**The Task**

This task comes from the Mathematics Vision Project, a free curriculum from some sharp Utah educators. This is the first task in their “Secondary One” curriculum, a variation on the classic Pool Border task.

**What I Did**

**Reduce the literacy demand.**There’s a lot of text here boxing out English Language Learners and obscuring the point of the task. Much of that text exists only to service the story surrounding the problem — this school administration replacing tile in the cafeteria. I’ll argue later this story buys us very little while the text it requires costs us quite a lot.**Start the problem with a concise, concrete question.**Let’s not bury the point of the problem. Let’s tell students what we’re doing here as soon as possible, as quickly as possible, with as little jargon and in as few words and syllables as possible. Keep it to a tweet.**Add intuition.**Let’s make that initial question something students can guess at. This is my go-to. I’d be surprised if it didn’t feature in every single makeover this summer. It’s the easiest, cheapest move I can make to get students to commit to a task.**Motivate the move to generalization.**Why should we move from counting by hand in part A (easy!) to generalizing to a formula in part B (hard!)? The problem just asserts that the contractor wants to. I’d rather put the student in a position to say to herself, “My word. I’d rather eat chalk than count up all these tiles. Does anybody have a faster way?”**Scaffold the move to generalization.**The task moves straight from “count the number of tiles in a single small instance of the pattern” to “generalize the pattern” and I’m not confident the pattern will be clear to the student. Bree Murray thinks the domain of the pattern is only even-numbered squares, for instance. She may be right but I interpret the problem to include all whole numbers. This is not the good kind of ambiguity.**Open the problem up to more than one possible generalization.**I can see at least two ways to generalize the pattern. One defines itself in terms of the large boundary square and the other defines itself in terms of the small inset square. Defining the pattern by the large boundary square felt most natural to me. The task authors force the student to define the pattern by the small inset square. We’re losing mathematical richness and student creativity there and, aside from easier grading, I’m not sure what we’re picking up on the other side of the trade.**Raise the ceiling on this task.**In addition to lowering the floor with all the guessing and scaffolds we’ve already added, if a student figures out how to generalize quickly, can we raise the ceiling with extra questions that will provoke her to develop the concepts further?

Let’s try this.

Tell your students, “I’m going to show you a picture for only a few seconds and I want to know how many blue squares you see. Just a guess. Don’t overthink it. Go from the gut.”

Show this image for a few seconds.

Ask them to think-pair-share their guess.

Ask them to write down a number of blue tiles they know is too high and a number they know is too low.

Tell them, “I know that if you *had* to, you could count up those squares one-by-one. But that sounds like a *lot* of work. Let’s look at some smaller versions of the pattern and see if we can think smart rather than working hard.”

Pass out this handout [pdf].

It has the four earliest iterations of the pattern. They have to count and/or construct each one, which should give them a better intuition for the pattern than the original task does.

Ask them to write down and share a fast way to figure out the number of blue tiles in the twentieth iteration of the pattern. (I initially wrote “Ask them not to draw it” here but I don’t think they’ll find drawing it all that enticing. That’s the point of using larger numbers.)

Ask them to share how they’re thinking about the number of blue tiles. They’ll help each other here.

Then show them the twentieth iteration and see if they were right.

Now we tackle the huge square. We give students the information they want. The huge square is the 100th version of the pattern.

Some may generalize to a formula now. I imagine most will still just apply the informal verbal method they created earlier. Say, “If you can turn this sentence into variables, then a computer can handle any problem of this sort instantly and we’ll be done with them forever.”

Ask them to take their answer for the huge square and reconcile it against the error boundaries they set up earlier.

If students generalize to a formula in different ways, show those different formulas and connect them to the ways of thinking they shared earlier. Ask students to reconcile the differences. How are the formulas different? How could they turn one formula into the other?

As students finish, offer some follow-up questions:

- Tell them “One version of this pattern has one million blue tiles. Tell me everything about that version of the pattern.”
- Ask them to find the number of white tiles in the huge square. (Computationally simple, but it’s a bit of conceptual leap to realize you’ve already done the hardest work.)
- Ask them what numbers of blue tiles they’ll never ever see.

We’re trying to challenge students who finish quickly here and buy ourselves some time to help students who need it.

Show the answer. Find out who had the closest original guess. Have the class give that student one clap.

