Put Your Marks Where Your Mouth Is

I was struck recently when reading Robert Kaplinsky’s post, Why Are You Using That Problem?.  In this piece, he articulates 3 different reasons that we might choose to use a particular problem: to introduce a concept, for productive struggle, or for problem completion – each of which has its own value.  Kaplinsky convincingly argues that we should be purposeful not just in which problems we choose, but in why we choose a particular problem.  I have been considering his thoughts in reference to productive struggle, and specifically in how to best to support students when we choose a problem with this as our goal.

I have always talked about encouraging and nurturing a classroom culture in which it is safe to take risks, and to experiment.  When I introduce a novel problem, I tell students that my expectation is that they try – that making an attempt is what I value.  My marking scheme always always includes credit for getting started, and for each step of a problem.  But in some ways, my marking scheme has been generally geared toward a sequence of steps leading toward a correct answer.  In awarding credit for specific answers, I have been communicating to students that I value those answers.  If I want students to put value on productive struggle, I need to demonstrate that that is what I value – and by extension, that is how they can earn marks.  (For now, I’ll leave the question of whether earning marks in general is productive at all…)

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I worked with Dr. Andreas(@DrAecon), my excellent colleague in the chemistry department, to create a set of guidelines and a rubric to support students in becoming independent and confident problem solvers.  In our guidelines, we tried to give students some explicit ideas to try when they are presented with a novel situation, and in our rubric, we tried to express criteria that values the process more than the answer.  We want students to make valid arguments, to justify their reasoning, to persevere in problem-solving,  and to feel that they can take risks, including taking a path that might not lead to a correct solution.  I hope that our guidelines and materials communicate these values.  I want to be explicit and transparent with students about this.  We’ll need to set aside time where we are not focused on learning specific content, but instead are focused on becoming strong and reflective problem solvers who know that they can earn respect and grades with this focus.  I want to put my marks where my mouth is.

Here are the rubric and problem-solving guidelines, based on Polya’s four steps.  I’ve included both Word and PDF versions, and a blank version, which is intended for students to use as an organizer for their work.  These are written with 10th grade students in mind, but I plan to create simplified versions of these for middle and elementary school students.  Please let me know if you can use these, and if you have any suggestions for improvement.

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PDF: Problem_Solving_Rubric_Grade_10

Problem Solving Protocol_Grade 10

PDF: Problem Solving Protocol_Grade 10

Problem Solving Protocol_Grade 10_Blank

PDF: Problem Solving Protocol_Grade 10_Blank

There Can Be Only One (Marker)

Observing a student working on a whiteboard is the best way that I’ve found to get immediate insight into his or her thought processes.  Perhaps because of the impermanence of the medium, students act much more freely than when working on paper.  They are more willing to take risks and to potentially make mistakes.  Even when writing in pencil on paper, the act of erasing is slower than it is on a whiteboard – it seems like they can think more quickly and freely on the whiteboard, which leads to a more fluid thought process, and less barriers between their thinking and their writing.

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I’ve been a huge advocate of students’ use of whiteboards since I began teaching.  One of the very first things I do when I move to a new classroom, is to cover as many surfaces as possible with whiteboards.  Asking students to stand up and work in a visible way has the immediate effect of increasing sharing of ideas and showing thinking in a public way.   And it’s fun and they just really like it.

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John Orr’s whiteboarding protocol in his recent “My Favorite” post (Week 2 of the MTBoS 2016 blogging initiative), has filled in a piece that was missing from the work in my classroom.  When he groups kids at the whiteboard, he gives each group only one marker.  Every few minutes, he calls “marker switch” and whoever has the marker has to give it up to someone else in their group.  Sounds simple, and I know that I have read about this somewhere before (I think maybe in Henri Picciotto‘s blog but I couldn’t find the reference), but I never thought that it would have the profound effect that I observed when we tried this.  When the kids all had markers, some would inevitably be drawing instead of  mathing,  most would be doing their own thing, and they would occasionally talk to each other.  Providing only one marker forced communication and collaboration in a way that I just hadn’t anticipated.  If they wanted to express their idea but it wasn’t their turn to draw, they had to argue for it verbally.  I can’t recommend this strategy highly enough, and it has had a very positive effect in my class.

