Let’s just call them “theories of engagement” for now. Every teacher has them, these generalized ideas about what engages students in challenging mathematics. Here’s the theory of engagement I’m trying to pick on in this series:

This theory says, “For math to be engaging, it needs to be real. The fake stuff isn’t engaging. The real stuff is.” This theory argues that the engagingness of the task is directly related to its realness.

This is a limited, incomplete theory of engagement. There are loads of “real” tasks that students find *boring*. (You can find them in your textbook under the heading “Applications.”) There are loads of “fake” tasks that students *enjoy*. For instance:

No context whatsoever in any of them. Perhaps the relationship actually looks more like this:

I’m being a little glib here but not a lot. Seriously, none of those tasks are “real-world” in the sense that we commonly use the term and yet they captivate people of all ages all around the world. Why? According to this theory of engagement, that shouldn’t happen.

Here are fake-world *math* tasks that students enjoy:

- Ihor Charischak’s Jinx Puzzle.
- David Masunaga’s Magic Octagon.
- Malcolm Swan’s Area v. Perimeter.
- NRICH’s Factors and Multiples Puzzle. (Megan Schmidt: “… I have one student in particular who is not particularly motivated by much … when I bust out a puzzle, he’s all in.”)
- Matt Vaudrey’s Magical Triangle Theorem.
- Andrew Stadel’s Weekly Puzzle.
- The meaning of the sequence 3, 3, 5, 4, 4, 3, … , which drove kids bananas the day I wrote it on the board at the end of a test.
- The proof that 2 = 1.
**2013 Dec 3**. The four fours puzzle.**2013 Dec 5**. Timon Piccini’s Broken Calculator.**2013 Dec 10**. Patrick Vennebush sends along “How much is your name worth?”

My point is that your theory of engagement might be limiting you. It might be leading you *towards* boring real-world tasks and *away* from engaging fake-world tasks.

We need a stronger theory of engagement than “real = fun / fake = boring.”

**Homework Time!**

Choose one:

- Write about a fake-world math task you personally enjoy. What makes it enjoyable for you? What can we learn from it?
- Write about an element that seems common to those enjoyable fake-world tasks above.

## 33 Comments

## Jared Cosulich

December 3, 2013 - 6:57 pmWhen you look at what is going on in a puzzle (everything from a jigsaw puzzle to Tetris to Angry Birds) you see some very consistent patterns:

1) A clear goal that is neither too easy or too hard.

2) Immediate feedback loops enabling someone to develop and quickly test hypotheses

This also happens to be the definition of “Flow”. Which is the very definition of engagement (you lose track of time, hunger, etc).

I also think there is another pattern, which is actually pattern-matching. The human brain is a pattern-matching machine. The most abstract mathematics usually comes down to recognizing and leveraging patterns in unique ways and yet we generally don’t approach mathematical learning from the perspective of patterns. It may seem like patterns to an expert, but feels like memorization to a beginner…

## David Coffey

December 4, 2013 - 5:18 amDan,

Thanks for taking this on. John Golden and I often talk to teachers about making math the context. Here’s an example of one of our favorite activities, When will it end?

http://deltascape.blogspot.com/2012/11/when-will-it-end.html

## Chelsey Mayo

December 4, 2013 - 6:14 amConfession time…

In those glorious hours of freely surfing the web, I love to play games on Neopets. There are so many of those games that have a math component. It’s definitely one of my favorite time wasters. I think what I enjoy most is the instant gratification of beating a level or winning a game, or my competitive side that comes out when I fail or get beat. I think the scaffolding of the games also has a lot to do with engagement. I know it gets harder as I progress, but since it was so easy to beat the first few levels I should be able to beat the harder levels as well. I also think a factor is the pressure associated with these tasks, the fact that is no pressure makes them so engaging. There is nothing riding on whether or not I get three stars on a particular level of Angry Birds, but when something is for a “grade” I always spazzed out about it.

## Stan Yoshinobu

December 4, 2013 - 7:33 amI agree with your overall thesis. “Real” is not necessarily good or important. If only real-world things mattered, then much of what the ancient Greeks accomplished that is the bedrock of many modern ways of thinking in math and other subjects, would not matter. Context-based and inquiry-based math requires suitable contexts for grounding the mathematics. With that said, it is our ability as humans to abstract and generalize our ideas, finding math as a context itself, that allows us to be creative and insightful.

