## Dear Mathalicious: Which Of These Questions Is “Real World”?

An ongoing question in this “fake world” series has been, “What is real anyway, man?”

Are hexagons less real-world to an eighth-grader than health insurance, for example? Certainly most eighth graders have spent more time thinking about hexagons than they have about health insurance. On the other hand, you’re more likely to encounter health insurance outside the walls of a classroom than inside them. Does that make health insurance more real?

I don’t know of anyone more qualified to answer these questions than our colleagues at Mathalicious who produce “real-world lessons” that are loved by educators I love.

I’m sure they can help me here. Here are three versions of the same question.

Version A

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle.

Find P such that the area of the square and circle are equal.

Version B

When do the circle and the square have equal area?

Version C

Where do the circle and square have the same number of candies?

Version D

[suggested by commenter Jeff P]

You and your friend will get candy but only if you find the spot where there’s the same number of candies in the square and the circle. Where should you cut the line?

Version E

[suggested by commenter Mr. Ixta]

Imagine you were a contractor and your building swimming pools for a hotel. Given that you only have a certain amount of area to work with, your client has asked you to build one square and one round swimming pool and in order to make the pools as large as possible (without violating certain municipal codes or whatever), you need to determine how these two swimming pools can have the same area.

Version F

[suggested by commenter Emily]

Farmer John has AB length of fencing and wants to create two pens for his animals, but is unsure if he wants to make them circular or square. To test relative dimensions (and have his farmhands compare the benefits of each), he cuts the fencing at point P such that the area of the circle = area of the square, and AP is the perimeter of the square pen and PB is the circumference of the circular pen.

Dear Mathalicious

Which of these is a “real world” math problem? Or is none of them a real-world math problem?

If anybody else has a strong conviction either way, you’re welcome to chip in also, of course.

Featured Mathalicians:

These are the two ideas that seem to be confused here are just “real world” and “interesting”. There seems to be an inherent assumption that they are somehow related when I doubt they even should be. Sometimes they will overlap, and sometimes not, but it is a bit random and quite personal in my opinion.

I caught the maintenance staff at my high school using “real-world math.”

The problem probably lies in the many ways that one could define “real world”.

What qualifies as “real world”?

• A problem simulating a project/job/task being performed by someone performing their normal job duties, such as the example of the contractor building a pool to meet municipal code?
• Is a problem involving objects or tasks that would be considered an experience students are likely to have “real world”(dividing up the M&Ms to be equal size)?
• Actual evidence of math in the “real world “(video or otherwise) being applied as a part of someone’s job (the maintenance staff example above from @Kevin Polke) could qualify, or perhaps the application of math to a problem that is foreseeable in that person’s job duties.

I think it’s best to take the more pluralistic viewpoint on this one, as it would be quite a task to attempt to statically define the exact nature of “real world” math, as there are always countless more examples lining up to disprove whatever narrow definition you choose.

Liz:

“Real world” is just a means to an end. The goal is interest.

What matters for a ‘good task’ is not whether it’s real, it’s whether
1) its meaning is clear right away
2) kids want to solve it
3) have the mathematical tools to solve it (even if not very sophisticated tools)

I’ll say again though: I don’t really care if a problem is real-world. There are so many great problems that aren’t, and so many terrible problems that are. I don’t think it carries huge added value. Everyone decides what’s “real” to them (as Jeff P said). Right now for my kid, 7-5 and 5-7 being related to one another is plenty real, even though there is no connection yet to physical objects or money or any of that.

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2014 Mar 26. Fawn Nguyen asked her eighth grade geometry students which version they preferred.

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1. #### Jeff P

March 10, 2014 - 3:54 pm -

None of them are real world. And the second two hurt my eyes. But the first one hurts to read so I guess it evens out.

If somehow the candy one involved two kids each getting candy and the goal is to make it shared equally, that’d be getting closer to something that might actually be intriguing. But I’m not sure how to link that up with circles and squares. Those seem rather arbitrary. You and your friend have a piece of paper, how can you use it to create two baggies for candy? See that doesn’t work. If I’m a kid I just stuff the candy in my pocket / mouth, Real World Problem solved.

2. #### Dan Meyer

March 10, 2014 - 4:17 pm -

Jeff P.:

If somehow the candy one involved two kids each getting candy and the goal is to make it shared equally, that’d be getting closer to something that might actually be intriguing.

This paragraph raises lots questions that interest me.

One, so if a problem doesn’t intrigue a student, then it isn’t “real world”? Are all intriguing problems also “real world”?

Two, taking your suggestion, if I use the candy task and tell students, “You and your friend will get candy but only if you find the spot where there’s the same number of candies in the square and the circle,” is that problem then “real world”?

3. #### Mr. Ixta

March 10, 2014 - 4:28 pm -

I used this exact problem with my students and applied it this way:

Imagine you were a contractor and your building swimming pools for a hotel. Given that you only have a certain amount of area to work with, your client has asked you to build one square and one round swimming pool and in order to make the pools as large as possible (without violating certain municipal codes or whatever), you need to determine how these two swimming pools can have the same area.

