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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?

July 16, 2012 / Jeff d.

Random Problem Idea: Butt Dialing 9-1-1

Okay, so not much thought went into this one, but it might work well as a starter to a unit on probability (it has zero engagement value if the students already know how to “calculate” it).

A co-worker told a story yesterday about how the cops showed up at his house because his young son had been playing with the phone and accidentally called 911.

The phone in my hotel room right now.

So what are the odds of dialing 911, in that sequence, on accident?  It could be very simple (don’t forget to include the * and # keys), or there could be more to it: wouldn’t you think the odds of accidentally hitting the same number twice in a row would be a little higher (an argument espoused by EmergentMath last night)?  And what if there are more buttons, like my hotel phone?

July 15, 2012 / Jeff d.

How Do We Grade? – Grading and Reporting in a PBL/PrBL Math Class

It isn’t possible to describe how I graded in my math classroom last year without giving a little more background about the New Tech model that we use.  Two things are important here.  First, for everything we do we use Project-Based Learning (PBL) and Problem-Based Learning (PrBL), and it is all cooperative (group work).  Second, we use an interesting grading system that puts “soft skills” – like communication and work ethic – explicitly into the gradebook.  I’ll tackle these two issues one at a time.


Assessing Mastery with Cooperative Learning:

So our school uses all PBL and PrBL, working through complex problems and projects in small groups.  So how are we supposed to assess individuals for content mastery?  Well, the only time we work individually is when we need to demonstrate content knowledge (i.e. a quiz or other form of assessment).

So what does that look like in my math classroom?

1. We work through a problem or project in groups (though we’ll often start the problems thinking about them individually).  After we have “solved” the problem, thoroughly discussed it, made all of our thinking visible, and presented it one way or another, the learners will get another similar problem (or a problem derived from the project) to work through by themselves.  This is where the content grade comes from, while the group time is graded on things like Work Ethic, Communication, Critical Thinking, and Collaboration.  I think of it as being similar to the Formative Assessment Lessons from the Shell Centre, which I consider to be good stuff.

2. To better assess the mathematical skills (think “solving linear systems” or “factoring trinomials”) we use short content quizzes.  The quizzes almost always have two or three concepts on them, spiraled back to give learners automatic retake opportunities.  (See this presentation from Dan Meyer for where this idea came from).  Of course, there are procedures in place for learners to get extra help and instruction, and to request a retake on their own, regardless of how long ago the concepts were discussed.

As far as what constituted proficiency, it varied from quiz to quiz.  Generally I would create the quiz (usually about four questions), then make a judgement call on what would let me know that they “got it.”  It could be that three were correct, and there was a small mistake on the fourth.  It could be that they needed all to be correct, or (rarely) two of the four would suffice.  Next year I plan to create a simple rubric for each quiz that lets the students know ahead of time what will be considered proficient (I regret not having done this last year).

3.  Last year I used BlueHarvest to help the learners keep track of the concepts they had mastered and to manage some feedback.  I used it primarily because it was cool looking – there were no grades attached.  But it did help shift conversations from “I need to get my grade up, what can I do for extra credit?” to “I need to learn how to solve systems by substitution” – which is always a joyous feeling as a teacher.

Here you can see BlueHarvest in action for one of our quarters.  The blue boxes indicate Proficient, while the yellow boxes mean they have not yet mastered that concept.  (No, not every student mastered every concept – don’t judge me, I’m still learning this whole teaching thing!).

Separating “Soft Skills” in the Gradebook

So as part of the New Tech Network, we have the ability in our gradebook to separate content from other attributes that we would like to measure.  For example, this past year Content Mastery was 50% of the grade, while five school-wide learning outcomes like Collaboration,Work Ethic, and Communication were 10% each.  See as an example (though obviously this isn’t a student at my school, just a stock pic):

This has the benefit both of measuring “21st Century Skills”, and allows us to separate content mastery from behavior (something I believe strongly in – read some Ken O’Connor, et. al., if you want more convincing).

So what does this look like in my math classroom?

