Reality Is Relative

Al Cuoco, responding to Sol Garfunkel and David Mumford’s op-ed in the New York Times which called for “a math curriculum that focused on real-life problems”:

This issue of viable and engaging contexts is complicated for a couple reasons. Many of the students in my high school classes came from situations that many of us would find hard to imagine; the last thing they cared about was how to balance a checkbook or figure the balance on a savings account. But they loved solving problems. For another thing, reality is relative.

Also, Deborah Loewenberg Ball on real-world context:

So I do think, on the question of context, it’s worth remembering that mathematics itself is a context and that puzzle-like problems are often both very engaging for kids and good equalizers because kids looking at those diagrams aren’t shaped by some of those same inequities about kids’ experiences.

Featured Comment:

Paul Wolf:

A facilitator at one of the workshops I went to this summer, Laura Kent, said that context is “anything that gives the students access to the math.”

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Reality is also changing. This is an important argument (which Cuoco makes) in favor of “habits of mind” over “special-purpose techniques.”

    Regarding context, whether it feels real/authentic to students often hinges on whether it “makes sense” — that is, whether it reflects a coherent set of concepts. Cuoco makes this point eloquently as well. Together, these points make Garfunkel and Mumford’s proposal (finance, data, engineering) a non-starter.

  2. It strikes me that we in math education often feel compelled to choose between two poles: skills vs. applications; pure vs. applied. This seems silly.

    Certainly Al Cuoco and Prof. LB are right: students come from different backgrounds, engaging is relative and puzzles are worthwhile. At the same time, certainly Mumford & Garfunkel are right, too: the world is full of opportunities to use math, and if we learn language by having conversations then we should learn math by doing applications. It’s not either-or but both-and.

    That said, it’s not so post-modern, and I think there actually is a right answer. Teaching math is similar to cooking: we have to use the right ratio of skills:applications and in the right order. Too many math classes use a Skills first (and foremost), and applications if there’s time approach. I’d argue that this is backwards, and that we should lead with the application, use it to contextualize and develop skills, and then extend these skills to other applications. Put another way, real-world math should play the lead role and puzzles/brain-teaser-type problems the supporting role, and for the following reasons:

    Abstract follows concrete by definition. If we lead with, “Write 3/24 as a percent,” then a bunch of kids will fall off almost immediately. But if we frame it as, “Of the 24 pieces of the Wheel of Fortune wheel, 3 are bankrupt. How many bankrupts would you expect if you spun the wheel 100 times?” Not only does this keep everyone in the game (both in terms of accessibility and engagement), it actually allows us to get more math out of the problem: ratio tables, proportions, theoretical vs. experimental probabilities. Whenever someone advocates for “real-world” math, there’s always someone else who pushes back saying this approach is less authentic, less rigorous. This is understandable. It’s also wrong. Students will be more successful in calculating percents in the abstract if they know what they mean, and that starts with a concrete application.
    A majority of secondary students dislike math because they think it’s irrelevant to their lives. If we ignore that, then we’re like a pastor assuming that everyone is in the choir. We can preach about the virtues of math all we want, but it’s reminiscent of Einstein’s definition of insanity: Doing the same thing over and over again and expecting different results….
    Speaking of Einstein, he was awesome at math. He also had a reason to learn it. Mathematics has historically been about real-world applications: understanding how the stars moved and, yes, how to balance a checkbook (see Fibonacci). Sure, there will always be mathematicians who explore math for math’s sake, but the reason Galileo built telescopes was not to look at the telescope but at the stars. Which is to say: the purpose of math is to focus on something else. When we use skills to support applications (not vice-versa), we’re actually being more authentic to mathematics itself.
    Puzzles are great. But when you get the NYTimes, how much time do you spend reading the news vs. playing Sudoku?

    Skills vs. applications? Pure vs. applied? This isn’t the right debate. Instead the questions should be How much? and In what order?

  3. Thanks Karim! I agree completely about moving from the concrete to the abstract. “Write 3/24 as a percent” is a capstone to some experience. Saying “you’re learning this so you can solve these cool problems later” is junk — start with the cool problems! If a cool problem turns out to be too hard, and needs some math background, kids will want to learn how to write 3/24 as a percent…

    I feel these important concrete experiences can come from a variety of sources, and they don’t all have to come from the real world. Here are some examples of great opening activities I’ve seen in classrooms:

    – Here’s a list of the ages of the last 21 Academy Award winners, both male and female. Is there a significant difference between them?
    – Here are walking directions from A to B. Write walking directions from B to A.
    – What’s the distribution for the sums when you roll two number cubes (dice)? Three? Four?
    – If everyone in the room shook hands, how many would that be? Oh, and how many diagonals are there in a regular polygon with that many sides?
    – Here’s a multiplication table. Fill in the missing entries. Describe and justify any patterns you find.
    – Here’s a huge pile of input-output tables. For each one, describe or find a rule that matches the table.

    A good opening activity needs to have plenty of legs, like the one you describe for Wheel, or all of the ones listed above. But their level of direct real-world application varies wildly.