**What You Did**

Over on Twitter:

- The usually stalwart
**Chris Lusto**threw in the towel immediately. **James Cleveland**thought the problem could stand as is.**Jennifer Silverman**created a Geogebra applet. We had a split decision, she and I, on whether students would get better intuition for the pattern by having Geogebra create it dynamically or drawing it themselves.

Over on the blogs:

**Beth Ferguson**motivates the move to generalization by posing a school administration that quickly and repeatedly changes its mind on the dimensions of the cafeteria. That’s great. I’m curious what sending students to the cafeteria buys us here in exchange for the time it costs.**Mary Dooms**pulls a nifty trick by taking a task that’s fully prescribed and makes it both harder (students have to ask themselves “What information is important here?”) and more fun (students get to be creative and autonomous) at the same time. That isn’t easy.**Jonathan Claydon**modifies another task that moves to generalization way too quickly.

**Open Question**

**What does the real world buy us here?** I know what the cafeteria context costs us. It costs us an extra page to include all that text. It costs us some participation from our English Language Learners. That’s fine if we get something worth all that cost, but I can’t see what we’re getting.

Lately, when I work with math teachers, I encounter a particular theory of student engagement that says, “Make the problem about something the students are familiar with or which is close to their everyday lived experience.” On Twitter, for instance, **Sarah Lowe** suggests students may be more intrigued by stones or gardens.

I don’t think this is wrong. But I think we overestimate the effect of swapping the context from checkerboard cafeteria tiles to a checkerboard dance floor or a checkerboard stone walkway or a checkerboard pizza platter while leaving the rest of the task intact. I just don’t think students are that easily placated, so I haven’t focused on context at all in my makeover. We’re just counting up abstract blue tiles in my task. Instead I’ve tried to start the task easy enough to bring in a lot of students and end hard enough to be worth their time, with or without any context.

I’d like someone to convince me that adding a context would raise a student’s heart rate over this problem, but I have a hard time seeing it.

**Featured Comment**

I just did this activity in my College Algebra course (I teach at a two-year college) as an introduction to sequences and series.

19 students in class, 18 participated. (1 student walked out and came back when the activity was over.) I teach in an ‘active learning classroom’ with desks set up in groups. 4 groups of students working the problem together and individually.

3 different models for the pattern were given to me. I had already created my own model, and the 3 in class were all different from mine.

Each group explained their thinking. Most had not generalized their approach into algebra.

I helped them put it into algebra. I also showed that we could simplify the algebra for each approach and end up with the same thing!

Comments on the lesson:

- I was surprised how long it took them to come up with the pattern. I had to ‘nudge’ one table along and point out an error in their thinking at another table. The process of coming up with 88 tiles for the 20th iteration took over 10 minutes.
- I was happy to see so many different approaches to solve the problem. Things also ‘felt different’ in the classroom. Working to figure out a problem like this is much different than working on the process of completing a square. Different people were taking the lead and speaking up to help their group mates. It was great!
- I was able to use this task as a reference right away. For example, when introducing the idea about the domain of a sequence, which normally is very confusing, I referenced the iteration # in this task and the students seemed more able to make the connection.
- I didn’t realize the importance of telling the students the handout referred to the first, second, third and (draw-in) fourth iteration until after I handed it out. I had to go to each table numerous times and show them which one was #1, #2, etc… In the future, I would label the handout first.
This activity took about 20 minutes from start to finish. I definitely think it was time well spent! Thanks!

## 30 Comments

## l hodge

June 24, 2013 - 7:37 am -I agree the literacy demand gives almost no payback for the context chosen. I doubt that many students would find this context credible, it is not interesting or amusing, and it is not needed to clarify the question.

My natural inclination is to focus on this as an algebra or sequence problem — how do the number of blue tiles change as the size of the inner square changes. But, looking at this in terms of geometry is simpler and more interesting.

I would like to nudge them towards looking at ways to re-arrange the blue squares to make them easier to count. Take the original picture and slide the inner ring outward. The number of blue squares is obviously the perimeter of the large square — 4. Is this true for different sizes (not so obvious)?

To give students a bump in this direction I might ask them to come up with other designs that use the same number of blue tiles. I think providing a 5×5 and a 6×6 almost guarantees a sequence approach & does not leave many options for re-arranging the blue tiles – don’t like it.

As an extension, I might ask what happens to the ratio of blue to total as the size increases. Both in the original pattern and some of the patterns they come up with. What can you say about designs where the ratio does not change?

