The Real Worlds

In part of his ongoing “real world vs. fake world” series, Dan Meyer offered three versions of the same problem, and asked us at Mathalicious to opine on which we considered the most “real,” and how we evaluate the real-ness of a mathematical task:


Version A

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle.  Find P such that the area of the square and circle are equal.

Version B

When do the circle and the square have equal area?

Version C

Where do the circle and square have the same number of candies?


As always, Dan asks an excellent question. Perhaps in response to what they view as an over-emphasis on rote skills, many people – from students to teachers to administrators to the public at large – believe math should be more “real.” The Common Core Standards document for mathematics mentions “real world” 28 times, while the NYTimes Editorial Board asserts that “millions of students…could benefit from real world” activities.  Heck, we even put “real” on our t-shirts!

MLshirt

Yet for all the talk of “real world math,” Dan’s right: there’s no clear consensus on what real even means. Is real the same as relevant? Is it the same as engaging? It’s like Nietzsche, Euclid, and every New Age philosopher rolled into a single pedagogical question: what is real, man?

In order to decide whether a particular math activity is “real-world,” it’s important to first determine the world in which the activity is intended to exist. From our perspective, there are three different worlds that constitute the universe of math instruction: the world of procedural fluency; the world of conceptual understanding/problem solving; and the world of applications. Only once we understand how each world works can we determine whether an activity within it is “real” or not. In other words...reality depends.

To help illustrate the differences between each world, we’ll focus on three factors: a world’s overarching goal/purpose; the types of activities one might encounter there; and problems that arise when task designers misunderstand (or don’t consider) its rules.

World 1: Procedural Fluency

In the world of procedural fluency, math serves itself, which is to say, the purpose of math is to do more math.

MATH → MATH

There’s typically no context – or if there is, it’s so generic as to be swappable without losing much – and the goal is for students to practice skills in order to strengthen their math muscle memory. What is 12 × 9? If you have ten apples/suitcases/pillows and take away four, how many do you have left? Turn to page 30 and do problems 1-43, odd. Examples include workbooks, the computer-generated exercises on Khan Academy (and many of the videos), and pretty much every “adaptive learning” app ever made.

Here, the role of the task designer is fairly limited. If the goal of a task is for students to practice rote multiplication, there’s very little risk in creating a “bad” question. He or she may order the questions in increasing difficulty, but the creation of the questions is pretty simple (which is why computers can do it). Indeed, questions aimed at procedural fluency usually aren’t questions at all, but rather commands with question marks.

Even though it’s relatively simplistic, the world of procedural fluency is incredibly important; without it, students would (and often do) spend all their time focusing on the procedural trees, rather than exploring the larger forest. However, rote tasks aren't terribly helpful in furthering a discussion about the nature of “real.” In the world of skills questions, there is no distinction between real/fake or better/worse. There's just process: steps to follow. It’s pretty Julia Child like that.

World 2: Conceptual Understanding/Problem Solving

In the world of conceptual understanding, the goal is for students to engage in problems that help them develop a deeper understanding of what math means: what it means to multiply; what it means for a relationship to be proportional; not that the vertical line test can be used to identify functions, but why.

Here, a teacher or task designer may use a context that might appear to some to be “real world.” An example of this is Dan’s Water Tank problem, in which students estimate how long it will take to fill a tank with water. Other examples include his Penny Circle collaboration with Desmos, and Andrew Stadel’s giant wheel activity.

Yet while these activities all involve a context rooted in the external world, their focus is actually on something else: the mathematics that underlies it. It’s unlikely that students will be motivated by the question, How long will it take to fill up the water tank? But that's okay. Because the activity isn’t really about water at all; it’s about rates and proportional reasoning. The water is merely a vehicle by which students can engage in problem solving and critical thinking. Put another way, even if students aren’t motivated by the question, they are motivated by the process of answering it.

