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.

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.

…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!

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.

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.”

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.

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.

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?

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?

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

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.

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?

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

“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?

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?

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.

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?

(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.

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.