## newman

June 24, 2013 - 7:44 am -bravo.

## Andrew Busch

June 24, 2013 - 9:35 am -I’m not so certain about ditching the context. I think you all did great with reorganizing it to help student inquiry. With just a bit of work we can turn this into a bona fide modeling problem. My neighbor installs flooring (no really). If he’s hired by the school to install the cafeteria floor, he needs to know how many boxes of each type of tile to purchase. I think you can push students with a “real-world” context which forces them to think about the important aspects of the problem: 1) how big is the cafeteria? 2) how much does a box of tiles cost? 3) how many tiles are in each box? 4) do colored tiles and white tiles have the same price?

We can get around the non-reader issue by presenting the problem to the class. This is done as a group anyway so what does it matter if I have a few students who can’t decode the intro paragraph? These are often my building trade students anyway. Context is the only thing that matters to them.

They don’t want trite. They don’t want veneer. They don’t care how long it takes Jimmy to shovel a driveway at a rate of 5ft/min because Jim is going to stay outside until it’s done anyway. They want to become convinced this is worth learning because they can make money with it or apply it to their lives now. That’s not my final goal but it is what it is as a starting place of motivating my self described “I hate math” students.

## Andrew Busch

June 24, 2013 - 9:41 am -After rereading my post, that was too simplistic of a description of my students. But I do find it on the ‘more true’ side of the scale.

I would change these things in my previous post:

-Context matters to my struggling students but it isn’t the only thing.

-Students do want to become convinced this is worth learning but money isn’t the sole or even the primary motivation.

## Beth

June 24, 2013 - 10:11 am -I agree that the context of the cafeteria is not going to light the fire of any of my high schoolers. And changing the location of the checkerboard pattern isn’t going to help.

I included the trip to our school’s cafeteria not so much as a motivating factor, but because in our curriculum we have standards that require gathering and recording data.

– Algebra 1: Gather and record data and use data sets to determine functional relationship between quantities

– Algebra 2: Collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments.

In each of our units of study we spend one class period collecting/organizing/interpreting data … shooting rubber band cannons, measuring the wing span of our classmates, determining the height of a dropped bouncy ball, measuring Barbie’s descent, etc. Collecting data does take time — and there were times last year when I thought how much easier it would be if I gave students nice, neat, clean data to use.

I agree with Dan that we are going to lose class time by going to the cafeteria. So if this isn’t one of our purposeful data collection activities, then it would be better to work in the classroom with graph paper.

As I ponder this problem today, I wonder about using it as a first day activity since no specific high school mathematics is needed to solve it. Watching how teams might work through this problem would be insightful and helpful to me — especially if I had a strong observational tool for noticing leaders, problem solvers, those who are persistent in the work, those who assist even if they appear somewhat lost, and those that simply hang back.

## Chris Lusto

June 24, 2013 - 10:23 am -Yes Ã— 1,000,000 to reducing the literacy requirement. By the time I was done reading the entire question, I was already opening another blog post. Hence the early towel-throwing.

As an aside, if we’re interested in a kid’s metacognitive process (and we should be), we have to do better than, “Track your thinking.” That’s the psychological equivalent of saying, “Solve this problem.”

## Steven Shirley

June 24, 2013 - 11:13 am -I feel like this follow-up question is begging to be asked:

What if that center region was a rectangle, not a square?

Especially if the problem eventually becomes contextualized by my class in such a way that the center region is a school cafeteria, a garden, a bedroom, or whatever, it seems very likely that any of those spaces would not be perfect squares.

While I have not tried the math yet, my intuition tells me that opening up the constraints to include rectangular regions will make the problem much more challenging. Probably require two variables? I might start with a Sx(S+1) class of rectangles (you know… 1×2, 2×3, 3×4, etc.) out of a need to keep one variable and in the interest of not biting off more than I can chew.

Thoughts?

## Aaron B.

June 24, 2013 - 11:24 am -@l hodge touches on the issue of “what is the mathematics we want the kids to learn by doing this problem?”

If I read Dan’s makeover correctly, the problem has essentially become an “answer getting” exercise. The problem as the authors intended it was about the mathematics of equivalent expressions.

If I were to make this problem over, I would lean in Dan’s direction but build toward asking the students something like: A student last hour came up with the generalization 4s + 8 and another student came up with [(s + 4)^2 – s^2]/2. Explain how to “see” the tile pattern in each of those generalizations.

