I’ve been truly enjoying Dan’s blog post series (from last year!) on the “Fake World.” I’d highly recommend you go read those posts if you haven’t already. It deftly exposes the fallacy of authenticity-as-engagement. I would like to offer a defense (three, really) of applied mathematical tasks.
1) Developing Tasks
There are basically two ways teachers generally develop a mathematical task for a K-12 math classroom:
- Identify an intriguing scenario and map it to a mathematical standard or concept.
- Identify a mathematical standard or concept and attempt to come up with an intriguing scenario.
(Aside: I suppose there is a third – to modify an existing mathematical task, but we’re talking from scratch here.)
The first of those two modes of task development almost necessarily requires a real world experience. I mean, hop over to 101qs.com. These are things that interested teachers mathematically and most of them are, in fact, real things.
As for the second mode of task development, when you’re staring at a mathematical standard and attempting to think of how to make it engaging and interesting for kids the very first thing I ask is “in what contexts are this standard applied? What’s a real-world scenario that you can develop a problem around?” It’s the first, easiest in-road to designing a task.
2) Delivering the necessary shock to the system.
My own experience necessitated that I get some sort of EMP shock that scrambled my circuits. You see, I was kind of getting in to my groove as a teacher after about 5 years or so. I was beginning to recycle lecture notes and assuredly improve them. Then our entire school (or rather, our academy within the school) began doing Project Based Learning, which does mandate that you employ authentic tasks. While I’ve discussed some of the mediocre and ill-spent time I implemented while struggling with PBL, the most important thing it did was force me to start from scratch. I was forced to actually try to make meaning of the chapters I had been teaching for years. That’s huge.
Many of those reading this, Dan’s and other blogs have probably already gone through that experience or something like it. Or maybe many they were already awesome teachers to begin with who started out at the tender age of 22 and was fully bought in to having students solve non-routine complex tasks straightaway. But I wasn’t there, and I’d bet that a vast majority of math teachers working today aren’t terribly familiar with designing high quality tasks from scratch. Let’s be honest and pat ourselves on the back a bit: folks who are so invested in their practice that they’re reading and writing blog posts and interacting on twitter after hours – these are your honors kids. The real world may be more engaging to teachers than it even is to students at the outset.
Dan writes:
My point is that your theory of engagement might be limiting you. It might be leading you towards boring real-world tasks and away from engaging fake-world tasks.
We need a stronger theory of engagement than “real = fun / fake = boring.”
Again, no argument here. But most math teachers are, in fact, giving fake, boring instruction and will continue to do so unless something breaks the cycle. For me, it was a fanatical devotion of my school to developing real-world projects. For many of the teachers I’ve had the opportunity to work with, PBL was similarly their “Road to Damascus” moment as well. Now that I’ve had years to look back and reflect on the mediocre projects I implemented I can happily develop and employ non-authentic tasks.
3) It’s a good thing if students experience math in hands-on, novel, and concrete ways once in a while.
To be clear, there’s nothing wrong with beautiful, abstract math for math’s sake. Bryan Meyer’s session on Mathematical Play was phenomenal (and potentially just as earth shattering to his audience as a session on real-world math would have been to me 10 years ago). But there’s something to be said for students experiencing authentic scenarios, researching real-world topics, presenting findings to panels of experts in a professional manner, and so forth. It would be kind of a bummer if a kid left a math class without having experienced some of that.
Also, by dismissing the “real-world” as a lever to engagement, you’re giving teachers a kind of out. I’ve had conversations with sanctimonious math teachers and district instructional coaches that cite Paul Lockhart as a reason to keep doing what they’re doing. I’ve read Lockhart. I love Lockhart. But his books aren’t about instructional practice. While much of “Measurement”, say, can and should be handed over to students to explore, it’s frustrating to kids who have only experienced math in the abstract.
I don’t know that 100% of tasks should smack of authenticity, but it should be more than 0%.
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I don’t have other specific blog posts to cite, but I’d suggest there is a general muted backlash against authentic tasks swimming around twitter these days. Not that there’s anything wrong with that. I love the conversations, iron-sharpening-iron and all of that. Dan’s visible thinking on the subject has really challenged and inspired me to reconsider theory of engagement. Consider these defenses a backlash to the backlash.
A unifying theory of these three defenses of the real word might be “first steps”. When designing a task from standards, the first, easiest, and sometimes best tasks can be started by asking “where in the real world does this apply?”. When eradicating your old, often-ineffectual instructional modes, the real world is a good place to begin building yourself back up. And finally, when students experience math in authentic situations for the first time, it may be the first connection the student makes with an until-now boring subject.