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A related, but maybe non-mathy aside: I was an art teacher, both at the college and the community level for years before I got into math teaching, and have continued that work along with teaching math.  Most of my life as an artist has been focused on making sculpture, but I did some animation and installation work for a couple of years, which involved a technique I learned from studying the South African artist, William Kentridge.  The process involved making a charcoal drawing and taking a photograph of the drawing.  The drawing would then be erased and/or altered slightly, and then photographed again.  This process was iterated again and again and again (this must be related to my interest in fractals…).  The photographs could be played in a sequence, which gave the illusion of motion.  Here is an example of an animation of some flying bats, which I used as a projected component of an installation piece.

For me, this process of animation was extremely freeing.  I was not afraid to make marks on the page because I knew that whatever I did would be erased soon.  There would be a record of the act of making the drawing, and all of the pieces would come together to form a whole, but each individual drawing would only be seen for fraction of a second.  I think that kids experience something similar to this when white-boarding.  They are more inclined to take risks and just try things because there is no danger of permanence.

Try giving them only one marker!  And let me know how it goes.

Bullseye

This is a short reflection from a lesson focused on solidifying understanding of linear and absolute value equations with Grade 8 Algebra 1 Students.

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I created a game, based on the Green Globs software.  I’ve never actually used the original materials, but it looked like it would be a highly motivating activity, and being on a tight school budget, I decided that since I wouldn’t be able to make the purchase, next best thing would be to use Geogebra to make my own materials.  I called my game “Bullseye.”  I bet that the original version is slicker and more complex than mine, but it worked pretty well for us.

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Here is what a “game board” looks like.   The basic idea of the game is that you need to write equations which, when graphed, hit the green dots.  Your team scores points based on how many green “orbs” your graph hits.

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I grouped students in pairs and gave them whiteboards.  I handed out the rules, and projected the game board.  Students had 2 minutes to decide on their best two equations.  At the two minute mark, we called “markers down,” and students held their equations in the air.  We entered them into Geogebra and calculated their scores for the round.  I also stole the scoring from the Green Globs people: for each equation, 2 points for the first orb, 4 for the second, 8 for the third, etc. doubling for each additional orb.  Asking them to work in pairs was key.  They were forced to talk and argue about the best two equations to choose.

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Students who spotted the absolute value equation in this one ran the table!

The “Expert” games included “Shot Absorbers.”  If your graph hits a shot absorber, you don’t score any points.  When these were on the board, I also allowed inequalities, but you might want to allow piecewise functions or Domain or Range restrictions if that’s where you’re at.

My 8th grade group this year is by far the most competitive group with whom I’ve worked.  They are just dialed-in when they are competing against each other (There is a total ruckus in the room when we play Grudgeball!).  I have to admit that I am not much of a gamer.  I don’t really play games, and I’m not a very competitive person.  But we need to adapt to the group that we have.  These kids are really pretty good sports.  They desperately want to win, but they are also good losers.  Even though Nathan Kraft has decided that it is potentially destructive to his classroom culture, it just works for my kids.  And as long as I have them playing in pairs or groups, at least there’s collaboration in addition to competition.

Here are about 12 game boards along with my instructions.  These could be very easily modified to work for quadratics or whatever functions you’re studying.  Let me know how it goes if you try this out, or if you have ideas for improving the game.

Bullseye Game Files and Instructions

UPDATE (2/7/2016): Of course several better versions of this activity surfaced quickly.

Generating questions

MTBoS blogging initiative, week 3!  This week’s prompt focuses on questioning.

betterquestionsMy colleague, who teaches the grade 6 and grade 7 math courses at my school is in training to run a marathon.  He has put together a training program for himself, which includes a schedule of endurance-building, and he has been collecting data with a GPS watch.  As he examined the data, he thought that this might make a rich exploration for his students and we have been working together to set up a project  for them.

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Here is what the raw data looks like

We started with the driving question: How long will Mr. Feutz take to complete the Limassol marathon? and then we began by brainstorming questions together.

  • How long will Mr. Feutz take to complete the marathon?
  • How many steps will he take to complete the marathon?
  • How many calories will he burn during the run?
  • What percentage of his overall time will be spent moving?  (Compared to taking breaks)
  • What will his average heart rate be during the marathon (In B.P.M.)?
  • What will be the shortest/longest mile time, and what is the range between these?