The goal is good, rich math and good delivery (i.e. teaching methodology, class environment, etc.). The form this takes depends on the mathematical landscape and the creativity of teachers.

Cheers!

Stan

## Andrew

December 4, 2013 - 7:38 amFor me? It’s the game called Flow Free!

A wonderful little game that draws in 30-year-olds, high schoolers, and 4-year-olds alike. It seems like there is a lot going on in the game that can translate back to lesson design. In general, there is a simple goal with few instructions to get started, increasing difficulty throughout, and unlimited do-overs.

If you want to read more, I talk about it in more detail here.

http://thegeometryteacher.wordpress.com/2013/12/04/fake-world-math-real-world-engagement/

## Brendan Murphy (@dendari)

December 4, 2013 - 8:19 amI’ve always felt that the inquiry or context based math was perfect for introduction to the concept. A way to draw the student in and get them interested. But the most important part of the lesson was the discussion. The translation of the guessing and checking and fits and starts we used to actually solve the problem into standards, rules, and procedures that most people see as “actual math”.

Once the students understand the relationship between the real and the pure anything can be engaging.

Teaching then becomes the work being done to help students connect fun math into “pure” math.

A fake world math task I always kind of enjoyed are those find the measure of all the angles in this shape type of problem. I suppose it could easily be made into a game by allowing hints to appear when a player clicks on the angle. And perhaps an explanation of proofs after completing levels. Someone should create something like that.

## William

December 4, 2013 - 8:52 amI agree completely that there are plenty of REAL tasks that aren’t engaging, but in my personal experience as a math student and as a teacher that occasionally creates and teaches math lessons, I find the most engaging problems are those that have a real application to my personal interests and life. Personally, I believe that if teachers present tasks to the students that they are passionate about and have fun teaching, that rubs off on the students, even if the students themselves don’t find the task very real or relevant.

As an example, I helped to teach this lesson yesterday:

http://blogs.henrico.k12.va.us/21/2013/12/03/race-training-and-slope/

The students were a mixed bunch, but when I asked if any were runners, barely any students raised their hand. However, at the end of the period, nearly all of the students were saying things like “man, this is really interesting,” “this is pretty cool,” or even “this is fun.”

## Kenneth Tilton

December 4, 2013 - 9:25 amExactly. http://stuckonalgebra.com/gaming_misunderstood.html

We even used the same examples: “…who has not become addicted to Tetris or Sudoku or something along those lines?”

I like especially your point re Sudoku et al: “No context whatsoever in any of them.”

Ironically math has the win here because even if kids (or grown-ups) do not like math, they understand it is important.

## Michael Pershan

December 4, 2013 - 9:30 amI’d say that all of these fake-world tasks are both doable and unusual. We study area and perimeter all the time, but how many times have you plotted them against each other? We do arithmetic all the time, but how often do we have the constraints of a four-fours puzzle?

My explanation would predict that one’s level of interest in these tasks would decrease as your exposure to them increases.

## wwndtd

December 4, 2013 - 9:59 amI have a similar problem in chemistry. Based on amounts of reagents, you can calculate the amount of product using some conversions (or vice versa). Stoichiometry is one of the basic calculations in chemistry (so I can’t skip it or water it down), but it’s hard to motivate students to figure out something like, “If 8.23g of hydrochloric acid react with 3.52g of zinc metal, how much zinc chloride will be produced?” I can make practice problems about explosives and poisons and antidotes, but that’s in the “fake math” category… pretty disingenuous. I’ve had them predict lab outcomes (and then do the experiment), but that’s still mostly like work rather than an interesting problem.

But I think something that a lot of the posted math-tasks have in common is that students know when they’re done. They can solve a puzzle (a la, Ben Orlin’s distinction between game and puzzle) and know that the answer is done. My chemistry students usually like balancing equations because they can tell when it’s correct and it’s finished. I get the highest homework turn-in rates, as well as the highest homework accuracy during this unit. Especially with stoich, it’s often hard to tell if the weird, funky decimal answer is what they’re really after or not.

