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September 14, 2012 / Jeff d.

Finding the Good in the Common Core

In general, I don’t like the Common Core State Standards (#standardizethat), but today I think I may have found a use for them.

I was looking for a student-friendly reading on what makes a good mathematics learner.  My plan is to run a Socratic seminar prior to developing class norms and beginning our real mathematics work (we don’t start school until after Labor Day, and spent the first week and a half building school-wide culture…so we won’t even be starting “math” until Monday).  I felt like my search was coming up short.

Then I remembered the Standards for Mathematical Practice within the CCSS.  As it turns out, that part is not too bad – although it is a bit lacking on some things I consider important, like learning from mistakes and risk-taking.

I re-phrased, re-worded, and deleted my way to what I think will be a decent, short read for learners leading into a text-based seminar.  Here it is…I’d love to hear what you think (is it still over a ninth-grader’s head? will it help spark meaningful discussion?  is it just garbage warmed over?).

(the formatting is a little screwy…I think because I wrote it in Pages).

August 28, 2012 / Jeff d.

The Original Random Problem Idea

As part of the Math Blogger Initiation Project, Sam Shah, in his classic avuncular style, offered us the prompt to share anything that we are proud of.  Wait – of which we are proud.  Whatever.  Anyway, I’ve decided to share the very first random problem idea that I ever had.  Here it is – pay no attention to the horrid diagram, not to mention the ridiculous context.  (And here’s the google doc version, which is where all my work resides).

This was an introductory problem to a unit on circles, and the goal was simply to review basic concepts like radius, diameter, circumference, and area of a circle.  It is by no means the greatest problem, and it would be vainglorious to claim it as such.  However, I am proud of it for this reason: it marked a personal turning point for when I finally started to “get it.”

I mean “get it” in two ways.  First, I realized that it is far better to create your own problems than to steal them wholesale.  Believe me, I am not creative and I spend a great deal of time taking Henry Wong’s advice to beg, borrow, and steal.  I steal a ton of ideas from all over – colleagues, the interwebs, books, strange cloud formations…but there are some things that are so much better if you create them yourself.  And problem ideas are one of those things.  (To avoid turning this into a white paper, I think another post may be in order for this topic).

Second, I started to understand what it takes to make a good problem – things like multiple entry points that are accessible to all learners, multiple solution paths, allows opportunities for extension, etc.  If I could take a mundane review topic and create a decent problem out of it…well I can do anything, right?  And the thing is, the problem is almost Hemingway-esque in its simplicity.  I mean, there’s not much to it…and there doesn’t need to be.  (By the way, if you’ve never read Hemingway’s “Cat in the Rain”, do it.  It takes all of three minutes and is the best short story of all time according to this math teacher).

Since that time, my mind has shifted to be on a constant quest for problem ideas.  Most of them hit me at the most inopportune times…which means just at the moment that I’m about to fall asleep.  At first it was kind of a rare occurrence, but now I see them all over the place: the label on a bottle of Mountain Dew, some fried okra, a dirty windshield.  Not to sound trite, but math really is everywhere…especially when you’re on the lookout for random problem ideas.

August 16, 2012 / Jeff d.

Eight (yes 8!) Goals for This School Year

Every year I make two or three personal teaching goals for the coming year…but I never write them down and I rarely tell anyone.  This year I have told many people, and I have more than a few goals – eight in all!  Some of them are new for me and some of them are simply a desire to continue what I’ve been doing, but do it better.  Still others I could really use some help with, so I’d love any feedback.

Here goes:

1. Step up my SBG game

Last year I tried my best to implement at least a version of Standards-Based Grading.  It certainly wasn’t “pure” SBAR, although I don’t think that I want it to be.  I gave a presentation at New Tech’s Annual Conference about how we graded and ended up learning a lot from other teachers (go figure).  One element I want to add this year for skills assessments is a simple rubric at the top (or bottom) of the quiz that lets learners know up front what will be considered “Proficient”.  Last year I made the determination about what “Proficient” meant after the quiz was already done.  Big mistake.

