Category: loa

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[LOA] The Real World Multiplier

Geoff Krall examines different treatments of “real world” math – the lame kind and the good kind – and concludes that the self-awareness of the task is important. He excuses preposterous applications of math if they’re aware they’re preposterous. This is interesting, but his horizontal axis kicks a serious question down the road:

How do we gauge the “real-worldiness” of a task? Whose world? Is that scale absolute? How is “weather” more real world than “blueprints”?

Nevertheless, Krall’s post is important because, for one, it’s always useful to have new ways of talking about old things. For another, his post usefully highlights our total bedwetting panic over the whole real world thing.

“When will we ever use this?” is a question that’s Kryptonite for a lot of math teachers. Some have managed to script out answers in advance along the lines of, “Math is PE for your brain,” or, “You never use history in your day-to-day life either,” or, “Next week on the test.” But the fact that they’ve prepped themselves for an inevitable attack indicates a serious issue that needs more exploration.

So let me sketch out a different way of thinking about “real world” math. First, I’m convinced that the adjectives “real” and “fake” obscure a lot more than they reveal. They tap into an emotion that many of us intuitively understand but they aren’t persuasive to those who don’t. I’m going to swap out “real” and “fake” for “concrete” and “abstract,” which will be a little more helpful.

Here are two ways to think about the “abstractness” of mathematics. There’s what the context is and what you do with it. Let’s put those on two axes and watch what happens.

Math teachers pick away at the horizontal axis relentlessly, seeking newer, realer contexts for the same old tasks, but most of the gold is in the vertical axis.

One reason for this is that different things are more and less concrete to different populations. Concreteness is subjective. Teachers in Kansas were much less interested in measuring Garrett McNamara’s big wave ride than teachers in Honolulu. Teachers in Grand Forks were much more perplexed by these hay bales than teachers in urban Atlanta.

The other reason we should focus less on the concreteness of the context and more on the concreteness of the task is, as Bryan Meyer succinctly put it, “Kids don’t like feeling dumb.” Working at abstract levels without having worked at the concrete levels beneath them is like starting out lifting enormous weights without having worked up from smaller ones. It doesn’t matter if the weights are barbells, sand bags, or jugs of water. You’ll still feel helpless and small.

Our goal, of course, is that students will eventually work at higher and higher levels of abstraction. That’s where much of math’s power lives. But that doesn’t mean we should start there.

Let’s look at the four quadrants.

Abstract contexts with abstract tasks.

We can argue whether or not this context [pdf] is concrete or abstract. To me this context is concrete. Squares and diagonals and line segments are concrete to me, but I understand that this is what we often mean when we call a context “abstract.”

It’s easier for me to argue that the task – what you do with the context – is more abstract than it could be. Important features have already been highlighted and named. That’s abstraction. That’s work the student should participate in. Instead, we’ve started at a heady place, one that’s bound to make some students feel helpless and small.

Abstract contexts with concrete tasks.

Math teachers grossly undervalue these tasks.

Take the same task from the previous quadrant. Remove the labels. Take away the names. Students decide what information is important and what to name it. They get to guess. Estimation is a concrete task – something you do while just poking at the surface of a context – one that students don’t experience often enough in math class.

Concrete contexts with abstract tasks.

Math teachers grossly overvalue these tasks. Math teachers are eager for new contexts, new reasons for students to evaluate y = 2x + 4 for x = 50 (for example). The student asks where she’ll use this in real life so the teacher panics and swaps in another context. iPads. Basketballs. Fast food. Anything. Barbells. Sand bags. Jugs of water. It doesn’t matter. The trouble is that evaluating y = 2x + 4 for x = 50 is an abstract task. The abstract equation y = 2x + 4 came from somewhere and that place has been hidden from students. It doesn’t matter that the context is concrete.

What’s important here? Why is a linear equation the best representation of that important stuff? What do we do with that representation?

These are concrete questions the students might need more experience answering before we move onto that abstraction.

Here’s another example.

Money may be a concrete context but this task (from COMAP) is already abstract. The important information (the principle, the duration of the bond, the interest rate) has already been abstracted. It’s already been represented as a table.

Concrete contexts with concrete tasks.

Don’t throw the task away. Just table it for a second. Ask students first, “If I put $100 in a savings account and walk away for 30 years, what will I find there when I get back?”

Students have a chance to guess here. Save those guesses and credit the closest guessers later. Some may say, “$100,” offering us an quick formative assessment of their understanding of savings accounts. They’ll have to decide what information is important and where to get it, like the interest rate at a local savings bank.

After they’ve participated in that abstraction, they’ll be much better prepared for COMAP’s abstract task.

My scientific evaluation.

My scientific evaluation is that concrete contexts (what it is) buy you a 2x multiplier on student engagement while concrete tasks (what you do with it) buy you a 5x multiplier. Concrete contexts with concrete tasks? You know how to multiply.

So take something that’s concrete to your students and give them concrete tasks before you give them abstract tasks:

  1. “What’s your question?”
  2. “What’s your guess?”
  3. “What would a wrong answer look like?”
  4. “What information is important?”
  5. “That’s a pile of information there. How should we represent it?”

Et cetera.

I’ve been exploring that kind of task for awhile now but I don’t think the “concreteness” or “realness” of the context matters anywhere near as much as the fact that those tasks all start with guessing and other concrete tasks.