When it comes to real world application, this was the best I could come up with. Interesting enough, my students did discuss the effects of floor plans for things like homes, parks, playgrounds, and many other things and how to maximize area using squares and circles.

4. #### Bowen Kerins

March 10, 2014 - 4:30 pm -

None of these three problems strikes me as anywhere near “real-world”, unless nobody ever taught me that squares and circles always have candies.

5. #### Nico

March 10, 2014 - 4:33 pm -

I saw you playing around with this problem earlier, and the visual (Version B) does help me bring it closer to life. It is more real, but not necessarily more real world. It is less abstract if I see the circle and the square. Even better when I can move it myself (I hope that is an option soon).

In Version A, once I read the words ‘Given an arbitrary…’, I was hit by the wave of mathiness in the question. By the time I read ‘…circumference of a circle’ I was drenched. Don’t get me wrong, it’s the kind of mathiness I love, and wished my students loved as well (and some do). But some students still prefer to stay on dry land as we all know.

Version C (if that is candy) seems like it is trying too hard. The candy seems out of place.

So my answer is B…if the question ‘which version do you prefer’.

But I feel inclined to reduce some of the mathiness to make it more real.

“Draw a circle and a square that have the same area” is something I would like to try with my students. And although it doesn’t have the line, or point p, I think it sufficiently gets the student’s feet wet before being hit by an arbitrary tsunami.

And now I feel I’ve taken the water metaphor too far, so, I’ll stop there.

6. #### Dan Meyer

March 10, 2014 - 4:42 pm -

It seems as though our determination of whether something qualifies as “real world math” or not is, as Justice Stewart said about pornography, “I know it when I see it.”

I had hoped for something less subjective.

7. #### Bowen Kerins

March 10, 2014 - 4:58 pm -

My own determination of whether something is “real world” is based on where it came from. If it came from a math teacher it’s basically not real-world, unless it was a math teacher doing something in the outside world leading to an interesting problem in mathematics.

Problem E is pseudocontext to me, even though it can legitimately be solved and the entire problem is presented. It’s a contrivance. The work to bring the problem from that context to mathematics doesn’t add anything to the problem, so I feel it should probably be presented without context.

None of A, B, and C (to me) come from anything that happened in the real world. Equally so, this (to me) has no impact on their viability as math problems. A, B, and C all have viability in the classroom. I prefer A, I feel like some students will find the right point for B using technology, but then won’t transfer that to the discussion about why it is correct. Just a gut feeling that field testing could answer…

8. #### Josh Bobbitt

March 10, 2014 - 5:00 pm -

I always have a hard time with this, especially when it’s “real world” vs. just plain interesting.

I feel like that top question is just interesting–I want to solve it right now, and my brain is a little antsy that I haven’t yet.

The second question feels like a model of the first, and helpful for understanding the abstraction.

That doesn’t however, mean it’s always going to be interesting for kids.

(And perhaps the point you’re making, Dan, is that “real-world” isn’t the thing the whole enterprise rests on…)

9. #### Michael Pershan

March 10, 2014 - 5:05 pm -

There seems to me little point in searching for the true meaning of “real world,” since it’s a messy human term that’s applied by all sorts of people in all sorts of situations. There are times, though, when it’s valuable to sharpen our use of language, and I take this post as an attempt to do that.

Really, what you seem to be getting at is the fact that there are two different common ways that educators use “real world.”

1) Something is “real” to a student if it’s concrete, attainable, comprehensible. A teacher might make quadratic functions “real” by asking students to find the next step in a squaring pattern.

2) Something is “real” to a student if it has a non-mathematical purpose. A teacher might make quadratic functions “real” by showing students how to predict where a golf ball will land with a quadratic function.

There are interesting things to say, I think, about the coherence of the second usage. What counts as a non-mathematical purpose? Does joy or pleasure count as a non-mathematical purpose? There’s always going to be an alternate, non-mathematical approach to solving any non-mathematical problem, right?

But not everyone is interested in investigating the coherence of commonly used terms, and that’s OK too. Language doesn’t need to be precise until it has to be. We all know what a non-mathematical purpose is, right? It’s the sort of thing that people solve out there, in the real world.

10. #### Emily

March 10, 2014 - 5:37 pm -

“Real World” to me equals “application in realia,” where segments, points, planes, variables, and equations are everyday concepts. When I read this Stanford problem originally (and subsequent to your post bought the book), I was thinking of AB as, say, a length of rope or wire fencing.

A variation of Version E: Farmer John has AB length of fencing and wants to create two pens for his animals, but is unsure if he wants to make them circular or square. To test relative dimensions (and have his farmhands compare the benefits of each), he cuts the fencing at point P such that the area of the circle = area of the square, and AP is the perimeter of the square pen and PB is the circumference of the circular pen.