## Mike Lawler

December 4, 2013 - 10:26 amI’ve done a few things recently with my kids that fall into the “fake world” definitely (hopefully I’m understanding the definition correctly!).

(1) The Chaos Game

After seeing Conrad Wolfram’s talk at the computer based math summit, I thought it would be fun to introduce a bit more computer stuff with my kids. I happened to be reading a book about Fractals at the time and chose to introduce them to the so-called Chaos Game.

http://mikesmathpage.wordpress.com/2013/12/01/computer-math-and-the-chaos-game/

As for your questions:

(i) Why is it fun? Even now, more than 25 years after seeing this result for the first time, it still surprises me. If you watch the video, you’ll see that the kids were floored. The fact that you get more or less the same picture from any starting point is also incredibly surprising.

(ii) What can we learn from it? At least a couple of things. First, since the code itself isn’t too hard, it lends itself to kids playing around with a math-related computer program. Maybe they’ll ask questions like – what happens with a square, or some other shape? Second, the there’s a nice tie back in to other simple fractals that kids can play around with using pencil and paper.

(2) Talking about pentagons

The first time around with my older son was just sort of a whim. We’d spent several weeks talking about quadratic equations and I wanted to spice things up a little and show a neat math result that used quadratic equations. The result I chose was talking about pentagons and, essentially, why cos(72) = (sqrt(5) – 1 ) / 4.

Several weeks later my younger son asked me how to draw a pentagon, so I talked about it with him, too:

http://mikesmathpage.wordpress.com/2013/12/01/conrad-wolframs-computer-based-math-education-summit-talk-and-perfect-pentagons/

As for the questions:

(i) Why was this enjoyable. For me it was really fun to have an example that took so many seemingly different parts of elementary math and produced a nice and easy result. For my kids, I think they had fun seeing how some math that is still way over their heads came into play making a shape as simple as a pentagon.

(ii) What can we learn from it?

First, even with elementary math, seemingly different ideas can tie together in unexpected ways and kids might find that to be really interesting. Second, after I finished making the pentagon with my younger son, I mentioned how the square root of 5 came into play. Wish I would have kept the camera running since his question back to me was something like – how can you take the square root of 5, 5 isn’t a perfect square?

## Amy Zimmer

December 4, 2013 - 12:43 pmI agree with Stan and Brendan if I understand what you are asking correctly. Today I had two students walk into Advanced Algebra and say, “the homework was fun!” (What?!) All the students were doing were practicing some complex manipulations with rules of integer exponents. (Can’t think of a real world use for simplifying 〖4x〗^(-3)/(8〖(xy)〗^6 )). Yet the conversations were rich, as in, “will 3^4/5^4 ever have common factors? And finally seeing ∛125 as 〖〖(5〗^3)〗^(1/3) and just swooning with the excitement of seeing something so gratifying!

Sometimes you just never know what will light a fire under their butts!

## Cathy Yenca

December 4, 2013 - 2:36 pmI think the “obtainable challenge” idea that makes gaming so appealing has a lot to do with students’ motivation and “buy-in” to the math we sell, regardless of it’s “real-world-ness”.

Here are a few samples I used just yesterday in class with my 8th graders:

http://www.mathycathy.com/blog/2013/12/sort-of-real-world-math/

## Diane

December 4, 2013 - 8:59 pmI love this post and the comments. I’m glad to hear this question discussed by experienced teachers as I’ve wondered about it during my teacher training (just finished and am set to begin teaching in January). Throughout the teacher training, there is huge emphasis on the “real world application” theory of engagement but I know it’s at odds with my own experience so I think there must be more to it. For myself, real world applications can be tricky because Bring in one that involves something I’m not interested in and I’ll switch off no matter what the maths is. Give me a puzzle or abstract maths and I’ll be happy.

Two of my favourite elements in problems are:

– pleasing unexpectedness (like discovering a pattern or a connection to another concept)

– changes in pace eg Slow hard going followed by a flurry of activity as you ‘get’ something and a few different parts fall quickly into place.

