or changed over the years?
Our goal was to help geometry teachers use the emerging technologies of the Internet (which then included email, newsgroups, and file sharing) to extend the learning of their students and their own professional development beyond the walls of the classroom. We also imagined that these tools might increase communication between math learners (and their parents), math teachers, and math and math ed researchers. We were also planning to write the “Forum News Gateway”, a software package which would allow the inclusion of pictures and sounds and other media within text. The introduction of Mosaic, the first popular “web” browser, in 1993 made our fancy news reader unnecessary, and we instead focused our energies on helping educators understand why they might care about “the Internet”.
We supported many teachers through local Saturday workshops and national summer workshops. As more organizations and individuals found their own way online, we shifted our focus to providing increasing amounts and types of content so that when math teachers came online, they would have worthwhile things to use and an active community with which they could engage.
Common Core is a hot button topic with teachers and parents alike. Can you give an example of how the PoWs can help teachers implement the Mathematical Practices.
MP1, “Make sense of problems and persevere in solving them,” is the focus of the activities in our Understand the Problem strategy. Often students are encouraged to reread a problem or read it more carefully. But that’s not so helpful for many students who aren’t really sure how to engage with something that didn’t initially make sense. What does it mean to make sense? What should they pay attention to? Encouraging students to re-tell a problem in their own words, act it out, or pull out quantities and their relationships gives a purpose to their rereading. MP3, “Construct viable arguments and critique the reasoning of others”, has been a focus of the PoWs since the beginning. We used the Internet to provide problems for students to solve and write a text-based explanation of their work for someone who wasn’t in their classroom. Changing the audience from their teacher to someone on the other end of the computer gives incentive to write about mathematics. So even if you just use the PoWs to explicitly implement those two practices, it will pay off in many other areas of the classroom.
You recently traveled to a classroom to take part in the video filming of the scenario: Campfire Camaraderie. Why are Scenarios* an important part of the learning/problem solving process?
Yea, I got to be a bear! See the picture at the top of this column. Scenarios are important because they present mathematical situations in which students can look for math (and other things) without feeling like they are supposed to find an answer as fast as they can. To the detriment of our students, we have long rewarded speedy answer-getting in math, instead of sense-making. By presenting a situation that doesn’t actually have a question, we slow down the process of engaging with the mathematics and encourage students think of all the math they can that might be related to the situation. It is a very powerful way to make a statement about what you value in your classroom. It also gives all students a better chance to contribute to classroom conversation, since the fast answer-getters can’t be “done” (because what does it mean to be “done”?), so everyone else gets a chance to think about the situation and share their ideas before “the answer” gets yelled out.
It also means that most of the mathematical ideas that drive the learning with and about the problem come from the students, which communicates to them that you’re interested in their ideas. When I model a lesson using a Scenario and the I Notice, I Wonder activity, once I have read the story or drawn a picture on the board, I’m done contributing “math”. The students do all of it from there on out, including making sense of the situation, pulling out any and all mathematics that might be involved, and even coming up with questions they are curious about. I’m just facilitating conversation, making sure ideas get recorded and revisited, and keeping things moving.
Can you tell me about a person who most influenced you and the way you think about teaching math?
Wow, that is a tough one. I actually did an Ignite talk a couple of years ago titled “The Teacher I Would Have Been“. It wasn’t my best Ignite talk, but I tried to get at how my teaching philosophy developed. I’ve been fortunate to work with great people at the Math Forum, but I’m actually going to pick a non-human. The Geometer’s Sketchpad software has been the vehicle through which I’ve interacted with hundreds (thousands?) of teachers (and students) over the last 20+ years, and those interactions have shaped my thinking immeasurably. I’ve worked with a lot of awesome teachers who totally get Sketchpad and instantly have ideas about why and how they would use it, but I’ve also worked with teachers who can use Sketchpad but can’t begin to envision teaching with it because they simply can’t imagine what their classroom would look like if they weren’t controlling everything. A big piece of my work has been helping teachers think about what a more student-centered, discovery-oriented classroom would look like and how you might move in that direction, even if it’s in baby steps, such as doing one Sketchpad activity in the lab one day.
*Ready to try a free scenario? Find them here.
]]>Steve recommended starting out by stating some aspect of the student work that you find interesting, intriguing, reasonable, useful, etc.
Then raise questions like these:
Let’s use the Elementary level PoW, “Horsin’ Around” to illustrate.
Zachary travels on a journey of 50 miles. He spends half of his time riding his horse and half of his time walking. When he rides his horse, he covers 9 miles every hour. When he walks, he covers 3 1/2 miles every hour. How long does it take him to complete the journey?
Often times a student fails to connect two or more parts of an explanation, so ask them to make that connection.
All of those can present the student with an interesting new task to think about. It can also reveal more to us about how they think and give them a chance to reflect on the thinking they did.
]]>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.
]]>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:
Students’ Concrete Action |
Teachers’ Abstract Response |
Student’s Concrete Response |
Mention 36 meters of fence | Re-state the idea as “the perimeter is 36 meters” | Ignore the word perimeter, not use any of the teachers’ taught strategies for finding side lengths of a given perimeter. |
Use guess and check and drawing pictures to try to find different shaped rectangles that would use 36 feet of fencing; it’s taking a while. | Remind the student of the “hint” that the first step is to “divide it [perimeter] in half. What is half of 36? Can you find two numbers that add to 18?” | The students can, but as soon as the teacher leaves, they start looking for 4 numbers that add to 18 because they look at the picture and remember that rectangles have 4 sides. |
Mention that each frog needs one square meter | Ask, “great, what do square meters measure? Area? Yes! Now you need to find the area of each pen you came up with in part 1.” | Ignore the suggestion to find area; give up on the problem; raise their hand to ask for more help. One student tells me, “I know how to find area, but I don’t get what that has to do with how many frogs can fit.” |
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.
Students’ Concrete Action |
Teachers’ Organizing Response |
Student’s Concrete Response |
Mention 36 meters of fence | Great, that’s one of the requirements the farmer has | Check their guesses against the 36 meters of fence constraint |
Use guess and check and drawing pictures to try to find different shaped rectangles that would use 36 feet of fencing; it’s taking a while. | Organize the guesses that worked into a chart with the columns Length and Width | Immediately generate all of the other missing fence shapes that work, and confirm they had them all. No one explicitly mentioned that L + W = 18, but it was clear from the speed of their mental math they were using some version of that pattern. |
Mention that each frog needs one square meter | Diagnose student understanding by asking, “how many frogs do you think will fit in one of your pens?” | Make guesses using reasoning that shows they aren’t making sense of the area the frogs take up: 36 frogs or 9 frogs (each square meter uses 4 of the meters of perimeter). |
Assume that 36 meters of fencing means 36 frogs will fit in each pen | Invite students to use a drawing to show how many frogs will fit in a pen | Suddenly blurt out, “I can just multiply these! 6 rows and 12 columns of frogs is 72 frogs!” and even “that’s just the area!” One student who filled her 3×15 pen with lots of small squares (over 100) suddenly said, “I did it this way but I wasn’t supposed to. It should be 45 frogs but I drew the boxes too small. All I had to do was multiply.” |
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:
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!
]]>