The Third Millennium of Those “Find My Age” Algebra Problems

I’d explain the title of this post, but you already know what I’m talking about. I refer to questions like this one, from the 4th century:

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I, for one, pity old Demochares—enumerating the fractions of his life, yet unable to recall his own age. It’s a bizarre, selective senility, like something from an Oliver Sacks book: “The Man Who Mistook His Life for a Math Problem.”

Or consider this problem, from the 21st century:

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Over the last three millennia, much has changed. Civilizations have risen, collided, and fallen. Revolutions have left legacies in blood and ink. There have been, for good and for ill, 417 million Marvel films. Yet somehow, these age-based math puzzles have remained a constant.

What’s the case for them?

Well, they’re easy to state and tricky to solve. They take a naked mathematical structure and give it a fig leaf of narrative—just enough to require some imaginative effort. They’re a convenient variant on an algebraic theme.

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And the case against them?

Well, they’re artificial. If you’re presenting such a problem to a pupil or a pal, then you’d better hope they’re already invested in the project of mathematical puzzle-solving. If not, a stilted find-my-age puzzle ain’t gonna reel them in.

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I recently came across a “real-life” (well… “fictional-life”) instance of such a problem on the first page of Lolita, Vladimir Nabokov’s classic novel about a child predator who becomes infatuated with a twelve-year-old girl:

In point of fact, there might have been no Lolita at all had I not loved, one summer, an initial girl-child. In a princedom by the sea. Oh when? About as many years before Lolita was born as my age was that summer. You can always count on a murderer for a fancy prose style.

For what it’s worth, our narrator does not give quite enough information to determine the age gap. (You can count on a murderer for an under-determined system of equations.) But a few additional facts—for example, that he was 13 years old that initial summer, and 37 upon meeting Lolita—suffice to fill in the gaps. (The composition and solution of such ghastly problems is left as an exercise for the novel’s reader.)

Should we forswear such problems as carrying the ineradicable stain of Nabokov’s protagonist? Or embrace them as carrying the indisputable glow of Nabokov’s prose?

Cards on the table: I’ve rarely used such problems in my own teaching, though I have nothing against them on principle (icky Lolita associations notwithstanding). My own taste is towards heightening the weirdness and trying to nudge them towards a more open-ended form. Something like this:

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Do these have the same spirit as the classics that open this post? Not really. But then again, those two openers don’t have quite the same structure, either. Love ’em or hate ’em, these age problems will stick around because they’re convenient hooks for hanging all kinds of algebra on.

What say you, my fellow jurors?

NOTE: I’ve made a few edits because people weren’t loving this post’s winning combination of jokey unhelpfulness and pedophilia references. I can’t imagine why!

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29 thoughts on “The Third Millennium of Those “Find My Age” Algebra Problems

  1. Hmm. I’ve had math major students who liked composing their own problems like this. And I’ve always thought of them as riddles to inspire some number play. They’re best before algebra, when learners are poking around in guess and check. Somehow had forgotten that opening of Lolita, though.

    1. Yeah, I think the closed-ended questions are decent puzzles for pre-algebra students (I’ve used them once or twice in that setting) and composing them is probably more fun than solving them for students with more algebraic fluency.

  2. Why not future-oriented “find my age” questions, as Version 2 is hinting at. For instance: “Figure out how long it will take until you’re half your father’s age.” Here the unknown is genuinely unknown.

    Thanks for this post, Ben.

    1. Mmm, I like that – it’s a pretty simple entry point that leads to a sort of cool regularity (that you’re half your parent’s age when you reach the age they were at your birth).

  3. Perhaps the advantage of age problems is their lack of correspondence to real world issues. Correspondence to real world issues can cause problems that are a bit like algebraic PTSD. For example, I have met persons who start to get a bit angry if they hear about two trains that are traveling towards each other. This talk triggers in them bad memories from algebra.

    1. That’s an intriguing point – although it seems that in the train case, the bad feelings flow *from* the algebraic context, rather than *into* it! I guess algebra is less a trigger for bad memories than it is sometimes the source of them.

    2. The lack of real world issues is one reason I use age puzzles. “Real world” problems suffer from significant issues:

      They date quickly. Write a question about Minecraft or Facebook and modern students will think you are even older and more out of date than they do already. I have some linear algebra problems about mobile phone costs that no longer make any sense in an era where no-one pays by calls made.

      They can distract rather than focus. Ask a question related to baseball and a significant number of boys will now be thinking about baseball, not the question.

      They’re often so fake to the real world that kids just think they are pathetic. No-one actually works out a basketball’s parabolic trajectory, so a “real world” question about a basketball’s parabola is totally fake. (Dan Meyer is good on this.)

      So age problems are fine. I didn’t tick that in the poll though because it is only in extreme moderation. I will use one or two age problems a year only, because any more and they become stale and boring. Similarly I will only use one four legs/two legs style one each year — they are intriguing the first time you see one, and quickly become tedious.

    1. I can’t tell if your comment is sarcastic, but then again, I think it was pretty hard to tell which parts of my original post were sarcastic, so I’m definitely Patient Zero for the ambiguity here.

  4. Lots of algebra word problems are artificial and contrived; why single out age problems? And what’s the alternative? Natural, real-world applications can be complicated and cluttered and require lots of set-up or background knowledge.

    But I do think ages can provide a natural, useful (if simple) example of the power of algebra: I have trouble keeping track of how old my brother is–his age keeps changing every year! But I can remember that his age is x–3, where x is my age. (And if I have trouble keeping track of my own age, I can remember that I’ll turn y–1968 years old this year.)

