Total 18 Posts

## Real Work v. Real World

“Make the problem about mobile phones. Kids love mobile phones.”

I’ve heard dozens of variations on that recommendation in my task design workshops. I heard it at Twitter Math Camp this summer. That statement measures tasks along one axis only: the realness of the world of the problem.

But teachers report time and again that these tasks don’t measurably move the needle on student engagement in challenging mathematics. They’re real world, so students are disarmed of their usual question, “When will I ever use this?” But the questions are still boring.

That’s because there is a second axis we focus on less. That axis looks at work. It looks at what students do.

That work can be real or fake also. The fake work is narrowly focused on precise, abstract, formal calculation. It’s necessary but it interests students less. It interests the world less also. Real work – interesting work, the sort of work students might like to do later in life – involves problem formulation and question development.

That plane looks like this:

We overrate student interest in doing fake work in the real world. We underrate student interest in doing real work in the fake world. There is so much gold in that top-left quadrant. There is much less gold than we think in the bottom-right.

BTW. I really dislike the term “real,” which is subjective beyond all belief. (eg. What’s “real” to a thirty-year-old white male math teacher and what’s real to his students often don’t correlate at all.) Feel free to swap in “concrete” and “abstract” in place of “real” and “fake.”

Related. Culture Beats Curriculum.

This is a series about “developing the question” in math class.

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I would add that tasks in the bottom-right quadrant, those designed with a “SIMS world” premise, provide less transfer to the abstract than teachers hope during the lesson design process. This becomes counter-productive when a seemingly “progressive” lesson doesn’t produce the intended result on tests, then we go back not only to square 1, but square -5.

I love this distinction between real world and real work, but I wonder about methods for incorporating feedback into real work problems. In my experience, students continue to look at most problems as “fake” so long as they depend on the teacher (or an answer key or even other students) to let them know which answers are better than others. We like to use tasks such as “Write algebraic functions for the percent intensity of red and green light, r=f(t) and g=f(t), to make the on-screen color box change smoothly from black to bright yellow in 10 seconds.” Adding the direct, immediate feedback of watching the colors change makes the task much more real and motivating.

## The Money Animal Marketplace Was The Most Fun I Had Doing Math This Summer

In my modeling workshops this summer, we first modeled the money duck, asking ourselves, what would be a fair price for some money buried inside a soap shaped like a duck? We learned how to use the probability distribution model and define its expected value. We developed the question of expected value before answering it.

Then the blogosphere’s intrepid Clayton Edwards extracted an answer from the manufacturers of the duck, which gave us all some resolution. For every lot of 300 ducks, the Virginia Candle Company includes one \$50, one \$20, one \$10, one \$5, and the rest are all \$1. That’s an expected value of \$1.27, netting them a neat \$9.72 profit per duck on average.

That’s a pretty favorable distribution:

They’re only able to get away with that distribution because competition in the animal-shaped cash-containing soap marketplace is pretty thin.

So after developing the question and answering the question, we then extended the question. I had every group decide on a) an animal, b) a distribution of cash, c) a price, and put all that on the front wall of the classroom – our marketplace. They submitted all of that information into a Google form also, along with their rationale for their distribution.

Then I told everybody they could buy any three animals they wanted. Or they could buy the same animal three times. (They couldn’t buy their own animals, though.) They wrote their names on each sheet to signal their purchase. Then they added that information to another Google form.

Given enough time, customers could presumably calculate the expected values of every product in the marketplace and make really informed decisions. But I only allowed a few minutes for the purchasing phase. This forced everyone to judge the distribution against price on the level of intuition only.

During the production and marketing phase, people were practicing with a purpose. Groups tweaked their probability distributions and recalculated expected value over and over again. The creativity of some groups blew my hair back. This one sticks out:

Look at the price! Look at the distribution! You’ll walk away a winner over half the time, a fact that their marketing department makes sure you don’t miss. And yet their expected profit is positive. Over time, they’ll bleed you dry. Sneaky Panda!

I took both spreadsheets and carved them up. Here is a graph of the number of customers a store had against how much they marked up their animal.