    With any real-world context, if you hang the problem only on the context, you will lose some people. Al’s response to your comment would be “What’s Wheel of Fortune?” because he has never watched it. (Seriously.) It’s the same with kids, though. There’s a danger anytime you lead with a context that’s outside their experience, and it’s very difficult to judge what those contexts are and aren’t.

    Two quick examples from dead-tree curriculum writin’:

    1) I wanted to write about mortgage payments, but many people go through their life without ever having a mortgage. We used car payments instead, but there are still kids who get annoyed at these activities since they don’t have a car.

    2) I wrote an activity relating musical scales to ratios and fractional exponents. The entire activity was skipped by a large number of teachers, who did not feel confident enough about the context to use it. The activity was eventually rewritten as a project.

    To sum up, I agree with you that we should lead with concrete applications, which provide motivation for learning skills. But I also think that application can also be mathematical or puzzle-y in nature. It works!

  4. Bowen! We meet again. You’ll be happy to know that I “fixed” the Wheel of Fortune lesson per your totally legit razzing.

    So I think we may agree more than it appears. When I read your list of possible prompts, they all seem contextualized to me (well, maybe not the last one…okay, two). Academy Awards. Hand-shaking. Google Maps.

    Context is what allows for the abstraction. Here’s an example. I’m finishing up a lesson on Body Mass Index, or BMI. The formula is:

    Weight times 703…divided by…703-squared

    So much math to be had! For instance, How many ways could you write the formula using parentheses to get the right/wrong results? Or, Which has a larger effect on the BMI: adding an inch or gaining a pound? Or, If you gained a pound, how many inches would you have to grow to maintain your BMI?

    Are these applied? Are they puzzled? Whatever we answer, surely kids can engage with them because they know why they’re doing what they’re doing. If a kid gets confused about why an increase in w causes the BMI to go up, we can address it mathematically (numerator) and logically (the dude got heavier). Absent the context, though, and you risk losing the point.

  5. Karim, I think there was a typo in your BMI formula. I found a source saying that it’s “Weight times 703 divided by height-squared”. My question is, where does that 703 come from?

    Also, you clearly improved on the “write 3/24 as a percent task.” But I think the salient feature of your improvement is that you made it into a problem. The Wheel of Fortune connection might get your students’ attention, but the fact that it happens on a “real” game show isn’t what’s doing the motivating here. Like Bowen said, an application that is mathematical or puzzle-y in nature can motivate further learning. I would argue that the puzzle-y side is more useful for the average secondary mathematics classroom, as many real-world phenomena are deeply complex.

    And back to Karim, you also stated that, “Mathematics has historically been about real-world applications.” Maybe I’m just showing my true colors here but I have to take issue with that! Sure mathematics is often a powerful tool for understanding the real world, but I think mathematics is more broadly driven by the urge to figure things out. Most of these “things” are abstract in nature.

  6. @Joe:

    Thanks for that correction. Yes, height is in the denominator. Apologies for the typo.

    Re: 3/24, this goes back to the ordering. I didn’t start by saying, “I have to teach fractions to percents. Let me find a real-world example.” Rather, I said, “Hmmm…I wonder whether Wheel of Fortune is rigged,” and the math flowed naturally from that. The outcome may be the same, but the motivation is very different, which means the conversation will be different. In the context serves math approach, the emphasis will be on the numbers themselves. In the math serves context approach, on the other hand, the emphasis is on whether the game show is legit or not, which makes for a very different kind of classroom experience.

    Anyway, this isn’t to knock puzzles for puzzles sake, math for math’s sake. Returning to the BMI:

    B = (703w) / (h^2)

    We can ask, “If you increase h by 1, how much w change for B to stay the same?”

    Or we can ask, “If you grow an inch, how much weight do you have to gain/lose to maintain your BMI?”

    Either way it’s going to be (703w) / (h^2) = ((703)(w + a) / ((h+1)^2)), but I’d argue the second formulation will resonate more while still maintaining 100% of the puzzle-y nature.

  7. At the heart of this debate is how we view education. That’s not the tone of this blog and will be forever debated anyway.

    But I’m not sure dismissiveness of the Times piece is appropriate.

    We take issue with the iniquities of kid’s experiences by not exposing them to said experiences? Garfunkel and Mumford don’t propose anything radical. There is nothing culturally elitist about this:

    ‘Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers.’

    In Cuoco’s piece, he responds to this: ‘I just did the exercise and concluded that we’d end up with a program that equipped students with a set of special-purpose techniques that would likely be out of date by the time they graduated college.’

    Studying the budgets of people, companies and governments is going to be out of date by the time they graduate college?

    And further, what does the present sequence equip students with? Later in his post, Cuoco relates number theory to a house-building project. It was an astonishing insight. It was also one that I would expect from a very specific sub-sect of human: math and science teachers.

    Garfunkel and Mumford’s proposal isn’t perfect, isn’t fully fleshed out, etc. But their sequence is not biased towards anybody except those who want a head-start on a cool career. Forget about abstraction or application or whatever. Designing curricula around these themes (finance, government, engineering) feels appropriate for this specific time.