## Jason Dyer

December 5, 2013 - 9:20 amPart of what turns a “classroom math problem” into a “puzzle” is simple presentation. A Professor Layton game for the DS can get away with a system of linear inequalities if it is presented as different colors of cats.

## Chris Hunter

December 5, 2013 - 10:09 amYou call this a limiting theory of engagement. I call it simplistic. Potato, potahto. (Sad to see Diane’s experience of this theory being the dominant one in her teacher training. Probably not atypical.)

Race to 20, a Nim variation, is enjoyable. No context, just naked numbers. Taking turns, players count on one or two numbers (i.e., you say “one, two”, I can say “three” or “three, four”); the player who says “twenty” wins. Students enjoy the challenge of finding a winning strategy. Some form of success comes fairly quickly: most key on the importance of being the one to say “seventeen” which presents a new challenge (and pattern to discover).

Sometimes, I think teachers mistakenly believe it is the context, rather than the math, that engages students. Area vs. Perimeter problems, brought up above, are examples of this. I can present the problem as “You have 24/240 m of fencing. Build the largest pen for your pet dog/dragon.” Real? Fantasy? Dunno. Doesn’t matter. Spot/Sparky are quickly forgotten. What engages students is the challenge of finding the largest area given the perimeter constraints. I’d say it’s competition, but it’s not really about having the largest area within the class– more about bettering one’s previous attempts. By the way, this elicits argument (“Does 6 by 6 count?) further engaging students.

## Josh Britton

December 6, 2013 - 8:29 amRe #8: Thanks, Kenneth, for the link to your essay. Nicely succinct skewering of ‘everything’s better in a game’ blather.

## Josh Britton

December 6, 2013 - 8:35 amI recently spent several hours obsessing over peg solitaire. I enjoyed the trickle of pattern revelations found through experimentation, but I suspect that the most powerful motivator was the desire to end up with one peg all alone in the middle. Never happened.

Competition helped: my daughter was playing, too, and we frequently compared scores.

## Kevin Hall

December 6, 2013 - 9:58 amI think student engagement depends on specific characteristics of the connection made between something already in students’ minds and something new. The thing already in their minds could be the memory of an experience, an preference or attachment they hold, an opinion or idea they have formed, etc. The term “prior mental construct” or just “prior construct” may capture what I’m describing here. This is completely subjective and has little to do with what’s in the real world.

Using that term, engagement is proportional to: i) intensity of the prior construct in my mind, ii) the number of connections made, iii) the conceptual distance between the prior construct and the new experience, and iv) the strength of that connection. Something like Engagement = (Intensity of prior construct)(Number of connections)(average Distance of connection)(average Strength of each connection), or E = k(I)(N)(D)(S). Here is what I mean:

i). The intensity factor means my engagement will be higher if you make a connection to something I care more about, remember more vividly, etc. Connecting quadratics to snowboard sales data won’t work for me, because snowboard sales data is not something I’ve ever thought about before.

ii). The number of connections is important because I’ll care more if you can connect one math concept to many prior constructs. You have to make me feel that when I’m in my mental world, I’m somehow surrounded by applications of this math concept.

iii). The average distance is basically the surprise factor, and it’s subjective. It measures how crazy I find it that the thing you taught me is actually connected to something else. For my students, realizing that completing the square allowed you to derive the quadratic formula was an experience like that. They had no idea those two concepts were going to be related. In addition, for many students, the experience of doing math is so unlike the experience of living that anytime math can predict something real, it’s a surprise. (How long will it take the water tank to fill up? I can’t believe math actually works to answer that! I thought math was just math).

iv). Finally, the stronger the connection, the more I’ll be engaged. When it comes to 3-Acts, this is why it’s so vital to show Act 3 rather than just telling students the answer. Playing out the tape shows that this connection really works, and isn’t contrived with a mere “textbook answer”. The strength of a connection also provides satisfaction to students as they move up the Van Hiele levels in Geometry. What originally seemed like a bag of unrelated characteristics (4 right angles, two pairs of parallel sides) begins to have some coherence and necessary dependence. That’s satisfying, as long as the connections actually make deep sense.

Anyways, that’s my theory.

