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Christopher Danielson finds a text in his college library called How to Solve Word Problems in Algebra: A Solved Problem Approach (Johnson, 1976).

A sample problem:

Mrs. Mahoney went shopping for some canned goods which were on sale. She bought three times as many cans of tomatoes as cans of peaches. The number of cans of tuna was twice the number of cans of peaches. If Mrs. Mahoney purchased a total of 24 cans, how many of each did she buy? (p. 14)

From Johnson’s preface:

There is no area in algebra which causes students as much trouble as word problems…Emphasis [in this book] is on the mechanics of word-problem solving because it has been my experience that students having difficulty can learn basic procedures even if they are unable to reason out a problem.

Danielson:

And here is the crux of the matter. I have already argued that the very nature of word problems is such that people’s actual experience has no bearing on solving them. But in this preface is the rarely stated truism that we can train students to work these problems even when we cannot teach them to think mathematically. Entire sections of textbooks are devoted to the translation of word problems into algebraic symbols and Ms. Johnson has written the book on it.

2011 Mar 07: Christopher Danielson responds to some of our commentary at his blog.

46 Responses to “[PS] The Progress We’ve Made In 34 Years”

  1. on 05 Mar 2011 at 10:13 amDavid

    My question would be, why would we want students to solve these problems? What would be the purpose.

    I think that what happened is someone said, “let’s make sure we make math relevant.” Someone else said, “okay so we need to discuss real life problems.” Another person said, “well we can’t trust teachers to have the time or dedication to figure this out for themselves, so we’ll create word problems for them,” and the rest is history.

  2. on 05 Mar 2011 at 12:01 pmWaldo

    Relevant real life problems are messy. They’re usually too messy to be solved using simple high school math like linear systems or quadratic functions.
    I’d be perfectly happy teaching pure mathematics. The perception is that students and parents want relevance so we crowbar in word problems.
    It’s my experience that if the students understand the theory behind the manipulation of symbols, that’s relevance enough.

  3. on 05 Mar 2011 at 12:15 pmChris Sears

    I’m not sure if you are using “progress” ironically. This is a good example of the inertia in the textbook industry. I’ve been thinking about that a bit because I’m working with a small group in AMATYC about online educational resources. Your article gave me some good things to think about.

  4. on 05 Mar 2011 at 12:29 pmChris Sears

    Waldo:

    Relevant real life problems are messy.

    That’s why we need to teach about the messiness.

    It’s my experience that if the students understand the theory behind the manipulation of symbols, that’s relevance enough.

    I’ve been reading about Piaget and educational psychology recently. People build knowledge on existing knowledge. To strip away prior experience by focusing on the symbols is to rob students of the raw materials for learning.

  5. on 05 Mar 2011 at 1:05 pmjg

    …And here is the crux of the matter. I have already argued that the very nature of word problems is such that people’s actual experience has no bearing on solving them. But in this preface is the rarely stated truism that …

    Well… if we stick with peoples’ actual experience (especially kids’), then there will be almost no problem solving at all. They’re not used to actually thinking. We definitely need to try to model and teach reasoning, but the lion’s share of the difficulties isn’t the silliness of the problems, no matter how silly they are – it’s the illogical, poorly defined, and trite mental worlds that most folks live in! _That’s_ our challenge!

  6. on 05 Mar 2011 at 2:23 pmPedagoNet

    I liked that problem.
    Have you asked students for their opinion?

  7. on 05 Mar 2011 at 3:58 pmDan Meyer

    Some pretty good pushback in this thread on the pseudocontext meme.

    Waldo: Relevant real life problems are messy. They’re usually too messy to be solved using simple high school math like linear systems or quadratic functions.

    Depends on what we mean by “solved.” If you’re looking for the quadratic model of projectile motion to be accurate to the nearest thousandth of a centimeter, then I agree: the real world is too messy to be solved. But if the only precision you’re after is, “Will the ball go in the hoop?” then the quadratic model works just fine.

    Waldo: I’d be perfectly happy teaching pure mathematics.

    Me too. Mostly. Math has had some extremely satisfying and profitable applications in my day-to-day life. It’d be a shame not to put students in a position to experience those for themselves.

    Waldo: It’s my experience that if the students understand the theory behind the manipulation of symbols, that’s relevance enough.

    Not that either of our anecdotal experiences should be the ultimate arbiter of the matter, but I’m curious what your experience is. Public math classroom? You’ve never received the question, “When will I ever use this?”

    PedagoNet: I liked that problem.