Were this an activity for students, I would probably include some follow up questions about the benefits of choosing these metrics (why circle? why square?) and seeing if it makes a difference if we say that Area circle = Area square, or if we say that AP = PB. Would Farmer John be able to make pens of greater area if he used AB just to make 2 circular pens or if he used AB to make 2 square pens? How might P or area or dimensions change if one of the shapes was an equilateral triangle? As a reflection: how might choosing to construct this model (the original problem: 2 pens, one circular, one square, Perimeter = Circumference) help the ranch hands decide what is best for their animals? (MP4, right there, baby.)

11. #### Chuck Collins

March 10, 2014 - 5:55 pm -

Version A is the only real problem.

Version B is a nice visualization but it reveals too much, and in my mind, makes the requisite math too technical and uninteresting. I see the answer in B; I don’t want or care to solve for it.

Versions C, D, E, … feel like the added elements are just there trying to make them more real in a textbook sort of way.

So, maybe a definition of a real world (or just real) problem is that one you come to as you move from the pure math/abstract version towards the dreaded over-developed textbook version, where anything else you add is perceived as extra.

12. #### Karim

March 10, 2014 - 6:24 pm -

Great question. I’ve actually been thinking for some time about the nature of “real,” and how it translates into how one evaluates what can/should happen in a math class. I’d like to be the [first] one to respond to the “Hey, Mathalicious” invitation. However, I’m still in New Zealand and will be at a conference until next week. May I blog a response on our site next week?

In the meantime, I really do think it’s one of the most important questions that’s currently being asked in math curriculum land: how much time to spend on topics that one legitimately encounters in the real world, and how much to spend on the more problem-solving type. My personal take is that it comes down to a question of balance, and this is a question that I am also really curious about. I think the reason your question is so valuable is, as you seem to be suggesting and other posts, there’s a real risk of sending the pendulum too far in one direction the neglecting the other, which of course down the road will just result in exactly the same in the other direction. Hopefully we can nip that silliness in the bud.

13. #### Dan Meyer

March 10, 2014 - 7:26 pm -

Bowen Kerins:

My own determination of whether something is “real world” is based on where it came from. If it came from a math teacher it’s basically not real-world, unless it was a math teacher doing something in the outside world leading to an interesting problem in mathematics.

I’m seeing two criteria here.

One defines itself by a negative: “Not from a math teacher.” The other criterion seems to beg the question: “What makes ‘an interesting problem’?”

I think I’m following your line but I could use some more clarity.

Josh Bobbitt:

(And perhaps the point you’re making, Dan, is that “real-world” isn’t the thing the whole enterprise rests on…)

I’m guessing that if anybody can articulate what makes a task “real world,” it’s the fine folks at ML. Otherwise it might be time to call off that hunt and chase some other criteria.

Chuck Collins:

Version A is the only real problem.

Hey whoa now. You’re going for extra credit there, trying to define “real.” At this point, I’d just settle for a definition of “real world.”

Michael Pershan:

Really, what you seem to be getting at is the fact that there are two different common ways that educators use “real world.”

On the evidence of this thread alone I’d say we’re seeing many more than two meanings for “real world,” to say nothing of all the reform documents, speeches, and op eds that invoke the term.

Michael Pershan:

We all know what a non-mathematical purpose is, right? It’s the sort of thing that people solve out there, in the real world.

You devil.

Karim:

I’d like to be the [first] one to respond to the “Hey, Mathalicious” invitation. However, I’m still in New Zealand and will be at a conference until next week. May I blog a response on our site next week?

Nice! I promise on behalf of everybody here that we’ll still be interested in your response next week.

14. #### David Taub

March 11, 2014 - 12:04 am -

I think that part of the problem is the mixing of two separate but important concepts here. This seems to come from the lack of a clear goal as to the actual reason to make or define a problem a real world or interesting.

These are the two ideas that seem to be confused here are just “real world” and “interesting”. There seems to be an inherent assumption that they are somehow related when I doubt they even should be. Sometimes they will overlap, and sometimes not, but it is a bit random and quite personal in my opinion.

There are many very important applications of math that easily qualify as real world I think in most people’s minds that are not at all fun. A good example would be doing mortgage calculations when buying a house. Anyone who is buying a house, or a car, has to do (or get help with) a lot of important math that will have a giant impact on their lives for many years to come. I doubt anyone argue that this is not “real world”, however very few people find this “fun” or “interesting”. Most people consider it a necessary evil. I believe the majority are “happier” when they have math skills to understand what is going on and participate in the process more, but that does not make it “fun” or “interesting” for them.

I love math, I think it is fun and interesting. I find nothing at all interesting in mortgage calculations. I think problems like the circle and the square one are very interesting and as soon as I read a problem like that I want to solve it, regardless of how it is worded. The way it is presented makes not difference to me personally at all, as long as I can understand what is being asked (okay, not entirely true, some attempts to force an abstract problem to appear more concrete can annoy me personally). But that is just me and I know I am not representative.

