• Exactly 1/3 of the books on the shelf are mysteries.
• Aubrey has read 10 of the mysteries on the shelf.
• The number of mysteries Aubrey has read is greater than 1/5
of the number of mysteries on the shelf, and less than 1/ 4
of the number of mysteries on the shelf?
Which could be the number of books on the shelf?
Most people read a problem like this and their initial reaction is, "WHAT?" Teachers read this problem and the first thing they think is, "How the heck am I going to break this down so my students can solve it?"
Solving open-ended, higher order math problems is messy business for a lot of reasons. First of all, these kinds of problems really highlight the range of abilities in a classroom. You present this type of problem to a class and some kids have the answer before you've even finished reading the problem, and other kids will stare at the paper for as long as you leave it in front of them because they haven't the foggiest idea where to even start. Then, there's the fact that by the very nature of their design, these problems are not cut and dry. There may be only one correct solution, but there can be as many strategies and methods students use to get to that answer, as there are students in your class.
As teachers, we know it is our responsibility to scaffold instruction for students and gradually release responsibility for learning to them, with autonomy being the ultimate goal. This is easier said than done in the best of circumstances, but can seem impossible when you have 20-30 students with a wide range of cognitive abilities and different learning styles who have all been given a problem that is intended to stretch their understanding and push them to notice obtuse patterns and relationships.
It's no surprise that teachers get intimidated by higher order, open-ended math word problems. The problems are HARD, and they're so unpredictable. I've always struggled with finding the best way to scaffold open-ended problems for my own students. For most of them, solving higher-order math problems is a battle, but I am bound and determined to arm them with as many weapons as possible so they can be victorious.
I am pleased to announce that I have finally found a problem-solving template that is working in Room 202. At least, it gives all my students a common starting point and a reliable framework for dismantling these complex problems into smaller components that they can tackle incrementally. We have been working with the template all year, and I have seen some measurable growth in most of the children's problem solving skills. My revised version of the template looks like this:
We solve problems like the "Aubrey" problem on Fridays in Room 202. The problems we work on are aligned to whatever eligible content we are covering in math that week. Initially, the lessons were entirely teacher-led and featured a lot of me "thinking aloud" at the SMARTBoard. At this point, we only work on Step 1 together as a class. After we have read and scrutinized the problem carefully, my students now work through Steps 2-5 independently. There are still several students who are not able to move passed Step 2 on their own. I provide very targeted, explicit 1:1 instruction for the students who still need it, as I circulate during problem solving time.
I have also developed a rubric for measuring my students' implementation of this problem solving platform. The rubric is tailored to the steps on the template and it looks like this:
My goal is to practice these problem solving strategies with my students frequently enough that they become automatic for them. (I do see the kids underlining the question and circling key information in other classwork problems, so I know there has been some transfer.) Ultimately, I want my kids to feel confidence rather than intimidation when they read an open-ended math problem. I want them to intuitively apply the strategies we have practiced together so they can systematically get to the point of what the problem is asking, make a plan for how to answer that question, and be able to explain why they did what they did. Sounds simple enough, but we all know it's NOT! It's actually about as complicated as it gets when it comes to math instruction, and often seems utterly impossible, but I refuse to throw in the towel. It's when the work is the hardest, that our students need us the most, and this is really a life skill the kids need.
Problem solving is a fixture in life, and it is my goal as an educator to prepare my students for LIFE. Problems pop up everyday. Sometimes they are small and sometimes they are large. You run into problems everyday, from flat tires to a failing product line at work. Sometimes solving a problem is a matter of life and death, and other times it is merely a matter of keeping your sanity. Regardless of why we need to use problem solving, we can not deny that we do need it. There is also no denying that the best problem solvers become the most successful and productive citizens, and that's ultimately what I want for the kids I teach.
If you want your students to be good problem solvers, too, you can get my problem solving template and rubric along with 15 problems (and answers) aligned to the fifth grade Common Core Math Standards in my TpT store. Click HERE if you'd like them for your classroom.