    Me too. But I had really strong mathematical schemas (for a high school student) to fall back on. I wasn’t still trying to figure out the answer to the question, “Where does this math stuff figure into my life?” If I were trying to answer that question, this weird contrivance where Mrs. Mahoney knows all these relationships and facts about the cans of tuna, tomatoes, and peaches she bought, but not the number of cans themselves would pretty well answer the question for me.

    PedagoNet: Have you asked students for their opinion?

    Not explicitly, if only because the answer has been really implicitly clear.

    jg: Well… if we stick with peoples’ actual experience (especially kids’), then there will be almost no problem solving at all. They’re not used to actually thinking.

    This mistakes cause and effect. If students are told on a daily basis that they have to shut off their prior experience and common sense to succeed with pseudocontextual assignments, then they’ll oblige. It’s the wrong response to then say, “Look at them! They don’t think! It’s pointless to assign them thoughtful problems!”

  8. on 05 Mar 2011 at 4:20 pmWaldo

    Dan,

    I tell the kids pure math is strength training for the brain. Just because you never do bicep curls while in a “real sports” situation, it doesn’t mean that doing those curls is irrelevant to your sporting ability.
    I also tell them that most universities use math marks as the best indicator of students’ ability to think logically. That’s why universities don’t seem to care much about how the topics in a high school math course match up with the program for which you’ve applied.

  9. on 05 Mar 2011 at 7:26 pmMark Schwartzkopf

    There seems to be some idea that word problems are not math, but rather, a demonstration of the intersection of math with real life. Under that view, yes, this problem is pseudocontext. But I don’t believe it.

    From Danielson: “But in this preface is the rarely stated truism that we can train students to work these problems even when we cannot teach them to think mathematically.”

    I’m not sure what could be meant by “thinking mathematically”. Translating words into algebraic expressions, without the need to understand the situation fully IS thinking mathematically.

    Before 1500 or so, the science of math was developing at a snails pace. It was extremely hard to think about. So much so that people would have to travel to other countries in order to learn the arcane skills of multiplication and division. At this point, math and algebra texts were pretty much exclusively word problems; algebraic notion had not been invented yet. As the mathematical community began to develop the means of translating word problems into algebraic notation, math became way easier, and began to develop at a faster and faster rate.

    This primary concept here is the idea of taking a problem that is hard to understand, and translating it into a language where it is much easier to think about. This idea could be argued to be one of the very most important ideas in math.

    To me, word problems are where you practice this translation process. Word problems have no obligation to be relevant to the real world. Many of the contrived conversations that you translate in Spanish class are not particularly true to real-life Spanish speaking, but they do help develop the skills necessary for real-life Spanish speaking.

    I don’t see how creating problems (and in this case, books) to allow you to practice this translation from word problems into the easier-to-think-about algebraic expressions is pseudocontext.

  10. on 05 Mar 2011 at 7:39 pmChirs Sears

    Waldo:

    Just because you never do bicep curls while in a “real sports” situation, it doesn’t mean that doing those curls is irrelevant to your sporting ability.

    Actually, in strength training circles, there is the concept of “functional strength.” Functional strength is like the strong man competitions you see on ESPN2 with people pulling cars and carrying heavy stones. These are the events that evolved from doing real work

    The opposite of functional strength is “beach muscles” from bodybuilding. This is lifting heavy weight just to build muscle size, and nobody will notice if you can’t open a jar of pickles.

    The problem with your analogy is that bicep curls are the classic example of a beach muscle exercise.

    I also tell them that most universities use math marks as the best indicator of students’ ability to think logically.

    I’m not a fan of this line because students feel fatalistic about their ability in mathematics. They already feel like they are doomed to be bad at math and no amount of effort will improve their ability. Adding to their math funk the belief that their success in college will be torpedoed by their genetic inability to do math is a recipe for student disengagement.

    I personally think that the best way to get the most students involved with math is to get them to believe that math will help them today. The students who will respond to learning math for its own sake will seek you out for support.

  11. on 05 Mar 2011 at 9:17 pmDan Meyer
    Mark: I don’t see how creating problems (and in this case, books) to allow you to practice this translation from word problems into the easier-to-think-about algebraic expressions is pseudocontext.

    It’s pseudocontext if the expressions have nothing to do with the context from which they’re derived. For instance, what is gained by using “peaches, tomatoes, and tuna” instead of “secret number one, secret number two, and secret number three?” What is lost?

