In a recent blog post, Dan Meyer describes his discomfort with the expression "real-world," writing *I understand what it means. I know it's code for something that basically everybody understands. But I'm not comfortable with the implication that if the mathematics won't help you build a deck or make payroll or beat the odds at a card table that it's "fake-world math" (or, even more unfortunately, "fake math") and without value.*

I've not heard anyone argue that applied math and pure math are mutually exclusive--indeed, I don't recall ever hearing the expression "fake math" until I read the post--but Dan does bring up a very good question: how do we as educators balance the need to make math concrete on one hand, and abstract on the other. We want students to recognize how math relates to "real" life (as they experience it), but also to be able to extend it in hitherto unknown directions: known-knowns, and also unknown-unknowns.

In this sense, perhaps it's a bit like parenting. You provide a structure and framework to help children make sense of the world. You make the act of growing up familiar and comfortable, gradually introducing elements of adulthood. What you're *really* doing, though, is preparing them for the day you show them the door. It's an imperfect metaphor, of course, but not too far off. For the real question isn't *whether *we should teach pure vs. applied math, but *when*. And I would argue that there are a number of factors which compel us towards a *concrete, then abstract* approach:

- Real-world (again, as experienced by the student) scenarios helps students get their footing, and act as a sort-of trail map should they get lost later. In the iCost lesson, students use iPad pricing to come up with the equation of the line between (16GB, $499) and (32GB, $599). First, they calculate the cost per additional GB of memory, i.e. the slope. At $6.25/GB, they know that for the 16GB model they're spending $100
*on memory*; for the 32GB model, $200 on memory. In either case, the final price is $399 more, which must be the base price of the iPad, i.e. the y-intercept. And so assuming iPad pricing is linear (more on that later), Apple appears to be charging a $399 base price, plus $6.25 for each additional GB: y = 6.25x + 399. Of course, we may not really care about iPad pricing per se, and certainly we want students to be able to extend beyond this limited example. But what's useful about this upfront contextualization is that, when students confront something like (18, 200) and (25, 235) later, they can say to themselves,*I haven't been here exactly, but I've been somewhere like here, and the first thing I did was...* - Real-world scenarios allow us to ask more questions--to cover more math--in a way that makes sense to students. If we want them to solve 6.25x + 399 = 1000, how do we explain "subtract 399 from both sides" without it sounding totally random, like some trick we just made up? But ask, "What size iPad could you get for $1000?", and a student may instinctively say,
*The first thing I need to do is get rid of the $399 base price, leaving the $601 I'm spending on memory. At $6.25/GB...*Other questions we can ask include:*If you double the size of the iPad, does the price also double?*(proportions).*Would it make sense for the 16GB version to cost $499 AND $529?*(vertical line test).*For the same size, will the Wi-Fi model ever cost the same as the 3G model?*(parallel lines).*Assuming linear pricing, how much should the 64GB model cost?*(evaluating equations).*According to website, the 64GB model costs $699, $100 less than we'd expect. What does this mean?*(linear vs. non-linear functions...and marketing to boot). The list goes on, and a small amount of contextualization allows us to do more math. Indeed, the iCost lesson addresses approximately 30% of the Common Core standards for Grade 8. From the teacher's perspective: a third of the year in 90 minutes. From the student's:*I get it*. - In addition to being more accesible to students, real-world math can also be more accessible to teachers. This may not matter to everyone, but it's worth remembering that in many states 6th grade teachers can teach on an elementary license, while many middle school teachers teach on a Grades 6-8 license (which doesn't necessarily include Algebra). To the extent that many teachers themselves feel a certain math-phobia, contextualizing skills can assuage their angst and make them more comfortable, confident and effective. Again, this doesn't concern everyone, but it absolutely must concern curriculum developers whose content can act as both a tool for instruction as well as professional development.
- Not only is the
*concrete, then abstract*approach more effective--especially for students yet to fully develop the capacity for abstract thinking--it's also more reflective of the K12 trajectory. In elementary school, kids use coins to understand place value, marbles to learn addition and make groupings, etc. Topics become a bit more abstract in middle school--we begin to substitute one thing for another, variables for values, etc.--though it's mostly pretty concrete. It's really only in high school that math starts to transition towards the pure, until eventually it's almost exclusively abstract. Of course, even before this happens there's some serious number theory going on beneath the surface, and we would do well to develop this type of thinking as early as possible (which I imagine is even earlier than we tend to think possible!). Still, it would be strange to teach modular arithmetic*before*addressing "Tell and write time...to the nearest five minutes" (2.MD.7), which is why even in college it's called "clock." - Not only does the
*concrete, then abstract*parallel the K12 trajectory, it more importantly parallels the evolution of mathematics itself. Just as humans developed a screwdriver in order to build something, humans developed mathematics as a tool to construct a more comprehensive understanding of how the world works. Put more simply: we've always had a practical/utilitarian relationship to math. The Egyptians wanted to calculate field dimensions and army formations. The Arabs, Indians and Italians: to improve commerce. This is not to minimize the contributions of pure mathematicians such as Pythagoras, Cantor & Euler, but simply to highlight that whoever notched the Ishango bone was probably motivated less by some fascination with prime numbers, and more by the immediate need to count his sheep and make sure they all came home at night. We went to the moon because it was there, and there's something profoundly beautiful about exploration for exploration's sake. Still, aviation began when some dude strapped feathers to his arms, which is to say: it was pretty concrete (and flew like it, too). - By using real-world topics to teach how math works, we end up teaching something else, too: how the world works. A student goes home and her mother says, "Hi, honey, what'd you learn today?" And the girl doesn't say "how to write the equation of the line between two points" (in all of human history, has that conversation ever happened?), but rather "I learned how Apple prices its products." Or,
*I learned that...video game consoles haven't followed Moore's Law, but still may be changing exponentially*;*...even though people with small feet pay more for shoes, Nike probably still shouldn't charge by weight; ...Wheel of Fortune may be rigged, but I have to watch more episodes to find out for sure.*And isn't that what's so unique about math: it's ability to shed light on...everything? And isn't that what makes being a math teacher so cool: our ability to ask...anything?