  8. What a great discussion! As a fisheries scientist–definitely no lack of applied math here–and as a parent, I find myself keenly interested in math education. (For one, we need more natural science college grads with strong math skills.) Interest of course is no indication of expertise, however, so I’ll leave only the following observation: I’d hate to see things go so far to the applied side of the spectrum that all opportunity for experiencing the joy and beauty of “pure math” is lost. After all, wouldn’t that be something like abolishing poetry from the English curriculum on the grounds that it ain’t applicable to completing a job application or writing a software user’s manual?

  9. Peter,

    As a ‘pure math’ enthusiast, I am with you in heart.

    I have started to question why, though. It is intellectual vanity? I imagine myself as a parent, regardless of SES level or where I live, and I’m presented with this choice:

    algebra- geo – algebra 2

    finance- data – basic engineering

    Hmm, I think. The cool stuff about the former will certainly be incorporated into the latter (it has to be). So, uh, yeah, I’ll take the one that will prepare my kid to work, give them access to current events and ideas, and not focus on units with problems like this:

    8. What is the exact value of cos(arc sin(1/2))?

    (source: regents high school exam, mathematics b, 6/15/2010)

  10. The 703 comes from conversions from metric to U.S. measurement. In metric, the formula is simply

    BMI = (weight in kg) / (height in meters)^2

    So you have to account for converting kilograms to pounds, and meters to inches… twice.

    About 2.2 pounds in a kilogram, and about 39 inches in a meter, so the conversion factor is (2.2…) / 39…^2, which ends up being about 1/703. Multiplying by 703 brings it back to the original.

    Never mind that the actual numbers in BMI are just benchmarks anyway… wouldn’t it be more awesome to have a BMI of 0.041?

    If we did change to a curriculum with courses in finance, data, and engineering, we’d still have to prepare students for other careers. Giving students ways to think about and solve problems is the real goal, not cos(arcsin 1/2), which is an assessed skill — I’m confident we could build an equally ugly-looking test with nothing but finance formulas or z-scores.

    Also, an entire YEAR on finance? What would that even look like?? There are a lot of really deep, good applications of mathematics to finance, and at many layers of difficulty. I’d rather see finance, data, and engineering appear in all courses appropriately, and let students who want to specialize in those careers choose to take an elective course in that direction.

    So, why do we tell every student in the country “you must take algebra” instead of telling them “you must take (your favorite application here)”? That’s a long conversation, but I believe the thinking skills that can be developed in mathematics have a greater pull than any specific application we could teach students. Common Core’s mathematical practices do a better job than I can on this, but it’s still a long road to change attitudes about mathematics, pushing more toward those goals than the content and skill-based goals often cited as the purpose of taking mathematics.

  11. A facilitator at one of the workshops I went to this summer, Laura Kent, said that context is “anything that gives the students access to the math.” I’ve thought about that since then, and I’m starting to think that we can’t ever really avoid context, we can only hope to put problems in the right context the right way.

    I identify heavily with the second quote, too. I think one of the things that drew me to math as a student in the first place was the fact that no matter how crazy my over-dramatic high school life was, math was always the same, which was comforting.

    Another thing I think about is how it seems that over time topics that at one point seem to be “pure math” ideas end up having unforseen applications decades or centuries down the road.

  12. Again, I think we’re getting lost in the duality that doesn’t exist. This conversation seems to have strayed towards the “pure vs. applied” poles with no mention of the middle ground. For me, the question isn’t whether we should spend all year teaching decontextualized functions (difficult to apply, but maximally flexible) or turning algebra into a finance course (directly applicable but possibly narrow).

    Rather, it’s HOW to get kids comfortable with the abstraction. Given the way students react to math every single year, it’s really hard to make the case that the traditional (which is to say, decontextualized) approach is the best one. We may love pure equations ourselves, but we also read math blogs.

    Every year students ask, “What does this mean?” and “When will I use this?”. These questions are related. If we want students to apply math to something else then we have to give them the something first. Back to the Wheel of Fortune example from earlier, students walk out knowing a lot about game shows, and that’s great. But they also walk out better at fractions, better at percents and better at probabilities than they would have been otherwise.

    It’s not a choice between pure vs. applied. It’s a question of order. Do you learn grammar rules before having a conversation, or do you have a conversation, learn and refine the rules, and then apply them to a better conversation later?

    I think we’re making this way too complicated.

  13. As a middle school math teacher and creature and user of real-life based math instruction the solution to this seems rather easy and is steeped in education theory already.

    You do both.

    I have students who love solving math in the traditional “text book” style and I have students who would avoid touching one at all costs. So I differentiate and pull in real-life based instruction fairly often, but it is not my only source of math instruction. Some students respond amazingly to it, some students could care less.

    Reach all types of learners and vary your styles.

  14. Well put, Matt! You never know what will grab students.

    Karim, I didn’t want it to be a pure vs. applied argument either. What I think does matter is experience before formalization, and that experience should be in context.

    But I also believe mathematics can be the context.

    One of my favorite open-ended activities in Algebra 1 amounts to “Here are some number tricks. Why do they work? Now you make some.” It’s a real grabber, but has absolutely nothing to do with any application or “real-world” context. Still it serves the same purpose: opening the door for further study and conversation.

    Thanks to you both!