## David Wees

June 24, 2013 - 12:30 pm -Is there a way we can research the effectiveness of these tasks?

I mean, we could argue all day about how important the context is for our students, but each of us is relying on our intuition.

Maybe what we should really try and do is take some very similar tasks like this, find a way to assessment/measure the effectiveness of the task in some way that any classroom teacher can use, and then crowd-source some research around the issue of “how important is context in a mathematical modelling task”? We could even compare different contexts, etc…

## Isaac D

June 24, 2013 - 2:40 pm -1. The most confusing thing to me about the original task was the second diagram. The seemingly inconsistent decisions about which tiles to show and which to leave off created a red herring that functioned like an anti-scaffold … it took what should have been a simple generalization and made it more difficult because it took five minutes of staring at the diagram before I realized that most of those tiles were within the edges of the inside square. To me a scaffolded version would have shown the 4 tile (2 blue and two white) squares at each corner set apart from the remaining (rectangular) portions of the pattern. A less-scaffolded version would have shown me a larger version of the first image with no hints.

2. Describing larger and smaller variations by number of iterations in your version of the task was also distracting. Rather than talk about the 20th iteration, I would rather talk about the inside square having a side length 20 or the outside square having a side length 24. I’m be curious whether this is unique to the way I approach the problem (geometrically instead of as an iterating sequence) or if most students would feel the same way about it.

3. There is definitely a trade-off in terms of providing a concrete context vs. leaving this as an abstract problem. If there is a compelling enough context I think it would be easier for some students to engage with, and definitely easier for younger students to think through the principles involved. On the other hand, this is a case where context-verging-on-pseudocontext can make the task less relevant and more boring. The sheer competition factor brought in by guessing and comparing guesses about an abstract image on the screen is certainly more interesting than most of the contexts (tiling floors or otherwise) that we could apply this pattern to.

4. The question of where on the context-pseudocontext spectrum the original task falls is an interesting one. I am reminded of an experience I had while planning a corner bookcase with an Amish carpenter. A problem involving the length of a diagonal facing came up, and while I was solving it using a calculator and the Pythagorean theorem I looked up to see that he had already solved the problem using a ruler and a square to essentially create and measure a scale model. Even when a created context does a wonderful job of illustrating a mathematical principle, and even when the principle can actually save time and work, there is often a faster and easier way to cut the Gordian knot with a less mathematically-involved solution. To me as a non-carpenter a word problem involving use of the Pythagorean theorem to find the length of a diagonal on a bookcase might sound like a very plausible scenario, but to a professional carpenter it might seem absurd to muck around with square roots when the problem can be solved more quickly and easily in some other way.

## Paul Bogdan

June 24, 2013 - 2:49 pm -Bravo. I love it. Please do it to the rest of the book.

## Jennifer Silverman

June 24, 2013 - 4:19 pm -I don’t think I conveyed what I was seeing in the applet strongly enough. I don’t count past the number 8, I see the pattern in chunks, like this: http://bit.ly/14tGVXI

Steven Shirley, my follow-up questions were about rectangles, too.

Isaac D, yes. That’s what led me to create the original applet.

Here it is again: http://bit.ly/10ddh9x

## James Cleveland

June 24, 2013 - 8:17 pm -Okay, yes, while I think the problem is a good problem as is (which I did not think about the last makeover problem), I totally agree with reducing the literacy demands. I guess I assumed that automatically as I would present the problem to the class, not just hand out a sheet if paper.

## Diana Bonney

June 24, 2013 - 9:01 pm -I prefer your way of showing them the image quickly and asking them how many tiles there are. It caught my interest, and I think it would catch the interest of my middle schoolers as well. I like giving problems context, but in this case, who cares about a school cafeteria floor?

Can anyone think of a context that might interest my middle schoolers more?

## Dan Meyer

June 25, 2013 - 3:19 am -Beth:That’s the placement of the task in the MVP book. I think the authors intended it to be used as you suggest.

Aaron B:From my version of the task:

The difference is that I’m offering some kind of motivation for those generalizations (large numbers, too big to comfortably draw and count). If the fact that a problem also resolves to a numerical answer means it’s necessarily fixated on “answer getting,” someone please fill me in.

Isaac D:Glad I’m not alone there. Isn’t there some kind of rule that if a group of math teachers are flummoxed by an aspect of your problem, that aspect is going to run over actual students like a train?