We tried to analyse which questions are actually interesting, and what might we be able to ask kids to do with them.  While running, he found that he was constantly doing math of one sort or another.  How much further will I run today? When will I arrive back at home?  Things that he had genuine curiosity about, and questions that math gives us the power to answer.

We set up a graphic organizer, and decided to ask our driving question directly.  Here are some of the kids’ initial responses.

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They always manage to think of something that we haven’t anticipated (…which is why we love kids!).  “Will you be listening to music, because if it’s like… Taylor Swift, you wouldn’t be as inspired as if it was like… hard rock.”  Students were given a graphic organizer and asked to write their first guesses.   As they acquire more information, they will refine their estimates.  Part of the beauty of this work is that they will get to actually test their prediction, and compare their answer to what actually happens.

Students will revisit this project over the next weeks, and will be asked to refine their work.  They have already studied unit rates, and are moving into work on ratio and proportion next.  We are hoping that more questions will arise as we continue this work.  My favorite so far is, “Given a start time, time spent running so far, and a map of the run, can you figure out where Mr. Feutz is now?”

Here is our Graphic Organizer – totally open to your critique and suggestions.  What questions can you add to our list, and how do you come up with your project questions?  We would be most grateful if you share your curiosities or strategies in the comments.

Related:

Create Suspense – MTBoS Week Two: My Favorite

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For week two of our blogging challenge, we were asked to write about one of our favorite lessons, games, resources, tools or strategies. It was tough to pick one.  I have so many excellent resources and tools, that as I reflected on what to write about, it made me once again realize what a great time is it to be a math teacher and just how lucky I am.  What a hard but awesome job, and what a generous and sharing community we have.

I really like keeping students in suspense.  If I can set up a situation where students want to know what’s coming next, that often translates into engagement and the desire to learn.  When you watch your favorite TV show, and it ends on a cliff-hanger, you make predictions and you think about it in between episodes.  You are connected to and invested in the story, and you can’t wait to see what happens next.  I want my classes to have at least some of this kind of anticipation.

I also like to create some public presence for math in my school, and I try to create a bit of suspense around this as well.  Typically, a few days before we publish our work on the math wall, I will put up a provocative question, or something to generate interest. This week I just put up a funny title with a big question mark, and listened for the buzz..

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As a culminating activity of learning about graphing linear equations, I asked Algebra 1 students to create “math faces” through drawing with graphs. They used Desmos (…it was very tempting to write a series of “my favorite Desmos” posts – everything those guys do makes my classroom better!) to create their works of mathematical art, to practice transforming linear equations, and to solidify their understanding of domain and range.  I ask students to make sketches ahead of time to ensure that they are purposeful in manipulating their equations.  This is an activity
conceived of by the incomparable Fawn Nguyen, and one that I use every year.  I have written about it before as well.  This kind of task gives all students an easy entry point, but allows for real complexity for those who are ready. This low entry, high ceiling aspect of drawing with graphs makes it a rich and motivating activity that we can return to with students again and again.  Although the Des-Man activity is not currently available through the teacher dashboard at Desmos, I have heard that it is getting a make-over and that they will be bringing it back again.  Each time, I am amazed at how motivating this activity is for students.

After a couple of days, we published our process and our results on this year’s “Sweet Wall of Math!”

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How do you create suspense and anticipation in your classroom?

2016 MTBoS Blogging Initiative: The One Good Thing Glow

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As a math teacher, it’s really fun to work with students who come to my classes already loving the subject, students who have already mastered the content from last year’s course so they are ready to dig deeply into novel problems in our courses, and students who feel really good about themselves as math thinkers.  Doesn’t this describe most of our kiddos?

The reality is that a large part of my job ends up being working to alleviate the trauma that students bring with them to Algebra 1 or Algebra 2.  They have known since second grade that they are not a “math person.”  They feel that they are not as good at math or as fast at math as others around them.  Many of their mathematical experiences have left them feeling inadequate, and math has been a place where their self esteem has been eroded.  Not a big surprise that they have trouble accessing the beauty of the subject.

One of my grade 10 students showed up this year with all of the marks of earlier trauma.  She was reluctant to speak in class or even when I worked one-on-one with her.  When she did answer a question, it always sounded like she was asking rather than answering (…y’all know that “I have no idea if what I’m saying is right” tone of voice).

This week, she had a perfect score on her linear systems and inequalities assessment.

