Making Problem Solving Part of the Math Curriculum
I also want to start them immediately on the weekly Problem Solving (aka PS — our parents call them PMS, haha, such funny parents we have). I might write a post on this because it's really at the heart of why I love teaching math.So, this is a post about my favorite geeky mathy thingy that I call a PS.
I love teaching problem-solving for a very selfish reason: I always learn something new from it. I learn from struggling with the problem, I learn deeper mathematics, I learn from my students' different solutions and their non-solutions, I learn from other teachers, I learn that I have soooooo much more to learn.
I am not a math major. I didn't study beyond calculus. I teach algebra and geometry. But I read a lot of math books. I even buy books that are way over my head and have no clue what I'm reading two pages in. I'm mostly overwhelmed by my ignorance.
I don't believe one's love of math is innate, thus I'd be remiss if I didn't mention all the people who generously and lovingly share their love of problem solving that directly or circuitously have touched my life.
- My father: He was a math teacher for over 30 years. When he passed away 13 years ago, I was still teaching science, not math, but I knew he was proud of me because I'm the only one in our family who's a school teacher. I'm also the poorest one.
- Michael Shaughnessy: I took a summer course from him in 1996 called Problem Solving for Middle School Teachers. It remains the best class ever. Dr. Shaughnessy was NCTM President in 2010-2012.
- Sabrina: My daughter has such a beautiful, efficient way of seeing and solving math puzzles. But she's more passionate about drawing and dancing. Go figure.
- Martin Gardner: The wealth of his knowledge and his love of recreational mathematics are second to none.
- James Tanton: One of the coolest things about Twitter is the ability to follow, in real time, someone whose work (videos, blog, books) you've stolen over the years. @jamestanton's tweets are like little chunks of chocolate.
- Joshua Zucker: Biggest treat for me at last month's Math Teachers' Circle training was meeting Josh. I first learned about Josh through the Julia Robinson Mathematics Festival where he's the director.
These are the main online sources for my PSs:
- The Math Forum Problems of the Week ($25 teacher annual subscription)
- The University of Mississippi Internet Math Contest
- MathCounts Problem of the Week
- Problems to Ponder from Dr. Michael Shaughnessy
These are books on my bookshelf that I absolutely treasure because they are loaded with great PSs:
- A Passion for Mathematics, Clifford A. Pickover — Amazon, $13.13
- The Colossal Book of Short Puzzles and Problems, Martin Gardner — Amazon, $23.21
- Problem Solving Through Recreational Mathematics, Bonnie Averbach and Orin Chein — Amazon, $9.91
- The Ultimate Book of Puzzles, Mathematical Diversions, and Brainteasers, Erwin Brecher — Amazon, (only used books available from selected sellers)
- A Moscow Math Circle, Sergey Dorichenko — AMS, $35
- Mathematical Circles (Russian Experience), Dmitri Fomin, Sergey Genkin, Ilia Itenberg — Amazon, $36.86
- Thinking Mathematically, John Mason, Leone Burton, Kaye Stacey — Amazon, $61.59 (Holy moly. I don't remember the book costing this much; it was a required book for my course with Dr. Shaughnessy as mentioned above.)
- Solve This, James Tanton — Amazon, $37.34
- How to Solve It, G. Polya — Amazon, $23.40 (You know you MUST own this book, right?)
- Updated 09/10/12: Sadie @wahedahbug reminded me of a small book that we both love Fostering Algebraic Thinking, A Guide for Teachers, Grades 6-10, Mark Driscoll — Amazon, $22.20. Driscoll also wrote Fostering Geometric Thinking — Amazon, $26.75
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If I didn't hand out a PS on the first day of school, then it'll definitely go out the second day. I have three types of PSs — weekly, in-class, and group.
Weekly PS
The first time that I introduce this, the process roughly takes on this form:
- I pass out the PS. (To save paper and photocopying, student just gets a strip of paper that has the problem on it.)
