The other day, a friend of mine who does not consider himself to be a “math person” and who is somewhat amused that I edit books about math, sent me this question, which supposedly came from a 5th-grade classroom:
This is not a trick question.
There are 7 DNR wildlife managers in the woods in
Each manager has set up 7 elk traps.
In each trap, there are 7 elk.
For every elk, there are 7 deer.
How many legs are there?
Ten years ago, being someone who would have told you I was not a “math person,” I would have looked at this problem, started thinking about all those legs in traps, and lain down my pencil in defeat. Today, I look at a problem like this and can’t wait to get to work (although I would beg to differ that it isn’t a trick question. It doesn’t tell you how many whose legs you’re counting. Does it expect you to include the wildlife managers’ legs? Is it total legs or legs in each trap? It could have been worded better, but we’ll assume it means total human and nonhuman legs). Anyway, my guess is that the majority of you are having the same reaction I would have had ten years ago (if you're even still with me at this point), but I don’t believe you need be defeated. I am absolutely confident you can do this problem (unless you suffer from dyscalculia, which is highly unlikely if you can do things like tell time and read a train schedule).
What happened to you, then, to convince you that you can’t do it and that you are not a "math person?" Most likely, you were presented with horribly-worded questions such as this one when you were in 5th grade. It probably showed up on a test or work sheet, and you were told to solve it in isolation. If you bothered to ask for help (which, if you were me, you most certainly didn’t), your teacher made you feel like an idiot when you didn’t even know where to begin to answer this question. Most likely, you couldn’t care less about wildlife managers and elk and deer (or, again, if you were me, you were distracted by the fact that the poor things were all caught in traps, and you hoped they weren’t going to be killed). If you managed to get away from those traumatic numbers (7 is a bad number, all around, as those who study numbers and cognition can tell you) long enough (me again. Anything to distract me from the actual equation I was magically meant to create), you were wondering what kinds of traps these were. How could you possibly fit so many large animals into one trap?
Then, having automatically marked this question wrong on your paper without even waiting for the answer, you heard the teacher ask if anyone had the answer. Some Smarty Pants you always despised raised a hand and before being called on, blurted out, “10,990 legs.” “Correct,” your teacher said. What? You were flabbergasted. Where was the potion that had been dumped on that huge number to bring it into sight? If you were lucky, maybe your teacher (or Smarty Pants) explained a little more:
“You’ve got 14 human legs
1,372 elk legs
9,604 deer legs
14 + 1,372 + 9,604 = 10,990.”
You were none the wiser and hadn’t learned a thing or understood the problem any better than when it had been handed to you. Nobody discussed the problem. Nobody was encouraged to show how it could be done another way. You just moved right onto more problems with invisible answers for which you were given no revealing potion.
Today, I’m going to put what I learn from the books I edit into practice. I’m going to see what I can do to help you understand this problem better. First of all, we’re going to take into consideration situated cognition, which is learning and problem solving in particular contexts. Even those who flunked math in school can develop powerful problem-solving skills when they are doing so in their own contexts, like when they're camping, if that's what they like to do, or cooking, or building a doll house. In fact, even when doing things we might not like to do, but need to do, such as attending staff meetings, we can be pretty good at it. If not, most of us wouldn't last too long in our jobs.
I’m pretty sure that you, my blogging audience, is likely to have the same reaction as my fifth-grade self would have to the elk and deer problem. Interest is half the battle, and most of you don’t find yourselves in situations in which you have to (or care to) count large animal legs. So, what interests you? Let’s take a wild guess here: books? And if you are interested in books, you’re familiar with walking into bookstores and skimming pages of books you might want to buy. Forget elk and deer then. Here’s my problem for you:
(First of all, before we get started, get out your calculators, because, even though I am going to make these numbers much easier to work with than Smarty Pants did, there is absolutely no reason for those of you who had that sort of teacher for fifth-grade math to try to add and multiply numbers greater than 100 without the use of a calculator.)
7 book addicts belong to a book swap group. Every so often, they get together, each bringing 2 books they’ve read to swap with others in the group (isn’t that a neat idea? I just made it up, but I’d like to belong to a group like that). Everyone gets to choose 2 books from the pile of fourteen. Today they’ve met for lunch at Good Enough to Eat in NYC. They all trust each other by now, confident no one is bringing a real dud to this meeting, so each one skims one page of each of the books he or she has chosen, and then they get down to the business of eating and catching up with each other. Near the end of the meal, one of them says, “Hey, why don’t we all go to The Strand today?” They all agree this is a great idea, and they make their way to the bookstore with “18 miles of books.” Once at The Strand, they go their separate ways, and:
Each addict visits 7 sections.
