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Why Math Word Problems Fail

• namelezz 11y ago
  Does anyone know how the second expression in the picture is evaluated? "12 - 3 = 15"
  • blahedo 11y ago
    That's meant to be "12 + 3 = 15" (or maybe it's an intentional error to make the thing look more confusing?). The idea behind this particular subtraction algorithm is to reinforce two number-sense ideas: 1) a subtraction problem like "32 − 12 = ?" is equivalent to an addition problem "12 + ? = 32", and 2) we can group additions in ways that are convenient to use numbers that are easy to add or to use convenient intermediate results that are multiples of 5 or 10 or 100. EDIT: to fix my own math typo. :P
    • namelezz 11y ago
      LOL, "New math" was misleading. Thanks for your explanation.
  • gohrt 11y ago
    The illustration was copied from the corner of the Internet where innumerate wingnuts tried to discredit Common Core but ended up highlighting their own ignorance. Google search link, sorry the URL stinks: https://www.google.com/search?tbs=sbi:AMhZZis86j61bP06Kk5rlrdY2JVS0H-D1Xsn5sK08f05G1wSqOdLQ5g8T9U_1yF-w7sv38paVFKOdSVHysQNKwWbK4YDdxZm6MqKqCux3W0dwtWLAZKHw1U8MhpBvinHzxDzAyIouinbahNg99ZAqVBD1jJV-GM--c4QnTZdrhFnOqEBzoEaeHDQuDtqrAudr4p7CHLUuhVGEn-XZYm3Ns4GHlGIvrR5s8bhkocwdS0z0vxdtaZo96q6jygbSB8hh9jQuCMFrPYUY4_1I0R-D8Ie79Kx35Yyy_1MUq_1zPo38apyjf67JWzxPIdyqEmm8Vjhl_1Yn-rj_1Bm9_1JfERtUpViA9XMmn5x8S1e-Zqte1UCnXDVmpA0lUGw2GKyJ4ljKlwBblXwsGEzVbnM1mky-unilzl8UJxXE1lfTstS2StECOyckCd6Q1sribDT5_1QCLcpAbQ6HR1ZKKYk74m9UInbq9Q-oydbkgXu617TJRTkboBgzo3tdXcqEsj5EK-djbD7iCeTWggLgSKyQWZv9wragMzWzMr0N6fPIxwal-V9Aj3zCKkYHxkFXPVPqNHA9RqRqCLnmCLSHDIAL1LgPArxqL97Nc4N9O0KLgyqIkDDc-BizjT4M_1PvoIiwdiAsKiJDj_1QjoYqssEazUalAs_1jwJbj1h3k2qbF-u5ddWVuqb-Xfh98I8G_1nJjZIBsaWTBZo35aS2yT7z2N5NLbZU2GITFtnKQoZeyS0TkRivDhrVUx_1NzlMbnYZwW_1K4Wu5jLKWSgCPztn4TSLWVKfFREc2zdvVX2vsQLKmxkg1M_1mjFEoOWmmp8BHT6qC6_122vjSbT5iPbq5Nx35a-n0fIRhfDTYmcfzTWBqCO4pfp5u3lmFSpSJc4BUf4Zg6ey9o3nEMzpCynNaudsn7JcNvBq17PtZ0rwpKtChCeHoDSqnZszvf0t97tblltV2Qt35Q8Panot0ns2ko5jZBlZQQDZs6kpmo4J1P0Od4dKym-w14G54poln_1SQiIcqx4VLJbSLpkhBa4dYUnmkbYNXCWGjoczmmRR6NRYb8D--or9tvaJyjbqTDEBBzWihYEuDPKShccwNamxoxt0PqFddSAlVEC4V8vGoEyOqluHxzElKqxL3cXCYoajTT8CU1wdU7ALhrAvTX797WOqSuSWLNu8QBW3SSsz9ekMX6MOTD-gFplMBSemfYuD5QBEt_1sUD-scqHz4cwUab6Pv4JajRCUQkNBhyGmYDN8STXCNprUaXZwPEGsQTyW1GjTepCnbsKyA8Hl1TMwiCjVYP5g7QHtxu2853xJMsAEYpjoMf_1F6fvClxcDfYx8DqdPGq5JGBEQpwsCcHqUwyWI-Tzw7xni19StRMKcFlIIUo0SxUn3aSmUgV23sCFBV7pnTgzl16cKopXBlBkFPGq-nu93jfVJvLmBAVE_1ZSOhIiOsfY8eQKBEtCxeZycUJ9zMO0b5CkmwKTGj43UYYyjsercPXWaal6GwheFSgs4iND33oU7Vp8XPl4Bw121R5G6ZzdxSJ0pHEc_1x3xhy1IPV2Hp7HI-iQo8z0GQ01lYp77yr8pQ&btnG=Search%20by%20image&hl=en
• mjevans 11y ago
  Maybe a math book should start with a few problem questions, built around every day scenarios. Then it should break down the skills that are needed to solve each of these problems, and cover those skills and the smaller steps of those problems. Finally building to a cohesive solution to the initial problem.