1.  Homework (practice) – given only when necessary, and after its purpose has been made clear by working on the problem/project – is graded strictly on Work Ethic.  This is practice for the game, and I do not believe in counting it toward the content grade.

2.  “Participation” (in the form of class discussions, working in groups, etc.) is graded strictly on Collaboration.  Again, this should not affect their content grade since it says nothing about their mastery of the content.

3.  Critical Thinking is a difficult one to define and use schoolwide (something we are tackling as a staff this summer).  In my classroom it included things like making thinking visible, taking multiple approaches, giving proof, explaining work, revising when new knowledge is acquired, and risk-taking.  Again, these only went into their Critical Thinking grade and had no effect on their content grade.

4.  Writing and presentations are graded on Communication.  I can’t imagine docking a learner’s content grade for failing to use a complete sentence.  But I have no problem giving them the feedback to improve and including this in their separate Communication grade.

5.  And if it hasn’t been made clear yet, Content is Content – it reflects nothing but what the learner knows.

What would I do if I didn’t have this grading system?

Obviously not everyone has the benefit of being able to report grades this way.  I often think about what I would do if I were to go to a different school that graded the old way.

Barring district or department mandates, this is what I would change: nothing.

At this point, I can’t imagine polluting Content grades with things that clearly reflect Work Ethic.  I can’t imagine telling a learner that they only know 75% of the math when they scored 100% on their test but didn’t turn in their homework.  And I can’t imagine killing their Content grade for failure to use commas correctly in their writing.  I just can’t do it.  I would simply have to find a way to report it this way myself.

*Special thanks to my teaching partner last year, Kelley Watson, who shared in the creation of these ideas.

July 13, 2012 / Jeff d.

Let the Random Problem Ideas begin…

One of the ways that I want to use this blog is to throw problem ideas out there and get feedback.  I don’t want to make a big deal out of any of them, like putting them in immediately usable form, because if I require myself to spend inordinate amounts of time to publish them, this blog will die a very quick death.  So I will just toss ideas out there and see what happens.  A lot of them will suck, by the way.  Hey, I sound like EmergentMath right now.  I hope he won’t mind.

So here goes my first one.  I was at a baseball game yesterday and in the parking lot saw this vehicle with a dirty rear window:


So, simple enough, here’s the problem:

  1. How well-designed is the wiper system? (or what is the ratio of clean:dirty)?
  2. Design a better wiper system.

I could also take many pictures of different vehicles to give variety (I like problems to be messy and to not personally know any “answers” ahead of the students).

One other note: when I worked for Motorola developing automotive software, I drove a Mercedes E320 for the weekend, and it had a single-wiper system for the front windshield.  There could be something there too.

My own critique would be that it seems too easy for a full-on problem, but too hard for a bellringer.  It might not be engaging enough either.  But I’ll leave the majority of bashing to others…please.

[Also, I know that I have absolutely no following on this blog, and what I do have is probably more interested in SBG than math at this point…so feedback will be close to nil, but I have to start somewhere, right?]

July 7, 2012 / Jeff d.

Learn to Love Standards-Based Grading in 4 Easy Steps!

Alright, so maybe it’s not that simple.  But I love Standards-Based Grading.  Love it.  However, any time I try to explain to others why I love it, I can never seem to get my point across in a convincing manner.  So I’ve done a lot of reflecting on how I came to love it and this post will be an attempt to piece together a timeline of how my beliefs came about.  I’ve parsed it into four easy steps.  I know there has been a ton written about SBG, but this is just the path that I personally stumbled through, mostly by accident.

Step 1: Visit Dan Meyer’s blog.

I mentioned in my inaugural post that about a year ago I signed up for Twitter, and that’s where it all began.  One of the first people I followed was a math educator by the name of Dan Meyer (if you’re into math, I’m sure you’ve heard of him…he’s basically a math rock star).  At the time I was simply looking for cool math ideas, and believe me, there are plenty on his blog.  Then I stumbled on this post that describes how Dan assessed in his math class.  It included a mini-thesis that spelled things out in further detail.

