So, I’m wondering what you make of the liberal use in CCSS-M of the phrase “real-world problems” and the possibly even more ubiquitous star (modeling) symbol? Applications is 1/3 of the three-legged table of “rigor” as defined here and also every time I’ve seen Bill McCallum address more than 4 people.

Here’s my perspective: real-world vs fake-world is not an argument worth having. Neither one is necessarily good or bad. Publishers have done a monstrous hack job of applications and real-world “problems” because what they publish are not really problems – they’re exercises dolled up with stock photography and decimal coefficients. What “applications” and “real-world” /ought to/ mean are that the symbols and numbers and diagrams a class is working with represent something that means something, and the relationship expressed means something, ideally because students generated it themselves. I don’t think it particularly matters if those things are instantiated by a question that civilians might legitimately wonder or if they spring more fully-formed from a puzzle/video/word problem/pile of sticks/whatever contrived by a math teacher. Contrast your lame textbook example with any of the modeling tasks on IM. here or here or here. Are these real world or fake world? Who cares? If used appropriately they’re legit problems (not exercises) and the symbols and relations mean something tangible. That’s a win.

I am focusing on math as puzzles to be solved in my geometry class. Circles, in degree and radian, circumference and area, all present linguistic and conversion issues that kids struggle with. There are puzzles there – keys are found in the patterns and relationships of formula. I relate these measurements to round foods, or percentage discounts, or sectors of a particular group – women who buy high heels as part of a group of all women who buy all kinds of different shoes. No matter – what does matter is how interested is the student in solving the puzzle. How curious or challenged is he/she.

Dan,

In previous posts last year, you and your community came up with the idea of “is it perplexing?” as the proper test for the value of a prompt. I really like that expression. It covers all the bases as to why anyone should want to learn math.

The word “puzzle” used in this post conjures up a narrower view of valuable prompts.

“The premise of this (hopefully brief) series is that we don’t have a lot of well-articulated ideas for creating fake-world tasks that students enjoy. We also have a lot of poorly-articulated ideas about the real world that result in tasks students don’t enjoy.”

Agreed on real/fake is not correlated with enjoyable/joyless. Interested to see where this goes.

That dissonance settles once I ask them to write down their best guess of how many stickies cover the thing.

What do we think is going on when you ask that question? How does an estimation question change how puzzling and interesting the question is? What’s the process?

In the past you’ve answered that question using the concrete/abstract framework, but now I’m struggling to apply it. That question makes the task more concrete? So the theory is that people find concrete things interesting?

At times, we’ve also suggested that offering a guess allows every student to have ownership over their particular answer, and then drives them through the rest of the task. So the increase in interest comes from a desire to show themselves be proven correct. Here the theory would be that people find opportunities to be correct interesting?

I would (very tentatively) suggest another theory. How interesting we find a task depends on two factors: (a) How valuable we perceive the outcomes of the task to be, and (b) The likelihood of reaching that outcome.

In this framework, what makes something like the File Cabinet problem interesting comes in two parts. First, there’s the perceived value of figuring out how many notes it would take to cover the cabinet. If you could answer that question, you would be really, really smart, because it’s a tough question. But, at first, the question seems really difficult, and the outcome seems unavailable. But then the question (“What’s a guess that’s too high?”) makes the task much easier by suggesting its connection to other things we know how to do.

That rush is perplexity. It’s the release when you’ve gone from jealously looking on an outcome that’s incredibly difficult to obtain to seeing a way to go about achieving it.

So what’s an uninteresting task? One that maximizes difficulty at the front, killing interest, and then progresses to make that knowledge extremely easy, common and available by being didactic. A bad task runs from one limiting case of this framework (high difficulty) to the other (low value).

Good & somewhat irrelevant information. The main question in my mind is still “The Point of Math Class”????????????

I am also searching for the meaning of math class. Love to hear more thoughts on that.

I can see the point of math class for kindergarteners – basic addition, measurements, clocks, money. After that, fractions are interesting and useful.

Rest seems a bit made up. Maybe we should be teaching programming instead. But, maybe programming is only fun because the standards writers haven’t got a hold of it yet.

Since I asked my students to make up silly scenarios as a means to an end, that end being to contextualize what and why we “need” or rather, “want” so many forms of quadratic equations, I would be interested in your opinion if the assignment “deadens” curiosity. For zeros, roots, x-intercepts we finally settled on this analogy: There are many ways to greet one another, and how you would greet your grandmother is probably not the way you’d greet your friend, but you KNOW how to greet each person. In the same way, I want you to KNOW why I may ask for the solution is so many ways./

I think math is beautiful and love the patterns and aesthetic of doing it. Is that enough to make it a worthy task? Not really, but my students certainly gain an appreciation for math and know their brains are more capable at the end of class than when they walked in.