- I call on one student to read the problem aloud. I then ask everyone to read the problem again quietly on their own. If it's a particularly lengthy one, I ask them to read it again.
- Questions I tend to ask, "What are you being asked to find out?" "What information do you already know?" "What are your immediate thoughts about this problem?" "Have you seen a similar problem before?" "Do you have ideas on how to start the problem?" "Wanna give me a guess on what the answer might me?"
- I tell students that since this is the first time they do a PS in my class, I'll walk them through the process of writing up a PS. I walk them through Polya's four steps:
- Understanding the Problem
- Devising a Plan
- Carrying out the Plan
- Looking Back
- I blah-blah-blah about the importance of writing in mathematics, and that I don't ever want to hear any whining, especially of this sort "but this is not an English class...," because help-me-God if I do hear it.
- I emphasize that all write-ups must be on notebook or grid paper.
- Students have one week to turn in the PS. They get a new PS each Monday, it's due the following Monday. This is the only assignment that I do not accept late.
That's roughly my introduction. Then I help them begin the problem in class. This should take up a full period. Before I dismiss them, I hope for this exchange:
Me: When is this PS due?
Ss: Next Monday!
Me: When next Monday?
Ss: At the beginning of class!
Me: What if you don't turn it in next Monday?
Ss: That's too bad!
Next Monday comes around...
True to my promise, I ask the kids to pass forward their PS write-ups. As I quickly leaf through the pile of papers in my hand, I can expect (and you should too) to witness the following:
- Their papers: ripped holes, ripped corners, half-sheet, unlined, spilled cappuccino.
- Their write-ups: red inked, work written sideways and upside down (a complete mess), Dad's handwriting, four papers are identical (including the non-legible and nonsensical steps), only two papers include the "looking back" step, Mom's writing at the top of paper asking for extra time.
- Their solutions: all the numbers in the problem got added, all the numbers got multiplied, oh look, this student performed all four operations on the numbers, $850,000 for the bicycle, Victor is 48 years old (while Victor's father is 36), the building is 756,411 feet tall.
- Their reasons for not having it done: I don't get it, I forgot it at home, my Dad accidentally threw it out, my sister who's in calculus couldn't even do it, my uncle who's an engineer couldn't figure it out, I don't think there's an answer for it, I was absent when you gave it out, remember? I'm-sorry-I-forgot-to-do-it-but-I-love-your-dress-Mrs-Nguyen!
I take a deep breath. Mentally embrace these precious children. Remind myself that I'll be with them all year. Worse, they have to be with me all year.
So at this time I pass out a PS scoring rubric and carefully go over it. I've always used a 6-point rubric but I really want to change it to 4-point.
I give them a new PS for the coming week and only do the first three steps — reading the problem and checking for understanding — as I outlined above. I remind them that I offer PS help at lunch time: Wednesdays for Math 6, Thursdays for Algebra, and Fridays for Geometry.
In-Class PS
I don't grade these. Because I encourage kids to do math with their family and anyone with a pulse, it's nice to learn once in a while (about one per quarter) what they can do completely on their own. One class period.
Group PS
I don't grade these either. Of course this is my favorite type of PS because I get to watch the kids do the math, ask them questions, and listen to their discussions. Before getting into their groups (I almost always assign kids in groups randomly), students have about 10 minutes of quiet individual time to work on the problem. I do make a conscious effort to follow the 5 Practices whenever kids work in groups. I'd asked my principal for funds to purchase large whiteboards so I can copy what Master @fnoschese does with his kids — and I think my principal said yes!
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I'm writing this post with the genuine hope that you'll incorporate problem solving in some fashion, if you haven't already, into your curriculum. If you don't think you have time because you feel you have to cover x-y-and-z content standards, then please make the time. Learning math is a social endeavor, a really fun one, please provide students with lots of opportunities to think critically and struggle productively. I think it's a beautiful thing when we can develop a classroom culture of doing mathematics so contagious that it spreads beyond school boundaries.