The thing that clicked for me was his example of a gradebook that shows “Chapter 6 Test – 63%.”  Sixty-three percent of what?  Chapter 6 probably has six, seven, ten different concepts in it; how is the student or the teacher supposed to know exactly where they struggled?  Compare that to a gradebook that shows “Operating with Integers – 2/4; Solving Two-Step Equations – 4/4.”  Here the student and teacher knows that the student needs help with operating with integers, but dominated solving two-step equations.

I also found this presentation that helped explain it in a more visual way, and his Comprehensive Math Assessment Resource.

Step 2: Visit Shawn Cornally’s blog.

The next thing I came across was Shawn Cornally’s blog, ThinkThankThunk, and in particular this post describing his SBG manifesto.  He describes the thrill of having a student approach you with something like “I need to work on writing linear equations” rather than the previously ubiquitous and generic “How can I improve my grade?”  I remember the first time this happened to me as a teacher, and he was not exaggerating the thrill.

Step 3:  Read up.  (You know, like books and stuff).

At about this time I began a master’s course on assessment, so various texts entered the fray (although I continued to devour anything on grading that I could find via Twitter).  Some of the books were required and some I sought on my own.

  • Ahead of the Curve: The Power of Assessment to Transform Teaching and Learning – Douglas Reeves (ed.) – A compilation of goodness, especially the chapters on using assessments to improve teaching and learning (Guskey), assessment for learning (Stiggins), and tackling the grading dilemma (O’Connor).
  • Classroom Assessment & Grading that Work – Marzano – Although I’m not always the biggest Marzano fan, this book lays the foundation for standards-based grading.
  • Educational Leadership, November 2011 – Effective Grading Practices – This issue of Ed Leadership is fantastic (and recently back in print after being sold out).  I used the articles by Guskey, Wormeli, Marzano, O’Connor, et. al. in nearly every paper I wrote for the class and they gave me a solid base of understanding.
  • Elements of Grading: A Guide to Effective Practice – Reeves – This book emphasizes feedback, which is inarguably important to student learning.  He spells out how feedback should be accurate, fair, specific, and timely (I still struggle with the timeliness of my feedback – a goal for next year).

There are many more books and articles, but these were the most important for me.

Step 4:  Visit Frank Noschese’s blog.

Next I came across a comic that I first saw on Frank Noschese’s blog, Action-Reaction.  The comic laid out, in a very simple and understandable way, what is wrong with the way we often give grades.  His blog is awesome, by the way, and he has a whole section for Standards-Based Grading.

After all that, I was hooked and things get hazy.  Suffice to say there are plenty of other places to get great SBG knowledge dropped on you.  I’m actually a little embarrassed that it took me so long to find Matt Townsley’s blog, which is extremely comprehensive and pre-dates many others.  It’s my go-to now, and the first place I would start if I were to do it all over.  Actually, maybe not, because I like the path I took.

July 7, 2012 / Jeff d.

Well, I finally started blogging.

I’ve wanted to start blogging for quite some time now, and I think I have finally found a reason worthy of putting forth the effort. In about a week I will be presenting three sessions at the New Tech Annual Conference in Grand Rapids, Michigan and I want a more conspicuous place to post resources (i.e. I want them available to anyone, not just network conference goers). Plus, I think it’s time I get over my biggest hangup for starting a blog: not being able to think of a clever name (I’m still not happy with it, but life goes on).

As a little background, I am a high school math teacher (actually a facilitator, as we in the New Tech Network call ourselves). I’ve been teaching for six years, but this past year has been by far the most eventful, thought-provoking, and rewarding of my short career. Just over a year ago at New Tech’s New Schools Training a friend of mine and former principal of the school (@ptrkmkl) introduced me to Twitter. Not that I hadn’t heard of Twitter, but at the time I had no online presence whatsoever and preferred to keep it that way. But when he explained to me all the potential professional benefits, I signed up. I began following some awesome educators (more on that to come). Combine that with the fact that we were starting a New Tech high school, and a journey began for me that can be summed up as follows:

I have a lot of passions and they are changing and growing every day – which was another hangup for starting this blog. But I figure I’ll find my niche as the blog progresses.