CONTEXT → MATH*

The reason Dan, Andrew, and others in the Three Act vein are successful is because, in their activities, the context serves the math…and then gets out of the way. Their activities put students in a position to wrestle meaningfully with mathematics, and approach problems from various angles. While the questions themselves may be relatively limited in scope and their answers closed-ended – How long to fill the tank? How many pennies in the circle? How tall is the wheel?  – the path in between is characterized by an “open middle” that may be unique to each student. In this sense, effective problem solving activities might be thought of as being inwardly focused, since a student’s ultimate interaction is with his/her own mathematical thought process, i.e. him/her self. (Sorry, Buddha.)

As Dan has described on his blog, effective tasks reveal their purpose quickly; just as the goal of Angry Birds is immediately obvious, so too is that of an effective problem solving activity. So in this world of conceptual understanding and problem solving, what does it mean for a math activity to be “real?” We’d argue that, here, realism is subjective; it’s not an inherent characteristic of the problem – a question isn’t by itself fake or real, interesting or boring – but depends on how meaningfully a student uses it to progress towards the intended mathematical understanding/goal.

Still, this isn’t to say that all problem solving tasks are created equal. In the circle-square problems above, the goal is for students to determine the point at which each area is the same. Even if students find the problem difficult, the question is clear. Version B maintains this clarity; the question is about a circle and a square, and students see a circle and a square. Even though students are unlikely to encounter “equalize the areas of a circle and square” in everyday life, Version B is very real in that it provides students a real opportunity to engage in real mathematical thinking (in the same way that Sudoku is a great way to practice real logic, even if the context is entirely contrived).

Version C, however, obfuscates. In this version, students see a circle…and a square…and candy. But when did this become a problem about candy? You can imagine the task designer saying, “Calculating the areas may be difficult for some students, so I’ll give them something discrete like pieces of candy to help.” It’s an understandable decision, but one that ends up muddying the problem by preemptively incorporating the scaffold into the building itself, and making the desired question about area even more difficult to ask: the following question, rather than original one. Even though the designer may have intended the candies to make the task more “real,” he ends up making it less so by adding an element that has no business being there. If a teacher wants to add candies after the fact to help concretize the task for struggling students, that’s fine. Preemptively changing the task, however, is not, and will almost certainly result in students concluding, “Why would anyone put candy in a square? This is stupid. Math is stupid.” And once that happens, any potential for students to engage meaningfully with mathematics towards an understanding of areas – and any notion of “real” – goes out the window.

(By similar logic, Version A also falls short, though not from an excess of scaffold but an absence. To many students, the wording is likely to be so obtuse as to be unintelligible. If the goal is for students to discern what the author meant, then it might work as a reading comprehension/interpretation task. If the goal is for students to compare areas, though – and if those areas are specific to these exact shapes along this line exact segment (which is to say, if the goal is not to generalize) – it seems unnecessarily complicated, like forcing students to wrestle their way into the ring before the main event.)

In the world of conceptual understanding and problem solving, the distinction between “fake” and “real” may be less about inherent quality than about cleanliness and purity. Versions A and C are distracting – albeit for opposite reasons – and muddy what may otherwise prove a good opportunity for problem solving. On the other hand, Version B maintains its integrity; it follows Einstein’s [apocryphal?] dictum to make itself as simple as possible, and no more. Of the three versions, Version B affords students the greatest chance of successfully engaging with the mathematics of shapes and measurement and in the cleanest way, and is therefore the “realest” of the three. Of course, whether Version B ends up being real relies on how meaningfully students engage with it, and to what end. In other words, its nature depends. It’s pretty Schrödinger’s Cat like that.

(*Note: there are many examples of effective conceptual understanding tasks that do not include any semblance of context, e.g. multiplying binomal factors to identify patterns useful for factoring trinomials, looking for number patterns in the Fibonacci Sequence, or applying sub-set counting techniques to explore the Binomial Theorem. In these cases, math serves math, albeit to a very different end than in the world of procedural fluency. However, since Dan’s original question is how we define “real world,” and since nobody is likely to characterize these as such, we will ignore the Math → Math subset of conceptual understanding activities.)