Look at that downward trend! Even though customers didn’t have enough time to calculate markup exactly, their intuition guided them fairly well. Question here: which point would you most like to be? (Realization here: a store’s profit is the area of the rectangle formed around the diagonal that runs from the origin to the store’s point. Sick.)

So in the mathematical world, because all the businesses had given themselves positive expected profit, the customers could all expect negative profit. The best purchase was no purchase. Javier won by losing the least. He was down only \$1.17 all told.

But in the real world, chance plays its hand also. I asked Twitter to help me rig up a simulator (thanks, Ben Hicks) and we found the actual profit. Deborah walked away with \$8.52 because she hit an outside chance just right.

Profit Penguin was the winning store for both expected and actual profit.

Their rationale:

Keep the concept simple and make winning \$10s and \$20s fairly regular to entice buyers. All bills – coins are for babies!

So there.

We’ve talked already about developing the question and answering the question. Daniel Willingham writes that we spend too little time on the former and too much time rushing to the latter. I illustrated those two phases previously. We could reasonably call this post: extending the question.

To extend a question, I find it generally helpful to a) flip a question around, swapping the knowns and unknowns, and b) ask students to create a question. I just hadn’t expected the combination of the two approaches to bear so much fruit.

I’ve probably left a lot of territory unexplored here. If you teach stats, you should double-team this one with the economics teacher and let me know how it goes.

This is a series about “developing the question” in math class.

## Developing The Question: Bike Dots

Let’s look at an example of developing the question versus rushing to the answer.

First, a video I made with the help of some workshop friends at Eanes ISD. They provided the video. I provided the tracking dots.

To develop the question you could do several things your textbook likely won’t. You could pause the video before the bicycle fades in and ask your students, “What do you think these points represent? Where are we?”

Once they see the bike you could then ask them to rank the dots from fastest to slowest.

It will likely be uncontroversial that A is the fastest. B and C are a bit of a mystery, though, loudly asking the question, “What do we mean by ‘fast’ anyway?” And D is a wild card.

I’m not looking for students to correctly invent the concepts of angular and linear velocity. They’ll likely need our help! I just need them to spend some time looking at the deep structure in these contrasting cases. That’ll prepare them for whatever explanation of linear versus angular velocity follows. The controversy will generate interest in that explanation.

Compare that to “rushing to the answer”:

How are you supposed to have a productive conversation about angular velocity without a) seeing motion or b) experiencing conflict?

See, we originally came up with these two different definitions of velocity (linear and angular) in order to resolve a conflict. We’ve lost that conflict in these textbook excerpts. They fail to develop the question and instead rush straight to the answer.

BTW. Would you do us all a favor? Show that video to your students and ask them to fill out this survey.

This is a series about “developing the question” in math class.

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Bob Lochel, with a great activity that helps students feel the difference between angular and linear velocity:

I keep telling myself that I would love to try this activity with 50 kids on the football field, or even have kids consider the speed needed to make it happen.

Without some physical activity, some sense of the motion and what it is that is actually changing, then the problems become nothing more than plug and chug experiences.

## Developing The Question: Ask For A Sketch First, Ctd.

Kate Nowak, on my recommendation that teachers ask for informal sketches before formal graphs:

I agree with everything you say here. However, I think you will get silent resistance on this because teachers don’t know what to do next if their students can’t sketch a graph. But they know their students can follow mechanical instructions, so they’ll fall back on that.

Let’s say you’re working on Barbie Bungee. You’re tempted to jump your students straight to the mechanics of collecting and graphing precise data but you decide to develop that question a little bit first. You ask them for a sketch and the results come back:

A is (basically) correct. With zero rubber bands, Barbie falls her height and no further. Every extra rubber band adds a fixed amount to the distance she falls.

So what would you do with each of these sketches? Me, I think I’d say the same thing to each student.

BTW. Kate is back in the classroom after a short hiatus so there’s never been a better time to watch her think about teaching.

I’d need to think about it in context of the lesson and course flow. What happened before? What was done to orient them to the problem; do they have any concrete experience of the situation or is this more like just get something down, and then what kinds of things would they be basing their response on? What were your reasons for anticipating these 4? Are these kids in Algebra 2 or 8th grade? So I have more questions than answers.