  12. on 05 Mar 2011 at 9:31 pmR. Wright

    The most depressing thing about “word problems” is that they have apparently always been around, in all their nonsensical glory. Google the phrase “word problem as genre” and you’ll find a paper on the subject that contains, among others, the following example from 4000+ years ago:

    “I found a stone, but did not weigh it; after I subtracted one-sixth and added one-third of one-eighth, I weighed it: 1 mana. What was the original weight of the stone?”

    Sometimes I despair of trying to combat a tradition basically as old as recorded history.

  13. on 06 Mar 2011 at 6:28 amYaacov

    I’m a newer teacher, and I’ve been trying different things to engage my students. I’ve tried some WCYDWT lessons with success, but I’ve had more success with taking an extremely respectful approach to the questions and answer from students. My rule of thumb is to ask “How would I react if an esteemed colleague with background in a different subject said what my student just said?” and try to react the same way.

    I’m finding this approach leading to a high level of engagement among my students compared to the other things I’ve tried, including the WCYDWT lessons. As an example of the level of engagement, every Friday they break into groups of five or six and discuss the work they’ve done on our puzzle of the week. They stay on topic without my intervention for twenty minutes, regardless of whether they managed to solve the problem or not, and regardless of whether they are friends with the other people in their group or not.

    I’d be interested in hearing from other teachers who’ve tried this method about what benefits and limits they’ve found with it and if they have any suggestions for how I can improve on this method. I’m only one month in, so I’m well aware that I may still be in the honeymoon phase.

    p.s. Sorry if I’m taking this thread on a bit of a tangent, the comment that got me thinking about this is the one from Chris Sears that “…the belief that their success in college will be torpedoed by their genetic inability to do math is a recipe for student disengagement. I personally think that the best way to get the most students involved with math is to get them to believe that math will help them today.”

  14. on 06 Mar 2011 at 8:29 amWaldo

    Seriously Chris? You really think my analogy isn’t sound because I said bicep curls instead of a kettle bell workout done on bosu balls?

  15. on 06 Mar 2011 at 8:39 amSam Snoddy

    Being relatively new to teaching math, I have struggled with these dilemmas. Coming from a computer science background I’ve tried to distill my philosophy down to the simplest ideas.

    Math is
    1. Finding patterns.
    2. Describing the patterns
    3. Make predictions from the patterns

    My first assumption is that we are all born with the ability to look for patterns in the world around us. It is what allows us to survive as a species.

    Describing patterns is esential in order to share knowledge. We have to be able to communicate that information to others. The best way for most people to acquire that knowledge would be direct observation but in many cases that is impractical or impossible. So what is needed is a common language to succinctly describe the patterns observed.

    Lastly we take the knowledge of the patterns we recognize and apply it to the world around us to make predictions. I don’t care how “math illiterate” a person says they are, if they could not recognize the patterns around them and make predictions based that information they will not succeed or survive. We are genetically encoded with the ability to observe and make predictions from patterns.

    Where most people have difficulty is describing the patterns. Because of the quirks of language, interpretation of someone else’s statements can be challenging. I see word problems having two purposes. First, they are simple exercises in taking a limited set of details and writing them down mathematically to practice our precision in mathematical language. Next, word problems allow the students to take the prior knowledge of mathematical patterns and apply it to an imagined problem. I think this is critical because it allows us to take prior experiential knowledge and apply it to similar systems and make predictions without prior knowledge.

    Now, having said all that, I still have a heck of a time convincing my students. Having watched Dan’s TED lecture this morning I am enthusiastic about trying more experiments first to build more experiential knowledge and then moving more to the theoretical WCYDWT afterwards.

  16. on 06 Mar 2011 at 9:39 amKathy Clark Couey

    Relevance is required if you want any engagement from your students, and the relevance better reach into careers that are appealing to the wonderful dreamers in our classrooms.

    I can’t let go of my frustration when Danielson wrote:

    “even when we cannot teach them to think mathematically”

    If we can’t teach them to think mathematically or reason why are we bothering at all? How many times have we taught a concept and died inside when students couldn’t generalize. Are we teaching them before their brain is ready? I think we are missing something.

  17. on 06 Mar 2011 at 9:42 amKathy Clark Couey

    Relevance is required if you want any engagement from your students, and the relevance better reach into careers that are appealing to the wonderful dreamers in our classrooms.

    I can’t let go of the frustration when Danielson wrote:

    “even when we cannot teach them to think mathematically”

    If we can’t teach them to think mathematically or reason why are we bothering at all? How many times have we taught a concept and died inside when students couldn’t generalize. Are we teaching them before their brain is ready? I think we are missing something.

