Open-Ended Math Problems

Project Details

Introduction:
I cannot count the number of times that I have heard from parents and students alike, “Oh, I’m just not a math person.” For many people, math class is a series of lectures on procedures, followed by lengthy and repetitive problem sets. While some may appreciate this approach, for many, it invokes negative feelings.

During my early years of teaching in a traditional school setting, I often ended up perpetuating these negative feelings. Although I taught with a lot of enthusiasm, the content was just dry. I would move from one chapter to the next, helping students master one set of rules after another with little or no heed for relationships, patterns, or applications. I now realize that the way I was teaching math then was stripping it of its beauty and relevance.

Getting Started:
I initially felt apprehensive about using open-ended problems in my classroom. If the answers were not immediately evident to me, a self-proclaimed “math person,” then I couldn’t imagine how my students would react. However, after researching the literature on the benefits of open-ended problem solving in the math classroom, I decided to give them a try. During that first year of implementation, my apprehension turned to appreciation and excitement for the freshness and authentic conversation these problems brought to my classroom. Open-ended problems allowed me to teach math with relevance, through multiple strategies. They also helped me to challenge the students who were high achievers and to develop the confidence of the ones who needed more support.

An open-ended problem that I often use at the beginning of the year is “The Weather Problem.” It is fairly straightforward and gets the students’ number sense gears turning, which is pivotal at this point in the year. The problem also requires the students to work with time, which is a great review.

Given that this is the first problem of the year, I take great care to be positive while getting the students started. I know the power of first impressions, and I want the students to be energized and excited for more. I have the students work the problems in a special “math journal” which is saved for only our most challenging problems. It makes them feel as though they are true mathematicians whenever we get this journal out.

When I implement this problem, I allow the students to work in self-selected groupings. I also give them a freedom of space, meaning I allow them to choose whether they will work inside the classroom or outside in our sixth grade commons area. For some reason, the space inside the classroom becomes the “quiet” work environment and the commons area becomes a space for those who prefer to work more interpersonally.

Something I learned from my first year of implementation is that students need time to have productive conversations when discussing their problem solving strategies; after all, this is where much of the learning takes place and it cannot be rushed. Now, when I teach this problem and other “longer” problems, I am careful to allow two days for the work. At the beginning of the first day, I go over my expectations for students’ work and spend a bit of time reviewing basic concepts. Then the students get to work solving their problem, and I step back into the role of facilitator. For homework that night, the students finish their work and complete a short reflection on the problem (what was hard/easy about it, etc.). The next day students share their solutions either in groups or out to the class. I try to pinpoint a couple of students who solved the problem differently but arrived at the same answer; these students are highlighted on the board. Sometimes students who have the wrong answer will also put their work up, and the students in the class are challenged to find the error.

Students often express feelings of satisfaction upon completion of one of these open-ended problems. One student stated, “When you first look at it [the problem], it looks scary. But when you’re done, you feel great.” Another student elaborated on these feelings exclaiming, “I was excited when I heard the problem because I had some ideas, like I was a detective trying to solve a mystery.”

The Weather Problem:

The weather is reported every 9 minutes on ABC and every 12 minutes on CBS. Both stations broadcast the weather at 1:30. When is the next time the stations will broadcast the weather at the same time?

a) When was the last time (before 1:30) that the weather was reported at the same time?

b) List the next 5 times that the weather will be reported at the same time. How often does this happen? Explain.

c) Suppose CBS changed to reporting the news only every 24 minutes. When is the next time that they will report the news together?

d) Since CBS has been getting low ratings on their weather program, they have decided to only report the news every 45 minutes. After 1:30, when will they report the weather together again?

e) Are there any numbers you could choose that would make it so that the news programs were never reporting the news at the same time?

Advocates of open-ended problems cite three general benefits: they develop problem-solving skills, they allow for natural differentiation, and they help students make connections across mathematical concepts.

