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The internet has failed me.

In spite of following 150 creative math teachers on Twitter and subscribing to 750 creative math teacher blogs (including one blog that’s dedicated exclusively to creative math), I’m only now learning about Gordon Hamilton’s Unsolved K-12 Project. It’s creative. It’s math. It’s almost three years old!

Better late than never.

See, Hamilton convened a bunch of creative math types in Banff in November 2013 to a) select unsolved math problems and b) adapt them for use at every grade in K-12. Not a simple task, and I’m enormously impressed by their results. You can watch videos introducing the problems at this page or read about them in these slides.

Here are two of my favorites. (Click for larger.)

Grade 3: Graceful Tree Conjecture

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Grade 10: Imbedded Square

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These two problems have the capacity to develop fluency just as well as any worksheet or worksite. In working out their solutions, students will perform the same operation dozens of times – subtracting whole numbers in the third grade task and calculating slope and distance in the Cartesian plane in the tenth grade task. But these problems ask students to think strategically and systematically in addition to practicing efficiently and accurately. That’s no easy feat, but Hamilton and his team pulled it off thirteen times in a row.

Related:

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If you tell me you’re a fan of real-world math, I know almost nothing about what goes on in your classroom. That’s because there’s enormous variation within real-world tasks. Almost as much as there is between real and “fake” tasks. (I’ve written about this before.)

This summer we’ll interrogate that thesis. Every week I’ll post three versions of the same real-world task.

Please tell me: who wore it best?

  • In what ways are they different?
  • What do their differences say about their authors’ beliefs about students, learning, and math?
  • Would you make changes? Which and why?

We’ll begin with Barbie Bungee, a lesson which is as old as math teaching itself. (The earliest reference I found in an exhaustive #lazyweb search on Twitter was this 1993 Mathematics Teacher article. Thanks, Norma.) If you’ve never heard of it, here is a video summary from Teacher Channel.

Click each image for a larger image. Or click through for the PDF.

Version #1

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Version #2

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Version #3

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I previewed these problems on Twitter a week ago.

A number of people noticed that Version 3 asks for a lot of literacy in addition to numeracy. “Example 3 is too wordy for me and the students that I work most closely with,” said Bridget Dunbar.

I’m sympathetic. I was initially repelled by the dense text, but several educators I respect came along and noted that Version 3 leaves a lot of room for the teacher to develop the question along with students. Andrew Morrison said that “the structure of the activity is a lot more open ended than I expected based on the amount of text I initially saw.” Paul Jorgens used some of my favorite advice to support Version 3: “You can’t subtract but you can always add,” continuing to say that “the third one seems the easiest to start thin and add as necessary.”

“A thin start.” Great description.

A number of participants in the discussion said variations on, “It depends on the student.” That seems hard to falsify. Even if it’s true with these three worksheets, though, I don’t think that advice extends to any version of this task. Some are probably just bad.

For my part, I look at each version and try to imagine the verbs, the mental work students do. In each version of the task, the work becomes formal and operational very quickly. Version 1 has students measuring precisely in its first step. Version 2 has students graphing precisely in its first step. That’s important work but once the task has been formalized like that, it’s very difficult to ask students to do informal, imprecise work, which is just as important and often more interesting.

Like wondering what kinds of questions a bungee jump operator would wonder.

Like estimating how many rubber bands would be ideal for a given bungee jumping scenario. (Bridget Dunbar with the eagle eyes: “Version 2 misses estimation while Version 1 asks for it, but at the end of all of the directions.”)

Like abstracting the world of bungee jumping into a few manageable pieces of data which we can measure and track. (eg. The temperature outside probably doesn’t matter. The number of rubber bands probably does.)

Like sketching the relationship between rubber bands and fall height before graphing it.

It’s difficult to load all of those tasks onto the same piece or pieces of paper. Perhaps impossible, as later tasks will provide the answer to earlier tasks. My ideal Barbie Bungee task (and modeling task, in general) requires a dialog between teacher and students, with the teacher adding context, questions, and help, as the situation and students require it.

Watch Twitter for next week’s preview. You should find three versions of a task and play along at home.

Featured Comments

Chris Hunter:

I had a similar reaction to Version #3: all that text. Marc often asks “Who’s doing the math?” This, like “You can always add. You can’t subtract,” rattles around my head when designing or evaluating tasks. Version #3, for all that text, probably wore it best; in both Versions 1&2, the answer to Marc’s question is “The author.”

Elizabeth Raskin:

I wonder if the “step by step” worksheets that educators can be so fond of stems from the fear that students wouldn’t know what to do to solve the problem or from the idea that teachers have a vision for what the students’ output should look like. In versions 1 and 2, students will most likely have much cleaner products than those of version 3. I assume it’s some combination of both.

