Christopher Danielson, recounting an imagination exercise with his son:
Me: Now … the strange thing about this elephant parade is that each elephant is a bit smaller than the one in front of it, and each one has an additional leg.
Griffin: How many legs does the last one have?
Let’s pause for a moment. This was his question, not mine. Real world be damned, this is a habit of mind thing.
Jimmy Pai, on students who dislike applied math problems:
Have I been focusing too much on applying mathematics and expanding the concept of “relevance?” Have I been expending too much energy on looking for relevance when I should play off of the interesting and awesome world that is mathematics? I had one student who became more disengaged throughout the year as everyone else was loving the relevance and exploring their own questions.
These are important posts. They remind me that it’s a flawed theory of student engagement that leads you to draw two circles, one labeled “real” and the other “fake,” and put material into either one with any kind of objective confidence.
Seven-legged elephants are real to Griffin. Numbers are real to Jimmy Pai’s student. Star Wars isn’t real, but for millions of viewers, it feels real. The techniques of storytelling can make the unreal seem real, but let’s all agree to bear in mind that what’s real to one student isn’t necessarily real to another.
My favorite quote [from this article] is “the quality of a task need not be judged by its relation to real life but in relation to how it engages students’”
BPLAFebruary 19, 2013 - 3:20 pm -
Many things that will eventually become ubiquitous are not ‘real’ now. Being able to logically demonstrate an idea is what matters. Can students mathematically construct their imaginations into mathematical arguments, and make the impossible, possible?
I would argue, that problems should be a mixture of ‘real world’, futurist, and fun.
Making a Strandbeast sounds like fun to me:
However, also being able to calculate the partial pressure on oxygen in my air tank relative to my depth while I scuba dive, so not to suffer from oxygen toxicity, is neat too.
Bill McCallumFebruary 19, 2013 - 3:39 pm -
I love the elephant parade. There are two aspects to the use of context here. First, the point you make, that imaginary contexts can be just as real and more engaging than real contexts. In this case the question Griffin asked leaps out from the context; it’s impossible not to think of that last tiny multi-legged elephant. As Christopher says in the original post, the context creates the intellectual need.
The other aspect is that “fake” contexts can be crafted to a mathematical purpose, without the dreary requirements of reality to add more stuff than you need. The context makes this into a very different problem than if you had just described the pattern. For example, thinking “start with 4” becomes an act of the problem solver rather than a given in the problem. I call contexts that have this second aspect thin contexts.
Chris RobinsonFebruary 19, 2013 - 3:43 pm -
I’ve been trying to wrap my head around this issue lately: pure vs. applied math, student engagement, interest and intellectual need. They’re a constantly moving target, but well worth exploring for the benefit of our students’ learning. Here’s a couple of good reads I’ve came across recently: Audrey Watters — “Is Math Education Too Abstract?” and Nikhil Swaminathan — “In Abstract: Avoid Concrete Examples When Teaching Math”
CraigFebruary 19, 2013 - 4:22 pm -
Here is a great article on the topic:
Exploring Mathematics in Imaginative Places By Cynthia Nicol from U of British Columbia and Sandra Crespo from Michigan State
My favorite quote:
“the quality of a task need not be judged by its relation to real life but in relation to how it engages students’”
Chris RobinsonFebruary 19, 2013 - 5:02 pm -
This is the biggest hurdle/hardest task we have as teachers when implementing tasks into our classrooms. 3-Act tasks are great, but I wonder if my students (rural Pennsylvania) would be more interested and engaged and have greater intellectual need if “Taco Cart” was instead “Deer Stand” or “Cow Pasture.” Context is so regional/cultural/SES that we have to be careful about the problems and tasks we pose to our students and the general curriculums/textbooks we purchase to use in our classrooms. And as Dan mentioned, context is relative to the student sitting in front of us.
David WeesFebruary 19, 2013 - 5:09 pm -
I think that context is important, but that you can build context from an engaging abstraction.
You have to be very careful of contexts that don’t match your students’ experience at all. See https://twitter.com/davidwees/status/301036990986592257/photo/1 for example.
When I define “real world”, I am careful to point out that not every activity should be based on the physical world the kids inhabit, but instead activities should have stories, and students should understand those stories. The stories provide the context, and in some cases, those contexts come from the physical world, and in others they come from descriptions of relatively abstract ideas.
GeoffFebruary 19, 2013 - 7:39 pm -
This is one I wrestle with quite often. I love Chirstopher’s blog post about his kitchen conversations. Narrative and the way a task feels to students is critical for engagement, far more than the pretense of an applied situation.
