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[Makeover] Marine Ramp

Previously

Makeover Preview: Marine Ramp

The Task

The British Columbia Institute of Technology explains the origin of this task in their Building Better Math project:

Many high school students are avoiding math and cutting off pathways to exciting technical careers before they even know about them.

Their solution? More real world problems. Specifically, job world problems, problems that relate to “areas of geosciences, health care, engineering, renewable resources, oceanography, forensics, architecture and other industries.”

The BCIT has a very shiny coin here. They know better than anybody else – better than most teachers and curriculum developers, certainly – where our mathematical models are useful. I was blind to the mathematical modeling essential to the construction of a ramp at a boat dock, for example. BCIT helped me see it.

The BCIT knows that “trigonometry lives at the boat dock!” but without very careful curriculum development and very careful enactment by teachers, students will only experience the opportunity to calculate at the boat dock! This context offers many other opportunities to think mathematically besides calculation.

Here is one way to exploit them.

Show your students this video.

I begin so many of my applied tasks with video not because “kids love their YouTubes” but because multimedia allows me to de-mathematize a context that has already been heavily mathematized, leaving information, formulas, and other scaffolds to be revealed at an appropriate moment, and involve students in that process.

Ask your students, “What’s wrong with this scenario?”

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A: Without a ramp from pier to dock we can’t get on the boat.

Then ask students, “Which of these four ramps is best? Which is worst? Why?”

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A: The shortest one is lousy because it’s too steep to safely cross. The longest one is lousy because, while it’s safe enough to cross, it’s longer than it needs to be, which is wasteful. The best is probably one of the other two and there may be one that’s even better.

This is an important moment for student learning and for student interest.

Learning. There are cognitive gains to be had by showing students contrasting cases of the same question and asking them to invent a measure to describe them. Here is an example from Schwartz and Martin (2004).

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One group attempted to invent a measure and another group simply received instruction on the canonical measure. (“Variance” in this case.) Both groups then saw a worked example, after which the “invention” group outperformed the “tell-and-practice” group on a battery of measures. The invention activity helped students transfer in knowledge that prepared them to learn from explicit instruction later.

These multiple contrasting cases also allow me to ask students, “What measurements stay the same in every case? What measurements change?” That sets us up to assign variables to the changing measurements and quantities to the fixed measurements. The original problem offers only one case – one single ramp – offering us none of those cognitive gains.

Interest. As I summarized earlier, Sung-Il Kim’s research predicts that students will find this makeover more interesting than the original. Rather than explicitly stating the question and all of its relevant information, we’ve shown something incongruous and stated just enough that students will have to make the inferences that drive interest.

We should mathematize the context further now, assigning quantities to the measurements we know. (The distance the boat dock drops and the distance from the dock to the pier.) We should tell students the crucial constraint that the ramp can’t be any steeper than 18° as it meets the dock. We should model for students how a mathematician takes a context full of useless noise (eg. the color of the water, the shape of the hills) and draws a new version that includes only the useful details.

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The problem is now where we started, fully mathematized. The goal of our previous work was to expand student access to the mathematics and also broaden that mathematics to include more verbs than just “calculate.”

Let’s not stop there. Let’s head to Chris Lusto’s Boat Dock Generator (source code).

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This allows us to extend the existing problem. Hit the refresh button and get a new boat dock. Another one. And another one. Can students turn their one correct answer into a method for quickly calculating the best ramp length for any boat dock? Can they write it in algebraic language?

Concluding Remarks

I realize the new problem is more difficult to implement than the old. This new problem requires the teacher to involve herself in the posing of the problem and not just the assignment of the problem. It’s relatively easy to say to students, “Head over to this link and do the problem. I’ll be around to help if you need it.” It’s rather more difficult to embed yourself in that problem, to see yourself as an agent in the posing of that problem and the development of its question, even if the upside is better learning and more interest. This makeover is high reward at a high cost. At the moment, the reward interests me more than the cost.

You can download the problem at 101questions, but my main intent here wasn’t to create a problem we could use in the classroom. The point of a math problem isn’t just to get an answer, it’s to learn about math. And in the same way, the point of a math problem makeover isn’t just to get a better math problem, it’s to learn about learning.

What You Recommended

Dawn Burgess:

I have also been rolling this same problem in my head, but I didn’t know about the Vancouver version. I teach on an island in Maine, where the tide swings are larger, and these kinds of contraptions are everywhere. I’ve thought about making a three-act type problem, but can’t wrap my head around the best application. I was thinking of doing it for more advanced trig in precalculus: Here’s the ramp, here’s the dock, and for what portion of the day will the ramp be usable? For walking up and down? For hauling a hand-truck? For a wheel-chair? How could you change it to make it usable for more of the day? How might the harbormaster foil your plans? This is a great problem for my context, because many of my less mathy students know more about harbor restrictions and practical “dockery” than I do.

Justin Brennan offers a word of caution about these job-world applications:

After spending 8 years as an engineer prior to teaching, I always felt that I’d include all kinds of stuff from my engineering life into teaching. However, now that I am slightly wiser and more humbled, that stuff is too specialized, only interesting to me and maybe 2 other kids on a good day.

I appreciate Justin’s testimony that “math + jobs = fun!” is too simple an equation. But rather than give up the “jobs” part altogether, I have attempted here to bring students into the job in a particular way. Not all job math problems are created equal, in other words.

Jonathan Newman made a simulator in Desmos. My concern with every simulator is that the person who made the simulator uses more math than the students do. Scaffolding questions around the simulator to simulate mathematical thought, as Jonathan does, is no small task.

I have been rolling the same math problem around in my head for the last two months. Here is a link and a PDF.