I’m an engineering professor, not a math teacher, and my courses are built around design projects. What I’d tell the students is probably what I usually tell the students in the lab: “Try it and see!”

All four of these kids appear to have slightly different models for understanding how this graph relates to Barbie falling. I’m assuming that we are just asking for a rough sketch here, as per your previous post.

#1 seems to indicate some important understandings of the relationship between the two variables. It is hard to come up with that graph by accident. My feedback to this kid would be to ask her what else could be modeled with this graph.

#2 seems to know that the more rubber bands there are, the longer the distance is. This is a pretty key understanding. I am curious about why they chose to start their graph at the origin, and I would ask them to explain their reasoning behind their creation of this graph. Either they will notice their mistake themselves, or I will have more information with which to ask a better question. One possible response would be to ask kid #2 and kid #3 to justify their graphs and defend them.

#3 seems to be confusing the graph as a map of the actual fall itself, but there could be other explanations for their choice of graph. For example, they could be interpreting distance fallen as just distance, in which case they might be thinking that this means the distance from the ground. I need more information about their thinking, and so I would ask them to explain to me what they have done, and then depending on their response, I ask another question.

#4 did not do the question. There are many reasons why this could be true. They could not be able to read, they could not have a starting place for figuring this out, they could be unwilling to make a mistake, they could be still thinking about the problem by the time I get near them, and more. I need to know more information. Is this a typical pattern from this student? Have they produced similar graphs in the past? What socio-emotional concerns do I need to be aware of? Based on my understandings of these questions, I would ask a question like “Can you explain to me what the problem is asking?” Ideally I have already spent enough time clarifying the problem before everyone started that this particular question will not give me much information (eg. the student does know how to explain the problem) and I will likely need to ask another question. Maybe I need to ask them to describe the relationship between rubber bands and falling bands in words first.

My second reaction, when I read a few of the Barbie PDFs is that these things are so longgg …. I was a middle school science teacher and my ideal worksheet was a one pager. We did a lot of context building by talking through the prompt, what we needed to know and the experimental design. I didn’t always pull it off well, but I also didn’t have kids mechanically following my directions.

This is a series about “developing the question” in math class.

## Developing the Question: Ask For A Sketch First

This is a series about “developing the question” in math class.

I’ve been a bit obsessed with “Barbie Bungee,” a lesson on linear regression which you’ll find all over the Internet. It’s the kind of lesson that doesn’t seem to have any original mother or father, only descendants. (Here is NCTM’s version as well as a video from the Teaching Channel.)

Search the Internet for “Barbie Bungee handouts. I have. Invariably, the handout asks students to collect data for how far Barbie falls given a number of rubber bands tied around her ankles and then graph the results precisely. Often times those handouts include a blank graph with precise units and labeled axes.

Developing the question means starting from a more informal place. It means asking the students, “What do you think the relationship looks like between the number of rubber bands and Barbie’s distance? Sketch it.”

Asking students to sketch the graph serves so many useful purposes.

• It helps us clarify assumptions. What do we mean by “distance”? Barbie’s distance off the ground? The distance Barbie has fallen?
• Predicting the relationship makes it easier to answer questions about it later. This is from Lisa Kasmer’s research. It’s productive for students to decide if they think the relationship is linear, constant, increasing, decreasing, etc. What is its general shape? How do these quantities covary? As rubber bands increase, what happens to distance? Later, when students start to graph data precisely, the fact that the shape of their data matches their sketch will help confirm their results.
• It’s great formative assessment. Do your students even know what a graph represents? Find out by asking for a sketch. If they can’t sketch a graph, their later precise graphing is likely only going to be mechanical and instrumental. (ie. “First number right, second number up.”)
• Comparing informal sketches, which may vary widely, will likely make for better debate than comparing precise graphs, which will largely look the same. And controversy generates interest.

Which would make for a more interesting classroom debate? These three precise graphs?

Or these three imprecise sketches?

If the answer is “make a precise graph of a real-world relationship,” then developing the question means asking for a sketch first. That’s my resolution.