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[LOA] Sam Shah’s Worksheet

Sam Shah’s been writing a lot of thoughtful material about calculus instruction lately, including this piece on related rates.

He includes a worksheet with that post and two items struck me. One, this is a pretty charming illustration of a rocketship climbing into space.

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Two, it asks students to climb down, not up, the ladder of abstraction. Check it out. It asks students to calculate a table of values for the rocket …

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… then it asks for a prediction about the graph.

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It asks students to calculate the instantaneous rate of change …

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… and then make a prediction about the instantaneous rate of change.

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Calculation is something you can do once you’ve ascended the ladder and turned a concrete situation (a rocketship lifting off) into an equation (h = 50t2). Prediction is something students can do while they mill around at the bottom of the ladder and it’ll make their eventual ascent up the ladder easier.

So I’m here, again, wondering what would happen if the worksheet had asked the prediction questions first and then moved on to calculation. Would the students be more successful? Would they have enjoyed the work more?

2014 Feb 24. Sam Shah updates us:

Yup. I introduced the rocket problem this year and I had each group make guesses for what the three graphs were going to look like. I loved hearing their conversation and their incorrect thinking for some of them. Tomorrow they are going to do the calculations and see what they got right and what they got wrong…

Thanks for pushing back in this good way. I’m glad I remembered to go back and reread this this year!

[LOA] Family Feud

Once you see the ladder of abstraction you can’t unsee it. Family Feud is a game show that’s played on the ladder. When Steve Harvey says, “Name something that gets passed around,” that’s a higher level of abstraction than all of the items listed: a joint and the collection plate at church.

Every other quality of the joint and collection plate is eliminated except their passed-around-ness.

Which game show works in the other direction, giving you lots of items and asking you to move one level of abstraction higher to the category that includes them?

2013 Mar 18. Andrew Stadel mentioned on Twitter that he gives students on level of Family Feud’s abstraction (the joint and the collection plate) and asks students what higher level of abstraction they all belong to (“things you pass around”). Great idea, easily adaptable to mathematics also.

[LOA] London Underground Maps

Here are two maps of the London underground railway, the first from 1928, the second from 1933.

1928

1933

I stipulated earlier that the act of abstraction requires a context (some raw material) and a question (a purpose for that raw material). These are two different abstractions of the same context. So what two different purposes do they serve? Rather, whom does each one serve?

BTW. If you’ll let me troll for a minute: aren’t we doing kids a disservice by emphasizing “multiple representations” rather than the “best representations?” Given that some abstractions are more valuable than others for different purposes, why do we ask for the holy quadrinity of texts, graphs, tables, and symbols on every problem rather than for a defense of the best of those representations for the job given?

BTW. I pulled those maps from Kramer’s 2007 essay, “Is Abstraction the Key to Computing?

2012 Nov 19. Christopher Danielson links up two examples of curricula (CMP) emphasizing “best representations” over “multiple representations.”

Featured Comments

Nik:

My intuition is the first (‘real’ scale, ‘real’ layout) is more useful to anyone who cares about how far it is between locations that are not connected, or how they relate to things not shown on the graph, while the second is for those who only care about connections.

Sean Wilkinson

I’m not sure that I agree that both maps are same-level abstractions of the real-world subway system. I would argue instead that the second map is an abstraction of the first.

In order to abstract away the lengths and shapes of the curves that connect the nodes, we need to have already interpreted the subway system as a network of curves and nodes – as the first map does – rather than as a three-dimensional physical structure.

Similarly, I would argue that graphs and tables-o’-values do not occupy the same rung; rather, a graph is an abstraction (and infinite extension) of a table-o’-values.

[LOA] What “The Literature” Says

If all of this ladder of abstraction material has seemed soft, fuzzy, and opinionated so far, I’ll offer up my summer project, A Literature Review of the Process and Product of Abstraction. Feel free to add comments or questions in the margins. I’ll try to get in there and chop it up with you. If you have written more than a handful of literature reviews yourself, I’d be grateful for your feedback on the format.

[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.