  • lotharbot 11y ago
    > "start with a few problem questions, built around every day scenarios" One of the difficulties here is termed "cultural competency". Textbook writers sometimes try to write in "every day scenarios" that end up being much more familiar to middle-class suburban white kids than to lower-class inner city latin-american immigrant kids. Math problems that are supposed to reference familiar everyday scenarios sometimes end up turning into lessons about language or culture.
    • vitd 11y ago
      In my opinion it's way worse than that. I was a middle class suburban white kid and it was not an everyday scenario to figure out when 2 trains were going to pass. While I do recall hearing a train somewhere in my hometown, I don't recall ever seeing it, let alone seeing more than 1 at a time.
      • lotharbot 11y ago
        in this case, I'm specifically referencing circumstances where they've tried to write problems based on "everyday experience" -- not trains passing halfway through Kansas, but perhaps things like going to a shopping mall or a museum. Things that you think "that's just an ordinary part of childhood" until you spend some time in an area where a lot of the kids have never been to the mall, and the only museum they've ever seen was on a school field trip. Perhaps they don't know the term "supermarket" but they know what a "mercado" is, and they might expect very different products to be on sale from what the textbook author expected. The idea of "cultural competence" is about figuring out how to eliminate unnecessary cultural barriers -- to make sure that a kid trying to solve this type of word problem is working through the problem, and not getting sidetracked by the fact that they don't have the context to make sense of what it's even about.
• delish 11y ago
  Here are two math problems. The first is from an article linked-to in the post. The second is from Challenging Problems in Linear Algebra. Two planes, which are 2400 miles apart, fly toward each other. Their speeds differ by 60 miles per hour. They pass each other after 5 hours. Find their speeds. Show that a Matrix A is of rank 1 if and only if A can be written as A = xy^t for some column vectors x and y. I'm more excited by the second problem. I'm going to use concepts I mostly understand to prove ("if and only if") a concept. In so doing, I'll understand all those concepts better. In the first problem, it looks too much like I'm trying to tell the teacher something he/she wants to hear: the speeds. It's hard for the reader to see how he/she learns something about arithmetic. When doing the math that mathematicians do, what you learn is often explicit: "prove that for all..." It's more motivating. Of course I imagine the makers of Common Core have more expertise and experience in teaching students; I have no plan to reform math education.
  • Espressosaurus 11y ago
    I'll be honest: I solve problems more like the first than the second. I have sensors for certain kinds of information. I would like to be able to process it in order to convert it into other types of information. That isn't to say the second type of problem isn't important: to be able to say "solution X is the same as solution Y, even though their forms are very different" is powerful when you're trying to analyze and evaluate solutions to a problem. It's just a less common problem to solve.
    • deleted 11y ago
      [deleted]
  • saretired 11y ago
    Why are you comparing a middle school problem with a college level problem? I believe the purpose of the first problem is not to teach something about arithmetic, but to test (or reinforce or expand) the child's understanding of d = rt, a pretty important thing to understand.
    • 11y ago
• analog31 11y ago
  I taught freshman algebra for a semester at a big ten university. Therefore it's also a safe bet that I did a shit load of word problems myself when in school. Here's how you solve word problems: 1. From the text of the word problem, remember what section of the textbook it belongs to, and the standard form of equation to be solved in that section. 2. Match the parameters of the problem to the parameters of the equation. 3. Solve the equation. For instance, "find the maximum" means creating an expression that has the form of a parabola (the only function they learned with an extremum, up to that point in the course), and solve for the vertex of the parabola.