## Michael Pershan

December 7, 2013 - 5:27 pmCheck out this quote from the NY Times editorial about how math teachers suck and all our kids are finding math boring but the kids would be stoked if they were being prepared for STEM careers.

When you started this series, I thought that you were attacking a scapegoat. Who really thinks that the only math that’s interesting to kids is “real world” math?

Answer: The New York Times, and maybe President Obama.

Anyway, the theory of engagement on display in the Times’ editorial is pretty clear. They think that kids are motivated to prepare for their careers, and they think that the closer school work hews to the “real world” (read: job world) the more excited kids will be for it.

There are so, so many problems with this idea that I’m pretty comfortable calling it stupid.

This is a different sort of “real world” than your snowboarder. But is it related? Does the idea that kids are into the real world spring from this notion that kids are interested in job prep?

## Bob Lochel

December 8, 2013 - 4:53 pmJust posted about a task I have used: The Hot Seat. The task is explained on my blog, along with a quick video example: http://mathcoachblog.wordpress.com/2013/12/09/my-fake-world-task/

What I see in these tasks:

1. They aren’t intimidating, and are equally accessible to many types of learners.

2. Their secrets are not immediately obvious, and invite us to experiment with out ideas.

3. Their solutions can be explained easily, once we become experts with the problem. Or the explanations can take different forms.

4. There is an opporunity to adapt the problem, or create a new problem based on our experience.

5. Finally, many of these problems are one which simply drive us crazy. We know the answer is staring us in the face, and there is inherent desire to be the “first” to get it.

## a Different Dave

December 10, 2013 - 12:12 pmI think comment #1 by Jared is absolutely the right direction to be going, in terms of discussing the actual pieces that make something engaging. A clear goal and a well-tuned feedback loop? That’s serious gold.

He mentions flow, though, and I disagree on that usage of the word flow. Cal Newport has made some really great progress in dissecting motivation in school and the workplace. He points out that flow is actually the lazy state. Flow is what happens after you’ve already learned what to do and internalized the process, and you’re just in plug-and-chug mode where you don’t really have to think.

A productive state that Newport strives for instead of flow, is “deliberate practice” – http://calnewport.com/blog/2011/12/23/flow-is-the-opiate-of-the-medicore-advice-on-getting-better-from-an-accomplished-piano-player/

The ability to use math concepts in the “flow” state might be a good end goal for our students, but we have to start with something closer to deliberate practice.

## Jared Cosulich

December 10, 2013 - 12:34 pmHow you specifically define “Flow” isn’t super-important, but I have personally frequently felt something that I would define as Flow (complete engagement to the point of losing track of time) when I program and it happens most consistently when I am tackling problems that are very challenging. In fact if they are not challenging enough then I will not get in to “Flow” (that’s literally one of aspect of the definition).

So I don’t think that the state of Flow as I think about it in any way is a lazy state.

That said, I would agree that deliberate practice may be a more efficient mechanism for improving, but it may also be more exhausting. If you’re making tangible progress, though, then that progress can provide a great deal of fuel.

I would certainly never discourage a learner who was engaging in deliberate practice, but I think you have to be really motivated as a learner to take that path.

## Dan Meyer

December 10, 2013 - 4:10 pmA few way-too-late notes on what I’m reading here.

Jared Cosulich:The first is pretty well-supported. I find the second one problematic, though. When you play Sudoku, you write a number in a square and there’s no immediate feedback that it was the right or wrong number for that square. The feedback comes much later when you’ve realized you’ve created a logical contradiction.

wwntd:So there is

eventualfeedback, not simplyimmediatefeedback. And that feedback has a certain character. Something to do with validation. The Sudoku book has answers in the back, but you can validate ityourselfby verifying that your solution meets every constraint.Mike Lawler:“Surprise” is surfacing a lot around here.

Jason Dyer:I can’t tell if this is a joke or not.

Chris Hunter:A couple of points worth highlighting here:

1.) Optimization problems have a certain appeal, not necessarily for their competitive angle, but because they give everyone a place to start and then improve.

2.) When I find a productive and fun real-world task, it’s almost always the case that the “real world-ness” is its least salient aspect.