In each section, each addict looks at 7 hardcover books.
For each hardcover book, each addict looks at 7 paperback books.
Each addict skims four pages of each book to decide which ones to buy.
How many total pages do all of the addicts skim that day?
Now, let’s pretend I’m one of the addicts. I head first to the mystery section, because I’ve had so much fun reading mysteries for my mystery book discussion group. I check to see if there are any Ian Rankins. I’m in luck. I find one hardcover Rankin and seven paperbacks I haven’t read (not hard to do, since I’ve only read one Rankin so far, and he’s an extremely prolific writer). That’s 1 HC + 7 PB = 8 Ian Rankins. I skim 4 pages in each for a total of 32 pages from Rankin. Next, I cruise the shelves for Dorothy Sayers and again find one HC and seven PB I haven’t read. That makes 64 pages I’ve now skimmed. I do this with 5 more mystery writers for a total of 32 x 5, which is 160. Add the 64 Rankin and Sayers pages to that, and I’ve skimmed 224 pages before I leave the mystery section. In other words 7 (authors) * 8 (books) * 4 (pages) = 224 pages in one section.
Next, it was onto literature. Again, I skimmed 4 pages from 8 books from 7 different authors for a total of 224 pages. Then I visited the cookbook section where I found one hardcover book of vegetarian recipes and seven vegetarian paperbacks to skim, one hardcover and seven paperbacks on chocolate, one hardcover and seven paperbacks on soups, one hardcover and seven paperbacks on Chinese food, one hardcover and seven paperbacks on Indian food, one hardcover and seven paperbacks on bread, and one hardback and seven paperbacks on chili peppers. (I also found a whole shelf of Rachael Ray books, but I purposely ignored those.) By the time I left the cookbook section, I’d skimmed another 224 pages. I then moved onto the psychology, science, history, and religion sections, where I again skimmed 224 pages in each.
Uh-oh, after all that, I realized I’d better get to the checkout, because it was time to meet all the other addicts (after all, we’d been here for nearly five hours). Lo and behold, when we all got back together, we discovered each and every one of us had skimmed the exact same number of pages from the exact same number of HC and PB books. I skimmed 224 pages in seven different sections, and so did each of the others. That makes 7 (addicts) * 7 (sections) * 224 (pages) = 10,796 pages. But don’t forget, before we even got to The Strand, we’d each skimmed one page from each of our two swapped books. In other words, we’d each already skimmed two pages before we'd even walked through the doors of The Strand. That’s 7 * 2 = 14. We needed to add that 14 to the 10,796 to get 10,990.
The formula for this is (7*7)(7 * 8 * 4) + (2 * 7) = (49)(224) + (2 * 7) = 10,976 + 14 = 10,990. If you were talking about elk and deer in that original problem, that 224 would be the number of elk and deer legs in each trap. The 49 would be 7 (traps) * 7 (wildlife managers). The 14 would be the number of wildlife managers' legs. As you can see by Smarty Pants’s answer, neither one of us did this problem the same way (and there are other ways to do it, too). However, aren’t my numbers a little more friendly? Wouldn’t you rather tap in 49 * 224 on your calculator than 1372 + 9604? And, even if you hate math, isn’t it easier to get that 49 and that 224 than to get that 1372 and that 9604? And that, my friends, is what a good teacher would have helped you see.
Oh, and for those of you who wonder how many books I actually bought, well, you know the answer to that: I bought all 56 books from each section and found myself bringing home 392 (Strand purchases) + 2 (swapped) books! And yes, of course, I know real life is rarely this un-messy. I’m sure all the Ian Rankins would have been gone, and I wouldn’t have been able to find a single hardcover Dorothy Sayers, and one of my fellow addicts would have found nine hardcover M.F.K. Fisher’s, while I found none, not to mention the fact that I can’t possibly carry 394 books around NYC by myself, but that’s where fantasy intersects with math.
This was fun to read. :) I'm really good at this kind of math, and I think it's because I tend to think outside the box. But calc and I are not close; I can't look at a graph and see the equation the way a lot of my friends could.
Lol! I cannot deny I engaged with the page-skimming much more than with the leg counting. Most of all, I wanted to BE one of the book addicts in the Strand. A trolley should do the trick for those 300 hardbacks, no?
I have always loathed math. And I'm a lousy speller.
But I assume it's just because I'm detail-averse.
My mother was also a bad speller but you can't inherit a pipe.
I'm sure you know that I found this enjoyable. Agreed that context is so important.