Problem solving. According to the National Council of Teachers of Mathematics (NCTM), “Problem solving means engaging in a task for which the solution is not known in advance. Good problem solvers have a ‘mathematical disposition’—they analyze situations carefully in mathematical terms and naturally come to pose problems based on situations they see” (NCTM, 2000). Open-ended problems support this type of learning because they require students not only to understand the problem, but also to think about how they arrived at their conclusion. I see my students’ problem solving skills increase on four major fronts. The first is that they are taking complex problems and looking for patterns to make them simpler. Secondly, students begin to see and embrace multiple strategies for solving each problem. Third, students start working collaboratively and embracing the help and opinions of students around them. And finally, students take on each problem with greater diligence and persistence, sometimes not giving up after an hour on one single problem.

Differentiation. Open-ended problems allow students to work in their own learning styles and at their own ability levels, making personal choices in their process (Hertzog, 1998). In my classroom, one of the most striking phenomena to come out of using open-ended problem is the way that they are able to challenge students at all levels of understanding. Whereas some of my students are very savvy with variables, others still struggle with the multiplication tables. Because students at HTM come from all over the county, they enter my classroom at many different places mathematically. Amazingly, the days when I don’t feel tension in the curriculum are when we were all working on an open-ended math problem together. They are all “thinking hard” on a majority of these problems, and they all have different opinions on which problems are most challenging. In general, these problems are an equalizer in the class, breaking down the often-present dichotomy of “smart kids” and “dumb kids”.

Connectivity and coherence. One of the beauties of math is the interconnectedness of topics. While fractions, decimals, and percents can represent the same value, each will be useful in a different context. A traditional math textbook may contain upwards of 15 chapters with six or more sections in each chapter, each section representing a different topic or subtopic. Teaching straight through some textbooks may exchange the beauty and connectedness of mathematics for rote memorization of seemingly disconnected topics. Open-ended problems encourage students to make connections because these problems often cover many different topics all within one problem. In my classroom, students reflect on their own work and often collaborate with others as they work. Through this reflection and collaboration, they construct a mathematical understanding of the problem that is both deep and flexible and enables us to connect topics from various problems and within one problem together.

I’ve been using open-ended problems in my classroom for three years now, and they have become a core learning element for my students. Although not all problems work for everyone all of the time, I’ve found that adhering to the following guidelines greatly enhances the opportunity for success.

Know your problems. It is important, before you give a problem to students, to be aware of the basics of the problem so that you can help struggling students gain a solid foundation. Similarly, it is good to know some related extension exercises to deepen students’ understanding.
Encourage group work and conversation. Because open-ended problems allow multiple approaches, they offer an ideal context for students to learn from each other and share responsibility for finding solutions.
Allow freedom of movement. When we work on these problems, I often allow my students to go anywhere in my room or our common spaces to work. Sometimes a change of scenery is what they need to jumpstart their thinking.
Refrain from giving hints for a set period of time. Some students are apt to give up the moment they feel confused. By holding off on giving hints, you are encouraging them to “think for themselves,” and this is where immense growth happens. After a set time, offer support by encouraging them to draw and label a picture of the scenario. After this, you could pair them with a partner or group who seems to have a good start on the problem and who can offer some advice.
Hold off on telling them if their answer is “right” or “wrong.” If students say that they are done, I like to ask them how much they would be willing to wager on their answer. This gives me an idea of how confident they are in their thinking and encourages “guessers” to keep working at it.
Have students post and explain work on the board.Articulating their process is often the hardest part for early finishers. Encourage them to work out their solutions on the board in such a way that younger students would understand their thinking. Sometimes even incorrect approaches are helpful to put on the board so students can see where their thinking went wrong.

Forsten, C. (1992). Teaching Thinking and Problem Solving in Math: Strategies, Problems, and Activities. New York, NY: Scholastic Professional Books.

Hertzog, N. (1998). Open-Ended Activities: Differentiation Through Learner Responses. Gifted Child Quarterly, volume 42(number 4), pages 212-227.

Jarrett, D. (2000). Open-Ended Problem Solving: Weaving a Web of Ideas. Northwest Education Quarterly, volume 1 (number 1), pages 1-7.

National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

Schuster, L., & Anderson, N. (2005). Good Questions for Math Teaching: Why ask them and what to ask? (Grades 5-8).Sausalito, CA: Math Solutions Publications.