[Update: It didn’t take long to get more info on this one.  The same day I posted it, I found this post from Dan Bowdoin…between his post itself and the plethora of links, I think I’ve got exactly what I need.]

2. Improved questioning and mathematical discourse

This is a repeat goal for me.  How I interact with learners and the environment that we set up as a class for mathematical discussion has always been important to me, but boy do I need a lot of improvement.  An eye-opener for me prior to last year was watching this video of Deborah Ball, dean of the School of Education at the University of Michigan, teaching a group of fifth grade students (scroll down to the BlueStream button).  The eye-opener: never once does she say that an answer is right or wrong…she just facilitates the discussion so that the students make that determination.  Recently, I came across David Cox’s post on Creating a Culture of Questions and was amazed at how similar his description was to how I tried my best to run class last year.  If anyone has other suggestions for resources on this kind of thing, particularly facilitating mathematical discourse, I would love to hear from you.

3. Continue having learners create tutorial videos

Last year we received some grants to have learners create tutorial videos (similar to Khan Academy only in the sense that they are videos – the value was in their creation, not their use).  I saw many benefits in this process, but I made soooo many mistakes in its implementation.  In fact, I have yet to upload any videos because I’m still deciding if I should – here is where they would go.  The problem: I did not build in enough time for drafting, revision, reflection, and checkpoints.  In other words, the final products ended up so full of mistakes that I may not be able to post them at all.  Big lesson learned.

4. Provide a video alternative to practice (aka homework)

As I reflected on what went wrong with the tutorial videos, I thought that it would have helped if I’d had groups create a simple video of themselves explaining the concept on a whiteboard.  Then I thought, why not do that for a regular assignment?  I (and the learners) could get way more information about their understanding and their misconceptions from them explaining a single problem than I could from seeing thirty problems they’ve done on paper.  The only issue is I haven’t worked out exactly how to do it.  We have mini-whiteboards, and every learner has a laptop with a webcam…suggestions???

(As a side note, I’d be remiss not to mention that I don’t give homework very often, and only after its purpose is clear.  Just saying.)

5. Give feedback in a more timely manner

Another repeat goal.  I’ve got to get quicker at giving feedback to learners…I’m talking same day or next day.  The value of it can’t be overstated.  I think I’ll be able to do a better job this year; last year we were creating a new school – year two will be easier…right?

6. Continue to celebrate mistakes

I try to create an environment where it’s understood that we are all learning together, and that mistakes are another tool to help us learn.  Last year after handing back a practice assignment, we would celebrate our “favorite wrong answer” – an idea that I stole from Leah Alcala’s “My Favorite No”.

I also love the idea behind Michael Pershan’s page, Math Mistakes, and he also tweeted this over the summer:

I haven’t figured out the logistics yet, but I’d love to do this in my math class one way or another.  Again, any suggestions would be awesome.

7. Math Whiteboarding!

Having followed Frank Noschese’s blog, I’ve been interested in using whiteboards more effectively…but it didn’t make it onto my plate last year.  But now I’ve seen how Bowman Dickson has used whiteboarding for math, and Kelly O’Shea’s mistake game sounds fantastic.  I can’t wait to try these out.

8. Incorporate more student passion into my class

As I found out through writing and reflecting on this post on passions, this is not so easy.  But I think I’ve got some good ideas…we’ll see what happens.

So there you have it.  I’ll definitely update as the year progresses…

August 13, 2012 / Jeff d.

Random Problem Idea: The Perfect S’more

I was scrolling through Twitter a few days ago when I saw this tweet from John Scammell (@thescamdog):

It caught my eye because I’ve had the same problem before – all excited about these new, crazy marshmallows only to be disappointed with the results.

The Problem Idea(s)

You bought the giant marshmallows because you thought it would be cool; then you realized it “messes up the chocolate to marshmallow ratio.”  So, assuming that the “perfect s’more” consists of two graham cracker halves, one normal marshmallow, and three “pieces” of chocolate (or 1/4 of a bar)…how do we fix it with these jumbo things?