Not bad. I certainly understand the context issue. In fact, I think it highly improbable that a book lover would skim exactly four pages per book and you are correct about the original problem being confusing and poorly worded. You're also right that the problems are isomorphic and isomorphism is a much more interesting mathematical concept than arithmetic.
I came across a similar math problem somewhere -- I think Hobgoblin forwarded it to me -- and figured out the answer without too much trouble -- but I like math and so was excited by the possibility of a challenge. I was a little disappointed by it, though, because I thought it would be less straightforward. It says it's not a trick question, but shouldn't there be some ... I don't know ... twist or something? Something beyond fairly simple multiplication? This is more a matter of logic and careful thinking, making sure you don't miss a step or move through it too quickly. It seems to me to be more about language than math. But maybe my idea about math is wrong, and it's really often about language.
You completely lost me. Actually, you lost me at the first mention of 'math' but then I persevered, and then I got lost again. There were numbers, and then there were numbers with symbols, and oh my god, I remember this sense of dawning incomprehension, it's every maths class I ever attended...
Elegantly done. I admit I followed the Smarty Pants method myself, but you've grouped the sums much more neatly. I'm glad you're helping to fight the good fight!
Reading your problem made my head spin because the first thing I start to do is draw - 7 people (is DNR for Do Not Resucitate - I am consumed by that already) then 1 deer in each of 7 boxes with four legs each. Then draw 7 elk for each deer with 4 legs each and realize my initial trap was drawn too small so I draw another box outside the deer trap box. Then I snicker at my drawing and start drawing ropes from the traps to the 7 DNRs. Then I wonder where they are dragging the traps full of a rediculous amount of animals...wait..what...time's up? Oh crap, I didn't even answer the first question. But I do have a pretty awesome drawing to turn in.
Eva, calculus was my downfall, but I'm convinced it was due to poor teaching. I can still remember friends of mine saying, "Now, can't you see this sphere spinning around on its axis?" and replying, "No, I can't." Luckily, that kind of stuff can be left up to engineers, etc. while we do our page-skimming in book stores.
Litlove, just hop on a plane to NY (I won't even ask you if the plane is traveling at such and such a rate, and NY is so many miles from London, how long will Litlove have to read on the plane?). I'm sure I can find five other book addicts to join us at The Strand.
Nigel, I'm a lousy speller, too. Being detail-averse probably means you hate arithmetic, but you don't fool me when it comes to math. There's proof all over your blog that you're good at spotting patterns and problem solving.
ZM, yes, I knew you'd like it.
Dorr, technically I think it IS a trick question, because most people would probably forget to count the human legs (just like most people would probably forget to count those pages skimmed during the book swap in my problem). However, you're right. It's more logic than anything else, but that's the sort of math most of us use most out here in the non-textbook world.
MFS, you might be one of those few who suffers from dyscalculia. I've never been sure. If so, it's not too severe, and you're lucky because you're basically so brilliant in every other discipline, it doesn't really show.
Lokesh, thank you. And if you used the Smarty Pants method, you must be very comfortable with large numbers.
Sara, LOL! My first plan of attack was drawing the animals, too. Then I decided, due to my horrible drawings, that I'd better use symbols for each (triangle for elk, and circle for deer)instead.
I am an aerospace engineer swimming all day in applied mathematics, scientific computing, calculus, linear algebra and other swear-word-like math regions, but I can never be bothered with vain problems like those, be it elk or books (or even gardening).
I think what a long scientific education did to me is give me a very sharp view to the backstage area of math teachers. I know how these problems are written. I know they are 100% artificial. I know I can solve them if I have to (providing that there is no ambiguity about whose legs we have to count, and about the statistics of maimed animals and disabled rangers). And I know the result is generally uninteresting (unless it goes a little deeper, as in finding an interesting relationship between prime numbers, closed sets, and multiples of seven - maybe that's where my true interests lie: math for the sake of math).
Conclusion: if I had to do it all again, I am afraid I would not be able to play along anymore. Maybe I'd even fail my exams.
Norman, I agree about the ridiculousness of skimming exactly 4 pages (as well as finding 1 hardback and 7 paperbacks), which is why most of these artificial problems are silly. Still, I like to solve them.
Mandarine, funny. When my friend first sent me that problem, one of my questions back to him, was, "Do all of the elk and deer have all 4 legs, and do all the DNR's have both legs?" I also asked, "What kind of traps are these? Do they have legs?" This is why I hate standardized testing. Your point is interesting, given that there's a big drive here in the U.S. to encourage mathematicians and scientists to leave their current positions and become teachers. Maybe they'll change the world of math ed?
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