Two Ways of Problem Solving -- Math Practices and Thinking

The teaching of math is something that I am always trying to perfect in my classroom.  I tweak things from year to year to better suit the growing needs of my students so that I can more effectively help them access the standards.  Over the years, I have found that my biggest problem when teaching math is the lack of thinking that happens.  Students can memorize rules and facts, but applying those rules to any given math situation is so difficult for them.  To combat this, I have had to take a look at how I teach math and change it.

Yesterday I told you about how I use Head Problems in my class to begin the process of getting students to think and talk about math.  Another thing I use, which I find VERY successful, are "Two Ways of Problem Solving".  The Two Ways Problem approach is something that I have used over the years and have found success with over many grade levels and many classroom makeups, so I thought I would share it with you here.

Great problem solving exercise.
The basic idea is that the students are given one open-ended math problem to solve.   They must solve the problem and explain how they came about their answer.  Easy peezy, right?  Wrong!  What sets this apart is that they have to solve this same problem in TWO COMPLETELY DIFFERENT WAYS!   They students need to think of a two strategies that will get them to the correct answer but are different in nature.

They must then explain their thinking in words.  This is the part that always gets them.  The students have the hardest time EXPLAINING their thinking (here is where the math practices come in!!)  It is really an eye opener how difficult it is for some students to solve what (in my mind) are simple problems.

The general guidelines of Two Ways Problems are:
*  Both strategies are solved on the same page.
*  They are ready for the kids to start immediately.
*  This is a quick "lesson".  It is not dragged out to infinitum. ;)
*  Can be read aloud to give access to math vocab and content.
*  Should be algined to the grade level standards.
*  MUST be debriefed.

I follow the 5-10-15 minute approach during my math period when using Two Ways Problems.
5 minutes of independent work.  Students must try and solve the problem alone, circling key words, figuring it all out, explaining the answers.
10 minutes of partner work.  Students are then allowed to work with their table partner.  This is also when I am walking around, listening to them discuss the math and ask guiding questions.
15 minutes of debrief.  While I am walking around, I pick up two or three sample pages that I want to share on the board.  There are NO NAMES on the papers, so when I put them up on the doc cam, the kids don't know who they came from.  (we call this Anonymous Sharing)  During this debrief, I am asking the kids to tell me what they see mathematically on the papers.  How is the second paper different than the first?  What strategies did the person use to solve the problem?

Now, admittedly, this approach takes a bit more time in the beginning of the year to get down.  BUT the time limits I mentioned above actually are part of the overall process.  They encourage the students to be quick thinking, try things out, take risks, and use past strategies to mastery.  Putting this time limit on the kids helps to alleviate the "I can't do anything so I will just sit and think for a very long time" problem.  They learn to get to work, streamline themselves, and produce quality work in a shorter amount of time.  It really helps so much!

When choosing papers for the debrief, I tend to find papers that are correct and use a multitude of different strategies.  The point of the anonymous sharing is not to see that the person got the final answer, but walk through the process of HOW that person did it.   THIS is the most important and successful part of the Two Ways Problems.  When the students can discuss the process of math, it is much more internal than just doing rote memorization of rules.  This is where math thinkers are made in my class.  This is why I love Two Ways Problems so much.

Now, if you would like to use this approach to problem solving in your class, you can definitely choose problems from your math series to give to the students.  But if you would like a more no-fuss approach, I have created a set of problems for you that match ALL of the Common Core Standards, as well as have a matrix of the math practices that you can use to help ensure you cover them throughout the year.  You can get those for the following grades:

3rd grade
4th grade
5th grade


  1. Thanks Stephanie, this is great. I really feel the same way about my class and it is very distressing. Do you work it into your math block or is it a morning activity? If you would ever be interested in sharing, I would love to read about your schedule--you do so many excellent things throughout the day.

  2. Kristin --

    For morning work (well, it is after recess work, as I do math then) I do Calendar...see the tab above for an explanation of that.

    I do Two Problems about once every 2 weeks as one of my math lessons. The week that I don't do a two problem, I either do a Concept Lesson (which I will write about soon) or a Discovery Lesson (again, that post is coming in the future). Basically, I try to have the students thinking about math, problem solving, and those types of strategies once a week.

    The other 4 days I am teaching content lessons, but I also have the math homework I create (a spiral review) and Calendar. Taking all of that together, I NEVER feel like I am behind in math or like I HAVE to teach a certain math concept.

  3. I agree - it is imperative for students to write about their thinking in math. Since I integrated math notebooks where students write about their thinking several years ago, I have had a serious upswing in student understanding of math concepts as well as performance on tests. Thanks for sharing the Two Problems and talking about math!

  4. I love this idea, thank you so much for sharing your example problems. Can't wait to try it with my own class. I started Calendar math last year with 4th graders and it was a big success. *My only problem was I felt like when I got to the 100th day it had kind of lost it's luster and I reverted back to my old math worksheets. Any suggestions? Thanks.

    1. I think it all has to do with the investment of the teacher. At least for me, because I am so invested in it, the kids are as well. Because *I* stress the importance of it, the kids see it as a valuable piece of math instruction that doesn't get old or boring. We are doing the hand motions. Kids get a chance to interact and lead it. The kids begin to own it and it honestly, in 10 years of doing this, has never lost any impact with my classes.

  5. Do you have any suggestions of inexpensive resources for good problem solving questions? We used The Math Forum at Drexel University last year but my school didn't renew the membership.

  6. Hi Stephanie, is there a resource that has two problems and head problems already created by grade level (K-8)?


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