I don’t know, Dan, none of them seem to involve health insurance…

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

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…)

“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.)

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.

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.

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.

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.

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.

“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.”

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?”

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.

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.

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.

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.

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.

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.

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 these 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.

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
adj.
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.

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.)

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.”