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“Obsessed” wouldn’t be too sharp a description. Not with the math, which isn’t more advanced than high school trigonometry. Rather with the problem itself, and the opportunities it offers students to think mathematically.

In its current form, those opportunities are limited. In its current form, the problem asks students to read given information (and a lot of it), recall a formula, and calculate the result. That’s important mathematical thinking but hardly the most important kind of mathematical thinking (a statement of opinion) and not the only kind of mathematical thinking the context offers us (a statement of fact). There are more mathematical opportunities, and more interesting ones, than the problem offers in its current form.

So change that! How would you makeover this problem and help students experience all those interesting opportunities to learn mathematics?

On Monday, I’ll offer my own thoughts, along with a collaboration with Chris Lusto.

Geoff Krall:

The crux of Problem-Based Learning is to elicit the right question from students that you, the teacher, are equipped to answer. This requires the teacher posing just the right problem to elicit just the right question that points to the right standard.

Our existing knowledge and schema determine what we wonder so kids wonder kid questions and math teachers wonder math teacher questions. Sometimes those sets of questions intersect, but they’re often dramatically disjoint.

Which makes Geoff’s “crux” a form of mind control, or maybe inception, which is impossible. Kids wonder so many wonderful and weird things. And even if that practice were possible, I don’t think it’s desirable, since it seems to deny student agency while pretending to grant it. And even if it were desirable, I wouldn’t have the first idea how to help myself or other teachers replicate it.

If PBL is to survive, it needs a different crux. Here are two possibilities, one bloggy and one researchy.

First, Brett Gilland:

[The point of math class is to] generate critical thought and discussion about mathematical schema that exist in the students minds. Draw out the contradictions, draw attention to the gaps in the structures, and you will help students to build sturdier, creatively connected, anti-fragile conceptual schema.

Second, Schwartz & Martin:

Production seems to help people let go of old interpretations and see new structures. We believe this early appreciation of new structure helps set the stage for understanding the explanations of experts and teachers – explanations that often presuppose the learner will transfer in the right interpretations to make sense of what they have to say. Of course, not just any productive experience will achieve this goal. It is important to shape children’s activities to help them discern the relevant mathematical features and to attempt to account these features (2004, p. 134).

Notice all the teacher moves in those last two quotes. They’re possible, desirable, and, importantly, replicable.

2016 Jan 12. Logan Mannix asks if I’m contradicting myself:

As a science teacher follower of your blog, I’m not sure I follow. Isn’t that what you are trying to do with many of your 3 act problems? Get a kid to ask questions like “is there an easier way to do this” or “what information do I need to know to solve this”?

I am interested in question-rich material that elicits lots of unstructured, informal mathematics that I can help students structure and formalize. But I never go into a classroom hoping that students will ask a certain question. I often ask them for their questions and at the end of lesson we’ll try to answer them, but there will come a moment when I pose a productive question.

The possibility of student learning needs to rely on something sturdier than “hope,” is what I’m saying.

2016 Jan 13. Geoff Krall writes a post in response, throwing my beloved Harel back at me. (My Kryptonite!) It’s helpful.

Marbleslides Madness!

Watch some students play our new Marbleslides activity.

In the first example, Maggie and Claire exemplify basically all of the mathematical practices and then some as they try and fail and try and succeed to set up their marbleslides.

In the second example, Mr. Bondley recorded his students’ free play levels, some of which were quite elaborate.

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As I mentioned on Twitter, I don’t want to overstate the matter, but I think we’re seeing a level of human creativity unknown since the time of da Vinci.

Start your own class!

2016 Jan 19. One Marbleslide, Five Function Families.

Show the following five sentences to one group of students:

  1. A newly-wed bride had made clam chowder soup for dinner and was waiting for her husband to come home.
  2. Although she was not an experienced cook she had put everything into making the soup.
  3. Finally, her husband came home, sat down to dinner and tried some of the soup.
  4. He was totally unappreciative of her efforts and even lost his temper about how bad it tasted.
  5. The poor woman swore she would never cook for her husband again.

Then show all those sentences except the fourth, italicized sentence to another identical group of students.

Which group of students will rate their passage as more interesting?

For Greg Ashman, advocate of explicit instruction, the question is either a) moot, because learning matters more than interest, or b) answered in favor of the explicit version. Greg has claimed that knowledge breeds competence and competence breeds interest.

I don’t disagree with that first claim, that disinterested learning is better than interested ignorance. (Mercifully, that’s a false choice.) But that second claim is too strong. It fails to imagine a student who is competent and disinterested simultaneously. It fails to imagine that the very process of generating competence could be the cause of disinterest. It fails to imagine PISA where some of the highest achieving countries look forward to math the least.

That second claim is also belied by the participants in Sung-Il Kim’s 1999 study who rated the implicit passage as more interesting than the explicit one and who fared no worse in a test of recall. Kim performed two follow-up experiments to determine why the implicit version was more interesting. Kim’s determination: incongruity and causal bridging inferences.

That fifth sentence surprises you without the context of the fourth (incongruity) and your brain starts working to understand its cause and connect the third sentence to the fifth (casual bridging inference).

Kim concludes that “stories are interesting to the extent that they force challenging but resolvable inferences on the reader” (p. 67).

So consider a design principle for your math classes or math curriculum:

“Ask students to make challenging but resolvable inferences before offering them those resolutions.”

Start with estimation and invention, both of which offer cognitive benefits over and above interest.

[via Daniel Willingham’s article on the brain’s bias towards stories, which you should read]

2015 Jan 11. John Golden attempts to map Willingham’s research summary onto mathematics instruction.

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