  18. on 06 Mar 2011 at 12:11 pmJosh W

    I believe that it is human nature to try and figure things out. When you have a student or group of students that are disengaged, either sitting quietly, texting, or loudly questioning why we have to do this, is because they have been told over and over, by tests’ scores and grades, that they are not good at _______(math, division, fractions, word problems, etc.). At this point the potential reward at solving a problem, or finding a pattern, or simply getting the answer, is overshadowed by the dread of hearing yet again that they are not good at something.

    I think that part of “hooking” students is giving them work that they can feel success with.

  19. on 06 Mar 2011 at 1:57 pmlouise

    If I responded the way I would to an esteemed colleague I would be fired for sarcasm and eye-rolling. I try to respond as if it were my own kid. Adults can take it, especially the esteemed kind. Kids are astonishingly fragile.
    Anybody else here tried Mrs. Lindquist?

  20. on 06 Mar 2011 at 2:47 pmDan Meyer
    Sam: … we take the knowledge of the patterns we recognize and apply it to the world around us to make predictions. I don’t care how “math illiterate” a person says they are, if they could not recognize the patterns around them and make predictions based that information they will not succeed or survive. We are genetically encoded with the ability to observe and make predictions from patterns.

    This helps me figure out what seemed so evolutionarily wrongheaded (not to mention condescending) about jg’s “illogical, poorly defined, and trite mental worlds that most folks live in.”

    Sure, people can be counted on to mess up the really complicated decisions. (The question of “whom should I love?” seems to involve a lot of illogic, for instance.) But we’re talking about the observation of relatively uncomplicated patterns in secondary math instruction. This is what our minds were made for.

  21. on 06 Mar 2011 at 2:57 pmColleen

    I have been teaching Algebra 2 for a few years now and have had plenty of kids ask me when they will ever use radical equations, logarithmic equations, composition of functions, etc. “in real life.” I’ve given them versions of all the answers mentioned above: Here’s a scenario in which you might…, I’m helping you to think mathmatically…, this will help you in calculus…, etc. Of course no one is satisfied with any of these responses. I’m not either!
    There is plenty of math that is directly related to their lives outside of the classroom (mostly in algebra 1, geometry, and statistics, I think) but that is not all they are required to learn in high school. At our school anyway, all roads lead to calculus and you co as far along this track as you can in your 4 years. I see that the students are MUCH more engaged when I use the WCWDWT-type scenarios, but it seems that for algebra 2 and precalculus it is more difficult to cover the required topics with this approach. Also, when I do less “I’ll-show-you-how-to do-it-then-you-do-lots-of-problems-like-it” teaching they end up doing poorly on the math department’s (traditional) common assessments. Does anyone have insight on making Algebra 2 relevant for this kids who are not necessarily headed for calculus?

  22. on 06 Mar 2011 at 3:18 pmBowen Kerins

    I really latched on to one phrase Danielson found: “the mechanics of word-problem solving”. I taught these mechanics to high school kids because it’s what was in the book: use this box for rate-time-distance problems, use this box when people are painting houses. I spent weeks teaching this because it seemed like the right thing to do… here’s one example from the Johnson book:

    Two women paint a barn. Barbara can paint it alone in 5 days, Sara in 8 days. After 2 days Sara gets bored and Barbara finishes it alone. How long does it take to finish?

    While this is a terrible problem and context, that’s not the worst of what I did wrong. The worst thing was focusing on the mechanical method instead of any way to “think through” the problem. It took me a long time to realize that there is barely any mathematics in these mechanical methods, and longer still to realize the mechanical methods don’t do kids any good in the long term.

    I’m not trying to say “ban word problems”. I think the issue is less about the problems themselves than about the strategies we teach students to solve them. Kids can learn general strategies that lean on their ability to solve nearly-equivalent problems in arithmetic. If we ban something, it should be the one-purpose boxes instead of the word problem!

    Ideally kids should be learning and applying the same mathematical thinking skills, ones they can use no matter whether the context is “real-world”, mathematical, or some crazy made-up situational junky-junk.

    It’s one thing that makes me hopeful about Common Core, because some of those thinking skills are spelled out in the Mathematical Practices. We’ll see if people decide to pay attention to ‘em…

  23. on 06 Mar 2011 at 7:49 pmChristopher Danielson

    Lovely discussion! I have more to say, and have said it here:
    http://wp.me/pAG7Q-6u

  24. on 06 Mar 2011 at 8:38 pmChirs Sears

    Waldo:

    Seriously Chris? You really think my analogy isn’t sound because I said bicep curls instead of a kettle bell workout done on bosu balls?

