Current Issue Back Issues Cards About Submissions Subscribe
Issue 2, Fall 2008

On Schools of Education, Theodore Sizer
Opening up to Math, Sarah Strong
In Over Our Heads, Stacey Lopaz
African Bushmeat Expedition, Jay Vavra
Learning as Production,
      Critique as Assessment
, Elisabeth Soep
Speeding Race Cars
      & Dying Embers
, Ashley Bull-Carrico
Messy Business: A Student's Perspective
      on Project-Based Learning
, Mollie Davis
The Great Lego Caper, Zoltan Sarda



Cards:
1: Writing on the Walls
2: San Diego/Tijuana Crossed Gazes
3: Blogging is Writing
4: Public Service Advertising Campaign
5: Science Friction
6: I Am an Artist

HTH GSE » UnBoxed » Issue 2 » field notes

Students solve open ended math problems using a variety of methods.

Opening Up to Math




“Questions may be one of the most powerful technologies invented by humans. Even though they require no batteries and need not be plugged into the wall, they are tools which help us make up our minds, solve problems, and make decisions.”

—Jamie McKenzie


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 was a series of lectures on procedures for solving a certain type of problem, followed by lengthy and repetitive problem sets. While some may have appreciated this approach, for many, it invoked negative feelings.

Throughout my earlier teaching experiences, I often felt that I was perpetuating these negative feelings. Although I taught with a lot of enthusiasm (as much as someone could have about the quadratic formula!) 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—oh, except for those three word problems at the end of the chapter, which I often skipped. I felt strongly that the way I was teaching math was stripping it of its beauty and relevance.

When I began teaching at High Tech Middle I was encouraged to explore other methods for teaching math. My hope was that in this freedom, I would find approaches that would allow my students to see the beauty in math and gain a fresh interest in the discipline. As I began my research into various teaching approaches, I was particularly struck by articles concerning open-ended math problems—problems that can be solved in more than one way or that have more than one solution (Schuster, 2005). These problems are generally just beyond the student’s skill level, have multiple entry points, can be done either collaboratively or individually, and involve real-life situations, often without the distraction of numeric symbols (Jarrett, 2000; Forsten, 1992). Here are some examples of open-ended math problems promoting number sense at the 5-6 grade level (problems adapted from Schuster, 2005):

“The students in Mrs. Strong’s class want to know how old she is. Mrs. Strong told them, ‘My age can be written as the sum of consecutive odd numbers starting from one.’ How old might Mrs. Strong be?”

“The weather is reported every 18 minutes on NBC and every 12 minutes on Fox News. Both stations broadcast the weather at 1:15pm. When is the next time the stations will broadcast the weather at the same time?”

While each of these problems involves a real-life situation devoid of numeric symbols, the first exemplifies a problem with multiple solutions and the second a problem with one answer and many possible approaches. Both problems can be solved by simply drawing pictures and making lists, or by taking a more mathematical approach.
Though intrigued, I felt apprehensive about using these problems in my classroom. If the answers were not immediately evident to me, a self-proclaimed “math person,” then I couldn’t imagine subjecting my students to them. Nevertheless, as I began experimenting with these problems, first with support while teaching summer school and then in my own classroom, my attitude shifted. Students seemed to be engaged in their work and excited about math. Even students who routinely labeled themselves as “non-mathematicians” would take on these challenges, pouring over them until they reached a solution. Throughout the year 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, and through discovery. They also helped me to challenge the students who were ready for it, and to help develop the confidence of the ones who needed more support. Sometimes I would use the problems to begin or end a unit of study, and at other times I would use one as a quick warm-up. I found myself wanting to use these problems more regularly, and so I decided to look further into their benefits and how I could use them to build a more effective classroom learning environment.

Why open-ended problems?

