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Saturday, November 26, 2011

Writing About Our Thoughts While Problem Solving in Mathematics

            Problem Solving means engaging in a task for which the solution method is not known in advance. In order to solve the problem, students must draw on their knowledge of the process, and through this process they will often develop new mathematical understandings.

This easy to use graphic organizer was created to assist students in writing down their thoughts as they solve a problem.  The assessment requires students to express not only how a problem is solved, but why each step was taken.  According to the text, “Mathematics Assessment: A Practical Handbook for Grades 6-8”, students “need regular opportunities to write about their mathematical thinking to become better writers and better thinkers. One good writing task is to have students explain strategies and solutions using words, mathematical symbols, graphs, diagrams, or other representations.”

            One of the reasons this particular graphic organizer was developed  was the high percentage of second language learners in our school district. They were so overwhelmed in expressing their ideas and thoughts in English and writing them down in sentence format that they struggled to express their mathematical thoughts. In addition, students participating in our Gifted Program had difficulty expressing why they preformed certain operations or utilized a particular strategy they just "knew it."    
           A four quadrant graphic organizer was developed with the work of Dr. George Polya is mind. First, Dr. Polya stated that in order to solve a problem, a student must take time to understand the problem. They must ask themselves: “What is the unknown? What is the data? What is the condition? or Can you restate the problem in your own words?” Therefore the first quadrant begins with understanding the problem. Students usually begin with…First, I know.

            Next, students must understand what they are being asked to find. In order to clearly state their answer with appropriate units, the second quadrant begins…The answer is.  Students will insert a blank where the amount will later be filled in, for example, “The answer is $_________ is the price at store A, and $_________ is the price at Store B.”

            The third quadrant lists what the student does in a step-by-step manner, in order to solve the problem.  Many students find that they like to number the steps numerically. Since Dr. Poly stated that they must begin by devising a plan, teachers in our district teach some of the problem solving strategies described: guess and check, make an organized list, make a picture of diagram, make a table, look for a pattern, use a variable, and use an operation or formula. So the third quadrant usually begins with the student writing…First, I decided to.

            The fourth quadrant lists why I did it that way.  Again students usually want to number the reason why to reflect the step that they took, though it certainly isn’t necessary.  Many of the statements on this side of the organizer seem to start with because, to show, or to find. The “why I did it” is written immediately after they write down the “what I did.”

            An example of two more completed four quadrant problem solving organizer is shown below.  The same problem has been solved using different problem solving strategies. This easy to use graphic organizer is used in Grades 3-8.

Click Here to obtain a copy of the organizer.

Also here is the student friendly scoring rubric provided by our state currently to score student's thinking on mathematical problem solving extended responses.
Score Level
Mathematical Knowledge
(Do you know it?)
Strategic Knowledge
(How do you plan?)

(Can you explain it?)
I get the right answer.
I label my answer correctly.
I use the right math words. (For example, I know when to add or multiple.)
I work it with no mistakes.

I find all the important parts of the problem, and I know how they go together.
I show a good plan about how I got my answer.
I show all of the steps I use to solve the problems.
I write what  I did and why I did it.
If I use a drawing, I can explain all of it in writing.
I do the problem, but I make small mistakes.
I find most of the important parts of the problem.
I show most of the steps I use to solve the problem.
I write mostly about what I did.
I write a little about why I did it.
If I use a drawing, I can explain most of it in writing.
I understand a little, but I make a lot of big mistakes.
I only give part of the answer.
I find some of the important parts of the problem.
I show some of the steps I use to solve the problem.
I write some about what I did or why I did it but not both.
If I use a drawing, I can explain some of it in writing.
I try to do the problem, but I don’t understand it.
I find almost no important parts of the problem.
I write or draw something that doesn’t go with my answer.
I write an answer that is not clear.
I don’t try to answer the problem.
I don’t show any steps.
I don’t explain anything in writing.


  1. Deborah, this is fantastic! I have bookmarked this page so I can be reminded to do this soon! Thanks so much for always sharing!!
    Kristen :)

  2. Dear Kristen,
    Just remember that students are not use to writing about how they think mathematically. It takes repeated experiences but you learn about how your students are thinking and that's what makes it worthwhile.

  3. Hi Deborah!

    Great stuff (as usual!)

    I gave you a Shout Out on my blog today...

    Finding JOY in 6th Grade

  4. the is going so well keep up the good work


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