So anyway, “first steps.” Not “last steps”. If you stop at application, you end up with low-level PBL or psuedo-context which can be as destructive as low-level rote instruction and path-paving. The real world is NOT the lever that’ll switch on engagement or performance. Nor is it the checkbox that should be ticked off for every mathematical task one implements. And anyone telling you it IS the lever is selling snake oil (or perhaps, curricula). Still, the most effective teachers I’ve worked with have universally devoted a considerable amount of time attempting to figure out how to make mathematics relevant to their student population, often via the “real world.”
Feel free to let me have it in the comments.
(See also: “The Skynet Line” and Mathy Cathy’s “Sort of Real-World Math“)
Update (3/2/2014): Michael attacks real-worldliness from a modeling perspective. Which is great, because I initially few actual pedagogical reasons for utilizing real world, or even fake real world tasks.
emergent math 
This is great for getting me thinking. I agree wholeheartedly about “iron-sharpening-iron” and having to refine my thinking to define why I believe what I believe. My biggest take away from all of this is that “real world” by itself does not necessarily imply that students will be engaged by the mathematics. It starts to get into clarifying the difference between “real world” and “relevant.” Real-world topics may be relevant to students, but not necessarily. Even topics that are relevant to students are not necessarily engaging and it is important to consider this when making a problem.
I completely agree when you state, “when you’re staring at a mathematical standard and attempting to think of how to make it engaging and interesting for kids the very first thing I ask is ‘in what contexts are this standard applied? What’s a real-world scenario that you can develop a problem around?’ It’s the first, easiest in-road to designing a task.” The honest reality is that it is quite challenging to find a great topic and then quite time-consuming to turn it into an engaging problem. Somewhat easier is starting with an interesting context and then creating the problem… except this only works when you are creating problems without the need to focus on a specific standard.
Thanks Geoff.
Geoff, for me the “shock to the system” was my own interest in getting away from the textbook. I’ve done the most toward that end in calculus. Some is real-world connections (I focus way more on position, velocity, and acceleration, and I ground us in more calculus history), and some is mathematical (working with circumference and area of a circle, understanding that pi=c/d is a definition, while a=pi*r^2 is a theorem that can be proved). Both are important.
Right now, as I prepare for the semester to start next Monday, I am trying still to re-frame pre-calc and calc II in ways that bring the whole course together as a story, the way I was able to in calc I. Then maybe I can improve the courses at a more detailed level.
Thanks Geoff. I go back to something that Dan posted a while ago, I keep it in the front of my mind when developing activities:
“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.”
We have to keep experimenting with different types of problems and tasks until we find what hits that sweet spot.
And thanks to Robert Kaplinsky, whose “highway sign” project is providing our fourth graders with a very engaging and exciting experience, I’ll be writing about that soon on my new blog:
exit10a.blogspot.com
Thank you so much for this post! I have really struggled this year with coming up with questions that are “relevant” and “contextual to real world.” I have looked at the posts by Mr. Myers, but I’m not sure how to use them exactly. Is there a “Dy/Dan for Dummies”???
On Mon, Jan 6, 2014 at 12:27 AM, emergent math
Hi Charlene, I’ll point you to a couple resources that might help:
1) Fawn Nguyen’s “Deconstruction a Lesson Activity” posts, parts 1 and 2:
* http://fawnnguyen.com/2013/07/06/20130706.aspx
* http://fawnnguyen.com/2013/07/08/deconstructing-a-lesson-activity—part-2.aspx
2) My Problem Based starter kit:
* https://emergentmath.com/2013/10/30/a-problem-based-learning-starter-kit/
Give those a looksee and let me know if those are helpful. I’m a huge fan of Fawn’s. She’s great at simply explaining what she did in her class. Great in a kind of “I do / you do” way.
Hi Geoff…
I appreciate the post and the way that your words always get me thinking this topic. I don’t have anything terribly insightful to add, I don’t think, but I’ve been thinking about this post a lot since you put it up last night, so I thought I’d lay down some ramblings.
The thing most on my mind is mathematical identity. Our classes, more than anywhere else, are the places where students form their definition of what math is….and, consequently, of whether or not they see themselves in that mathematics (where they begin to think that they are/aren’t “mathematical”). For me, that’s important. I believe that identity (at least who we THINK we are) is inextricably tied to issues of agency, confidence, etc…in short, the actions we will take when confronted with what we believe to be mathematical situations. Anyways, my point here is that I think there is something to be said for offering students a wide range of mathematical questions because it can only broaden their definition of mathematics and, as a result, provide more opportunity for them to “see themselves” in that activity.
I guess I’ll quickly add that I think the problem is a bit deeper than that as well. A mentor of mine once said something along the lines of, “the teacher might bring in a task/question, but looks for mathematically ripe opportunities in the activity of the students.” It was an important comment for me….a reminder that my job is to draw out their questions/ideas/ways of thinking rather than impose mine.
As a total aside, Cathy Humphries leads a pretty interesting lesson via the task “What is 1 divided by two thirds?” in “Connecting Mathematical Ideas” (video on the DVD accompanying the book).
