How-to Guide to Attacking Word Problems

Word problems don't all fit into the same neat little box. There is no one graphic organizer, or magical online math word problems solver, that works well for all word problems. Each word problem is completely unique. This has always felt so incredibly messy to me when it comes to teaching word problems.

Word problems confessional: I dread teaching them. They're one of those things I have had a hard time teaching. My personal word problem solving technique, if you can even call it that, goes something like: "Read it over and over and over again until it makes sense, cross out words I don't like because they're unnecessary and I don't like that they were added as a distraction, read it again, imagine myself in the problem, pull out the numbers, make them into an equation, does the equation make sense?, solve it, does the solution make sense?" How could I ever teach this? It's a heaping disaster!

Word problems don't all fit into the same neat little box. There is no one graphic organizer, or magical online math word problems solver, that works well for all word problems. Each word problem is completely unique. This has always felt so incredibly messy to me when it comes to teaching word problems. 

But this all changed when I came across this paper on George Pólya's Problem Solving Techniques. There are no graphic organizers, there is no overarching goal to fit everything into neat little boxes on a 1-pager. The messiness is expected and embraced and a welcomed part of it all. If you haven't heard of George Pólya's problem solving technique, here is a partial summary:

First: Understand the problem 

"This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they don’t understand it fully, or even in part."


Second: Devise a plan 

"The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy."


Third: Carry out the plan 

"Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don’t be misled, this is how mathematics is done, even by professionals."


Fourth: Look back 

"Doing this will enable you to predict what strategy to use to solve future problems."


Word problems become a sort of choose your own adventure of problem solving instead of a stale and rigid horror show. The goal is to get into the problem, empathize with the characters in it, imagine you are in the problem's situation and need to figure your way out. I love this approach to word problem solving so much and it has completely changed my outlook on teaching students to problem solve. 

If you're thinking, "Well, this may be great for some students, but what about my students who need scaffolding in order to solve word problems?" Pólya has that covered. Here he lists very concrete strategies under his 2nd principal: Devise a plan:


Before this technique is used in class, it may be nice to hold a discussion with students about what each of these bullet points means to them and create anchor charts to hang around the room. This way students become familiar with all of the techniques and have more to choose from when it's time to choose a strategy.


Let's try Pólya's word problem solving technique on an example:


"José is three times as old as his sister Ana. In 4 years, José will only be twice as old as Ana. How old are they both today?"


First: Understand the problem

Let's read the problem a few times to really get into it. Can we change names to friends' or family members' names? Do we know anyone who is 3 times as old as a sibling? "José is 3 times older than his sister." Now let's give ourselves some examples. If Ana is 20, he's 60. If Ana is 10, he's 30. If Ana is a, he's 3a. So we can write J = 3a ("José is 3 times Ana")

Now for the "in 4 years" part: If José is 40 now, in 4 years he'll be 44. It he's 10 now, in 4 years he'll be 14. If he's J now, in 4 years he'll be J+4. And Ana will be a+4.


Second: Devise a plan

My plan is to build equations and use them to solve the problem. Some students may choose to guess and check, draw it out, create a pattern, etc. What I love about Pólya's technique is that problem solving is not the same for everyone; the goal is as much to figure out your preferred methods and strengthen them as it is to solve the problem at hand:


"The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy." - George Pólya 2nd Principle


With a little thinking, I know some students could reason out this word problem faster than I can build equations to solve. And this is great! There are so many different ways to solve word problems that can all be celebrated with Pólya's method. A big part of Pólya's word problem strategy is figuring out our preferred problem solving methods and developing them farther. 


Third: Carry out the plan

Here I will stick to my plan to build equations:

Now:            J=3a (José is(=) 3 times as old as Ana")
In 4 years:    J+4, a+4
"In 4 years, José will be(=) twice as old as Ana": J+4 = 2(a+4)
Solve J+4 = 2(a+4) for J: J = 2a+4
If J=3a and J=2a+4, then 3a=2a+4 and a=4, which makes J=12
José is 12 and his sister Ana is 4.


These "in x years" problems can be tricky. If this was the first time I had ever seen one, I could imagine abandoning my strategy for another. And this is OK. This is where exposure to multiple strategies comes in. 

"Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don’t be misled, this is how mathematics is done, even by professionals." - George Polya's 3rd Principle 

If my equations weren't working out, I may try to reason it out. In the classroom, students can share their thinking and strategies with classmates, giving students an insight into different strategies that can be used to solve the same problem, much like with Number Talks.


Fourth: Look back

No matter how we come to our answer, we can look back and ask, "Does this make sense?" If Jose is 12, in 4 years he'll be 16. Will Jose be twice Ana's age in 4 years? She will be 8 and he will be 16, so yes! Our work makes sense.


Clifford Pickover, the author of the amazing The Math Book, tweeted out this word problem the other day.


I know that some people are able to solve this problem by reasoning it out. I first tried translating the word problem and solving my equation for x, but decided to abandon that method. By multiplying through by 2, like we sometimes have to do when solving systems of equations, the equation became a simple 1-step equation.

"If it continues not to work discard it and choose another. Don’t be misled, this is how mathematics is done, even by professionals." 


You can read all of George Polya's method here on the Berkley site: Polya's Problem Solving Techniques






scaffoldedmath.com home page


More posts:









4 comments:

  1. I did read about this system just this past summer. I too loved it for the same reasons. I have been toying with the creation of a scaffolded organizer for working through them. Here is a link to a copy of what I have been working on. https://docs.google.com/presentation/d/1yiYGcq138vzlxp1J3but3FxV-H12J75hdScAPpMoaXI/copy
    I hope you find it useful, I have found so much of your tools helpful. It's hard to find math resources to support students at the higher levels.
    I would love to hear any feedback!

    ReplyDelete
    Replies
    1. This is great! Thank you so much for sharing.

      Delete
  2. Thanks. This article is great.

    ReplyDelete