    Actually, I find your analogy to be perfectly sound. I think it supports my argument better. If you hadn’t mentioned curls, then I would have.

    I agree with you that doing math helps the students develop their brain power. It needs to be a part of a balanced curriculum that helps student function outside of the classroom. Teaching Soduko or chess will also strengthen students brains without giving the same false impression of relevance in their lives as the word problems sited above.

  25. on 07 Mar 2011 at 6:47 amMark Schwartzkopf

    I’m still not seeing why everyone seems to expect word problems to be true to life. Or why the skill of translating is thought to be less mathematical than other skills.

    Getting back to the question of pseudocontext, I realize that this problem somewhat fits the second working definition of pseudocontext, but it doesn’t seem to fit the primary idea; it’s not a problem that alienates students. It has no claim to be true to life. I know you can turn it into a “secret number A, secret number B, and secret number C” problem, but I think that makes it a less rich type of problem.

    Taking a collection of sentences and figuring out which ones make mathematical statements and which ones don’t is a valuable, real-life skill that “secret number” problems rarely require. If we were changing this into a “secret number” problem, what would happen to “Mrs. Mahoney went shopping for some canned goods which were on sale.” This is an important sentence that adds depth to the problem. If all the statements in this problem are relevant to the solution, this problem would lose depth.

    Also, implicit in a word problem like this is that you can take statements from real life and translate them into algebraic expressions in order to get real answers. Despite the fact that this problem is not a real life problem, it is clear that these skills could easily be applied to real life. I’m not sure that “secret number” problems would achieve that.

    Also, I wonder how many “secret number” problems can be done before you’d be bored to tears with them and want some more traditional word problems.

  26. on 07 Mar 2011 at 8:44 amShari

    I’m tending to agree with Mark. At first glance, the problem looks ridiculous with wording that is overcomplicated. The statements make about as much sense as a logic puzzle with a clue like, “The baker who made oatmeal raisin cookies doesn’t know how to do the hokey pokey or flamenco.”

    But I’m beginning to think that problems like this have a place in the classroom. The provide practice sorting through what’s necessary, what isn’t, how can you find what you need. Yes, in a limited way. Problems like this have more in common with the logic puzzle, and maybe they should be approached in that way.

    This type of problem does a disservice by labeling it a real-world problem. We would never solve a problem exactly like Mrs. Mahoney’s can goods problem in real life, just as we would never solve a logic puzzle involving bakers who can or cannot dance. These puzzle problems are the bicep curls or the free-throw shots taken to maintain skills that we’ll use when we approach a real-life problem.

    Speaking of skills, you would never ever find the common denominator of a set of fractions just for the fun of it. Even so, we need to teach that skill in order for students to solve real-life problems. It’s up to us as teachers to help students make the connection between the skills they’ve learned and their application to real-world problems.

  27. on 07 Mar 2011 at 8:49 amShari

    One more thing…. I had a few questions…

    How does this can goods problem differ from the Coke/Sprite problem other than engagement level? When will you ever have a Coke/Sprite mixture problem in real life? I’m sure there are a few cases, but does it eliminate the benefits of the Coke/Sprite problem if you don’t mention how that problem relates to the real world?

  28. on 07 Mar 2011 at 10:51 amRWW

    How would you pose the Coke/Sprite problem without a context?

  29. on 07 Mar 2011 at 10:53 amR. Wright

    (Sorry; I keep accidentally posting under different names. Perils of using more than one computer in my office at once…)

  30. on 07 Mar 2011 at 11:55 amDMT

    Has anyone else here read “The Teaching Gap”? I highly recommend it to all math teachers.

    A lot of this conversations reminds me of one of the differences highlighted between Japanese and American math teachers.

    American math teachers assume that students will think math is boring and so make a lot of effort to make it fun or interesting either by “being cool and funny” or by trying to embed the math problem in a “real life context” that the students might find interesting.

    Japanese math teachers assume that students will think math is interesting for it’s own sake – that they will think solving difficult problems and tyring to figure out the answer will be interesting.

    On a related note, another difference that also seems relevant to this conversation:

    American and Japanese math teachers were asked what they thought the most important of math was.

    The majority of American teachers answered “procedures”.

    The majority of Japanese teachers answered “relationships between ideas”.

  31. on 07 Mar 2011 at 12:24 pmDon

    “I tell the kids pure math is strength training for the brain”

    Then do pure math.