Categories: Teaching, Problem Solving, General Math
Tags: problem solving problem of the week PS Michael Shaughnessy James Tanton Joshua Zucker Martin Gardner Julia Robinson Mathematics Festival Math Forum mathforum.com MathCounts Problems to Ponder Math Contest math books weekly PS group PS in-class PS whiteboards Five Practices asking questions
Fawn, I loved reading this post, especially the deep breath you take when you get students' first attempts at mathematical writing back!
I wonder if any of your students have ever submitted their Math Forum problem-solving to you online? I'm wondering because I'm the guy who reads every single Algebra and Pre-Algebra submission and selects some to feature publicly and write about -- it's so fun to get to know the students of teachers I've met! If you're thinking of having any students submit anything online, if you give me a head's up I'll be on the lookout! I like reading math stories from novice to expert problem solvers and finding the delightful nuggets of thinking in everyone's work!
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I feel like I "know" all of you at the Math Forum because I've been at your site for years! It's definitely my #1 go-to site for problems. I know you read the kids' submissions because I always look through all the featured kids' solutions in the teachers' guides. You know, I really need to start having the kids submit answers online -- what a COOL service! -- so I'll run it by my principal to approve this added subscription. We're up for renewal anyway. We have another math teacher (she's awesome), so we'll set up for both of us.
Thank you so much, Max!
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It's not even an added subscription to have students submit anymore -- it's included in the $25 membership (that was new for last school year and it's been pretty successful). Excited to welcome the other math teacher too! It's so neat to hear from someone who "knows" us but we've never met. Twitterblogosphere for the win!
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I noticed that, Max. Yippee! But I do have three different set of kids, so we'll make sure to submit the proper dues. Same with my colleague, she has three groups also. I'd already self documented how jealous I was of #tmc12, so I hope to meet you and other wonderful tweeps at #tmc13! I very much appreciate all your thoughts on this post, Max.
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When Max pointed out your blogpost earlier today I thought "that name sure sounds familiar" and it turns out that you participated in some of the online workshops that I ran a few years ago!
Besides the idea that Max suggests of considering having students submit online (you have up to 36 student logins with your $25 membership) I thought you might find some of these resources interesting:
Teaching with Problems of the Week: The Rubrics
http://mathforum.org/pow/teacher/index.html#rubric
[you might get ideas for a 4 point rubric]
Problem Solving Articles
http://mathforum.org/pow/teacher/articles.html
Think You Don’t Have Time to Use the PoWs?
http://mathforum.org/pow/teacher/PoWsDontHaveTime.pdf
[this just describes a way to use less class time]
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Hi Suzanne! Yes, of course, and I wish I have more time to continue to take the online workshops available. You guys are wonderful with the support and resources -- keep up the great work!
I have read the resources you mentioned. I think I've read every article on the Math Forum site that I have access to!! You have no idea what a mathforum junkie I am
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I really enjoyed reading your clear description of the types of problem-solving exercises you use in class. I laughed at the section dealing with what you can expect come Monday. Thanks for posting and I will check out the resources you listed.
Cheers,
Blaise
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Thank you, Blaise! I love your name, one of my favorite mathematicians Blaise Pascal.
I wrote the what-to-expect part (because all of that and more do happen) to remind teachers not get discouraged when the initial products might be far less than what they'd hoped for. I know, I want to cry when I get some of these write-ups, but laughing might be better.
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I love problem solving and require 6 pow with write up per semester (they ate all due every two weeks, but I count their best six ).
I love your description of how they come into you. Complete sentences is my mantra baby! 42 what? 3 blue what? No he can't what!!!
Getting so into this and hope to contribute to twitter.
Amy
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I remember being in 9th grade and having a complete sentences math teacher for the first time. I didn't get it at all -- I was the "this isn't English class!" kid (of course, I didn't write complete sentences in English class either). Now that I help kids write about their math ideas for a living, I realize how important those complete sentences are: because they're naming *units* and *quantities*, the staff mathematicians, physicists, engineers, etc. care about.