    1. Yeah, you make a good point about the clutter of the real world applications – the few times I’ve tried (or seen efforts) to draw examples from engineering contexts, the set-up is so unwieldy as to be impractical.

  5. I thought you were going to talk about those awful “tricks” that circulate on Facebook and suchlike. The ones that always break down to “if you take your age away from the year, you get the year you were born” but claim to only work this year…
    I’m glad you didn’t though!

  6. Tired of math ed takedown blog posts.

    There is almost always a way to effectively use 95% of what is found in old math curricula.

    Some children (and adults) find these problems very engaging. There’s a reason why they’ve been around for thousands of years! If they are given as one of a set of non-routine problems students can choose from there’s no reason to keep them from those who love good, historical riddles. There’s no reason not to ask about what is weird about the problem, or how it was created to draw students’ attention to the fact that the creator had to have known everyone’s age. You could also bring up indeterminate problems and other related concepts.

    Obviously these are not a good way to teach algebraic concepts. But seriously, I’m annoyed you even set up that dichotomy.

    1. Assuming you’re the same Kate (whose work I love!) who was irked by my dreidel post, I think there’s a common thread: both posts originated with a thing I found funny and cool (the dreidel study; the Nabokov quote) which I then elaborated into a full post by embedding it within a contrived controversy (“dreidel is broken”; “age problems are evil”). If you read the posts as excuses for quoting the thing I quote, they’re just fluffy internet ephemera; if you read the arguments on their own terms, they sound deranged. (Ben wants to fix dreidel by making it insane? Ben hates this arbitrary class of word problems for no good reason?) Obviously it’s my own fault for hiding my ephemera inside weird cooked-up controversies. I’ve made some edits to this post, and will try to avoid that writing tic in the future.

      (That said, I’m keeping the poll because I’m finding the “other” answers interesting to read. I’m also surprised how many people are picking one of the first two – I expected “other” to dominate.)

  7. Yes, these mathematics problems are obscure and annoying. But so are many of the real problems mathematicians, engineers, and physicists face. Being able to untangle the obscure framing of a problem to get at the underlying mathematical structure is a good skill to have.

    1. Yes – the good feature of “word puzzles” is that they teach the habit of re-casting a problem into algebraic form, in which some easy tools rapidly enable us to solve it, then applying the solution back to the original problem. In the real world, we seldom run into puzzles neatly packaged as algebraic formulae; yet those who can map between real-world terms and algebraic can solve real-world problems. So the actual problems that tend to be expressed in this age-form (solving simple linear equations) are incidental to the skill of transforming a problem stated in the vernacular into one expressible in algebraic terms. At the same time, the relative ease of solution in the algebraic form may help students to learn to love algebra.

      One less contrived problem is to ask the class whether they know how old at least one parent is; if each does, get them to write down the parent’s current age and their own; with any luck, most of your class are younger than their parents were when the kid was born, so you can ask how long it’ll be until they’re half the given parent’s age; and you can ask related questions like whether they’re be more or less than half that parent’s age after that. That gives each pupil their own real-world application of the puzzle. Getting them to notice that, in fact, there’s a simple rule (you’re half your parent’s age when you’re as old as that parent was when you were born), can help them to see how algebra connects to reality.

  8. I have recently acquired a private pilot license. Some of the classic math word problems actually come up in flight planning.

    But what really bothers me are cases when the units are wrong. The plane is flying at 300 km / h — No! airspeed is measured in knots. At an altitude of 3 miles — Altitude in ft!

    But, of course, the point of these exercises is to figure out how to unpack the relevant information buried in the problem and present it in the familiar structure of equations, rather than algorithmicly plugging and chugging. Any resemblance to “real life” is incidental (or contrived). Although, when the same structures of word problems get endlessly repeated, then can be reduced to a plug and chug.

    I do like this one that has been making the rounds.

    If a ship holds 10 cows and 26 sheep, how old is the captain.

  9. I don’t think there’s anything wrong with “find-my-age” problems in particular, as much as there’s a problem with the state of word problems and math ed. in general. In moderation, combined with a variety of more interesting problems, sure, students will probably overlook the artificiality. But if ten or twenty of these, all of which have the same structure, are assigned for homework each night, who wouldn’t be annoyed? The point of word problems is presumably to help students learn to convert “real-world” questions into mathematical terms, so perhaps that’s where we should be looking. Instead of daily assignments with a bunch of uninteresting exercises, have students work on projects that are as closely related to the real-world as possible. Sure, some of it may be contrived and/or require background knowledge, but if our goal is to help students prepare for the real world, isn’t that exactly what they should practice?

  10. To reply to a previous comment, save the applications for calculus class. As a better alternative, don’t stress the real-world usage of math. And if students complain how math class is useless, show them “Why Do We Pay Pure Mathematicians?”

  11. I couldn’t find an age problem, but how about: “A quantity and its 1/7 added together become 19. What is the quantity?” That’s from the Rhind Papyrus, written about 3,500 years ago.

    It depends on what you are trying to teach. If you are trying to teach someone how to extract mathematical facts and relationships from natural language text, then age problems are great. They can be expressed in simple language and almost everyone in our society, even little children, understand age and the passage of time.

    Now and then the real world throws out a simply stated problem. For example, while the New York Times as asking if algebra was worth teaching, they published an example of why it might be:

    Usually, there is a lot of real world knowledge that needs to be added and a lot of stuff that needs to be discarded. I think the guy(s) who wrote the Rhind Papyrus understood this, so they structured their teaching accordingly, and some of their problems and their descendants have survived the millennia.

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