    But the initial word problem here is not pure math, it is an exercise in translating verbiage into symbols. To do THAT and use an unrealistic situation – both in premise and in data content – impedes learning.

    It impedes learning because it sabotages motivation by telling people “you don’t need this.” That’s crappy by itself, but it does something worse: it impedes learning because it doesn’t present realistic situations, robbing students of the opportunity to understand through context and to really learn how to throw away unnecessary information.

    We have no problem accepting that students learn vocabulary words by reading them in use and discerning them from context. Why are we adverse to providing them this aid in math? Why do we increase anxiety by removing believable context?

    The thing that makes the example at the top so offensive isn’t just that it shoehorns in unrealistic context (and then talks about teaching people how to solve these garbage presentations), it’s that it would be SO EASY to make a can problem that might present someone a challenge and could really happen.

    How about… Mrs. Mahoney has a recipe for fruit salad that serves 18 people. For every can of peaches it requires three cans of apples. It uses twice as much apple sauce as peaches.

    When she gets to the front checkout counter she realizes she needs plates to serve it on. The clerk looks at the cans and says “I didn’t think you could fit 72 people in your little house, Mrs Mahoney,” and sells her 5 packs of disposable plates.

    The same information is in there, but the items have some rational relationship. What was just mechanics is now sorta fun. There’s a multitude of possible questions –
    Is the clerk right about how many people it is? If you can fit 10 cans in a bag how many bags will the bagger need? How many plates are in a package?

    Or have Mrs Mahoney only use some weird brand of peaches from Outer Whateverstan that you, the shop keeper, have to special order. Mrs Mahoney is entertaining her entire church group of 129 people – how many cans of peaches do you need to order?

    It’s easy to think them up because they have some rational meaning. So even if you believe that sometimes you just have to make people sweat, why provide them ammo to argue you’re wasting their time when you don’t HAVE TO?

  32. on 07 Mar 2011 at 12:33 pmKarl M

    Lot of research suggests (mainly from the Netherlands) that having an actual real life example is good, however a situation which is more imaginary but can still be manipulated in a real way still allows much more learning to take place.

  33. on 07 Mar 2011 at 1:33 pmDan Meyer
    Mark: If we were changing this into a “secret number” problem, what would happen to “Mrs. Mahoney went shopping for some canned goods which were on sale.” This is an important sentence that adds depth to the problem.

    If that sentence had anything to do with the operations that follow, I’d agree with you. The sentence would add depth. As is, it adds dissonance. The students expect operations related to grocery shopping or sale prices but find simultaneous equations instead.

    Same with these:

    “Mrs. Mahoney went grocery shopping. The square root of the price of a can of beans was $0.16. How much does the can of beans cost?”

    “Mrs. Mahoney went grocery shopping. She creates a quadratic equation from the number of apples, limes, and pears on the shelf. The number of apples is the second-order coefficient. The number of limes is the first-order coefficient. The number of pears is the constant. The roots of that quadratic equation are 2.5 and 7. How many apples, limes, and pears are on the shelf?”

    Mark: Also, I wonder how many “secret number” problems can be done before you’d be bored to tears with them and want some more traditional word problems.

    It probably wouldn’t take many. Luckily, “no context” and “pseudocontext” aren’t our only options.

    Shari: Problems like this have more in common with the logic puzzle, and maybe they should be approached in that way. This type of problem does a disservice by labeling it a real-world problem.

    Agreed. I have no problem with Diophantus’ riddle, for instance, because it only represents itself as a riddle. And if you could promise me that disclaimers would accompany Mahoney’s problem to the effect that “this is not how people really apply mathematics to their lives,” I’d be less uptight about the pseudocontext thing. But those disclaimers don’t exist, either implicitly or explicitly, and the result is that students think what we do is a joke.

    Shari: How does this can goods problem differ from the Coke/Sprite problem other than engagement level? When will you ever have a Coke/Sprite mixture problem in real life?

    Great question. In point of fact, the definition of pseudocontext has nothing to do with people using the problem in real life. I dare say no one has ever taken a jump shot and computed a quadratic model in mid-air. It’s never happened. But the math of quadratic modeling derives directly from the context as presented. Same with the Coke / Sprite problem. But the math of simultaneously equations does not derive from the Mahoney problem as presented (Don has a very nice revision though). That’s why it’s pseudocontext.