Which gets us right back to the question-at-hand: real vs. fake. Pure vs. applied. Embedded in Dan's post are really two questions: what's the point of math; and how do students learn it? There's no easy answer to the first--certainly it's a tool *and* an object of inquiry in its own right--and how you address it depends on any number of things, from your own experience as a student to the grade level you teach (or are writing for). In terms of "how students learn it," though, I think the answer is a bit clearer cut: "The way we [humans] developed it in the first place." *There seems to be this rate of change phenomenon that keeps coming up. Let's call it something. How's "slope?"*

How do we learn math? By using it. And indeed, I think Dan's WCYDWT materials are wonderful examples of this. *Pure or applied?* Yes, and a scenario like the basketball parabola allows students to access quadratics in a real-world setting, while also providing them an opportunity for "pure" extension later (e.g. *prove why the parabola opens down iff a is negative*). Mathalicious lessons do this, too, albeit in their own way and with their own focus.

Again, the question isn't "real" or fake, concrete or abstract. It's not *drill & kill or open-ended exploration*, but rather: *Why do so many kids hate math, where does this drop-off happen and and how do we fix this?* And before we can answer *How do we get where we're going*, where must first ask, *Where do we start?*

"Real world" may be an overused expression, but it's definitely an underused launching point in most math classrooms. Yet just as a swim coach doesn't teach his kids to dive from the raft, we'd do well to begin on solid ground, get our footing first and go from there.

I tend to agree that for the majority of our students, it is better to begin with concrete examples, and the more realistic the better. But that may be a social issue rather than a cognitive one, conditioned on the society we live in. See my summary of the Davydov curriculum: http://www.maa.org/devlin/devlin_01_09.html . Note that in Davidov's approach, the students do begin with concrete activities, and a lot of them, but then develop mathematics in a highly abstract fashion. Whether this corresponds to the way our predecessors developed mathematics I don't know. My own experience in mathematics research is that new methods and results are discovered from a mixture or bottom-up and top-down thinking.

“And before we can answer How do we get where we’re going, where must first ask, Where do we start?”

I think 'real-world' just means needing to put math in context, and there are lots of ways to do that, but they take time *and* more teachers who really understand math. The issue is that too much of what kids learn (not just math) is not connected to anything else, including their own lives. Once you make that connection, it builds a platform for further inquiry and questions, knowing and understanding. This platform is built through active learning, project learning, integrated learning, interdisciplinary learning, hands-on learning, or whatever label you want to give it. For example, I think that creating scenarios is a great start, but I wonder who is doing the learning in that kind of situation? The teacher has compiled all the data and created a context for some interesting discussion, but…what about bringing in all the shoes to class and a scale or, even better, asking them to take their own shoes and weigh them and talk about what they cost, then compile the data? What about taking them to the gym to shoot the baskets themselves and figuring out how to organize the data as the first step? Maybe it would take longer, but it might be more engaging, and therefore even more memorable. What about asking the question “What do you want to know (or know more about)?” and then guiding students to the relevant math? Is that too open ended?

The other issue in my mind in the concrete math vs. abstract/formal math question is the need to build procedural skills vs. the need to see the big picture so the skills are seen as useful. You need both, right? But if you skip certain experiences, you really can't move any further (van Hiele, right?). Developing literacy is a good model – children start by developing fluency with the spoken language; a lot of listening to stories, songs, and conversations; hopefully adults who engage the child in conversations and discussions about what they see, think, feel and read; experience with writing utensils and exposure to letters, etc. The written word will make no sense if there is no fluency behind it, and many kids are forced to learn to read before their brains or their own interests are ready. This may not help the discussion much in terms of middle or high school math, but still... how many of those older kids still need remediation with the basic building blocks of math? Just some thoughts.

I was at MCTM yesterday and loved your talk. I am a second career math teacher and after finally realizing that I can get support and ideas from teachers all over the country/world without just using google and trying to sift through things, I have discovered blogs and your site. I wasn't sure how to use some of the lessons until yesterday after I went home and looked at them again.

Thanks for coming to MN!

Wait, did I take the red pill or the blue pill, I feel like I am in the matrix...

The "real world" examples matter too - relevant vs. useful. I can easily articulate this point with a science example. When learning about physics, basketball is a relevant topic to many students but not all. And those who would engage more with such an example wouldn't become better basketball players, better coaches, better referees, better fans. Alternately, it might useful to situate all of the examples in a useful context such as construction or cooking. Entire units (thermodynamics, electricity, and materials & mechanics), could be used to better understand one's home (heating and cooling, electric generation/transfer/wiring/safety, building structure).

Situating math in "useful contexts" may be more challenging, especially due to abstract concepts and "pure" math, but they exist. Algebra could focus on computing: coding, spreadsheets and databases.