Dan, you made my Quotes of the Week board with:
“Find those puzzles in the real world, the fake world, the job world, or any other world — it doesn’t matter.”

I couldn’t agree more. From my observations and experiences, students are (for the most part) magically intrigued by puzzles. I have baskets full of puzzles in my class. Students sometimes work with them and they yearn resolution. They’ll spend a few minutes on it, get stumped, give up, and then practically beg for the solution. It amazes me. I never get that reaction with a word problem displayed on the screen or a worksheet with practice questions. There’s something mystifying about students and puzzles. It’s a treat to watch students problem-solve, experiment, fail, try again, persevere and collaborate on puzzles. The trick (challenge) is finding mathematical opportunities/lessons where those same elements are infused within the process.

I appreciate your challenge (or charge): Find those puzzles. They’re definitely out there. Let’s now move to the next step: executing the delivery of the puzzle, the classroom facilitation, and possible resolution. There are many variables involved, and what might work in my classroom won’t work in someone else’s and vice versa. Knowing your students is important, that’s for sure.

Again, thanks for the challenge. Puzzles… onward bound.

I copied something I read on Dan’s blog about this, and keep it in front of me.

“The real test of whether a math problem is “relevant” is not “do you use this in ‘real life’,” whatever that means, but “do you want to solve it?” It’s not that you want to solve it because it’s relevant; wanting to solve it is what it means to be relevant.
It doesn’t matter how I reply when a student asks “when will I ever use this in real life?” I’ve already lost. Because that student isn’t asking a question about an uncertain future; she’s lodging a complaint about a frustrating present.”

What’s often motivating for my students is having them come up with the questions. Yesterday we gave the 5th graders a printout of all the movies playing at the Cineplex in town. It had all the movie summaries, ratings, times, etc. This along with ticket pricing info, and food costs and theater capacities we got from the manager. All they did was look at the info and start to generate some questions. They loved it. They had all been to the movie theater. There were movies they wanted to see. One kid noticed that there were 14 movies playing and only 13 theaters. They were interested that each theater had a different capacity. They loved reading the little “blurbs” and salivating over all the popcorn, soda, and candy choices. They laughed over the fact that the word “Jackass” was on the page. They wrote interesting questions that will involve addition, subtraction, multiplication, division, elapsed time, etc. One boy (who could be characterized as one of Dan’s “unengagables”) wrote a question about taking his family to a movie, and how much that would cost. At the end of the day he told me, “Mr. Schwartz, that’s going to really help me because we are all going to the movies! And there’s no way we should see Thor in 3D because that will be an extra $2.00 each!”
The questions they love and that they’re getting good at writing are the ones where the answer involves defending an opinion, for example comparing the relative values of food combinations and asserting that one was better than the other. They are also fascinated by the crazy questions like how much would it cost to buy all the snacks and drinks on the menu? One kid asked today, “How much time would be wasted if you saw Free Birds five times?” They will spend days solving each other’s problems, rewriting and revising problems that they realize are unsolvable, or are confusing because they are vague or have been poorly written, coming up with different answers, and resolving the discrepancies.
Their questions are much more interesting than most anything I could come up with. Sometimes they have to be tweaked a bit, or nudged in a certain direction, but what’s most important is that they own the questions. I find that then they are much more motivated to find the answers.

Dan:

The point of math class is to build a student’s capacity to puzzle and unpuzzle herself — no matter what form those puzzles take.

I’m wondering if by “puzzle” themselves, you’re really talking about helping students to build the ability to motivate themselves. Edward Deci, a psychologist at the University of Rochester, offers up this idea:

“The proper question is not, ‘How can people motivate others?’ but rather, ‘How can people create the conditions within which others will motivate themselves?’”

I suppose that as parents, teachers and curriculum designers, we should be focusing on the ability to place students in situations where they need to “unpuzzle” themselves, and then use mathematical concepts and skills to resolve their “puzzlement.” Maybe I am (also) showing bias based on my position as a parent with two young children and a teacher of middle school students, but I don’t readily see my own kids or my students willfully trying to “puzzle” themselves. But there definitely is engagement when they need to get themselves out of the “puzzle.” This is the time when they are ripe for being engaged in their learning, when they’ll do the heavy lifting to resolve their cognitive struggles.

Why are curriculum designers and teachers often willing to force the “real world” on concepts? I think Mr. Deci says it best:

“Because it’s easy. It’s much harder to work with people to get them motivated from the inside.”

It comes down to either fostering motivation or forcing motivation.