  34. on 07 Mar 2011 at 4:43 pmMatt McCrea

    The key tragedy in this book is what was touched on but not explored in depth yet – what she means by the “mechanics” of word problems. Teaching 7th grade, I’ve got students that were drilled on looking for words (less => subtract) instead of developing an intuitive sense for what the problems are asking. Of course, this is the origin of the ridiculously bad estimation techniques that start to pop up just before they get me and continue all the way through high school (10% of 50 = 500). Even now, there are lessons in my county that have teachers doing this.

    Give the kids the problems. Let their intuitive sense take them where they need to go.

  35. on 08 Mar 2011 at 12:13 amMark Schwartzkopf

    I’m not seeing such a disconnect of context here. Of course you would never worry about the square root of a price at a store. But I can absolutely envision a shopper noticing that there are twice as many tuna cans as peach cans.

    The Coke/Sprite problem is awesome. The escalator problem is awesome. The octagonal aquarium is awesome.

    But despite their undeniable quality, I don’t think there are enough WCYDWT problems to adequately practice translation, and my contention is that this shopping problem has several things about it that make it better than a “secret number”/ contextless problem. I also don’t see it as a problem that alienates students near as much as the previous pseudocontext problems, if at all.

    What alternatives are there for practicing translation when you run out of WCYDWT problems?

  36. on 08 Mar 2011 at 3:31 amlouise

    Maybe what we are discussing is the assertion without proof that “math is everywhere.”
    My daughter, bless her heart, graduated and has a job in a chemistry lab. She is heartened to find that all of the math is done by computer. She just puts in her data results and the program that someone else wrote does the math.
    This year when we did the high stakes tests, I noticed they included a table for each minute of the hour, so that the proctors could determine the length of time for each section of the test. Apparently adults are having difficulty figuring out how to add time correctly.
    I worked in engineering and never once saw someone with a graphing calculator. For all that we are being told that using such a calculator is an important skill, outside of high school and college, these skills are useless to most people.
    Sure, some people are writing programs to decide on where weapons will fall (quadratic equations and you have to allow for the Coriolis effect), landing vehicles on Mars (and you have to know your measurement system), modeling disease epidemics… but not very many. There’s certainly no justification for requiring that every high school student be proficient in the use of logarithms, manipulation of algebraic expressions, and conic functions. Every one. Including the future secretaries, lawyers, ditch diggers, crane operators, store clerks…
    When you have to figure out the amortization for your house payment, do you do it yourself, or do you use one of the hundreds of free programs on the internet? I know what I do. Stats are not done by hand – computer programs are used. I know that people wrote the programs. But it was just a few, not everyone.
    The vast majority of the population “gets through” math in high school, possibly college, and then never do it again, and lead perfectly satisfying lives. High end math is being used as a tool to define obedience and conformity for entry to highly popular education programs where there are almost no applications (family physicians, for example, don’t generally use calculus on a daily basis).

  37. on 08 Mar 2011 at 8:26 amShari

    Here’s one more issue that tells us why we will continue to see such problems in textbooks.

    This is from the 2009 Texas state text for Grade 10 students.

    Mandy bought a bag of peanuts to share with her friends.

    Trisha received 1/2 of the peanuts in the bag.
    Vince received 1/4 of the peanuts Trisha received.
    Ray received 1/3 of the peanuts Vince received.

    If Ray received 4 peanuts, how many peanuts were in the bag Mandy bought?

    If you want to see a huge improvement in the types of questions we see in textbooks, get involved with The Partnership for Assessment of Readiness for Colleges and Careers (http://www.fldoe.org/parcc/) and/or Smarter Balanced Assessment Consortium (http://www.k12.wa.us/smarter/). These two organizations are developing assessments for the Common Core Standards. The second group provides e-mail addresses for representatives from each state, so you can make your voice heard.

  38. on 08 Mar 2011 at 9:03 amDan Meyer
    Mark: I’m not seeing such a disconnect of context here. Of course you would never worry about the square root of a price at a store. But I can absolutely envision a shopper noticing that there are twice as many tuna cans as peach cans.

    Noticing there are twice as many tuna cans as peach cans is certainly plausible. Noticing that same fact plus another similar fact all without noticing the number of cans is incredible.

    But the square root problem triggers your pseudocontext sensor so I’m curious, Mark, where have you drawn the line? How have you calibrated that sensor?

    Mark: What alternatives are there for practicing translation when you run out of WCYDWT problems?

    Word problems derived from real context are fine also. (Of course, many of those can be refactored as WCYDWT problems.)

    What do you think the point of learning to translate is? If the point is to eventually translate actual problems of your own creation from your own world into mathematical expressions, then whatever fluency is gained through pseudocontextual translation must be judged against what is lost when students get the sense that translating the world into math is a joke.