The perfect s’more and the imperfect s’more

At this point, there are so many different directions we could take this:

Option 1)  Do we just cut the marshmallow? Where do we cut it?

To me this is cheating because it wastes marshmallow!

Option 2)  Without cutting it, can we add graham cracker and chocolate and find just the right number to make the ratios work out correctly?  Taking the problem this direction makes it very similar to Dan Meyer’s “Nana’s Chocolate Milk” problem – which I’ve used in class.

Option 3)  On the jumbo bag, there’s this gem:

Turn the marshmallow sideways…brilliant!

They are doubling the graham cracker and doubling the chocolate.  How close are they?

Option 4)  Alright, why don’t we just send someone back to the store to get smaller marshmallows.  Uh oh…they got the wrong kind again!  Now how do we fix it?

I’ve made a huge mistake.

Option 5)  There is another consideration here: the perfect amount of marshmallow roasteyness (that’s a real word, right?).  Dan Anderson (@dandersod) brought this up:

Now the relationship between surface area and volume comes into play.  Assuming that the perfect roasteyness is a golden brown outer layer of the regular sized marshmallow, how can we get the same surface-area-to-volume ratio with the giant (or small) marshmallows?

We could also complicate things further by using my favorite s’more-making method: catch the outside on fire, blow it out, then take that charred layer off.  I then eat it that way, but couldn’t you now roast what’s left?  Alright, I think I’ve gone far enough with this.

Yep…used a lighter. It’s raining right now, do you expect me to start a campfire?

I’ve said it before: I like messy problems, and this one fits the bill.  There are certainly multiple – if not infinite – solutions to most of these questions.  And, you get to eat the results (I’m full now).  Enjoy.

August 4, 2012 / Jeff d.

Random Problem Idea: Exploding Water Jugs

Or: How to Never Buy Bags of Ice Again

I am a country boy; this is important for two reasons.  First, our well water is so terrible that we have to buy our drinking water.  (Although I suppose a true country boy would choke that rust-laden crap down – and like it).  Second, this country boy doesn’t go to the store to buy bags of ice.  Instead, I simply take the empty water jugs, refill them and put them in the deep freeze.  When I need some ice for the cooler, I take a jug, bash it up and hack it open with a hatchet, and bam!  The equivalent of a bag of crushed ice.

I’ve done this for a number of years, so I’ve developed a knack for filling the jug to just the right level so that it won’t explode when the water expands as it freezes.  Sometimes, though, I’ll screw it up – especially with an unforgiving jug like this one:

So the problem is this: after figuring out how much water expands when it freezes (I believe it’s about 9%), let’s use our knowledge of volume of various figures to find the magical level to fill a jug so it will be full, but won’t explode.

One of my favorite problems when we are learning about volume is loosely based on this Formative Assessment Lesson from the Shell Centre.  The students are given a series of glasses of various shapes, and they figure out which glass will hold the most liquid.  Next I have them guess which glass holds the most liquid when it is two-thirds full (by height), and figure that out as well.  This jug problem is similar – the shape of the various jugs can be thought of as compound figures; we already know their volume (or do we? See picture below).  Now the trick is to figure out the point at which it is about 91.75% full (to account for the 9% expansion).  Depending on the shape, it almost certainly isn’t 91.75% of its height.

What about those weird circular indentations on milk jugs – won’t they expand outward? Or any unfilled space they leave at the top?

At this point, the problem can be extended to nearly any container.  A bottle of dishsoap?  Let’s give it a shot.  That bottle of Mountain Dew?  Go for it.  And what about this strange thing?

It’s got some properties of a cylinder…doesn’t it?


I like messy problems, and this is definitely messy (and I’m talking about the math here).  What exactly is the shape of a milk jug?  Can we consider the lower portion to be a rectangular prism?  How do we account for the curves?  The handle?  Is the top close enough to a truncated pyramid?  Part of a sphere?