However, it would bum me out if/when students *never* get a chance to actually use math in an applied sense over a year’s course. If students aren’t allowed to collect their own data, calculate the height of the tallest building in their town, mess around with sound waves, then that is a rich mathematical experience lost.
It’s unsatisfying for me to not have a formula or ratio as to how much applied math should show up in a course. I’d suggest the percentage of time spent doing real world stuff is between 1% and 99% of the time. Like I said, very unsatisfying. Also, it would probably depend on the mathematical value and narrative value of the applied tasks developed.
Obviously, for a single task, the “interestingness” of the task supersedes the “real-worldliness” of a task. But I don’t want to let myself off the hook by not constantly looking for ways to make mathematics concrete and tangible.
Robert KaplinskyFebruary 19, 2013 - 9:34 pm -
I really like what Jimmy Pai is asking. Is it more important to explore a real world context or something interesting with no direct real world application? I have recently been siding with real world contexts (much like Geoff describes), but I was concerned that I had lost perspective. I actually recently asked Dan for his thoughts on this balance and his reply resonated with me. He stated: “All I need, really, is a student with a question in her head that math can help answer.” To me this sums it up so succinctly and helps bring it back to what matters.
I think what Bill McCallum said reinforces this idea in that any interesting context is worth exploring; but that forcing a context onto a problem in an effort to validate it is not ideal. To me I see these beliefs showing through in the CCSS’ Standards for Mathematical Practice. Math Practice 4 clearly explains that students need to be able to “apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.” However several other standards such as 2, 7, and 8 would be addressed nicely by the elephant parade. Both types of problems seem to have a place.
I also agree with Dan and Chris Robinson’s point in that “context is so regional/cultural/SES”. We try to make the context something that is relevant to students, but that can be challenging depending on the experiences they come in with. If you can create an interesting context regardless of whether it is real world, then it will be something worth exploring.
Great discussion here. Thanks for sharing.
Ben GraberFebruary 20, 2013 - 7:48 am -
I can relate to Jimmy Pai’s comments. I feel like in the past I have tried too hard to find “real life” uses for math. Those types of “real life” problems that feel very forced can make the math seem less real rather than more real. I would rather invite students to explore the beauty of math rather than forcing a “real world” feel.
Matt EFebruary 20, 2013 - 8:22 am -
See also the linked PDF at Dan’s post from 2011: “Cornered By The Real World,” by Samuel Otten. Relevant here, and a great read.
SFebruary 20, 2013 - 9:07 am -
Did the world arround suddently stopped providing us with wonders?
JonathanFebruary 20, 2013 - 6:03 pm -
I enjoy treating math with irreverence from time to time. Some of these topics should get over themselves. Thus, you get calculating the total load on a hook being held by Optimus Prime. Or modeling the swing of Batman.
CathyFebruary 21, 2013 - 6:16 pm -
I enjoyed the article and the conversation. Engaging students in any format is what is relevant. Using either real or imaginative examples for the applications of math is insignificant if neither engages the student. What was nice about the article was that it reminds us that we are not confined to using “real world” examples and that imaginative examples can provide a host of potential benefits, must importantly student engagement.
GretelFebruary 22, 2013 - 8:03 am -
I think about this and have this argument with myself all the time. And long ago, I decided that I was just weird because I seemed to be the only student in my math classes, the only math teacher, the only teacher of math teachers that HATED real world applications. I recognize that some people really really want to apply math to “the real world” (because somehow math class exists in some parallel “unreal” dimension). But, for me personally, applying mathematics to the real world kills at least half of what I love about it.
For me, the real world is a stressful place and always has been. There’s error to account for. If you’re predicting when a tank will be filled with water, you have to consider who’s going to put the hose away when you’re done. You also have to think about all those people who don’t have clean water, how much water you’re wasting, and all those other thoughts about all those other things that are now connected to that context.
The line of elephants, on the other hand, is just fun. Something about including questions like that in math classes allows math class to be a place where kids are free from the cares of the “real world”. High school kids are under a lot of stress, and wouldn’t it be great if math could sometimes be an escapist activity, like reading a really great novel or painting or playing a game?
Michael SerraMarch 5, 2013 - 10:52 am -
Love to see the exchange of ideas about “real applications” versus “fantasy” or “abstract.” One of my favorite quotes on this issue is by British mathematician and puzzle creator A. Gardiner.
Good mathematical problems are necessarily artificial. In contrast, “realistic” problems tend to elicit “realistic” responses involving little or no mathematics. In mathematics teaching, what matters is not whether a problem is plausibly real or artificial, but whether it is such that pupils are prepared to enter into the spirit of the mental world it conjures up.