I wonder if telling kids "complete sentences" feels like another math class rule, like flip and multiply? How can we help students experience the value of naming quantities and units? One thing I've tried is when we generate that first list of noticings or what we know about the problem, asking, "did we notice all the quantities (e.g. hours spent walking, hours spent riding, miles ridden, miles walked, number of people, number of horses...)?" "do we know what units they might be measured with?" "which quantities do we know the value of?" or "what quantity is that value (42 what? 3 blue what?) measuring?"
If we name those quantities and units first, and they help us make sense of our calculations later, maybe it will become a habit... and at that point reminding ourselves "complete sentences! units!" is a reminder of an important math thing, not just English class skills.
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You just reminded me of something I see all too often: kids will take two numbers and operate on them, but they rarely know what the answer means! And even those who think they know it have a hard time articulating what that result represents. So I'm always asking, "This answer of 42 that you got... What is it? What does it mean? Why did you choose to divide this number by this number? What were you looking for?..."
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Hi Amy! I know, when I don't see the appropriate unit to describe a quantity, my usual line is: 45 cats or dogs or dollars? We all benefit when we are share, so twitter is my new favorite PD! Thanks, Amy.
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Thanks, Fawn!! Hilarious description and super organized list of resources - what more could I ask? I use these types of problems (I call them Problem of the Week or POW), but always struggle with, should they be in class? at home? individual? pair? group? I love your suggestion of doing all of the above! I'm curious as to why you don't grade the ones you give them to do in class.
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I was chatting with @wahedahbug and @mr_stadel that I don't use the POW acronym because coming from Vietnam, it only means one sad thing for me
I totally need to admit that I used to grade the in-class PS. But I noticed that it was too hard to take points away from students who were struggling and sharing. I wanted them to struggle. Because I couldn't stay with any one group all period long (no teacher can or should), I honestly couldn't tell if the student was earnestly struggling or just being lazy and chatty -- and from personal experience, being a quieter student doesn't equate to being a non-participant or non-thinker either.
I always believe it is my duty to engage the kids. If they aren't engaged, then I have to re-examine the task that I'd selected (too easy, too hard, did I not pick a rich "low entry, high exit" problem?) And if it is a good task, then I have to question if I'd failed in giving directions and/or scaffolding the problem to make it accessible to them. Poor class participation is a failing grade for me, so I just stopped grading the kids exactly for that reason. I hope this makes sense, Anna. Thank you!!
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I know it's still pretty close but because of the POW/MIA idea at the Math Forum we use "PoW" and we usually pronounce it as "pow" (as in "sock it to you!" We tried PotW (including the "t" for "the") but it's kinda hard to say!
I like your use of PS because I tend to think of "postscript" and that makes me think of how there's always just one more thing that can be added on to a thought -- reinforcing the idea that "the goal is not to be over and done. The goal is to reflect and revise".
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I like that, Suzanne! Thanks.
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Do you have a student example of what a good response would look like using polya's four steps?
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Hi Amanda, I do have great ones (earning 6/6 on rubric) during the year, but it's summer right now, and I'm not sure if all the kids took their work folders home at end of year. That's what I love about the MathForum where they post the some of the kids' solutions to the PoW; this helps with anticipating your own students' work.
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Amanda, I think Fawn is referring to the Teacher Packets that we write for each of the PoWs. You can find samples here:
http://mathforum.org/pow/teacher/samples.html
and if you sign up for a free Trial Account
http://mathforum.org/products/store/demoAccount.htm
you can view many more by using the "Calendar" link on the left sidebar of the PoW pages. And then if you find them useful and would like continued access, you might considering a $25 membership!
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Thank you again, Suzanne -- yes, that's EXACTLY what Fawn meant!