  • derefr 11y ago
    This is true for the majority of word problems that exist today. Insofar as this is true, word problems don't really do what they set out to do: make the student understand why they need the formula, instead of merely pattern-matching. I've always imagined a particular kind of "interactive word problem"—one you might be able to present in an interview, or on a computerized test—where the word problem itself contains no numbers that fit any sort of formula or pattern. Instead, the student is expected to notice that the real-world situation could be solved by analyzing it with the help of a certain algorithm—if they just had certain pieces of data. From this, they are then expected to ask for those pieces of data. The pieces they ask for might not all be available—in which case they're expected to come up with another method. (Also, data can be available that's entirely useless for any method of solving the problem.) Imagine an episode of CSI: you see the blood spatter on the floor, the bullet holes in the walls. A real "word problem" would just describe those, then leave it to the student to come to the conclusion that they could use some formula they know to triangulate a source trajectory if they could get positionings for those spatter-marks and holes. Really, the asking for that data—the analysis step—is worth 90% of the mark; successfully applying the formula is just a bonus.
  • HonestQ_902F 11y ago
    From your description and context, it sounds like you're describing elementary algebra and not abstract algebra. Honest question that sounds mean: isn't elementary algebra required for graduation from high school and admission to university? My university didn't offer elementary algebra, and I thought students without elementary algebra skills needed to at least start out in the community college system. In my public school system, the advanced math kids took elementary algebra in 8th grade, and the less mathematically inclined took it in 9th grade.
• sgentle 11y ago
  I think this whole thing is missing the point. People, even young people, are extremely savvy when it comes to figuring out what they need to do to get what they want. What they want is to do well in school. How they do well is to pass their tests. How they pass their tests is by learning the answers to the questions. Making indirect questions that "appeal to real-life problems" is obviously worse, because the main real-life problem students have is passing their tests. Don't get me wrong, I'd love for this to not be the case. If students could be assessed on their ability to actually understand and use the material to generate new insights and solve interesting problems, I think mathematics education would be a lot better. But as long as we're assessing students on equation-memorising, let's not pretend otherwise in the way we teach them.
  • gohrt 11y ago
    The standardized tests have word problems, so your objection doesn't make sense.
    • scotty79 11y ago
      I guess his point was that world problems don't describe situations that are problems for the students. So they can't relate them to their lives and it makes things harder because they add to the complexity of the task.
• force_reboot 11y ago
  From what I've seen of the common core, 90% is fine and just plain old math stripped of any historical baggage. The 10% that is bad is the ideology that math is more ambiguous or less certain than it has been portrayed in the past. This is partly based on politics, and partly on the misuse of mathematical philosophy. I remember being given problems that were plain wrong, e.g. Q: How many miles would a person walking 3miles/h walk in 2 hours? A: 6 miles. I always asserted that the answer is 6, not 6 miles, since the unit is given in the question. Small errors like this in teaching add up to math that looks like a collection of arbitrary commandments that must be memorized. It's understandable people want to move away from this. But this solution is to fix all aspects of teaching so that everything is completely correct, not to pretend that math allows multiple correct answers. We should be teaching that you can express a well formed question in words, to which there should be a unique correct answer. Word problems shouldn't be an added layer of obfuscation, but rather a map from the semantics of math to the semantics of the real world, to test that the student understands this mapping. It's a bit like talking to a backend programmer and saying "if a user did X, then they did Y, what would your function return, and is that correct"? It forces the programmer to consider not the internal logic of their own code, but how that ultimately maps to what the user sees.
  • lotharbot 11y ago
    from what I understand of common core, the correct answer to your example would be "They would walk 6 miles total." There's an emphasis on clear communication and complete sentences. Many of the textbooks would even encourage the student to write out the sentence stem "They would walk _____ miles total" prior to performing any calculations, and then at the end of the problem to check that the number was actually measured in miles and that the sentence made sense and was a reasonable way to answer the initial question. This seems to be what you're encouraging in your final paragraph. This brings up the question: why is the common perception of common core so different from what common core is designed to do?
