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.

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 …

… then it asks for a prediction about the graph.

It asks students to calculate the instantaneous rate of change …

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

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!

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.

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.

tl;dr version

Currently, online math websites comprise video lectures and machine-scored exercises.

For several different reasons, online math websites should add an introductory challenge that activates a student’s intuition and intellectual need. The video lecture should then be directed at satisfying that particular intellectual need.

Here’s an example. Let’s make this happen.

tl version

Online math sites are quickly defining math down to a) watching lecture videos and b) completing machine-scored exercises. I’m not going to re-litigate whether or not that definition of mathematics is as good as what we find in the best classrooms in the highest-performing countries. (It isn’t.) Instead, I’m going to take this online model for granted and ask how we can make it better.

What should we improve? It isn’t the lectures.

For some time there, I was meeting with founders who were pitching their startups as “Khan Academy plus [x]” where x was anything from better graphics, better lesson scripting, a face on the screen, or multiple choice questions embedded in the video. (Here’s basically the entire set of [x] at once.) I don’t believe there’s much value to add there. The Mathalicious lecture videos are beautifully shot. TED-Ed pairs their lecturers with world-class animators. Woodie Flowers wants to see Katy Perry and Morgan Freeman narrate these videos (I think he’s at least half serious) and my suspicion is that we have reached a point of diminishing returns on the efficacy of lecture videos. Once we passed a certain point of coherence and clarity, watching Drake rap over a combinatorics lecture animated by the Pixar team just isn’t adding a helluva lot. If math were only about clear and coherent lectures, we could probably close up shop here in 2012. Thankfully, there’s more interesting work to be done.

So what should we improve? It probably isn’t the exercises either.

The machine learning crowd seems very impressed by the millions of rows in their databases which represent the clickstream of hundreds of thousands of users. That clickstream can tell a teacher how many hints the learner requested, how long she spent on a given problem, whether she’s more apt to score well on machine-scored exercises in the morning or evening. But what the learner and her teacher would really like to know is what don’t I understand here? And machine learning has added very little to our understanding of that question. So there’s certainly value to be added there but I’m pessimistic that machines are in any position right now to evaluate a written mathematical assessment at anywhere near the skill of a trained human.

So what should we improve? We should improve what happens before the lecture.

Currently, the online math experience begins with a lecture. The implicit assumption is that students need to be talked at for awhile before they can do anything meaningful. Not only is that untrue but it results in bored learners and poor learning.

Dan Schwartz, a cognitive psychologist at Stanford University, prefaced student lectures with a particular challenge [pdf]. He asked students to do something (to select the best pitching machine from these four) not just to watch someone else do something. Those students then received a lecture explaining and formalizing what they had just done. Those students scored higher on a posttest than students who were pushed straight into the lecture without the introductory challenge.

I’ll show you an example of how this could work online. Head to this website and play through.

Let me explain what I’m trying to do there. First, any student who knows or can intuit the definition of “midpoint” can attempt that opening activity. It’s an extremely low bar to clear. The lesson will ultimately be about the midpoint formula but we haven’t bothered the student with a coordinate plane, grid lines, coordinate pairs, or auxiliary lines yet. Save it. Keep this low-key for a moment.

Once the student guesses, she sees how her classmates guessed, which queues everyone to wonder, “Who guessed closest?”

We’ve provoked the student’s intellectual need and set her up with the kind of introductory challenge that prepares her for a future lecture.

So we move into the lecture video, which has several goals:

1. It references the introductory challenge explicitly. The point of the lecture is to bring some resolution to the conflict we posed in the introduction: “Who guessed closest?”
2. It offers a conceptual explanation of the midpoint formula, not just a recitation of procedures.
3. It explains very explicitly why we use abstractions like x-y pairs and a coordinate plane. This satisfies John Mason’s recommendation that we become much more explicit about the process of abstraction.

After the lecture, the student sees the original problem, now with x-y pairs and a coordinate plane. No longer does she simply guess, aim, and click. She calculates. There’s are blanks for the answer now. We have formalized the informal.

The student calculates the answer and finds out how close she was. We should also throw some love on the closest guesser who may be a student who doesn’t usually get a lot of love in math class.