Thanks for getting me thinking.
Geoff,
I’ve been an avid reader of your blog for about a year now. You are always thought provoking and thorough and I really enjoy your contributions. So, thank you.
I agree with you that a real-world scenarios can be the trigger to get students to explore mathematics in the real-world. It’s too common for most teachers, however, to resort to the pseudo-context in the their textbook and called it real-life application of math. Have you ever had a student say, after “applying” polynomial functions by modeling school band membership “Hey, math really does apply to the real world. I want to learn more about polynomial functions!”? I’m glad you make a clarification for the intriguing scenario being the basis for a problem or task. I don’t think students are intrigued one bit by what most teachers call “real-world.”
I completely agree with Robert that the real-world does not guarantee engagement, as that was my biggest takeaway from Dan’s series as well. Additionally, the definition of relevance (the desire to solve it) is something I also keep in my brain. (Thanks, Joe.)
In my classroom, I’ve seen (and blogged about) some really awesome stuff that has happened with some very unrealistic tasks. It’s amazing and bizarre at the same time, but it’s wonderful to see students embrace abstract math in this way.
Megan
Thanks for the pushback, Geoff. This has been more monologue than dialog so far, unfortunately.
A task, sure. Just not necessarily a good task, is my point. Optimizing for “real world” is unreliable. Optimize for surprise, model-building, a steady ramp from easy to difficult, intuition, argument, mathematical identity, the minimization of extraneous cognitive load, and other variables instead. Maybe the “real world” winds up being the best option there but maybe the world of pure math accomplishes those goals even better.
The shock to the system argument is interesting, and several people vouch for it, but man if it doesn’t strike me as some magical thinking. It’s the same magical thinking that says the flipped classroom leads to PBL and that MOOCs lead college instructors to reconsider how they use their class time. Why not just start with a more robust theory of learning instead of subjecting teachers to ideological whiplash? (“Wait. The real world was the answer before and now it isn’t?!”) And how many teachers never make it past that intermediate step, content with problems like this their entire career because they’re “real world”?
These are all fair criticisms of “real-world” tasks: that they can lead (and in the worst cases, metastasize) into low-level, inauthentic, and ultimately not particularly interesting mathematics. It’s certainly not the correct way to analyze a task (“is this real-world or not?”), but rather the cognitive demand and dialogue it fosters. Believe me, I’ve been in many math classrooms with allegedly authentic products that had no or low cognitive demand asks (including my own, at times).
Still, while my real-world defense may be flagging – and it’s certainly evolved in the past couple years, both in definition and its importance – I *still* believe in my bones that there is a value-add when it’s framed in a way that is challenging and interesting. The fine folks at Mathalicious are my North Star on this one. Their framing of, say, the Affordable Care Act, in mathematical terms is an example of the potential of real-world tasks that I don’t think an abstract task can offer to some (many? all?) kids. If a kid went through a math class and never got to experience something like that, well, that’s sad. And it’s probably also true if a kid went through a class and never got to experience “mathematical play” or mathematical puzzles, that’s also sad. Again, somewhere between 0% and 100% is my target.
Were I to criticize my own second point, I’d point out that the need for that “system shock” isn’t really as fundamental as it used to be. Now I can watch a TED Talk online, or check out Robert’s problem bank, or as Sue suggested, chuck the textbook in the trash and start anew with all the other resources freely available.
One of my goals would be to introduce mathematics to students in a way that gets *them* identifying mathematical scenarios. To me, a platonically ideal classroom would entail students bringing in their own 101qs type artifacts, which are real-world things that we can use mathematics to abstract around.
I’m sure there are a few more rounds to go through here. If you haven’t checked out Michael’s response, be sure to do that: http://rationalexpressions.blogspot.com/2014/01/the-unhelpful-distinction-between.html
tl;dr version – “Real world” describes content. Content, on its own, is never engaging. Don’t talk about content, talk about teaching.
I like how you state that tasks should be analyzed through a lens of the cognitive demand and dialogue rather than if it is real world or not.
Given that I live in Minnesota and we have had Frozageddon over the last three days, I’ve become particularly interested in how air temperature and wind speed are used to calculate wind chill. I’m hoping some of my enthusiasm about this, plus the timeliness of the topic will help to motivate the students to see its relevance.
Thank you for this post and fostering this dialog. I think for me, as an algebra teacher mostly, it is “easier” for me to resort to the puzzling type tasks that I feel are engaging, but not necessarily showing the real-worldliness of the math we teach.
Roger that. In sum, I don’t to be limited by my own theory of student engagement. I don’t want to tell myself any lies that lead me away from Good Stuff and towards Bad Stuff. Many (probably most) of the ideas I hear floating around about “real world” math are self-limiting.
Michael:
Loving the 2014 Pershan model. Pithy. Hard-charging. All-weather.