This seems to be about student perceptions – and I think part of the problem with this discussion is the question is removed from the reason for asking the question too much, which makes it harder to answer (maybe this is a lesson in itself for teachers).

Why do want problems to be “real world”? Why do we want problems to be “fun” or “interesting”?

To me, these questions are separate and need to be treated that way.

I can see getting students interested in an abstract problem as game and having fun with it.

I can also see “motivating” students to learn how to do mortgage problems even if they hate doing it. I can see them understanding the eventual need for that skill and though they may grumble they will learn how to do it because they realize it will be useful. On the other hand, I can see having students in the same situation who come from very poor families or areas and so think they will never buy a house and get angry at learning how to calculate a mortgage payment. Or students who just aren’t mature enough to care about learning anything they might find useful ten years down the road.

All this rambling is to say I think the goals here are not clear, at least to me, which makes answering these questions more difficult.

15. #### Iain Mackenzie

March 11, 2014 - 12:27 am -

The question to ask is what math are the students we teach using to solve their day to day problems ? That would be a real world problem. After teaching quadratics do our students apply it to their real life. I doubt it. Though one student of mine used simultaneous equations to decide how much of each type of liquor to buy for her teenage party! Context can make it interesting but I am sure the students don’t believe the often contrived context that are suggested. Interesting…….. lets have loads more because the majority of students don’t really find math interesting or relevant to them unless it is presented in an interesting way. I think it is time to stop pretending it is “real world” unless it actually is!

16. #### Gerry Rising

March 11, 2014 - 4:08 am -

Great post, Dan. And I believe that it makes an important point about applications. A case could be made for “who cares” about these beautiful applications. They won’t solve any important “real world” problem. But they are attractive, they involve some good math and they motivate addressing that math.

On another site we have physics teachers calling for math to be taught to serve their needs. This is a good example of the kinds of things that go well beyond the applications that physics teachers need and want us to concentrate on in math class.

17. #### Gerry Rising

March 11, 2014 - 4:17 am -

As to the suggestions for different frames for the same problem, I consider C to be a significantly more challenging exercise, but A and B are equivalent. I see no special purpose in framing the original A exercise in many ways before addressing it. Rather, I think that suggesting those various applications once the math problem is solved, is a good way to point out how a single math exercise solves many problems.

18. #### John Chapin

March 11, 2014 - 5:07 am -

It seems the whole purpose of the “real world” debate is not to show the students how math can be applicable, but to have them sit on the edge of their seats (in a perfect world) and be eager to work through the problem and gain an deeper understanding of the concept. As opposed to giving the students the problem and watch as their eyes glaze over as they check out and they start to wonder how they will survive until the next class.

Each person responding to this post finds a different version interesting. Frankly, version A makes my eyes glaze over, but I find version F interesting.

What if it is the same in the classroom? What if you offered all of the versions to the students and asked them to solve one of them? Would they all pick the same one? I doubt it.

19. #### Kevin Polke

March 11, 2014 - 6:29 am -

Just for fun:

I caught the maintenance staff at my high school using “real-world math” — see image at link http://imgur.com/KV3IOoq

Love the blog, Dan, love these discussions, keep it up everyone!

20. #### Ben

March 11, 2014 - 7:22 am -

The problem probably lies in the many ways that one could define “real world”.

What qualifies as “real world”?
-A problem simulating a project/job/task being performed by someone performing their normal job duties, such as the example of the contractor building a pool to meet municipal code?
-Is a problem involving objects or tasks that would be considered an experience students are likely to have “real world”(dividing up the M&Ms to be equal size)?
-Actual evidence of math in the “real world “(video or otherwise) being applied as a part of someone’s job (the maintenance staff example above from @Kevin Polke) could qualify, or perhaps the application of math to a problem that is foreseeable in that person’s job duties.

I think it’s best to take the more pluralistic viewpoint on this one, as it would be quite a task to attempt to statically define the exact nature of “real world” math, as there are always countless more examples lining up to disprove whatever narrow definition you choose. If I understand mathematics correctly, then the ultimate goal is to reach a level of abstraction so that what you’re working with can be applied generally across many fields and situations. This is turn, makes it more difficult for learners to work with more mature mathematics concepts. Any examples of math being directly applied to a “real world” situation in which mathematics is being used to describe a physical (analog) relationship or digital (software) relationship would be fine in my book.

However, practically speaking, for the 99% of the world population that AREN’T Astrophysicists and Engineers, “real world” math would probably cover most of the type of math that the average person needs to be able to successfully make it through their day, whether it’s automatic to them or not.

March 11, 2014 - 7:35 am -

If you Google the result of the ratio of the perimeter to the circumference (or the radius to the side), that’ll give plenty of real-worldedness. But the fact that one can discover that the ratios are equal is real-(platonic)-world enough for me.

22. #### Liz

March 11, 2014 - 9:12 am -

“Real world” is just a means to an end. The goal is interest. So I’d make version B into a game where you can drag P around and mark when you think the areas are equal.