Isaac D:It’s a judgment call on the part of the author. I can see the upsides to naming one or both of the sides. The downsides are that the information given will then influence the model the student chooses. That, and if the goal is mathematical modeling, the CCSS require

studentsto identify essential variables.Isaac D:Diana Bonney:I was introduced to this problem by a group of 100 math teachers a couple weeks ago. I listed here the best contexts they came up with. (Dance floor. Cafeteria floor. Pizza platter. And now stone walkway.) I don’t see how any of those contexts are helping make it easier for students to think through the principles (though that’d be great if it were true). I don’t see how any of those contexts are going affect a student’s interest in the task —Â either negatively or positively.

## Santosh Zachariah

June 25, 2013 - 5:10 am -Does anyone have an image of a reasonably contemporary example of this in the real world? (Dance floor. Cafeteria floor. Pizza platter. Stone walkway. Something that might make it to http://www.101qs.com/)

My own feeling is that (some) students are not asking for the context to be relevant to them, so much as asking for the context to shown to be relevant to somebody in the real world (other than mathematics buffs.)

Dan, you’ve repeatedly shown that making a real-life problem tractable, but not trivial, takes a huge amount of creativity.

When you ask “What does the real world buy us here?” I am not clear if you are now backing off slightly. Or if you are just accepting additional constraints specifically for the Makeovers (e.g. it must fit on a piece of paper that can be handed out.)

## Sean

June 25, 2013 - 7:29 am -This is excellent and at the right level of prescriptiveness. Could be implemented tomorrow.

Compelling-

1. to see “reduce literacy demand” evolve into “if possible, eliminate literacy demand.”

2. to see 3 act methodology transition into “ordinary” classroom tasks. Was admittedly skeptical on feasibility of the transition, but I’ve found these to be almost

moreapplicable to everyday classroom use.## Kirschen Seah

June 25, 2013 - 9:24 am -Lovely problem! But wait – there’s a visual / pattern matching element. Consider this… A pair of adjacent columns or rows has _exactly_ the same number of blue and white squares – which is the number of elements in each column or row. How do you visualise this? In a pair of columns / rows, exchange the blue / white squares so that all the white squares are in one column and all the blue squares are in the other. And that’s it! So for a pair of columns of height 9, there a 9 blue squares and 9 white ones. :) Fiendish!

Another engineer advocating STEM education!

Thanks for the thought provocation.

— Kirschen

## blaw0013

June 25, 2013 - 10:11 am -@DavidWees. I like the question, can we research the effectiveness of these tasks. But not because I am interested whether or not they are effective; rather to answer “what are they effective at doing?”

Such an empirical study demands that we carefully define this thing we wish to measure. And then, to develop valid and reliable measures of that thing.

Further, I believe it is worth measuring multiple outcomes. For example, maybe attainment of some CCSS-M content standard, or maybe a CCSS-M practice standard. But what about children’s mathematical agency, or identity? Or sense of what maths is? Or what it means to do maths?

Maybe we ought to measure something related to the “relational equity” (cf. Jo Boaler) developed in the classroom, something that may go down as the need to interpret an incoherent textbook problem into an agreed-upon and worthwhile investigation is diminished by more careful “makeover” occurs.

Seems to me there are MANY questions worth asking about the EFFECT of curriculum design, not just the percentage of students / people that ask the same question as a result of a written, auditory, or visual image.

Provocative question David…

## l hodge

June 25, 2013 - 11:00 am -There really are some neat designs when re-arranging the 28 blue tiles in the original diagram.

If we give them the first four “iterations” they will simply count to get the sequence: 12, 16, 20, & 24 blue squares. That is low entry, which is good, but won’t almost all students see it as an “increase by 4” number pattern at this point? Is that what you want? It seems like this is directing students towards a non-visual approach for a visual problem (not necessarily bad) and away from creative thinking on a problem that is inviting creativity.

## Chris Shore

June 26, 2013 - 5:19 am -Offering context is useful in many cases, as when asking students to graph y = 3x + 1; the students don’t know why the 3 is the slope and not the 1 other than the Math God’s say so. However, in the border problems, the diagrams serve as the context. The students already can see that one variable represents the number of squares on a side, the other represents the number of colored tiles, and the equation implies that there is a relationship between the two.