World 3: Applications

Whereas the goal of problem solving activities is for students to use some context to better understand mathematics, the goal with [our] applied activities is the exact opposite: to use mathematics to explore how the world around us – the external world that we often think of as the real, real world – works.

MATH → WORLD

Here, math serves the world. In the Mathalicious lesson XBOX Xponential, students develop an exponential model to explore how video game consoles have changed over time, and in particular whether their processor speeds have followed Moore’s Law. Students are deeply engaged with high-level mathematics, but in pursuit of something else. In the lesson On Your Mark, students discuss whether taller sprinters have an unfair advantage, and determine what would have happened had the London Olympics been organized by height. They’re doing lots of excellent math. If you ask them what the lesson is about, though, they won't say “proportions.” They’ll say “Usain Bolt.” (Other good examples of application lessons include Robert Kaplinsky’s terrific lesson on skywriting – how do airplanes use reflections to create messages in the sky? – as well as those from our friends at Yummy Math.)

Unlike those in many problem solving activities, questions in application lessons often come from someone other than the student. How has the iPod depreciated over time? How much confidence should you place in online ratings? Is LeBron James still the best player when you consider all the points he’s missed? The downside of this type of exogenous questioning is that it may limit student buy-in, at least initially. On the other hand, the upside is that it exposes students to ideas that they might not have thought of on their own, and to engage in conversations that they wouldn't have had otherwise (which is one reason schools exist in the first place). While problem-solving tasks have an open-middle towards a closed-end, applications lessons often have a narrower middle towards an open-end: How have video games changed over time, and do you think they’ll ever be so realistic that you could choose to live inside of them?

In the world of applications, evaluating whether a particular task is “real” or “fake” is relatively straightforward; you simply have to ask whether the underlying question is legitimately related to the external world in which we find ourselves (real), or whether it’s a math problem hiding behind a façade (fake). The question, “How does your enjoyment of candy change over the course of Halloween?” is an example of the former; surely this question could arise as trick-or-treaters consider when to stop eating candy and save the rest for later. On the other hand, “When will the square have the same number of candies as the circle” is an example of the latter; this problem exists nowhere but in the mind of the task designer.

(Note: unlike in the world of problem solving, the question of whether an activity is “real” here is unrelated to whether it’s engaging. For instance, the Mathalicious lesson Licensed to Ill uses expected value to explore the mathematics of health insurance. Certainly this is a very real topic in the “does this occur in everyday life?” sense, though some seventh graders might still find it boring. However, this isn’t an argument against the lesson or a referendum on its realness, but an example of how in application lessons, the teacher often has an important role in “selling” students on why the topic is worth exploring. Of course, the task designer can help with that, for instance by suggesting preview questions to elicit prior knowledge and get student buy-in. Has anyone ever gone to the hospital? Does anyone have a family member who’s sick and can’t afford to go to the doctor or buy medication? Irrespective of how students respond, the task was always real. If they respond favorably, though, it can also become effective for classroom instruction.)

Just as it’s possible to muddy an otherwise effective problem-solving task by forcing a pseudo-context, it’s also possible to ruin a good application task by sidetracking it with unnecessary math. This is something we think about a lot at Mathalicious. In the first half of XBOX Xponential, students write an equation to model Moore’s Law, and use as their starting console the 1977 Atari, the first with an internal microprocessor. When they estimate the processor speed of other consoles – e.g. the 1983 N.E.S. – they’re asked to first reinterpret the year in “video game years,” where 1983 corresponds to video game year 6. As math lovers and teachers ourselves, we debated including a question about how the equation would be different if it used the actual calendar year, and whether this would be a good idea. (The equation would only be slightly different, but the y-intercept would now correspond to the processor speed in year 0 – that of Jesus’s console – rather than the processor speed in 1977.) While this question was interesting mathematically, it was tangential narratively. To preserve the integrity of the conversation and to minimize distractions from it – in other words, to preserve its realness – we ultimately decided against the question (and instead included it in the lesson guide as a possible extension question).

