**Previously**

**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 â€“ better than most teachers and curriculum developers, certainly â€“Â *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?”

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?”

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).

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 â€“Â one single ramp â€“ 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 18Â° 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.

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).

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**

I have also been rolling this same problem in my head, but I didnâ€™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. Iâ€™ve thought about making a three-act type problem, but canâ€™t wrap my head around the best application. I was thinking of doing it for more advanced trig in precalculus: Hereâ€™s the ramp, hereâ€™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 â€œdockeryâ€ 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 Iâ€™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.

## 12 Comments

## Dawn Burgess

January 18, 2016 - 1:00 pm -Here’s a video I found that might be fun motivation to the problem: “Kami goes up the dock at low tide… and succeeds!”

https://www.youtube.com/watch?v=jpOwSIAG7Hs

Puppy cuteness, humor, and suspense.

## l hodge

January 18, 2016 - 4:33 pm -I like the choices as an opener. The animation is a nice visual for extreme cases – a short ramp falls into the water and a very long ramp appears to hover over the dock.

The animation is somewhat confusing. There is no sense of a rising/falling water level. Instead, the dock post appears to be a piston expanding and contracting. Perhaps a 2D view is better than 3D.

I have the same concerns that you expressed regarding simulations. In this case, I think many students are capable of creating their own simulation (like Jonathon’s) with a small amount of assistance. The beauty of doing this is that it is fairly easy to create a static diagram (low entry). And, there is instant feedback on what is working and not working as they build their models in desmos. Five lines for a crude desmos simulator.

## Julia Rowe

January 19, 2016 - 6:18 am -Dawn, I loved the videos! (I watched the going down the ramp too. :)) For my students in the middle of the continent it gives some perspective. I would love if you would take a couple of shots at high tide so that we can see the difference.

## Michael Caputo

January 19, 2016 - 10:38 am -I love this problem and Chris Lusto’s Boat Dock Generator. Having worked in a portable that needed a ramp, I looked into ADA requirements for gangways and they put a kink into the problem that would be good after initial computations are done. The long one may be the necessary one, though I’m not sure. From what I read, the gangway doesn’t have to be accessible at all water levels, but needs to have been built with the general intent of accessibility.

## Dan Meyer

January 19, 2016 - 11:53 am -l hodge:But building the simulation often requires mathematics that’s way outside the objective I originally intended for the simulator. Sinusoidal modeling in my case and the Pythagorean theorem in yours. It’s a different objective.

## Chester Draws

January 19, 2016 - 4:18 pm -Dan, why did the maximum steepness of 34% of the original turn into 18Â° in this version?

I thought the use of % for slope, which is actually much more common in normal use (for example roof pitches, road signs warning of steep roads), was one more way they had to convert world into Maths.

True they can solve the problem using Pythagoras and never get to angles, but is that a problem in itself?

## Dan Meyer

January 19, 2016 - 4:21 pm -Chester:That was my decision. I wanted the problem to focus on trigonometry rather than similar triangles & Pythagorus. I wouldn’t get worked up if someone wanted to change it back though.

## lhodge

January 19, 2016 - 5:59 pm -I have made the problem less focused. You have made it more focused and less difficult (through repetition and using â€œxâ€ as the ramp length rather than the track length). Time and focus are always issues and I might well not have them build their own simulators.

Pythagorean Theorem is really only A LITTLE outside the narrow objective of using right triangle trig ratios, no? The big bonus in building this simulator is that it forces one to think about what the coordinates actually mean/represent and how they are related – applicable in many, many settings. But, I would admit, well outside your objective.

## Chester Draws

January 20, 2016 - 3:25 pm -In this case, I think many students are capable of creating their own simulation (like Jonathonâ€™s) with a small amount of assistance.It’s not about whether they are capable or not.

We need the technology to be as invisible as possible when we teach Maths — every time we direct them to working inside the technology, we risk losing their focus entirely.

## Kaisa

January 22, 2016 - 7:52 am -Chester, I think it’s interesting that you say, “We need the technology to be as invisible as possible when we teach Maths.” At least in some projects, I take the opposite point of view — I push them into the technology so that they can implement their math ideas and test them against something external. When I was teaching pre-calc, for instance, many of my students thought that insisting on use of parentheses and order of operations was just a grumpy personal choice of mine because I’m a crazy lady who likes math. But then I had them do some computer modeling projects in Excel, and they discovered that while I still may be a crazy lady who likes math, these parentheses and order of operations are not a personal choice!

On the other hand, I was asking them to apply concepts they claimed they’d learned. It may be that the tech is distracting during the initial encounter with ideas, putting too much cognitive load on students. In the boat dock problem under discussion, I would not ask students to create a simulation until we’d gone through the problem on paper and discussed it thoroughly. Then creating a simulation is a different task, but it is one that allows students to test what they claimed they’ve learned.

None of those address the question of class time, though! One way I get around this is to allow students to do extra-credit projects like building a simulation (for my current class such activities can replace a low quiz grade, actually, but are not required).

## Chester Draws

January 23, 2016 - 3:09 pm -Kaisa: I think we might be talking at cross purposes: I’m not against using technology.

Excel is, largely invisible, once you learn the primary operations and don’t get into macros etc. In your example your students are thinking about how to get the results they want, and little else. As it should be.

Making little floating dock simulations is quite different. Then they are worrying about getting things to happen at the right speeds, with the right colours, whether the dock moves in tandem with the ramp. None of those are relevant to the learning of trigonometry, and positively distract from it. (There is a particular sort who will ace the simulation part but learn no Maths at all because they will brute force the answer.)

It is why I never use PowerPoint in class. If it is plain, then I may as well write it up on the board, because at least it is more easily adjusted. If it is fancy, then kids will spend at least some of their mental time focussing on the choice of colours, backgrounds and movements (and any errors in those will definitely distract them).

## Kaisa

January 25, 2016 - 11:58 am -Thank you for your answer, Chester. The elaboration on what you’re worried about is useful and I think you have a point.

In class (if I bring a class to the computer lab) I focus on activities that do push that focus on the math, as you write.

Out of class especially in a terminal mathematics class on the college level, I do admit that I allow assignments that may take considerable not-quite-math thinking: I let students do design projects that illustrate math or make math look nice, like knitting an abstract algebra sock (from Making Mathematics with Needlework edited by belcastro and Yackel) to writing an integration piece for clarinet to making an attractive computer program that does calculations relevant to the class content. I figure if they’re doing something pleasurable outside of class that they associate with mathematics and they *produce the appropriate mathematical parts correctly*, I am happy to have them do it! Teaching a terminal college math class, though, has some different pressures than teaching one math class out of many in a grades 5-12 context.