How do we generalize the characteristics of tasks , “real-world” or “fake-world,” that will engage students for learning? Look to those that place students in a “puzzling” situation without the students realizing they are getting into a “puzzle.”

The biggest puzzle is will I as a student be ready for the next level? Will I be able to function, problem solve, communicate, and compete at the next level? It hurts when I get an advanced algebra student that does not “feel” the difference between 3X and X-cubed. It hurts when they cannot subtract one-eighth from one-ninth. There is no flow when a student must take out their calculator to find 8 times 60. It perplexes my students into learning when I tell them they really do not have to do any math since there are thousands of students out there already willing to do the math and willing to take their spot at the UC, CSU or JC…

Hi Dan,

Thanks for a thought-provoking post. I am currently a pre-service teacher, and I’ve been asking myself how to appropriately answer this notorious question from my students, “How am I EVER going to use this?” While I can turn to the last few pages of a lesson in the textbook and point to a few “real-world” problems, I know that’s not a sufficient or satisfying response when the “real-world” problems don’t seem relevant to the students, like you mentioned. Then, I could respond by trying to share that developing problem-solving skills is what’s most important, and though that may be true, I find it difficult to get students motivated in math unless they see some relevant or interesting connection to their lives. I feel that it does matter what types of problems we give them to consider. You mention that teachers can “Find those puzzles in the real world, the fake world, the job world, or any other world — it doesn’t matter.” If it doesn’t matter what types of puzzles we give them, how would you propose motivating students to engage if they do find the work terribly disconnected to their lives?

An appropriate number sequence on the chalk board is engaging for most people of any age. It is not externally verifiable and doesn’t have anything to do with day to day experiences.

My guess is that it is engaging because it is very obviously about figuring out how something works, it seems like it is possible to figure it out, and there is immediate and clear feedback.

With so much math, real or fake world, the problem or dilemma involved isn’t presented very clearly (or isn’t even there) and it is cumbersome to get feedback on your answer.

Sorry, Last Comment: The question “Where will I ever use this math?” is usually an immature question…

Imagine the kindergartner asking “Where will I ever use the vowels?” One could answer that they are part of a cute ABC song or the ABC’s lead to words which leads to phrases which leads to sentences which leads to paragraphs which leads to pages which leads to articles which leads to books… But what if the youngster does not understand words yet? How does this young student conceptualize pages of words? Truthfully, trust in a qualified/competent teacher or parent is required. At some point reading and writing is part of everyday living just like different levels of math is part of “everyday chemistry” or “everyday economics” or “everyday nuclear physics”. How high should I go in math? Maybe the better question is where do you want to stop and limit your career choices or limit your ability to function? At what level do you want to limit yourself? What are your transitional goals? Do you WANT TO BE READY FOR THE NEXT LEVEL? Practicing the ABC’s and words leads to a flow and understanding of reading and writing just like proper algebra understanding leads to a flow and understanding of higher level math, science, business, engineering, etc. I contend that the biggest puzzle to solve is will I as a student be ready for the next level?

When the questioning student looking for a way out of working asks, “Where will I ever use this math?” I often kindly reply that maybe they don’t need the math at all, but others need it to be ready for this or that or the next level…

Gary,
“Where will I ever use this math?” is not an immature question, and is not related to a question like “Where will I ever use these vowels?” The math that these disaffected kids are talking about is not what they learned in kindergarten, how to count or recognize numerals. They’re talking about the soul crushing exercises they are asked to complete in their pre-algebra classes, the inane word problems, or the torture they undergo dividing 4 and 5 digit dividends by 2 and 3 digit divisors over and over again. In that context it’s a perfectly reasonable question and one that has no good answer. They weren’t born hating math, but they really hate math now. That’s a shame.

I agree the “when will I use this” question arises because students “[feel] bored and[/or] stupid.” And as I was reading the comments to this post, I imagined myself teaching, but without enough engaging, puzzle-ful lessons to cover all topics and, therefore, still facing “the question.” The vowels vs. variables argument might be a useful way to help students put their math learning in context. As a matter of philosophy, I think it is important to introduce students to the idea that there is something valuable, something worth working for, outside the here and now. They may not buy it the first time, or the nth time, but you plant the seed. This is not an excuse for using boring, brain-deadening exercises and calling it teaching – not by a long shot. I’m only saying I think there is a balance to be struck. After (note the timing) you gain students’ trust by piquing their curiosity and showing them they can be successful, I think it’s also beneficial for them to learn to survive those possibly dry times between puzzle-ful lessons by realizing they are building up to something (a level of math fluency.) Maybe comparing this process with learning to speak and write would be useful in making that connection for students?