I’d introduce the game at the elementary level as a primarily visual thing, plus some number-matching. Have different shapes, have 3 shapes at a time, have one shape that’s just changed by the length of AB. Really let kids get a feel for perimeter vs area.

Then in algebra, it’d be “hey, remember this game from when you were a kid? Here’s a version where you can put in this point directly. Figure out a way to get the right value on the first try.”

23. #### Kevin Hall

March 11, 2014 - 10:23 am -

@David, adults sometimes have to deal with mortgage calculations, but students never do, so those are not real world computations for students. “Real world” is completely subjective. It basically means “connected to something I’ve thought about or cared about before”.

The tricky aspect is that because culture beats curriculum, you can get students to think about or care about whatever your classroom culture emphasizes.

Puzzlement is a better lesson design goal than real world, and fortunately, a desire for puzzlement is exactly the attitude needed to sustain a growth mindset in real life.

24. #### Matt

March 11, 2014 - 10:44 am -

At this moment, I would say the animation in B (or C) is closest to a “real world” problem, because it requires a sense of inquiry to come up with a question to pose, represent, and answer. Doesn’t mathematical thinking in the “real world” imply inquiry, whether abstract or applied? Does reducing a problem to application and calculation take out the mathematical thinking of the “real world?”

25. #### Jane Taylor

March 11, 2014 - 10:53 am -

I didn’t take the time to read everybody’s comments, so this may be a repeat. If so, I apologize for wasting your time :)
That being said, none of them are “real world” and who cares? The important question–Is the problem interesting enough that someone would want to solve it? I am personally more intrigued (as a math person) to just try to figure out when the areas are equal. I like the visual if I could just get it to pause! Other people may find it more interesting if candy or swimming pools are involved. So we work at finding the hot button that works best for our particular students. (I remember reading a anecdote about an elementary teacher who never could get a boy in her class to do anything until she incorporated his love of John Deere tractors and then he just took off learning.) A lot of times our own hot buttons and passions for learning will rub off on our students. So it’s OK to teach things in a way that interests us, too. Those are my thoughts.

26. #### Dan Anderson

March 11, 2014 - 10:53 am -

I’m just putting this together (weeks later), but I’m now sure that the candy version is much much lower down the difficulty ladder compared to the abstract gif. You could stop the candy gif at the right time and get the answer without having done any “math” at all. I too made the problem far simpler with my desmos example that calculated out the areas on the fly. Just thought I’d pipe in.

27. #### Neil Adam

March 11, 2014 - 3:04 pm -

Real world = “what I need to know now”.

Until then it is all contrived.

I *prefer*option B. Perhaps I am just more visual with some stuff. (Ack, ack, nearly mentioned VAK. Will now wash my mouth out.)

Option C looks like it could be solved by a 3 year old. Actually, said 3yo could probably solve it in 3 dimensions using a whole tube of Smarties (US readers think M&Ms), so long as you promised they could eat the Smarties when they had solved it.

Option C also worries me because it doesn’t allow for real (ie. non-integer) solution, which probably makes the whole thing more than a bit hard.

I am beginning to hate the way that questions are being framed to make the context more real. “A ladder must be set against a wall so that the angle at its base is about 75 degrees. If the ladder is 2.25m from….” So, in the real world I’m going to get the tape out and start measuring? Then find my slide tables and log rules…. No, I’ll just find the line printed on the label on the side of the ladder that some elf and safety wallah insisted must be put there and make sure it is about vertical when I place the ladder (bearing in mind one foot is on soft soil and must be supported by a brick).

Kids can spot this stuff a mile off. “Karim and Aieesha are playing a game….” The kids in my neck of the woods ask me “who the heck are Karim and Aie… what’s that say?” They don’t care about Karim and his putative GF. (And Karim and Aieesha are too fussed about Kian and Chelsea from their perspective either.)

If it walks like a duck, quacks like a duck, but craps in the toilet pan, then it’s political correctness gone mad.

PS. Excellent answer David Taub and rejoinder from Kevin Hall

28. #### Dan Meyer

March 11, 2014 - 4:52 pm -

Some great commentary here. I added excerpts from David Taub, Kevin Polke, Ben Rimes, and Liz to the main post, as well as a couple of useful tweets.

For my part, I think if I could get consensus between all math teachers, policymakers, and journalists that a) the “real world” is defined in lots of different ways (per Ben Rimes) and that b) the “real world” shouldn’t be equated to “interesting” or vice versa (per David Taub), I could probably stop posting provocations.

But of course I can’t and provocations are fun so they’ll probably run just a little longer.

29. #### Ben

March 11, 2014 - 5:06 pm -

Thanks, Dan!

I had to go with a more open definition of “real world” thanks in part to an experience I had just today helping a Spanish teacher introduce blogging to her students. She asked them all to take a week to find one example of Spanish in the “real world”, and post it to the blog. Within 5 minutes, one student had posted a YouTube video from Vevo with a Latin pop star singing in Spanish. Who are we to say that YouTube isn’t the “real world” to our students?