Did you know there's a MF weekly blog too? http://mathforum.org/blogs/pows/ And when you're there, in right margin, check out the blogs of the people who make MF as awesome as it is. I follow Suzanne @SuMACzanne and Max @maxmathforum on Twitter!
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Thank you both for the info! I'll definitely think about joining MF :)
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Yeah!!! Just do it, Amanda! Thanks for dropping in.
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I am in a very unique position and need some help. I teach math to all 600+ 6th and 7th grade students in our school in a computer room, with each class visiting the lab every week or two weeks depending on their math teacher's needs at the time. I do not use a "canned program" and create the days agenda by finding resources online. I also use online online resources to create my own online practices and assessments.
I work very closely with the math teachers to ensure lab days are an extension of their math class.
I really want to incorporate more problem solving and less "drill and kill" in the math lab, but am having a difficult time seeing how this is best accomplished in a computer lab. I have to be realistic. There is no way I can read and respond to 300 - 600 students each week. Any ideas?
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Myrna, just to give you a sense of where I'm coming from -- I taught middle school math (and computers and newspaper) in a computer lab before I left the classroom (June, 2000) to join the Math Forum full time. My situation was a little different since the students in my math classes came to my lab every day but I've thought about ways to use computers and problem solving and communication for some time.
I'm not sure what computer functionality you are considering to use for problem solving but since I work for the Math Forum and it's the "problem solving system" that I'm most familiar with, my suggestions will come from that viewpoint. If you like the ideas, there are some free (although limited) options, including tPoWs and Financial Education PoWs. And, I'm happy to provide pricing info if you're interested in a membership.
Idea #1
A. Math teachers present a problem using this "short" use of classroom time idea:
http://mathforum.org/pow/teacher/PoWsDontHaveTime.pdf
B. In your computer class, students would submit their work online.
C. You (maybe with input from their math teacher) would have the students work in pairs.
D. After each of the paired students submitted, they would view the Answer Check and click "revise" so that they're each back on their own submission page.
E. They would have the task of reading each other's submissions with the Answer Check "notes" as a guideline [you could make a "half class" set of those notes or display them on a screen or somehow make them available.
MAIN THOUGHT: You (and the math teacher) train the students on Mathematical Practice #3: Construct viable arguments and critique the reasoning of others.
TIP: Once this idea gets going, the PoW system keeps all of the students' work and they can revisit any problem at any time. In my mind, the goal would be for students to use your lab time to both draft initial solutions but also reflect and revise others that they started a week or even a month ago. It can be used as an ongoing electronic math portfolio
Idea #2
A. same beginning as above but instead or in addition to, you assign students in the class a number. At the end of class you use a random number generator (the most fun is if it's somehow visible and the students can see that it's random) and you chose 6 (or whatever number is reasonable for you to handle) students whose work you will respond to. You use the "teacher office" system that we have for the PoWs and you give those specific students feedback.
B. those students then have the responsibility of talking about your feedback with their classmates so that all of them benefit from the ideas and they all use the ideas to revise
POSSIBLE VARIATION: If you use the short type of feedback that I like to use when students are first getting into a problem solving routine mentoring all of the students doesn't take as long as one might think. I don't use the scoring rubric and I just write one sentence that starts with "I notice..." w
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Thank you so much, Suzanne, for giving Myrna and the rest of us ideas on how PS can be used in a computer lab. I really like your "short" use suggestions as I hear many teachers are hesitant for fear there's no time to include PS. I worry more when a math classroom is devoid of PS!
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I went down memory lane this morning and added a few ideas:
http://mathforum.org/blogs/suzanne/2012/08/11/computer-lab-ideas/
It was fun remembering my other life at Frisbie Middle School -- wow, it was a long time ago in "Internet" years!
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I just discovered your blog and have already added you to my blog reader. :-)
I have been incorporating weekly (at home) PS assignments in my classroom since I began teaching math in the early 1990s. Students have had a composition book where they solve 5 new problems (plus make corrections from the previous week) every week. Your description of the types of solutions you should expect had me laughing out loud; they are spot on!