And this wasn’t an easy test.  I always include some questions that ask kids to synthesize and apply, and to recall ideas from earlier work – typically, I have very few 100%s.  In fact, hers was the only perfect score in the class.  Although I generally avoid comparing students to each other, I couldn’t resist sharing this with her.  You should have seen her trying to pretend that she wasn’t beaming!  I wrote a note the her mom, to share how proud I was of this effort and of her success.  When I saw her mom the next day in the school lobby, there were tears in her eyes while we spoke about this.  She said that her joy had nothing to do with the score.  She didn’t care about the test score, but she could see and feel the difference in her daughter’s confidence and sense of self.  I couldn’t agree more.

What a wonderful way to begin my Thursday.  The One Good Thing glow stuck with me all day!

2016 Blogging Initiative

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I am participating in the 2016 MTBoS Blogging Initiative.  I am doing this in part in to open my classroom up and share my thoughts with other teachers. I hope to accomplish this goal by participating in the January Blogging Initiation hosted by Explore MTBoS

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I’ve just dusted off my “About Me” page to include the schools at which I’ve taught, and I’m excited for the next month.  You, too, could join in on this exciting adventure. All you have to do is dust off your blog and get ready for the first prompt to arrive January 10th!

The Thirsty Crow

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Not sure which one I saw first, but I got the idea for this lesson hook from at least two teachers: Jensilvermath, and Pam Wilson. Both are creative educators, and generous online colleagues, who share their ideas, resources, and materials.

Screen Shot 2015-12-03 at 7.33.57 PMOne of Aesop’s Fables tells the story of a crow who comes across a half-full pitcher of water in the desert. He cannot reach the water until he figures out that by dropping pebbles into the vessel, the displacement causes the water level to rise until he can quench his thirst. Using this narrative as our lesson hook, students were given a cup full of marbles, and a graduated cylinder partially filled with water. They were asked to predict how many marbles they would need to reach 2000 mL, and then how many more until the water overflowed.

I have found that giving too much structure can take some of the life out of a task, but not enough structure, and students flounder. In this case, I asked them a direct question, but did not suggest any methods at first.  As we were right in the midst of linear equations, my assumption was that they would jump right to dropping their marbles into the cylinder, creating a scatterplot, find an average rate of change and line of best fit.  But students always surprise me.  They asked for an extra graduated cylinder to do some experiIMG_4298mentation, and pulled out the scale to start weighing marbles.  They traced the cylinder base to see how many marbles fit in that circle.  As we had more than one color, it was important to them to see if the lighter blue marbles were consistent with the dark blue – something I hadn’t even considered.  One group even qualified their prediction with the caveat, “…if the ratio of light blue to dark blue marbles is consistent with our sample, then
this prediction should hold.” What a nice expression of understanding. Reminder to self: always give students as much freedom as possible. Let them run until they really need help.

The students who dropped their marbles into the cylinder one at a time collectIMG_4312ed data points as the water level rose. They created scatterplots of this data, and calculated an average rate of change. Next, they used this information to find an equation for a line of best fit, which helped them to make a confident prediction about how many marbles they would need to bring the water all the way to the top. We took out enough marbles to test their predictions, and added them to the cylinder until the water level reached 2000 mL and then until it overflowed. Cheers and groans for the accuracy of their predictions.

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Creating ways for students to create mathematical models and make predictions is one of the most important opportunities that I can set up for them.  These types of tasks help students to connect the math from their classroom to questions that they will come across in the real world. Even if they will not need to calculate the number of marbles to overflow a cylinder, they will almost certainly need to use similar problem-solving skills, and equally importantly, they will have to decide what math skills they need to apply to novel situations.  Students react very strongly when they see the “answer” to this type of task – very different from how most students react when looking up the answer in the back of the math textbook.  Even reluctant mathematicians couldn’t help but look closely as we counted the last few marbles out!

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I used this video to introduce the Crow and the Pitcher. It’s short but gets the point across. I shot a video of our cylinder, and edited it into a 3-Act format while I had the supplies out.  I think that if you can get your hands on some marbles and a vessel, you may as well do this hands-on, but in case you don’t have a bunch of marbles handy, or if video is your preferred medium, I’ve published the materials below for you to use.  Did I give enough information in act 2 or did I forget something?  Please do let me know if you use any of this, and how it goes …and don’t forget to check out the Action Version…

Thirsty Crow Act 1

Thirsty Crow Act 2

Thirsty Crow Act 3

Thirsty Crow Act 3 Extended (Includes Action Version!)