After that resolution, we ask students to practice their skills, but not just on automatically generated clones of the same problem template. We give them the midpoint and ask them to work backwards to one of the original points. That’s essential if you want me to have confidence in your assessment of my student as a “master” of the midpoint formula.

That’s it. An intuitive challenge that precedes a lecture video that explains how to resolve the challenge. That’ll result in more engaged learners and better learning.

Other examples?

• Ask every student to guess how long it’ll take to fill up the water tank before you explain to them how to find the volume of a prism. (See: lots of other examples just like that.)
• Ask every student to draw a triangle with given constraints before you explain why those constraints result in the same triangle.
• Ask every student to try to draw a line that’s parallel to another given line before you explain to them how you can determine whether or not two lines are parallel.
• Ask every student to guess the age of an individual before you explain the definition of absolute value and use it to figure out who guessed closest.
• Ask every student to take and submit a photo of stairs before you show your own photo and explain how we can figure out which is steepest.
• Ask every student to write down two numbers that add up to five before you explain why our pairs all seem to show up on the same line.

And on and on and on. There isn’t a recipe for these challenge but I know two things about all of them:

1. Math teachers have a stronger knack for creating these challenges than people who haven’t spent years fielding the question, “Why am I learning this?” fourteen times a day.
2. These challenges are more fun when they’re social. It’s one thing to see my own guess at the midpoint. It’s another thing entirely to see all my classmate’s guesses next to mine. We need the Internet to facilitate that quick, cool social interaction. It just isn’t possible with bricks and mortar alone.

Current online math websites have managed to scale up the aspects of decades-old math learning that few of us remember fondly. We can tinker around the edges of those lectures and exercises, adding a constructed response item here or a Morgan Freeman narration track there. Or we can try something transformative, something that draws from the best of math education research, something that takes advantage of the Internet, and makes math social.

BTW:

1. I made that site for my final project in Patrick Young’s summer Front-end Programming course at Stanford, which, as I mentioned previously, was a pile of fun. If I can make that site in a couple weeks with a thimbleful of programming knowledge, I’m eager to find out what your team can do with its acres of talent and piles of VC funding or non-profit donations.
2. This isn’t real-world math. I thought initially to pull in some tiles from Google Maps and set up a scenario where the student had to place a helicopter pad exactly between two cities. I don’t think it matters. Students ask “Why am I learning this?” because they feel stupid and small, not because they want you to force a context onto the mathematics. I’m trying to demonstrate that here. Everyone can click on a guess. No one feels stupid and small. Any context would be beside the point.
3. Cost-benefit analysis. Too often we apply a benefit-benefit analysis to edtech. But there are clear costs to the model I’m suggesting here even apart from the cost of the technology itself. There were at least five different moments over that five-minute lecture where I wanted to stop, pose a question, or have students work for awhile. We lose that here. I acknowledge those costs. We may still come out ahead on benefits if we can scale this up cheaply. “Pretty good” times millions of students may outweigh “great” times thirty.

2012 Nov 7.

A couple of useful tweets.

@ddmeyer #5 BEST GUESSER right here suckas wooooo

— josh g. (@joshgiesbrecht) November 7, 2012

I’m concerned the competitive vibe appeals only to males, but FWIW this is exactly the kind of reaction I’m trying to provoke.

@ddmeyer super neat (no surprise), but how is my alternative method of finding the midpoint encouraged?

— Avery Pickford (@woutgeo) November 7, 2012

It isn’t! It’s passively discouraged, which is a huge bummer. I can think of at least two ways a student might think about the midpoint and how to find it. You can take half a side of the right triangle and add it to the point with the smallest value or you can subtract that half from the point with the largest value. Those multiple methods and the discussion about their equivalence are to be prized and they’re lost in the video lecture format. They’re lost. That’s absolutely a cost and not a small one.

2012 Nov 12. Mr. Samson reminds me of the Eyeballing Game, which has been nothing if not an enormous inspiration for the work I’m doing here.

2012 Dec 7. David Lippman does this discussion a favor and creates an environment where the video pauses for student input. Discuss.

Featured Comment

Michael Serra, author of Discovering Geometry:

Curiosity and engagement will always trump “real world” applications. Games, puzzles, being surprised or caught off guard with something new and trying to find out why, these are big tools in our teacher toolbox.

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.

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 …

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.

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.

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.