Any student can have input, any student can find a way to justify their solution, and these are the kinds of discussions that I love.  And the best part is that the 3rd Act, if you will, can be easily tested.  Each student can pick a container from home, provide calculations to justify where they are going to fill the bottle, then throw it in the freezer.  Hopefully they get it right and I don’t have a bunch of parent complaints about explosions in their freezers.

July 26, 2012 / Jeff d.

It wasn’t Twitter that convinced me!

Today as I was rifling through papers in my home office, I stumbled upon a note I had written to my wife dated March 11, 2010.  Yep, apparently I used to write my wife letters – this was pre-texting (we were late adopters).  In this letter was revealed an opinion I had on standardized testing.  The interesting thing: I had no idea that I held this opinion in my pre-Twitter days!

I’ve said before that Twitter has given me far more “professional development” than I could get nearly anywhere else.  But I’ve also worried that I follow too many of the same kinds of people and read too many of the same kinds of articles – that it is a form of echo chamber for me.  I’ve always tried to get my information from balanced sources, but that’s not easy to do with Twitter where there is a natural tendency to follow like-minded folks.

With that in mind, I could have sworn that my anti-standardized testing opinion originated through Twitter, not before it.  I won’t get into my opinion itself, because that’s not the point – see Joe Bower or Diane Ravitch or Alfie Kohn for that kind of thing.  All I will say is I believe there are far better ways to assess learning.  It’s just good to know that my opinions can actually be my own…though obviously solidified through my Twitter echo chamber.  How do you use Twitter – are you able to find balance?  Do you try?

July 21, 2012 / Jeff d.

Passion-based Learning – A Call for Help

I just got back from attending and facilitating sessions at New Tech Network’s Annual Conference, so my mind is swimming with thought.  I have dozens of takeaways, but one that keeps popping back into my mind as I try to unwind is the idea of injecting passion into our school and my classroom.

The conference literally started and ended with this idea.  The opening keynote was given by Dennis Littky, co-founder of Big Picture Learning, a network of great schools that use PBL.  These schools ask students from day one about their passions.  They help students find internships based on these passions starting freshman year.  Two students accompanied Mr. Littky, and their stories made it clear that it was these passions that drove them to love learning.

The conference ended with a series of Ignite talks.  All of them were inspiring, but it ended with Mike Kaechele’s amazing talk titled #standardizethat, which challenged politicians and ed reformers (and everyone else) to think about the things in education that should be standardized.  One of these is passions.

As I think about this now, I find it very difficult to find a way to infuse more passion into my math classroom.  Many projects were discussed at the conference that gave students the kind of choice needed to be able embed their passions into them.  But these projects often come from Social Studies classes (or really anything but math), where I imagine the thematic structure makes this easier to happen.  How can I do this in my math class?

I’d like to think that I can help my learners appreciate math for its ability to help us make meaning of the world around us, its usefulness in solving problems, and its inherent beauty.  But that’s not what I’m talking about here.  I’m talking about the learners having true choice to pursue any topic and letting their passion drive the project.

Here is the best I can come up with, and honestly I think it’s garbage.  That’s why I’m asking for help.

  • Learners begin thinking about what they are passionate about early in the year.  In fact, a better question that I heard from Dennis Littky might be “What makes you angry, and how can we fix it?”
  • Learners then research and prepare for a 5 minute presentation at some point throughout the year (I’m thinking like every Friday, for example).  Of course, the presentation time is being influenced right now because of the Ignites – I’m open to whatever.
  • [wherein the plan falls apart] Somehow, learners find a way to incorporate math into their research.  I’m picturing using linear regression to make a prediction, or something like that.

That’s it.  The plan is terrible.  The math is an add-on.  I need help.  How can I give learners the kind of choice and voice necessary to bring their passions into the classroom?  How can I help them make meaning of a real problem that they want to fix, but still ties into math?  And – I hate to say it – but how can I bring this kind of project to them without infringing on too much other learning time?  At best, this plan can create a spark.  At worst, it can be a complete waste of time.  What can I do to make it better?