Michael Paul Goldenberg:

I wouldn’t say that it’s so much the case that “it depends on the student” as that it depends on the classroom culture. Teachers that have cultivated a culture of risk-taking, serious inquiry, and other habits of mind and practices that draw on the natural curiosity and need to know and understand that we all have (before it’s schooled out of us) are going to have enormous flexibility in designing or adapting tasks like this one so as to not wind up stifling individual thinking and productive struggle.

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Ann Shannon asks teachers to avoid “GPS-ing” their students:

When I talk about GPSing students in a mathematics class I am describing our tendency to tell students—step-by-step—how to arrive at the answer to a mathematics problem, just as a GPS device in a car tells us – step-by-step – how to arrive at some destination.

Shannon writes that when she used her GPS, “I usually arrived at my destination having learned little about my journey and with no overview of my entire route.”

True to the contested nature of education, we will now turn to someone who advocates exactly the opposite. Greg Ashman recommends novices learn new ideas and skills through explicit instruction, one facet of which is step-by-step worked examples. Ashman took up the GPS metaphor recently. He used his satellite navigation system in new environs and found himself able to re-create his route later without difficulty.

What can we do here? Shannon argues from intuition. Ashman’s study lacks a certain rigor. Luckily, researchers have actually studied what people learn and don’t learn when they use their GPS!

In a 2006 study, researchers compared two kinds of navigation. One set of participants used traditional, step-by-step GPS navigation to travel between two points in a zoo. Another group had to construct their route between those points using a map and then travel segments of that route from memory.

Afterwards, the researchers assessed the route knowledge and survey knowledge of their participants. Route knowledge helps people navigate between landmarks directly. Survey knowledge helps people understand spatial relationships between those landmarks and plan new routes. At the end of the study, the researchers found that map users had better survey knowledge than GPS users, which you might have expected, but map users outperformed the GPS users on measures of route knowledge as well.

So your GPS does an excellent job transporting you efficiently from one point to another, but a poor job helping you acquire the survey knowledge to understand the terrain and adapt to changes.

Similarly, our step-by-step instructions do an excellent job transporting students efficiently from a question to its answer, but a poor job helping them acquire the domain knowledge to understand the deep structure in a problem set and adapt old methods to new questions.

I’ll take that trade with my GPS, especially on a dull route that I travel infrequently, but that isn’t a good trade in the classroom.

The researchers explain their results from the perspective of active learning, arguing that travelers need to do something effortful and difficult while they learn in order to remember both route and survey knowledge. Designing learning for the right kind of effort and difficulty is one of the most interesting tasks in curriculum design. Too much effort and difficulty and you’ll see our travelers try to navigate a route without a GPS or a map. While blindfolded. But the GPS offers too little difficulty, with negative consequences for drivers and even worse ones for students.

2016 Jun 17. The two most common critiques of this post have been, one, that I have undervalued step-by-step instructions in math, and two, that this GPS study offers very few insights into math education. I respond to both critiques in this comment.

What does it take to ask students a question like this?

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A poker face? A bit of malice? Nitsa Movshovits-Hadar argues [pdf] that it requires only the ability to trick yourself into forgetting that you know every triangle has the same interior angle sum. “Suppose we do not know it,” she writes, which is easier said than done.

The premise of her article is that “… all school theorems, except possibly a very small number of them, possess a built-in surprise, and that by exploiting this surprise potential their learning can become an exciting experience of intellectual enterprise to the students.”

This is such a delightful paper – extremely readable and eminently practical. Without knowing me, Movshovits-Hadar took several lessons that I love, but which seemed to me totally disparate, and showed me how they connect, and how to replicate them. I’m pretty sure I was grinning like an idiot the whole way through this piece.

[via Danny Brown]

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Not easy for math teachers to do!

Kent Haines:

What if you asked two questions: which triangle has the longest perimeter and which triangle has the largest angle sum? It might clarify what can change in a triangle and what cannot. Also it hides the surprise better. If you teach via surprise consistently, kids start looking for the punchline.

Featured Comments

Jo:

Elementary may actually have an advantage here! We play these games all the time. Some favorites:

Draw me a two-sided quadrilateral
Draw me a triangle with three right angles (or three obtuse angles)
(We have a manipulative that consist of little plastic sticks that snap together to build things)–Build me a triangle with the red stick (6″), the purple stick (1″) and the green stick (2″ )

Once the whole class is convinced they can’t you can get at why and then writing a rule for it. There is nothing an 8 year old likes better than proving the teacher wrong.

Ethan Hall:

Theorems and formulae in textbooks should be marked with a “spoiler alert”.

I know. I know. Too early, right? But Ali Grace is a go-getter.

My contributions:

Help the rest of us out in the comments. What do you do on the first day of school?

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