30. #### Zach

March 11, 2014 - 7:31 pm -

The “real world”-ness is not the question; the question is accessibility. Students can access problem “B” much easier than any of the other problems, as “B” minimizes the amount of reading. Reading has affected my students, as I work among a high ELL population.

Dan’s “Pixel Pattern” task and “Nana’s Paint Mixup” task were very engaging to my students, but one is “real world” in the sense of a real situation and one is not. Yet, both were engaging because, I feel, they were accessible.

31. #### M Ruppel

March 12, 2014 - 2:51 am -

+1 to Zach’s comment above.

I’ll see if I can list some criteria.
What matters for a ‘good task’ is not whether it’s real, it’s whether
1) its meaning is clear right away
2) kids want to solve it
3) have the mathematical tools to solve it (even if not very sophisticated tools)

I think B and C meet criteria 1, but I like B better. A is an ok prompt, but there’s something about the wording I still don’t like.

Making it “real-world” as done in E and F seems like pseudo-context to me – it’s just a unnecessary layer on it to get students to do the problem that would be more understandable presented with a piece of string and a pair of scissors.

32. #### wwndtd

March 12, 2014 - 7:13 am -

I think there’s a big problem with what’s “real” to that person. Seems like kind of a cop-out, but my current needs/wants are probably pretty different from other people’s.

In chemistry, I have problems like, “If 50.0 grams of CaSO4 reacts with 50.0 grams of Na2CO3, how much product do you get?” Which is completely boring. But it’s a pertinent problem for someone actually working in a factory so they (probably really their manager) knows how many orders they can fill. But it’s absolutely not relevant to high school kids. And even when I do something like this in lab so they can make predictions, they’re still not motivated by it.

If my students actually go into industry, then they’ll use this stuff. I’m not convinced they’ll find it interesting even then, but they’ll use it.

33. #### Vishakha

March 12, 2014 - 1:01 pm -

I see this most useful as a progression.
1. Draw a circle and square with the same area. All kids draw a pair. Perhaps we use the desmos collaboration interface.
2. the desmos class framework – auto-measures perimeter, circumference and other things and area for all the figures from the class.
3. the students then investigate the relationship among these various things by simple two variable graphs

they then discover the relationship and as a abstract round off try to “derive” the theorem!

34. #### Kevin Hall

March 12, 2014 - 5:48 pm -

Believe it or not, there is empirical support for some use of pseudocontext. Here is a study that modified the cover stories for boring story problems to match students’ individual interests, and saw a statistically significant benefit. Here’s a more informalblog post about the same study. I know we’ve all agreed that calculating the areas of dog bandanas seems ridiculous. But perhaps stereotype threat explains the effect: if you’re a student who thinks math just isn’t for kids like you, maybe a little pseudocontext that relates to you is enough to make you say, “No, I guess this class is designed for people like me after all.”

Granted, it’s not going to make you suddenly become fascinated by the subject, but it does make a difference. This is what I meant when I said that the importance of “real world” is completely subjective–it really comes down to whether the lesson is connected to something I personally have invested some time/emotion in, or at least connected to something other students I want to be like have invested themselves in.

I guess I’m arguing that “real world” isn’t what we should try for in lesson design. We should try for “my world”.

Lessons can be successful with varying degrees of connection to my world. If you fully immerse the math in my world, I can become so hooked by the task that I temporarily forget I’m doing math–it’s like a movie with great special effects that you forget isn’t real. If the connection to my world is more tenuous, the effect is more like suspension of disbelief. I know I’m being tricked, but I’m willing to go along with it (I’m thinking here of the broadway production of The Lion King–you can see the puppeteers the whole time, but gradually you stop noticing them even though they’re always there). Not everything needs to be a Jerry Bruckheimer film in math class–puppets work too.

35. #### Jeff P

March 13, 2014 - 5:11 am -

Wow, leave the followup comments button unchecked and a ton of great ideas magically appear! Obviously whether or not something is “real-world” has a visceral meaning for math teachers (which I am not).

If I could, I would upvote Liz. When the choice is between something that’s strictly speaking real world (building swimming pools, ala Mr Ixta) and something “arbitrary” like a game that rewards kids for completing a creative challenge, I lean towards the game. You want this reaction in students: “how on earth can I figure THAT out?”, followed by attempts to do so.

Summarized by Chris Betcher:

1) Give them something to care about
2) Give them the tools they need
3) Give them choices
4) Get out of the way

People care about “real” things. They also care about Flappy Bird.

36. #### Bowen Kerins

March 13, 2014 - 9:52 am -

Rephrasing the “not from a math teacher”, I guess I mean that the problem isn’t invented out of a desire to meet a pedagogical goal. Problem E is clearly invented to real-world-itize something that would work equally well without the context. No one outside the world of math is wondering about Problem E, but there are probably similar problems in the world of architecture or product design. I’d prefer to research and see if any of those could lead me toward something like Problem A.