On another note, while “low-literacy demands” help give EL students (and many others) easier access to problems, these students need to be immersed in language in math classes in order to help them learn the English. Schmoker and Marzano have been pushing for the reading of more text across the curriculum, and now the CCSS is calling for 70% of student reading to be reading informational text from other content areas. The only place for students to learn how to read technical material is in math and science class. With that said, I can say as the father of a dyslexic child, that offering the border problem visually would engage my son more readily than a garden context. He would want to see how close his guess was, rather than solve someone else’s contrived problem. At the same time, I hope there will still be ample opportunity in his math class to develop his impaired reading skills.

## Dan Meyer

June 26, 2013 - 6:54 am -Chris Shore:I’m compelled by

Andrew Busch’svocational math class, a class which ispremisedaround math that is useful for professionals. I suppose there the math hadbetterbe.In general, though, I don’t find students so eager to know someone with a job uses their math as much as eager to not feel stupid, to not feel small, to feel intrigued and surprised instead.

I’m asking, How does the cafeteria context do any of that? How could any context do that when the rest of the problem is rather intractable?

## Kevin H.

June 29, 2013 - 2:51 am -Dan’s last comment (directly above) is interesting, because it seems to contradict what I take to be the general thrust of his work. As I posted on the Unengageables, the 1-tweet lesson I’ve taken from Dan’s work is that the quality of explanations doesn’t matter until students are on the edge of their seats wanting to know…and you CAN get students on the edge of their seats. I’ve understood this to mean that Dan looks at instruction primarily through the lens of motivation, not cognition (the biggest question isn’t whether their brains can understand the lesson, but whether they care enough to turn their brains on). For example, good 3-Acts engage students because they elicit puzzlement or curiosity.

But then in this last comment, Dan says, “Instead I’ve tried to start the task easy enough to bring in a lot of students and end hard enough to be worth their time, with or without any context.” Here it seems Dan views the level of difficulty, which I think is mostly a cognitive factor, to be the primary mediator of engagement, at least in this task. I actually agree that carefully calibrating the difficulty of a sequence of questions can lead to engagement without much context.

But this is exactly why Dan’s tech contrarianism seems way too pessimistic to me. A well-designed software program can individualize that difficulty level for students better than any one teacher can do it for a whole class. Granted, when you do this with software, you’re mostly doing it with drill-practice questions rather than modeling tasks. But doesn’t the same principle apply? Can’t a sequence of drill questions easy enough to draw students in and hard enough to make them feel they’re learning something produce engagement? The sense I’ve gotten from Dan is not just that no software currently on the market is good at this, but that computers are fundamentally unsuited for that kind of practice.

In fact, RAND recently released results of an efficacy study of Carnegie Learning Algebra 1, which found an 8 percentile increase in scores for high school students using that blended curriculum, though no impact for middle school students. The study was extremely large–$6 million of federal funding with 18,700 high school students and 6,800 middle school students as subjects. You can read the results here:

http://www.rand.org/content/dam/rand/pubs/working_papers/WR900/WR984/RAND_WR984.pdf

In any case, I’ve sometimes grown as a teacher by having a dialogue with Dan in my head, so I’m genuinely curious: is it that you think cognitive factors are really important, but you’ve chosen to specialize mainly in motivational factors? Or do you see level of difficulty as motivational, not cognitive? Or is my whole cognitive/motivational dichotomy false in your opinion? What got me thinking about this was when we disagreed about the significance of the worked example effect a long time ago. Since then, I’ve assumed that either you didn’t care much about the cognitive side or that you and I simply don’t agree on what the research says about cognitive factors.

## Dan Meyer

July 2, 2013 - 12:00 pm -Kevin H.:I blog a lot about motivation and engagement because I find it to be the tougher of the two lenses to focus. I tend to think teachers need less help determining a sequence of worked examples to show their students for a given topic than they do how to get students to care about that topic. I don’t think one is more important than the other. They’re both co-requisite. But one is harder.

I don’t know if it’s primary, but it’s a big one in these tasks. Three-act math is premised on the idea that the first act is really easy to jump into with intuition and guesswork while the sequels develop the skills to a much higher degree. This isn’t a new feature here, though maybe I haven’t cast it as a focus on cognitive development like I should have.

In theory, sure. But “individualization” in math education technology to this point has only been possible by reducing “math education” down to computational exercises. I’d be happy to be shown an example of mathematical modeling (for instance) that’s personalized, that keeps asking developmentally appropriate questions without constraining the student’s answers to a limited input set. It doesn’t exist, AFAIK.