In the end, the realness of an activity in the world of applications can be evaluated on two factors: the likelihood that someone would actually ask the guiding question (e.g. When should you buy health insurance? How has the iPod depreciated over time? Is the wealth distribution in the United States fair?); and the potential for it to help students better understand the external world in which they find themselves. While the purpose of problem solving tasks is to help students develop a deeper understanding of math, the purpose of application tasks is to use math to develop a deeper understanding of how the world works. It’s pretty Radiolab like that.

The Perfect World?

The world of procedural fluency. The world of conceptual understanding and problem solving. The world of applications. As with the circle-square task, we may be tempted to debate which of these three worlds is the best. But that’s silly; imperfect analogies notwithstanding, it’s like debating which genre of music – classical, blues, or jazz – is the truest. Instead, each world of math activity fosters a different aspect of a student’s intellectual/mathematical development, and all three are equally important. Students need to understand proportions, and be able to think flexibly in situations that involve proportionality (world 2). They need to be able to use proportions to ask and answer questions about the external world, and make determinations about how it works (world 3). And they need to have sufficient facility with the mechanics of proportions that they don’t get sidetracked and forget what they were thinking about in the first place (world 1).

Instead of discussing which type of activity – procedural, conceptual, or applied – we should use, a more constructive conversation would be about how often and when. For instance, is it more effective to start a unit on linear functions with an open-ended problem (e.g. Dan’s Groceries), an application activity (e.g. our Domino Effect), or a procedural task (e.g. Khan Academy’s activity on  slope-intercept form)? Also, within the unit, how much time should we allocate to each type of activity, i.e. how much time should we spend in each world?

Unfortunately, we too often fall prey to simplistic notions of mutual exclusivity, and miss the nuance that characterizes not just good task design, but the larger and more transcendent curriculum design. As Dan suggests, we get blinded by buzzwords, flavors-of-the-day such as “real-world math.” While this particular flavor may benefit “real world” organizations like Mathalicious in the short-term, decisions rooted in such a narrow understanding of math education are inevitably unstable, and ensure that it’s only a matter of time until the pendulum swings in the opposite direction. (See: Math Wars.) What this suggests, then, is that instead of debating which world is the best, we would do better to consider how to best integrate them: how to stop the pendulum from swinging and find its equilibrium (or at least limit the swing to a stabler range).

Of course, Dan wasn’t asking about which circle-square task was the best. He was asking which was the most real, and by extension what “real world” even means. Again, before we can discuss “real,” we first have to decide on the “world.” How real-world an activity is first depends on the world in which it exists, and the goal it’s intended to serve. If the purpose of an activity is for students to develop a deeper understanding of mathematics – including their thought processes around it – it must respect the rules of conceptual understanding. If its purpose of a task is for students to use math to develop a deeper awareness of the world around them, it must accord to the rules of applications. The better it does this, the realer it will be. It’s only when task designers misunderstand/don’t consider the world they’re serving that tasks go wrong. As with the square-circle Versions A and C, trying to live in multiple words ultimately means living in none.

Which is to say, the question of “real-world or fake-world” ultimately comes down to gnōthi seauton: know thyself. It’s pretty Socrates like that.


6 thoughts on “The Real Worlds”

  1. I think you summarized it best with, "What this suggests, then, is that instead of debating which world is the best, we would do better to consider how to best integrate them: how to stop the pendulum from swinging and find its equilibrium (or at least limit the swing to a stabler range)." I certainly believe that application lessons provide the context to establish the conceptual understanding needed to build procedural skill and fluency. They all live harmoniously together and I question the extent of a learner's knowledge when one of those three pieces is missing.

    Thanks for helping me reflect on my practice.

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