Lies and Collaboration

I actually enjoy the puzzle-like aspect of exponent rules, and simplifying radicals.  For me, there is something satisfying about learning ways to manipulate numbers and letters – probably why I love algebra so much.  But I am tuned-in enough to my students to know that many of them don’t find the same satisfaction from doing this work just for the sake of intellectual exercise.   And since calculators came into vogue, it’s been harder to justify the need for rationalizing the denominator or expressing the square root of 50 as 5 times the square root of 2.  But we are tasked to follow standards that often include these kinds of skills and it has been helpful for me to turn this into exploration or game learning as much as possible.

IMG_4135I did some mining of the MTBoS for ideas to teach rules of exponent arithmetic and came across this post, which includes a nice exploratory worksheet from Andrew Stadel.  He describes a similar issue with contextualizing exponent rules for middle schoolers – one of the really great things about our online community are these moments where we are reminded that we are not alone.  He asks his students to find the mistakes in the equations, to explain where the author went wrong, and to find the correct solutions.  He used a bunch of the common misconceptions found on mathmistakes.org to help students to catch themselves in the common errors.  Very nicely done.  This would have been a good lesson as is.

IMG_4147Then I remembered the Bucket O Lies protocol from Nora Oswald at Simplify With Me.  Nora manages to gamify math like no one else that I’ve seen.  She manages to add entertainment even to potentially dry topics like this one.  I combined Andrew’s worksheet with Nora’s idea to make a bucket-o-exponent lies.  I printed the 3 worksheets, cut them out into individual problems, folded them up, and put them into buckets (or baskets).  Voila! Drama and Motivation.  In pairs or threes, learning happened.

IMG_2885Of course, I hammed it up with the students.  There’s nothing like telling teenagers that someone is trying to get one over on them to motivate them.  This has worked well for me in the past, especially when it came from advertisements.  I riled them up by acting outraged that someone had created this whole set of math material, which was full of mistakes!  (Actually, I blamed Andrew :) ) …Lies I tell you… these baskets are FULL OF LIES!  Let’s find the mistakes so we can write a self-righteous set of corrections back to this author who was deliberately spreading bad math.

They quickly saw through my act, but it was enough.  They were already motivated in spite of themselves.  Andrew’s worksheet was just enough for everyone.  I started by coaching the groups who needed help getting started and moved to pairs who were making mistakes with fractional exponents.  For my honors group, I added a few more examples with rational exponents.

Thanks Andrew!  Thanks Nora!  Our generous community is the Best!

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Revealing Learning Targets

DSC06230I am not always sure about how explicit to be about learning targets.  I have seen some convincing research, which seems to indicate that letting students know exactly what is expected of them for each lesson helps them to take ownership of their learning, and to make sure that they are getting what we think they are getting during each class session.  I agree with this practice in general, and I believe that it definitely has a positive impact on some students.  My current school, as well as the previous one have required that we post targets each day, and there are many educators who I respect who advocate for this practice.  But sometimes, I feel like a learning target can put a limit on where we can go as a class, and can feel a bit stifling, especially when we want a problem or exploration to feel open-ended.

Lately, I have adopted a practice of “Hidden Targets.”  I do post the learning target, but I often leave it covered up during class.

IMG_4121As part of our end of class routines, students make conjectures about what they think that today’s learning target was.  We reveal the target, assess how well the lesson matched the target, and whether the learning matched what was expected.  Although I think that I am good at starting class off, and generating enthusiasm, I sometimes am not as good at synthesizing and wrapping up.  Being conscious of synthesis and wrapping up class in a richer way has been one of my goals for this year, and this routine has been a good protocol for me and for my students.  It quickly reminds us about what we learned during class, and how this lesson fits in to the bigger picture.  Students have been highly engaged in figuring out the day’s learning goals; I hear students talking throughout the class period about what they think is under the flap for today – and you know that they remind me if I forget to do the “reveal!”

IMG_4119…And it doesn’t hurt that we have created this sweet Appolonian Gasket on which to showcase the day’s targets.  Who doesn’t want to stare at this and contemplate infinity?!