If students are working on tasks that don’t make them feel stupid, tasks that make them participants in an abstract process rather than subjects of it, the “real-worldiness” issue all but evaporates.

Related: Bet On The Ladder, Not On Context; Cornered By The Real World.

[LOA] They Don’t Know Their Own Power

I was at South Dakota State University last week and I asked some future math teachers to define the word “abstract” in a sentence. All of them defined it as an adjective, not a verb. They were more aware of “abstract” as something you are, not something you do.

  • A thought or idea that cannot be made tangible or concrete.
  • Abstract is something that is different, non mainstream, and requires higher level thinking.
  • Anything that is out of the ordinary or requires creative thought.
  • A concept or idea that is not easily or not able to be put into concrete or physical terms.
  • Beyond the logical ways of thinking about problems and ideas.
  • Not concrete. Imaginary. Out of the box thinking.

John Mason, in a great piece called “Mathematical Abstraction as the Result of a Delicate Shift of Attention“:

When the shift occurs, it is hardly noticeable and, to a mathematician, it seems the most natural and obvious movement imaginable. Consequently it fails to attract the expert’s attention. When the shift does not occur, it blocks progress and makes the student feel out of touch and excluded, a mere observer in a peculiar ritual.

If they don’t understand their own power, how will their students?

BTW: Also great. Frorer, et al:

… we rarely find [abstraction] explicitly discussed let alone defined. You can pick up a book entitled Abstract Algebra and not find a real discussion of abstraction as a process, or of abstractions as objects …

[3ACTS] Taco Cart

This task is one possible response to this week’s check for understanding. It was a pile of fun to produce.

Release Notes

Real to me. My wife and I were on a beach recently and found ourselves in this math problem. This happens to every math teacher, I’m sure. We use our own product. We employ mathematical reasoning in our own lives in obvious and subtle ways. I’ve tried to discipline myself not to miss those moments, to instead write them down, photograph them, and turn them into a task where students experience the same dilemma my wife and I did.

Google Maps. The game here is to screenshot a bunch of tiles from Google Maps, align and stitch them together in Photoshop, and then fly around that large image in AfterEffects.

Use appropriate tools strategically. The sequels aren’t optional here. One sequel suggests that the cart will start moving towards you and asks “at what location will both paths take the same time?” The other asks for an even faster path than either of the two originally posed.

In both cases, I enjoyed setting up and solving the algebraic models.

But as I contemplated solving one equation and finding the minimum of another, symbolic manipulation never occurred to me. Without any teacherly presence hovering over me, nagging me to rationalize my roots, the most obvious, practical solution was Wolfram Alpha – no contest.

A teacher at a workshop pulled off a similar move this week and felt embarrassed. He said he had “cheated.” Tools like WolframAlpha require us to come up with a more modern definition of “cheating.” (And of “math” for that matter.)

The ladder of abstraction.

Referring back to the check for understanding, here are ways the original task had already been abstracted:

  • the dog and the ball are represented by points; their dogness and ballness have been abstracted away,
  • very little of the illustration looks like the scene it describes, for that matter; the water and sand are the same color; the image of a dog swimming after a ball has been turned into the remark “1 m/s in water,”
  • points have already been named and labeled,
  • important information has already been identified and given,
  • auxiliary line segments have already been drawn; the segments AB and BC and DC don’t actually exist when the dog is running to fetch the ball; they have been abstracted from the context later.

My version of the task starts lower on the ladder. You see the sand and the sidewalk. You see what it looks like to walk in each. They aren’t abstracted into numerical speeds until the second act of the problem, after your class has discussed the matter. I do draw a triangle on the video, which is a kind of abstraction. I didn’t see any way around it, though.

BTW. Andrew Stadel also has a nice task involving the Pythagorean Theorem and rates.

[LOA] Check For Understanding

Adapted from the May 2012 issue of Mathematics Teacher:

A dog is running to fetch a ball thrown in the water. Point A is the dog’s starting point, point B is the location of the ball in the water, and point D can vary. Given that the dog’s rate of swimming is 1 meter per second and its rate of running is 4 meters per second, determine where point D should be located to minimize the time spent fetching the ball.

Some questions to consider here:

  1. In what ways has this context already been abstracted?
  2. Can you de-abstract (recontextualize? concretize?) the context? Describe a task that would allow students to learn about the process of abstraction rather than just encounter its result.

[LOA] Lies We Tell Ourselves

  • The number of party guests increases according to the function g(t) = 2t + 4, where t is the number of hours after the party started and g is the number of guests.
  • The number of iPads sold increases according to the function s(t) = 2t + 4, where t is the number of weeks after the iPad went on sale and s is the number of iPads sold in millions.
  • The number of points the team scores increases according to the function p(t) = 2t + 4, where t is the number of minutes after halftime and p is the number of points scored.

Party guests. iPads. Points.

They’re all the same to the student who doesn’t understand abstraction, the process by which we turn those contexts into words and symbols. The idea that any one of those contexts will engage that student any more than another is a fiction.

BTW. Clarifying: the issue at hand isn’t that these three problems are simplistic or false abstractions of a context. It’s that they start at a high-level of abstraction. (This isn’t a revisitation of pseudocontext, in other words.)

2012 Sep 27. Nathan Kraft points us to some research that says, “This kind of superficial personalization, indeed, increases engagement and achievement.” So I may have overstated my case considerably. The point of this ladder of abstraction series, though, is that investments in making abstraction more explicit are way more worth our while, not that other investments aren’t important also.