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:

- “What’s your question?”
- “What’s your guess?”
- “What would a wrong answer look like?”
- “What information is important?”
- “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.

## 11 Comments

## Chris Robinson

October 18, 2012 - 4:46 pm -Can we give innate student interest a multiplier? For example, if a student is more interested in hay bales than waves, how much does that buy us if they get the choice of what problem to do? I’m guessing that their ladder of abstraction is more easily climbed the closer we can relate a task to their experiences and interests. Not easily accomplished, especially with large class sizes, but definitely something to chew on.

## Geoff

October 18, 2012 - 5:05 pm -This is great stuff. I definitely had the ladder of abstraction in the back of my mind as I was thinking about “real worldliness” and took to twitter. This is a fantastic re-framing of the question. One that I’ll be sure to share as we continue to explore this “authenticity” question, which, as you indicate, may very well be a bit of a red herring.

What I like about this framework is this: I’m fearful of saying “it’s ok if your tasks aren’t concrete” because that tends to let teachers off the hook. I say “hey! let’s use math creatively in fun and interesting ways of exploring shapes!” and colleagues hear “it’s ok if I keep having kids solve 25 equations”. This better parses the difference between alleged authenticity in a task versus authenticity in the student behavior.

## Philip Seris

October 18, 2012 - 5:34 pm -I am glad that you provide a framework for making these abstractions into something that they can guess at and contribute to, because as I see it, students do not just want to feel dumb, deep down, they (like us) want to have something to contribute to a conversation. Starting that conversation with something that they do not need to know “the math” behind gives them something that they can talk about. I am constantly working on my own classes, knowing that I can do this really well with many tasks in some of my classes, but very poorly in other tasks in other classes. It seems that the higher the level of math, the more the abstraction gets away from something I feel I can make “concreteable”, for lack of a better term, and I am in constant reflection as to whether it is me, the math or the curriculum that need the overhaul. I am glad that you do what you do and empower me and so many others to see a vision and work toward a collective uneasiness in the status quo to the end that something better is created that will ultimately benefit the kids.

## blaw0013

October 18, 2012 - 7:33 pm -Very interesting post–where you’ve gone with this idea of what is concrete / abstract. My thoughts after first read: it seems as though you may not define this quality of concrete or abstractness as belonging to the task itself, rather it is what the student makes of the task. I took this from the section around the hay bales… But maybe it speaks to the closeness the student feels to the task, what Turkle and Papert wrote about in “Epistemological Pluralism.” However, when you return to placing sample tasks into your grid, it seems you’ve removed the student’s experiencing of the task from the decision making again… Is that allowable? Maybe unavoidable for writing curriculum. But if unavoidable, is there value to determining the LOA of tasks?

I think so… In comparison to these Learning Trajectories I hear Jere Confrey insert into the CCSS conversations.

## Belinda Thompson

October 18, 2012 - 7:55 pm -I think this is related to Blaw’s comment: Each task has a certain(?) amount of potential. But we should alos somehow consider the interaction of the teacher and student with the task. There are teachers who can make a silk purse out of a sow’s ear by taking a not-so-great task and making it great. I’m convinced there are teachers who do this every day with the curricular materials they have access to. Dan is obviously an example of this. He notices the structure and intent of tasks in the upper right quadrant and rewrites them to be somewhere more interesting Then there are those teachers who do the reverse by taking a task with lots of potential and moving it to a different quadrant. The 1999 TIMSS video study found that US teachers started with basically the same proportion of tasks with good potential as high-achieving countries, yet US teachers were not able to maintain the potential. Have you thought about what happens when the task goes live? Does it stay in the initial quadrant or does it get shifted?

## Thomas

October 19, 2012 - 11:25 am -This idea is very similar to Dowling’s domains of practice. Have you ever read this Dan?

## Bryan Meyer

October 19, 2012 - 6:53 pm -Interesting post, Dan. I’m becoming more and more convinced that “real-worldliness” means very little (if not nothing) to students. Recently, in one of my classes I polled the students and 23/25 said that they would be happy doing our mathematical puzzles (similar to “Problems of the Week” from IMP) and nothing else. Ironically, these are the most challenging tasks we do and are close to zero on the “real-worldliness” scale.

This realization opens up a more interesting discussion, I think….one that you have started here on your blog. If it isn’t real-world context, what is it about a task that engages students in the type of rich mathematics/thinking that we know to be the true benefit of math education? My action research this year has led me to hypothesize that students enjoy tasks in which imposition from the teacher (in terms of expected content outcomes) is most minimal. They enjoy exploring, looking for patterns, making and testing conjectures, and pursuing individualized paths from an open/rich task. I continue to think that the less I dictate the path of their work, the more they enjoy it. The less I hold them accountable to some concept/idea, the more they feel a sense of agency and capability as a mathematician. The curriculum must be the product of THEIR work, not some plan of MINE. For me, this doesn’t mean we necessarily need to abandon a general outline of suggested standards but it does mean that we need to abandon this sense of accountability and “mastery” that (to me) are all false indicators of success and security.

I appreciate that you have created an ongoing dialogue for all of us about designing interesting tasks for students. You empower teachers to free themselves from crappy textbook problems (read: exercises) and create an opportunity for their students to become participants in doing mathematics again.

## Christine Lenghaus

October 19, 2012 - 9:54 pm -I teach students who are 12-15 at the moment. What I have learned over the past two years with this age group is what I have distilled into three words:

Do, Draw, Dream (aka concrete, 2D, abstract).

To be able to do abstract maths students need to experience it, before they will be able to represent it on paper and then as a process/formula. Of course they can go straight to abstract but I think we can safely say that most of our students won’t ‘get it’ this way. This means that if I really want to build a bridge for my students to get to me, I have to meet them where they are and bring them across – not yell from the other side ‘you should be here!’. We do lots of ‘do’ activities before giving them any recipe for something (ie this is how you multiply fractions). Another example with rename 327 at least 4 different ways allows them to show me they know: 327 ones, 32 tens & 7 ones, or 3 hundreds, 2 tens and 7 ones and then they must think and learn to rename eg 7 ones is 70 tenths because this is important when dividing or multiplying (not just carrying the number to the next column!) This is to build a robustness of their knowledge and understanding and allows them to be able to maths at a higher level later on with confidence.

## Dan Meyer

October 20, 2012 - 4:28 am -Brian Lawler:I’m admitting several times throughout the post that the horizontal axis is really, really subjective, but not so subjective it isn’t worth talking about broadly. I place Shaughnessy’s partitioned square task in the “abstract context” category, for example, even though for most people here it’s fairly concrete. It’s a useful frame for me but I try to keep its limits in mind.