I’ve definitely run across interesting situations or problems where I think “Oh man that would make a good math problem” and I don’t think very many other human adults would jump to that as their first observation. Things like “What value menu option gives you the best price on fries and a drink?” or “Is that super-large Nutella really a better deal?” or “Could you always win at the yodely game on The Price Is Right if you just said the items were \$20, \$30, and \$40?” or even “What is the probability that these seven strangers would be picked to live in a house together?” To me, all of those are real world problems, because they are inspired by something out in the world.

I’ll say again though: I don’t really care if a problem is real-world. There are so many great problems that aren’t, and so many terrible problems that are. I don’t think it carries huge added value. Everyone decides what’s “real” to them (as Jeff P said). Right now for my kid, 7-5 and 5-7 being related to one another is plenty real, even though there is no connection yet to physical objects or money or any of that.

37. #### Gerry Rising

March 13, 2014 - 9:59 am -

I don’t understand all this concern about which is the better way to represent this interesting exercise. I think that it is a great problem and Dan has posted a remarkable illustration of how the area varies. That’s enough for me. But I will add that i consider the problem with various internal circles is (a) much harder and (b) almost certainly has an answer range rather than a specific point.

More interesting to me are possible extensions for the original exercise. Here are a few:

(1) cube surface vs. sphere surface
(2) cube volume vs. sphere volume (noticing how pi/6 appears in both)
(3) equilateral triangle perimeter vs. circle circumference
(4,5) the same for tetrahedron vs sphere areas and volumes

38. #### Andrew Stadel

March 14, 2014 - 9:05 pm -

Accept my tardy pass.

I’ve never introduced a lesson saying, “Key kids, here’s a real-world lesson that uses math.” and I don’t plan on starting. Even if the lesson stems from a real-life story, why should there be some label stuck on a math lesson in order to sell it to my students or me?

I’ve lost track of the questions in textbooks that talk up a story like it’s happened in the real-world, let alone possible. I become jaded. Sometimes textbook questions are like reading about greek gods: I don’t believe this, yet it’s kinda interesting.

My students can enjoy Mathalicious lessons. They can enjoy interesting puzzles/tasks like File Cabinet or Shipping Routes. They can enjoy doing review questions when we play math basketball. Estimation 180 contains pictures and videos of things captured in the real-world that I’m curious about or I find fun. I’ve never thought to label them as real-world. I mean, who puts green marshmallows on a skewer for St. Patrick’s day?

I agree with Karim and not letting the pendulum swing too far in one direction. My role as a teacher is to provide students with opportunities to strengthen their brains by using lessons (real-world or not) to assist with this process. Back in the day, I used to practice guitar A LOT so I could play shows in the real-world. I no longer play shows in the real-world, but still practice guitar because I enjoy it immensely. I like stretching my brain in the same way. I might not use math in the real-world, but I enjoy exploring it and it’s great for my brain. Therefore, just make lessons that are great for the brain.

Stealing a line from the Mel Brooks’ classic Blazing Saddles, “We don’t need no stinkin’ badges!”
We don’t need no stinkin’ labels.

39. #### Kevin Young

March 15, 2014 - 12:10 am -

“What is real?” is a troubling question! Sunrise was at 7:02 AM. Really?? The sun “rising” is an illusion. The hour and minute are based on an arbitrary system to designate time, and the observation of 7:02 AM is based on a particular season at a particular location. What impressed me is that the problem was presented in multiple ways–each approach made me think a little differently and increased my understanding.

40. #### Bryan Anderson

March 15, 2014 - 1:53 pm -

These examples are what I have been questioning for the past few years. Looking at our state standards, the buzzwords of “real world” are overused and can trap educators. Intriguing is also a word that I don’t want to throw out casually. Motivating seems to be the real key with students, what will get students to work on a problem and perservere even though they may cone along obsticles? I can’t say intriguing in this case because it implies student interest in the content, not just working on a problem to completion. With my students today, the candy examples are the most effective example that would illicit work.
None of these are practical real world, the fencing one is the closest, but then again farmers will go to any lengths to create straight-edged enclosures because you do not want to waste land area and.creating circles do just that.
All in all, my question would be which of theseÂ examples will get students to struggle to find an answer, and make a strong enough connection that students will remember the process if a similar problem was posed to them.

41. #### James N

March 15, 2014 - 4:06 pm -

I’d like to propose a version D.2, based on the “pie rule” – to encourage fair portions/arrangements one person divides portions of a pie, the other picks which portion they want.

http://en.wikipedia.org/wiki/Pie_rule

Version D.2

You and your friend will split as many candies as you can fit into the circle and square in version C, but here’s the catch: you have to pick the point on the line – you’re friend gets to pick the candy in the circle or the candy in the square.

Assuming your friend also likes candy, and picks more candy versus less – where on the line will you get the most candy?