As the years have gone on, I have refined the way I respond to student's work (including some of Suzanne Alejandre's great suggestions). The trouble is, when I take the time to respond thoughtfully to that many problems for several classes of students, I'm not left with enough time to sleep at night!
As much as my students (and their parents!) have gotten out of my weekly PS assignments, I've realized that one thing that was lacking was student-to-student interaction after individual work had been done. They were refining their own strategies, but weren't learning from each other's ideas! So during the last quarter of last year I made some changes. They had 2 problems to solve rather than 5, the solutions were done on loose paper rather than in a composition book, and they had to turn them in 2 days before our "discussion day." This gave me time to sort through their work to create discussion groups based on (1) strategy used, (2) clarity and completeness of thinking/explanation, (3) correct/incorrect answer (less important a consideration than 1 and 2). I would put students into groups of 3-4 and gave them time to discuss their solutions and solution methods. I'm still refining the process (in my mind) for this year, but I like where it was going. My original intention was to use a rubric to assess first, but the rubric I created was way too involved to be practical. I don't always have the time to do the group discussions for both problems, but having the student work (and a document camera) is a good starting point for whole-class discussion (a la the 5 Practices' sequencing step) and an opportunity to model the type of productive discourse I expect when they're working in small groups.
One thing I'm struggling with is whether to tie the PS problems to the curriculum being taught at the time or have the problems be good (but random) problems. With my previous PS notebook assignments they've been random, but when I tried my new idea at the end of the year I used problems tied to the topics we were covering in class. I'd love to hear others' thoughts on this!
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Hi Alisan! Wow, 5 problems a week, yikes!! I'm glad you cut it down to 2 because we teachers not only need to sleep but also to eat sitting down. My 1 PS a week -- times 3 classes of kids -- is plenty for me. I don't know how you do it. But I love how you're having kids be in discussion groups. Curious: do you tell the kids reason for what groups they're in? What if a kid didn't turn it in? (How is the turn-in rate of 2 PSs/week?) I'm assuming you haven't graded these yet prior them having the discussions.
Do your students have other math homework during the week?
I would NOT tie the PS problems to the curriculum at all because this makes them "exercises" and these are what the textbooks are good at providing. A good PS should employ multiple strategies to solve it, and matching it up with curriculum immediately takes away rich exploration because kids already have a prescribed method of solving.
Thanks so much, Alisan.
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Fawn, just as your name sounded very familiar to me, Alisan's name also sounded familiar to me. I couldn't help myself and I just looked back at the participant rosters for both the Algebraic Reasoning online workshop and also the Probability online workshop (both offered during the time the Math Forum had an NSDL grant [2006 - 2008] -- http://mathforum.org/nsdl_mathtech/). You BOTH participated in those workshops but you weren't ever in the same group. Fun!
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Imagine that. Small world. Math binds us all
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Wow, Suzanne, you have a great memory. And those were really good workshops!
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My students *do* have other homework during the week (not too much), but they're gifted students (in theory, but for most of them, not so much in practice). Because they're part of the gifted program, the parent support is pretty strong, and they come into the program knowing the higher expectations and work load. The turn-in rate is pretty high, except toward the end of the year with 6th graders. (There is something about that age and time of year that cause even the best students quit working!) I do have them turn in 2 days before the discussion day so that I have time to sort the solutions *and* it allows time for late work to be turned in. For those students who still didn't turn in their work, I assigned them to a group and gave them a job to do to keep them involved (like being prepared to explain someone else's solution method when I came around to observe their conversations). In addition to talking through their solutions, I also have them discuss what makes some explanations and models more clear and thorough than others, and what each person could do to improve the communication of his/her thinking. I can't wait to see how the students progress in their explanations of their solutions this year, when they're communicating with each other this way, all year long!