Michael Pershan:Kind of a few. This is a recurring topic of conversation between Mr. Mathalicious and me also.

Michael Pershan:The job world is a subset of the “real world” but the idea that kids will dig into math if teachers enforce its connections to the job world is problematic.

## Jared Cosulich

December 10, 2013 - 4:41 pmDan,

I agree. The use of the word “immediate” is misleading as I think it’s more of a spectrum. With Sudoku I think the feedback is still immediate enough to create engagement on easy puzzles and the feedback becomes less immediate on harder

puzzles.

In fact I’d say that you can control the challenge of any given puzzle or challenge by varying the immediacy of the feedback loop. Creating a feedback loop that is instantaneous may actually make a challenge too easy, potentially reducing engagement.

That said I think most of the time your better off providing more immediate feedback if you are looking for engagement. You can tackle more challenging problems more easily with a fast and consistent feedback loop, so the threat of making a challenge too easy can be compensated for through a more challenging problem.

## Patrick Vennebush

December 10, 2013 - 6:13 pmI have three examples of things that look “real world” but really aren’t, though kids love them.

Paper Pool: I like the version from Illuminations, though it was in CMP before that, and the problem is at least a century old:

http://illuminations.nctm.org/Activity.aspx?id=4219

Product Value Problem: I first heard this from John Horton Conway on NPR:

Find an English word for which the product of its letters (i.e., the product of the letters’ positions in the alphabet) is 3,000,000.http://mathjokes4mathyfolks.com/problem_productvalue.html

Name Letters: A completely ridiculous 29-equation system that doesn’t LOOK like a system of equations problem, and kids just dig it:

http://mathjokes4mathyfolks.com/problem_namevalue.html

(For this one, I think the appeal is that the question asks students to find the value

theirname, which makes it personal.)## Dan Meyer

December 10, 2013 - 7:30 pm@

Patrick, nice examples. Name Letters seems to tip-toe the line between familiar and strange that a lot of people have mentioned. It has, like,names… which are familiar … but the names haveprices… prices which seem kind ofarbitrary… and that’s a littlestrange.## Federico

December 16, 2013 - 2:44 pmPerhaps I’m a bit late to the party here… but I’ll give it a go:

Collatz Conjecture is one of my favorite “Fake-math” topics to explore with students. I wrote about how I explored it last year with my 5th/6th grade class.

http://artofmathstudio.wordpress.com/2013/10/06/fun-with-collatz-conjecture/

Perhaps what make it most intriguing is that it is simple to state, yet has eluded some of the most brilliant mathematicians’ attempts at proof. I usually drop in the discussion that a proof of this conjecture could even earn you an honorary Ph. D, or at the very least instant rock-star status in the math world.

Also students can have minor victories along the way to keep them motivated to explore, like proving that all powers of 2 will eventually go to 1.

## Federico

December 16, 2013 - 7:35 pm…and by “fake-math” I meant to say “fake world” math.

## Joshua

December 24, 2013 - 2:55 amObviously there are other dimensions missing in your engagement model, but one seems especially worth highlighting: the most compelling problem can be the one that is not what you are supposed to be working on. An example from my office yesterday:

I was supposed to spend some time building a forecasting model for UK interest rates based on various economic indicators. Instead, I spent time formulating a model of the break-even cost for aged port (a problem hinted at by your post http://blog.mrmeyer.com/?p=18174). To be clear, this has no relevance for my professional work. Also, actually, I didn’t spend much time on the wine model, but I found it really difficult to stay away from it!

In contrast, another co-worker was meant to be recoding some of his old models and, instead, felt compelled to work on my interest rate/economics model!

I feel this is also somehow the punchline behind some of the Vi Hart videos: the math is really fun when done subversively.

I don’t know how to engineer this as a classroom technique.

## Paul Hutchinson

January 30, 2014 - 9:37 amSide note: there are many real world applications for Shikaku, the same maths for instance is used in determining how to position planes on an aircraft carrier.

This is all.

– Creator of shikakuroom.com

Paul.

## Chris H

March 21, 2014 - 6:08 am“real = fun / fake = boring”

Curious. Games are fun specifically because they are NOT real world. Instead games are abstractions, yet are “fun”.