42. #### Isaac D

March 17, 2014 - 10:36 am -

A. I agree with James Taub’s featured comment.
“Real-world” (whichever definition we use) and “Interesting” are two very different things. Both are objectives that most of us have as teachers of mathematics, and they very often overlap, but they are two different things.

B. My working definition of “real-world” (although I try to avoid the term whenever possible because of it’s ambiguity), would be something like:

Real-world
1. (applied to a skill, idea, theory, or procedure) Useful to most people outside the academic discipline from which the skill, idea, or procedure originates.
The Pythagorean Theorem is a real-world theory which is useful in a wide range of contexts outside geometry or mathematics (e.g. architecture, physics, computer science, etc. — its uses in algebra and trigonometry would not apply because those are within the broad discipline of mathematics).
2. (applied to specific student assignments) Reflects a plausible application of an idea/skill/theory/procedure to a situation outside the discipline in which it originates (e.g. a real-world problem which teaches the Pythagorean theorem must be a believable problem that a non-professional-mathematician would reasonably encounter and try to solve using the theorem and not some other method).

C. None of the above problems qualify as real-world.

Problems 1 and 2 are definitely interesting, and would be more effective if combined (multiple representations of the same idea). They (especially combined) help me understand the arcane vocabulary of geometry, appreciate the beauty of the relationship between area formulas, and apply algebra in solving a geometry problem (I would probably solve them by setting up the area formulae as a system of equations).

Problem 3 is less interesting to me because it is highly implausible, distracting (why are we putting candy in geometric figures?) and off-putting (you are insulting my intelligence by thinking I will only be interested in geometry if candy is involved).
It is also less effective at teaching geometry because it involves discrete integers which are only loosely related to the formulae for area of a circle and square.

43. #### Rene

March 23, 2014 - 6:49 am -

The question should be if the problem is motivating, not if its real world. That is not necessarily linked.

What is motivating?

– Acceptance. It this a problem that is accepted in my social environment to waste time upon? This many be connected to the realworldness of a problem.

– Reward. Do I earn respect for the solution I might get?

– Mastership. Is it a nice problem to try my skills on? Can I solve it? Maybe this is the strongest motivation. You can replace “I” by “we”.

Assume you tell your parents about this problem and the reaction is no reaction, plus someone in the class has solved it before, and you do not really understand the problem, nor do you have any clue on it. That’s a bad problem.

Sorry. You were asking about which presentation is more real world. Does not matter! All that matters is that you create an environment for motivation.

44. #### Ava Erickson

March 24, 2014 - 7:54 am -

I think Real World Math also has to so with how you solve a problem or how you allow or encourage students to solve a problem.

Here’s an example. You have a pizza cut into 8 pieces but only 6 pieces are are left. Four friends want to share the pizza evenly.

If you ask the question, “How much pizza does each person get?”, a student might figure: Give each person one piece. There are two pieces left. Cut each of those in half and give a half piece to each friend. Each person gets one and a half slices. (You could push students further to come up with 3/16.)

But if the question is phrased, “What fraction of the pizza will each person get?” Then you might get students to use fraction division. (6/8) / 4 = 3/16. The context might be real world but the solution will be artificial. Who does fraction division in the real world? It’s hard to think of genuine contexts. I usually end up doing some combination of addition, subtraction and multiplication before I pull out fraction division.

So I think how we encourage students to use their problem solving skills in a way that is natural and intuitive is also an important component of Real World Math. It’s not just about context.

(That said we do want to encourage students to grow their toolbox of math skills that feel comfortable and intuitive. We certainly don’t want our high school students to continue adding repeatedly instead of using multiplication.)

45. #### Stephen

March 25, 2014 - 6:15 pm -

It strikes me that the best way to get students saying “hey, I see the point of this” with regards to math is to give them a question they want to know the answer to FIRST, and then give them the math needed.

It also strikes me that the reason why we’ve been teaching it the other way around for so long (start from “here’s some math” and then “here’s a sucky question we can ask about it”) is that as teachers we DO start from content objectives that need to be taught, and planning backwards is hard. I don’t have good supplies. I don’t have enough time.

46. #### EB

April 6, 2014 - 12:11 pm -

There is another dimension to this issue of “real world” versus “interesting,” and a few have touched on it. Real world to whom? interesting to whom? even among a group of high schools students all more or less the same age, some problems will seem “real world” to one group of students, others to other groups. A few students will understand that pretty much all of the problems presented in Algebra or Geometry COULD be “real world,” in application for them, within a few years. Another group might find pretty much any problem that is new to be “interesting.”

47. #### Stephen Cavadino

April 8, 2014 - 8:42 am -

At something in history, someone decided “Maths needs to be more “real life”” and since then boards the world over have insisted on shoving context into questions for no good reason. If a question doesn’t need context don’t shove it on, and certainly don’t put ridiculous context in thar makes no sense! I’ve ranted before here: http://wp.me/p2z9Lp-9R

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