    • nevdka 11y ago
      Common core is a specification that is implemented many, many times. It's not just each textbook or school system - each teacher will implement the spec in a unique way. With so many implementations, some of them are going to be terrible. But the common perception isn't built off an average or large-scale aggregate, it's built on what people see and hear. A single comically bad implementation can be seen by millions of people thanks to Reddit, Facebook, and the like. A single good implementation will be seen by the students, maybe their parents, and maybe a few more teachers. Most examples people see are comically bad, so the common perception is that common core is comically bad.
• Animats 11y ago
  The Gary Larson cartoon version: [1] [1] https://s-media-cache-ak0.pinimg.com/736x/be/55/08/be5508f4cf2c4c9d79730e2c88fbcc2f.jpg https://s-media-cache-ak0.pinimg.com/736x/be/55/08/be5508f4c...
• bsder 11y ago
  Math word problems fail because math word problems are useless. Good word problems are science or engineering. Once you abstract abstract those away, there is just not a lot of anything actually useful or interesting left.
• unabst 11y ago
  Their value would be so much more heartfelt if students actually encountered math word problems outside class. But they don't. And part of it is because they don't exist. They are hostile games. By hostile I mean they are designed with gotchas and purposely ambiguous, and they have unnatural rules like the inability to ask for more information or to get help. In real life, problem solving does not present itself like this. Those that do well are the students that understand the game, and play along. They are not learning math, but do well "against" math word problems. The question as educators is simply, "is this the definition of good?"
  • EliRivers 11y ago
    Almost every situation for which I have needed mathematics outside the classroom has been a "word problem". Nobody has ever simply come to me and asked me to multiply two numbers together and then add the result to something else. Instead, I get word problems like "how long is this task going to take, if we can give you this much resource?" and "how long is it going to take to drive from A to B at this time, given this knowledge of traffic conditions?" and "what's the best tradeoff of time spent travelling against airfare, to travel from here to there, given these limiting conditions?" and so on and so on. Word problems don't exist? Nonsense.
    • unabst 11y ago
      Where are the gotchas and purposefully ambiguous terms? And none of those are difficult math problems either. Regarding traffic, a smart human (including students) would just refer to an app on their phone. There is a clear line between problems described with words, and these "word problems". The topic was about math, but physics is worse. My high school physics teacher had a hobby of writing to text book publishers about how "silly" some problems were. They're designed to be difficult and relevant to specific material, but in the process reality goes out the window.
• Sniffnoy 11y ago
  I have to disagree with this pretty strongly. The best thing I've seen on the subject of word problems is the following piece by Andrei Toom (warning, this is 98 pages long): http://toomandre.com/travel/sweden05/WP-SWEDEN-NEW.pdf http://toomandre.com/travel/sweden05/WP-SWEDEN-NEW.pdf The claims made in this article seem like the sort of thing Toom is decrying; see section IV in particular. To attempt a summary -- the way math is taught in America is broken, and word problems are horribly misused. Because neither the students nor the teachers really understand math, they attribute their distaste for these useless word problems to the inherent nature of word problems, or to them not being "real-world" enough. Word problems are not the problem; Russian word problems accomplish their goal quite well and are not held in such distaste by Russian students. And no, it's not because those ones are more "real-world". Basically, none of this is the real problem, these sorts of attempts to address it will not help, and attempts to address it by teachers and curriculum-writers who don't actually understand math can actually make the problem worse. But really, I recommend reading the whole thing.
• scotty79 11y ago
  Word problems shaped the way how I take in information. In any given uttering there's usually just few actual pieces of information relevant to what interests me. Word problems taught me how to filter out the gibberish about how Johny felt and whom he bought potatoes for.
• nitwit005 11y ago
  I see the whole word problem approach as a bit dubious. Usually the aim is to help students learn to solve "real world" problems instead of artificial math problems. And indeed, I would say that I've learned math much more successfully when I've had a real world math problem. The trouble is, real world problems tend to be horribly complex. You're in an awkward position where you don't want to give artificial problems, but you also don't want to give real problems. She recommends this expii site, but the first problem I clicked on (the most recent one) seems ridiculous: https://www.expii.com/solve/11/1 https://www.expii.com/solve/11/1