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Alisan, I love the ideas you've described that encourage communication. I think they will go a long way to support Mathematical Practice #3. I would suggest adding in a "revision" assignment. So, maybe instead of 2 PoWs or PS assignments per week you have 1 new PoW/PS assignment and one revision assignment. So, in effect, the students work on a problem/explanation for two weeks rather than just one. The Math Forum has found in our research that the greatest gains students have in mathematical thinking while problem solving is when they revisit their original explanation and revise it.
The other thought is that maybe quarterly or, at least, once a semester, I would have a "Portfolio Week" (or maybe some cuter title!) -- students pick any 1 problem from what they've done and they revise their solution. I would probably combine that with classroom discussions/communication in some way, too.
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Suzanne, I like the idea of a "revision" assignment. In previous years when my students have had 5 problems per week, they've had to correct their ANSWERS -- but idea of requiring them to revise their solution in general (strategy, solution, and explanation) really gets at what I'm trying to achieve with this assignment. And I'm thinking that even those students who really struggled (and may not have been successful) with the original problem will be more motivated to be actively involved in their group discussion if they know that they have to revise their answer!
One thing I forgot to mention earlier is that in addition to requiring students to show all of their work and write enough so that someone else could follow their thinking, I also require some sort of model to go with the solution. This could be pretty much anything (a chart or graph, a picture, a number line, etc.). The model may be something that the student did to help them solve the problem, or it could be something the student created after the fact in order to communicate their thinking more clearly.
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Alisan, your description of encouraging students to include some sort of model reminds me of a question that came in to T2T (http://mathforum.org/t2t/) earlier this month. A New Jersey middle school teacher was trying to make sense of asking students to "illustrate their thinking". She was hoping for ideas about a template page for students' notebooks that she was designing. I bring this up because the conversation I had with her was what "illustrate" might mean. She hadn't realized that it could just be students' explanations since she was assuming that it referred only to illustrations -- some sort of picture. Once she realized that an illustration could be a chart of graph or numberline or (as she was already thinking) a picture, then she relaxed. Encouraging students to include some sort of model sounds like Mathematical Practice #4 but also #2 and #3 -- nice!
Actually, however, I wondered if the work the Math Forum has done with rubrics and the vocabulary that we have associated with the "problem solving" aspects of doing a PS or PoW and the "communication" aspects, might be something to make use of with your students. I wrote this blogpost about unveiling the rubric to students: http://mathforum.org/blogs/suzanne/2012/02/05/introducing-problem-solving-rubrics/
Besides suggesting the order in which I'd roll out the rubric, that post has links to the general rubric (written for a student). As Fawn knows (and refers to on this blogpost) each of our Current Problems has a problem specific rubric. You can see specific rubric samples linked under Math Fundamentals, Pre-Algebra, Algebra and Geometry on this page: http://mathforum.org/pow/teacher/samples.html Both our Primary and Advanced level problems have only the general rubric.
Fawn, do you make use of the specific rubrics and/or the general rubrics? Do you have your students use either of them?
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I really like the idea of picking 1 PS to revise also. Heck, as a teacher, there are still plenty of problems in my pocket that I'd love to go back (and I do when time permits) to revise my own earlier solution attempts. I just use the general rubric that I made available on the post. But I am changing it for next year to a 4-point one.
Suzanne, I so appreciate your referring to the Math Practices with these PSs. We are all new to the math common core standards and practices, and any help out there to connect the things we already do to the "new" stuff makes it less daunting.
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You're welcome, Fawn. It's definitely going to take a village to implement the CCSSM and, in particular, the Practices. On another note, I posted some ideas on Anna's blog this morning that I think you might find interesting:
http://borschtwithanna.blogspot.com/2012/08/integrating-problem-solving-into.html
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Last Thursday the new season of Math Forum PoWs went into "preview" and day after tomorrow (Monday) the first FunPoW (Math Fundamentals), AlgPoW, and TrigCalcPoW will be open for student submissions. The value of the $25 "current PoW membership" has just gone up -- now you get more for your money -- an additional 20 problems a year!
