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When Will I Ever Use This?

Click through to read John Mason response to the age-old question.

I have used this response, or ones like it, for many years with teachers when studying mathematics courses at the Open University, and I have noticed that it is only when I feel I am lost, when I lose confidence, when I feel as though I have reached my limits, that I find myself asking “why am I doing this?”

Co-signed.

They don’t actually want to know. They’re tired of feeling stupid and small.

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Mr. C:

Why do we need to know this? 2 words: Robot Apocalypse!

Who will reprogram the machines? Whose calculations would you trust your life with? I was sent her from the future… to be your math teacher!

Matt E:

I still love Sam Otten’s exploration of this family of questions:

https://www.msu.edu/~ottensam/Otten_2011MT_reprint.pdf

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Easier said than done:

I asked my Twitter team to come up with an application of imaginary numbers to dolphins. My Twitter team did not disappoint:

In math education, the fields of handwriting recognition and adaptive feedback are stuck. Maybe they’re stuck because the technological problems they’re trying to solve are really, really hard. Or maybe they’re stuck because they need some crank with a blog to offer a positive vision for their future.

I can’t help with the technology. I can offer my favorite version of that future, though. Here is a picture of the present and the future of handwriting recognition and adaptive feedback, along with some explanation.

In the future, the computer will recognize my handwriting.

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Here I am trying hopelessly to get the computer to understand that I’m trying to write 24. This is low-hanging fruit. No one needs me to tell them that a system that recognizes my handwriting more often is better than a system that doesn’t.

But I don’t worry about a piece of paper recognizing my handwriting. If I’m worried about the computer recognizing my handwriting, that worry goes in the cost column.

In the future, I won’t have to learn to speak computer while I’m learning to speak math.

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In this instance, I’m learning to express myself mathematically – hard enough for a novice! – but I also have to learn to express myself in ways that the computer will understand. Even when the computer recognizes my numbers and letters, it doesn’t recognize the way I have arranged them.

Any middle school math teacher would recognize my syntax here. I’ll wager most would sob gratefully for my aligned operations. (Or that I bothered to show operations at all.) If the computer is confused by that syntax, that confusion goes in the cost column.

In the future, I’ll have the space to finish a complete mathematical thought.

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Here I am trying to finish a mathematical thought. I’m successful, but only barely. That same mathematical thought requires only a fraction of the space on a piece of paper that it requires on a tablet, where I always feel like I’m trying to write with a bratwurst. That difference in space goes in the cost column.

That’s a lot in the cost column, but lots of people eagerly accept those costs in other fields. Computer programmers, for example, eagerly learn to speak unnatural languages in unusual writing environments. They do that because the costs are dwarfed by the benefits.

What is the benefit here?

Proponents of these handwriting recognition systems often claim their benefit is feedback – the two-sigma improvement of a one-on-one human tutor at a fraction of the cost. But let’s look at the feedback they offer us and, just as we did for handwriting recognition, write a to-do list for the future.

In the future, I’ll have the time to finish a complete mathematical thought.

If you watch the video, you’ll notice the computer interrupts my thought process incessantly. If I pause to consider the expression I’m writing for more than a couple of seconds, the computer tries to convert it into mathematical notation. If it misconverts my handwriting, my mathematical train of thought derails and I’m thinking about notation instead.

Then I have to check every mathematical thought before I can write the next one. The computer tells me if that step is mathematically correct or not.

It offers too much feedback too quickly. A competent human tutor doesn’t do this. That tutor will interject if the student is catastrophically stuck or if the student is moving quickly on a long path in the wrong direction. Otherwise, the tutor will let the student work. Even if the student has made an error. That’s because a) the tutor gains more insight into the nature of the error as it propagates through the problem, and b) the student may realize the error on her own, which is great for her sense of agency and metacognition.

No ever got fired in edtech for promising immediate feedback, but in the future we’ll promise timely feedback instead.

In the future, computers will give me useful feedback on my work.

I have made a very common error in my application of the distributive property here.

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A competent human tutor would correct the error after the student finished her work, let her revise that work, and then help her learn the more efficient method of dividing by four first.

But the computer was never programmed to anticipate that anyone would use the distributive property, so its feedback only confuses me. It tells me, “Start over and go down an entirely different route.”

The computer’s feedback logic is brittle and inflexible, which teaches me the untruth that math is brittle and inflexible.

In the future, computers will do all of this for math that matters.

I’ve tried to demonstrate that we’re a long way from the computer tutors our students need, even when they’re solving equations, a highly structured skill that should be very friendly to computer tutoring. Some of the most interesting problems in K-12 mathematics are far less structured. Computers will need to help our students there also, just as their human tutors already do.

We want to believe our handwriting recognition and adaptive feedback systems result in something close to a competent human tutor. But competent tutors place little extraneous burden on a student’s mathematical thinking. They’re patient, insightful, and their help is timely. Next to a competent human tutor, our current computer tutors seem stuttering, imposing, and a little confused. But that’s the present, and the future is bright.

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Justin Brennan:

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.

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I spent an hour on The Kathleen Dunn Show on Wisconsin Public Radio earlier this week. I was disappointed we didn’t get around to my thoughts on #PackerNation and Steve Avery, but I enjoyed our conversation about math, education, and technology just the same. Even though Dunn admitted she’s uncomfortable with math, she was gracious enough to let me assign her a math problem. It was also my first time on a call-in show, and the callers did not disappoint. [Show link.]

2016 Jan 27. As long as I’m wearing my public relations fedora, EdSurge just posted my interview with Blake Montgomery.

[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.

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