  39. on 08 Mar 2011 at 5:20 pmMatt McCrea

    Mark,

    I’d also argue it’s a matter of efficiency and effectiveness. How many of us have given the psuedocontextual problems to translate to students, only to see them fail again and again on later assessments? They fail because they don’t matter to them, so they don’t learn the skills. It may very well be that there aren’t enough WCYDWT problems to cover every content area of math, but if infinitely many problems fail to consistently teach the students the skill in the first place, while WCYDWT problems tend to be more successful, what’s the issue with focusing our efforts on creating more WCYDWT problems?

  40. on 08 Mar 2011 at 7:15 pmChristopher Danielson

    I’m finding it very hard to let go here because it matters so very much.

    I dig the motivation/engagement angle that seems to be a major thread here, as in Dan’s “students get the sense that translating the world into math is a joke,” and Matt’s “They fail because they [the pseudocontextual problems] don’t matter to them [the students].”

    But “context” is about more than motivation. It is also about using students intuitions about the world to productive mathematical gain. People do think quantitatively outside of math class, they just don’t do it formally. If the math classroom can harness that extracurricular knowledge, then students can build coherent and connected formal mathematics. Plus we know that memory is aided by narrative and by connections.

    The “Young Mathematicians at Work” series of books by Fosnot and Dolk offers a really nice definition of “context” that works on this second argument in favor of context over pseudocontext (and offers some lovely examples of pseudocontext to boot-although without using the word, of course).

  41. on 09 Mar 2011 at 9:35 amDaoudaW

    (note: I attempted to comment on this thread several days ago, but for some reason it disappeared… so try, try again)

    I have been a HS math teacher for 16 years, teaching everything from pre-algebra to calculus, and believe we have done a disservice to our students and our profession by our insistence that math is useful in everyday life. Be it history, science, english or math, most of what kids learn in high school is not immediately useful to them. I tell students that math is a game invented by mathematicians which occasionally turns out to be useful in real-life.

    I grew up on a farm and learned us much math in that context as I did in the classroom. I watched my grandfather square up a gate and learned the pythagorean theorem. I tried to figure out an optimal low cost ration for high producing milk cows and re-invented the rudiments of linear programming. But in the evening after supper before bed-time, my dad taught me to play chess and I discovered the joy of pure logical thought.

    This is the pattern I try to bring to the classroom. I try to demonstrate by my own passion the joy of pure math, but also try to present problems with enough authentic context that students will see it as a potentially useful tool.

    The main problem I see with the “canned goods” problem above is that with a bit of modification, additional information it could become a rich for authentic problem-solving. Add in some nutrition facts, some recipes, a budget constraint and myriad of interesting, non-trivial problems emerge.

  42. on 09 Mar 2011 at 2:30 pmDan Meyer
    DaoudaW: This is the pattern I try to bring to the classroom. I try to demonstrate by my own passion the joy of pure math, but also try to present problems with enough authentic context that students will see it as a potentially useful tool.

    Passion: good. I wish we could just assume that about every math teacher.

    Authentic context: good, but pretty ambiguous. As I develop curriculum, I’m finding the representation of that context to be one of the trickiest nuts to crack. It’s easy for me to see that paper is a really poor container for what exists Out There. It’s trickier to take any of the examples from your childhood and say, “what’s the best way to bring this context into the classroom? what’s the best way to reduce the extraneous load on the student while at the same time delegating to her the responsibility of making sense of the math?”

  43. on 10 Mar 2011 at 5:59 amYaacov

    Kathy wrote: “Relevance is required if you want any engagement from your students”

    Thanks for the response, Kathy! Your main point is what I’m questioning. When I think of what my students engage with outside the classroom, it’s video games, sports and TV shows or music about lifestyles that my students will never lead. These things don’t seem to be relevant to them, but they still engage. I’d be interested in knowing more about what led you to the conclusion that relevance is required for engagement.

    My best guess is that students will engage if there is a challenge that stretches them but is solvable and they will also engage if there is excitement. I aim for the first one in my class because I think it’s provides more learning than the second.

  44. on 10 Mar 2011 at 6:30 amChristopher Danielson

    Yaacov: Your examples all have narrative in common. Perhaps this, rather than relevance is what engages. And perhaps this is what we should seek in our mathematics teaching.

    There is no narrative to the peaches problem. But Dan’s escalator video has it in spades.

  45. [...] Meyer linked to and quoted from my screed on the end of word problems the other day. This led to some robust discussion on his [...]

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