Next Thursday, the first PrimaryPoW, PreAlgPoW, and GeoPoW will be in "preview" and then the Monday after that they will be open for submissions. Then the cycle will continue so that there are 3 new PoWs each week but the services alternate.
So, in effect our PoWs are "open" for 2 weeks -- the idea that a student drafts their submission the first week and then reflects and revises the second week. A student might work on two problems at a time since one week a new FunPoW appears and then next week a new PreAlgPoW (or PrimaryPoW) appears. I wonder how many combinations there are. :)
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Fawn,
I really enjoyed your post. I always concentrate on problem solving with my students. I have most of your favorite books. I am surprised that you do not have any book from the Art of Problem Solving series, my favorite lately. I use them in class and for my Math club. I like that authors concentrate on problem solving, starting from the simplest ones and going to the level of top math competitions. Take a look at them, you will like them.
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Hi Boris, I'm not exactly sure which AoPS series you might be referring to. Like these? http://www.artofproblemsolving.com/Store/viewitem.php?item=ps:aops1
If so, I'm sure these are very good too. I coach a small group of kids in MathCounts also, so between all the problems from MC and my current resources, I have built up quite an inventory. The AsPS books are rather expensive. Thank you, Boris!
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Fawn - I loved reading your thoughts about weekly PS. I recently gave my first one two weeks ago, but I 'm still reading and grading them. Even as a fifth year teacher, I'm often working at school until 6 and bring home work on the weekends. I would like to do more, but with160+ students don't know how I would find the time to grade them. Any suggestions?
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Hi Heidi! Yikes, how do you have 160+ kids in 5th grade?
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Thanks so much for your feedback! I will definitely try the shorter rubric. I was using one with 20 different points and it was some detailed it took 6-7 minutes per paper to grade. You are so right that it is more about them getting to experience some great problems than about getting back overly detailed feedback from only me. I'm willing to try a PS again with these new suggestions. So thanks again!
Oh, and I actually teach 7th grade, but meant that this is my fifth year teaching.
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Hi, Heidi. Here are a couple of additional ideas to add to the suggestions that Fawn offered.
1. Groups -- Assign the weekly PS as you already have but give feedback on 1 out of 6 of the papers. The feedback (not a grade or a score) could be in the form:
I notice ... (and you point out one part of the solution that you value.)
I wonder ... (and you ask a question that will keep the thinking going.)
The student with the feedback on their paper, reads both their solution and the feedback to their group. The group talks about what you were valuing and what you were wondering -- as they talk the idea is that they will reflect on their own solution as well.
Followup task (either as homework or classwork) is that they individually revise their solution.
2. Another idea is to use Fawn's suggestion of having students grade each other's PS work but rather than using the 4-point rubric you might consider the "I notice..." (and say one good thing) and the "I wonder..." (and ask one question about something that isn't clear)
I suggest this for two reasons: (1) encouraging students to talk about their problem solving is an important part of the process (2) it's less threatening and easier for all students to complete the task.
3. "Random" grading -- again using a grouping strategy -- say you have 6 groups of 6 students in a class of 36 just for PS discussions (since groups of 6 may be rather large for groups in general). In those 6 groups each student is assigned a number from 1 to 6. On PS "turn in" day, you roll the dice. (I used to have those huge foam dice!) If it lands on a 2 then you will "grade" those six papers. Again, I wouldn't "grade" them but instead I'd give feedback on them. I would also give points (credit) to each student who turned in a paper.
As Fawn has mentioned your students' experience of the problem solving process is the most important thing.
~ Suzanne
PS. You might gather a few other ideas from:
Tips on Managing Mentoring
http://mathforum.org/blogs/suzanne/2012/02/25/tips-on-managing-mentoring/
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