The worked example research is useful but I struggle with its ecological validity. It’s a dangerous game, generalizing these one-off lab experiments to 180 days in the classroom. Sweller himself presumes motivation in his research. It only works if the students are already motivated. And there’s nothing quite so de-motivating, from my experience, than repeated daily doses of worked examples. That’s why I fix on motivation more than cognition.

## Kevin H.

July 3, 2013 - 9:37 am -Thanks, that’s interesting–you and I aren’t quite as far apart as I had thought. My goal in this dialogue is to get a sense of the parameters within which you apply the motto “less helpful.” I use worked examples as a test case because they are explicitly designed to ease students’ cognitive load, but it’s really “less helpful” as a stance that I wonder about. To be clear, my read of the literature is that there isn’t an answer yet on when to provide more help, but that there are definitely cases when “more helpful” is better. You mentioned that the worked example research has been in short-term lab-based studies rather than full-year classroom studies. That was true before 2008, but not anymore: http://pact.cs.cmu.edu/pubs/SaldenEtAl-BeneficialEffectsWorkedExamplesinTutoredProbSolving-EdPsychRev2010.pdf (see the section called “Recent Research on Worked Examples in Tutored Problem Solving”).

When it comes to technology, I don’t see the questions in math software programs to be much different than the ones you asked in your own concept checklist tests: http://www.mrmeyer.com/blog/wp-content/uploads/070830_4.pdf . Granted, these aren’t modeling, but they are part of math class. I guess I see software as automating your concept checklist. However, I wholeheartedly agree with your point about software that constrains students’ input too much, and my intent here isn’t to get dragged into a pro/con of software. Lots of software out there stinks, and none of it is what we wish it were.

What interests me is that this particular lesson make-over just feels different, like I could adapt its general strategy to teach anything, even lessons on formalism rather than modeling. Kind of like the difference between Sam Shah’s stuff and yours. With 3-Acts, among other things, I have a sense of what the strategy to overcome disengagement looks like. But I’m still really stumped on the strategy for overcoming confusion with math formalism. For example, how could the basic flow of this makeover be used to teach Precalc students that in the expression (x+5)/x , you can’t cancel the x’s? And then how/when would you interleave drill practice with the meaning-making? Not asking you to respond to this particular dilemma of mine, just showing you where my head is right now.

## Chris Perry

July 18, 2013 - 9:58 pm -Hi, I am a edm310 student at the University of South Alabama and I agree with you. I think if you give students something easy to start with and get them engaged, then by the end of it once you deliver the “hard part of the problem” they will have already invested a lot of time so they will work harder to finish. I think that if you start out with the hard part then most f your students will give up without even trying to solve the problem.

## Bethany

July 24, 2013 - 9:34 am -I just did this activity in my College Algebra course (I teach at a 2-year College) as an introduction to sequences and series.

19 students in class, 18 participated (1 student walked out and came back when the activity was over). I teach in an ‘active learning classroom’ with desks set up in groups. 4 groups of students working the problem together and individually.

3 different models for the pattern were given to me. I had already created my own model, and the 3 in class were all different from mine.

Each group explained their thinking. Most had not generalized their approach into algebra.

I helped them put it into algebra. I also showed, that we could simplify the algebra for each approach and end up with the same thing!

Comments on the lesson

1. I was surprised how long it took them to come up with the pattern. I had to ‘nudge’ one table along and point out an error in their thinking at another table. The process of coming up with 88 tiles for the 20th iteration took over 10 minutes.

2. I was happy to see so many different approaches to solve the problem. Things also ‘felt different’ in the classroom. Working to figure out a problem like this is much different than working on the process of completing a square. Different people were taking the lead and speaking up to help their group mates. It was great!

3. I was able to use this task as a reference right away. For example, when introducing the idea about the domain of a sequence, which normally is very confusing, I referenced the iteration # in this task and the students seemed more able to make the connection.

4. I didn’t realize the importance of telling the students the handout referred to the first, second, third and (draw-in) fourth iteration until after I handed it out. I had to go to each table numerous times and show them which one was #1, #2, etc… In the future, I would label the handout first.

This activity took about 20 minutes from start to finish. I definitely think it was time well spent!

Thanks!

## Dan Meyer

July 26, 2013 - 6:09 pm -Thanks for the feedback here,

Bethany. I boosted your comment up into the main post.