*They know the math but they can’t do word problems. *

*They can do all the problems on the page correctly, but they skip the word problems at the end. *

*It’s a reading comprehension problem. They don’t understand what they are reading. *

The dreaded word problem. In many classes I taught, students weren’t used to being asked to read in math class and they certainly weren’t used to thinking about what they were supposed to do. Math was about following a set of procedures and getting the right answer. Math wasn’t about thinking about what types of calculations might help you make sense of a situation.

Early in my career, I fell into the trap of teaching them to circle the key words. That became a problem very quickly.

Then I tried teaching them Polya’s four step model: 1. Understand the problem 2. Make a plan 3. Carry out the plan 4. Look back

This didn’t help either. If students managed to understand the problem and explain it in their own words, they didn’t know how to make a plan.

What finally worked, particularly with my struggling students, was encouraging them to use pictures. At first, students created elaborate drawings of the situations that often focused on superficial aspects of the story. Then, I guided them to use simple diagrams to represent the story– diagrams that helped organize the information given and what was missing. I found that after a bit of practice, students were able to solve a variety of word problems, including complicated multi-step ones, with a picture.

Recently, I’ve been working with teachers and students on drawing diagrams to solve word problem and I figured it would make sense to go to the research and see if there was evidence that supported my experiences. It turns out there is.

The benefits of using diagrams for story problems include:

- Reducing the memory demands
- Assisting in unpacking the situation
- Helping identify important information
- Focusing students on the quantities involved in the situation and the relationships between them
- Flexibility Diagrams can be used across grade levels and for solving both routine and non-routine problems

It is important to mention that these studies don’t encourage the use of any pictorial representation, but rather ones that highlight the structure of the problem. Bar models or part-whole diagrams are some of the representations I’ve found successful.

I’ll talk more about these particular models later, but I’d love to know about your experience with word problems. What have you found to be successful?

Want to know more? Check out:

Diezmann, C., and L. English. (2001). Promoting the use of diagrams as tools for thinking. In A.A. Cuoco and F. R. Curcio (Eds.), *The role of representation in school mathematics.* Reston, VA: National Council of Teachers of Mathematics, 77-89.

Yancey, A. V., C. S. Thompson, and J. S. Yancey. (1989). Children must learn to draw diagrams. *Arithmetic Teacher*, 36 (7), 15–23.

mslcbillingsIn our small high school math department we have talked about our students’ reluctance to draw pictures as part of the problem solving process. We have laid some of the blame at the door of computerized practice. We think students become accustomed to problem solving in their head at the basic level when they are working on computers and then try to carry that through to the more complex levels. We have observed that while students are working on computers, they are reluctant to use scratch paper, never mind making sketches. We think this builds a norm of not using scratch paper or sketches when problem solving.

I would be curious to know if you or other teachers have made similar observations.

Nicora PlacaPost authorThat’s really interesting. I’ve never really thought about the effect of computerized practice. I have noticed that students are reluctant to make diagrams or use any sketch paper sometimes because they think that being good at math means doing it quickly and in your head. I’d be interested to see if others have had similar experiences with computerized practice.

deb gaskinI teach elem 5th graders who tell me they can not “see” or visualize what they are reading in fiction stories….they only understand what they are reading if they have physically experienced the situation…..so drawing a math problem is hard…they aren’t sure where to start. What does help is getting students to “act” ouy the problem…then they can draw.

Nicora PlacaPost authorI love your idea of acting out the story. It allows them to build on what they know and then they can later represent the idea with a drawing.

jvbatkins“Being good at math means doing it quickly…in your head.” I agree that students seem to feel this way and yet, while they don’t want to solve complex problems on paper (especially when taking computerized tests), they invariably grab their pencils to solve something like 53 + 78, instead of using mental math.

I’ve been using ideas and resources from Singapore Maths to teach problem-solving with bar models. It’s still difficult for my struggling students, but I think continuing to wrestle this way with ideas of part & whole and what’s known & unknown will lead to big conceptual gains.

Nicora PlacaPost authorI love the Singapore Math bar models as well. I’m interested to hear more about what difficulties your struggling students have with using the models.

lori8740Computerized testing is absolutely a nightmare! I have always stressed highlighting, underlining, writing notes, drawing pictures — whatever will help my 5th graders understand what they are reading. Now that we take tests on the computers I hand out extra paper and encourage them to fold the paper into fourths and take notes on each question one-per-block. In theory, this should encourage students to write things down. In practice, they don’t have enough room on their desks with their laptops, or they don’t want to write all that they need to/want to. My special education and second language learners find taking their own notes tremendously time consuming. When students can mark up their test papers I see much less stress and more success.

Don’t forget, too, that most students equate technology with games. The point of games is often to react quickly without thinking through what you are doing. The point of test-taking is to stop, think, reflect, analyze.

Nicora PlacaPost authorThat’s a really good point about students making a connection between technology and games and the problems that causes when they take a test. I wonder how we change that association.

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Joe SchwartzDrawing pictures is a strategy that also allows you to see what the student has in mind (literally). We’ve tried to make some connections here with the language used by our ILA teachers by associating these types of word problems with texts they might encounter in reading class and using the word “visualize”, which is one of the “active reading strategies” they practice in ILA class. (We also do this when we ask kids to write their own questions. We’ve borrowed “thick” vs. “thin” questions, also from the ILA teachers.) They do lots of reading comprehension work in their ILA class and I think we can use those strategies to help us in math.

But I have come to believe that, overall, we can never prepare the kids to tackle every type of problem they might see. The more we put them in situations where they need to solve problems, all types, both traditional and non-traditional, the less chance they’ll panic and the more chance they’ll at least make an effort to work through it.

Joe SchwartzAlso, try to give the kids the problem without the question. Let them generate possible questions. Take the problem apart and put it back together. This is something I describe on my blog here: http://exit10a.blogspot.com/2014/01/what-would-happen-if-we-took-problem.html

We did this with a grade 4 PARCC prototype. I think it helps demystify these problems.

Nicora PlacaPost authorThanks for sharing the link. I like the idea of pulling the problem apart and putting it back together.

mrmillermathI’ve also tried a variety of problem solving strategies, usually based on some acronym to help students remember what the steps are. Those strategies generally bored me and them – and I felt like they made problem solving much more formulaic than it is supposed to be. Thus this year I have been requiring diagrams with each “word” problem. If I have time, the next day I pre-select a few of the diagrams we discuss which diagram was the most helpful, and what elements are present in useful diagrams.

I have not made an effort to curtain the superficial aspects of their diagrams, just allowing it to be part of the their creative process, but mostly because they just like to do it.

Nicora PlacaPost authorThat’s great to hear. I also became quickly bored of the acronyms and I fond that even if students remembered the acronym, they often still struggled with applying the steps. I also like how you have the students compare and contrast different diagrams–that must be very helpful for them.

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