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. When the National Council of Teachers of Mathematics (NCTM) released their Curriculum and Evaluation Standards for School Mathematics in 1989 and their follow-up Principles and Standards for School Mathematics in 2000, the aim was to reform math education to better meet the needs of our society. A common focus in each document was the development of problem-solving skills and the application of mathematical concepts to real-world contexts. According to the 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, thus moving toward the aforementioned “mathematical disposition.” Whereas students working on traditional math problems might reason with numbers to produce an answer, the process of working on an open-ended problem involves reasoning about numbers to produce understanding (Steen, 2007). According to Robert McIntosh, a Mathematics Associate for the Northwest Regional Educational Laboratory, “To develop these (problem solving) abilities, students need ample opportunities to experience the frustration and exhilaration that comes from struggling with, and overcoming, a daunting intellectual obstacle” (quoted by Jarrett, 2000, p. 5). Open-ended problems provide this opportunity.

Differentiation. As Hertzog (1998) observes, open-ended problems allow students to work in their own learning styles and at their own ability levels, making personal choices in their process. They also include options for students to interact with content and to elaborate and integrate knowledge across disciplines. Hertzog cites one of Passow’s frequently quoted guiding principles to differentiation, “Curricula for the gifted/talented should focus on and be organized to include more elaborate, complex, and in-depth study of major ideas, problems, and themes” (Passow, 1982, cited p. 214). When it comes to struggling students, open-ended problems offer increased emphasis on problem-solving and reasoning with a decreased emphasis on the correct answer. Open-ended problems based on real-life situations have been found to help struggling students have a better opinion of math and a higher self-esteem (White, 1997). A Bay Area study of a group of disadvantaged urban high school students found that learning through open-ended problems not only produced more positive feelings about math, but the students in these mixed-ability, detracked classrooms outperformed wealthier teenagers in tracked, traditional classrooms on various assessments (Trei, 2005). Appealing both to the gifted and the struggling, the multiple entry points of open-ended problems can address the needs of each student and help to create a differentiated classroom.

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 a textbook 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 the process, students are reflecting on their own work and often collaborating with others as they work. Through this reflection and collaboration, the solver constructs a mathematical understanding of the problem that is both deep and flexible by connecting prior knowledge with new concepts and skills (Jarrett, 2000).

Inside Students’ Experience

Despite all of the research findings and my own positive feelings, I wondered what my sixth-graders were thinking as they worked on these problems. What was their initial reaction to these types of problems? How did they approach solving them? How did they feel while working on them? Did they enjoy them or see the point of them? Did they feel like they were strengthening their math skills (like I did) and learning things that would help them with future problems? Or were they completely confused? A little nervous about what I might discover, I designed an action research project around the following question: How do students experience open-ended math problems? As I embarked on this journey into the minds and experiences of my students, my desire was to gain insight into their thought processes so as to discover and design more effective ways of integrating open-ended problems into my classroom.

I started by interviewing one of my current students, Jasmine. While Jasmine does not consider herself a strong math student, she puts a lot of effort and thought into each of her assignments. Jasmine told me that her initial reaction to reading an open-ended problem was to “freak out.” She then described several techniques she used to calm herself down and solve the problems. One strategy was to try the easier parts of the problem and hope that they would offer her clues to the harder parts. When asked what she felt she got out of these problems, Jasmine stated, “Actually, I think that after I do those open-ended questions, I feel like I have learned some kind of strategy that I keep on using with the other problems that you give me on the next homework assignment.” On the whole, Jasmine felt that these types of problems helped her in her future work. When I asked her about her current feelings about herself as a math student, she responded, “I’ll say I’m not the best math student, but I guess I’m all right.” In a subject where so many people are hot or cold, I take Jasmine’s lukewarm confidence as a step in the right direction.

For my second interview I selected Kathleen, an advanced and confident math student. Though not outspoken in class, Kathleen is very thoughtful in all of her work. When asked to describe herself as a math student, she admitted to “knowing a lot of math” and “getting advanced math at home in elementary school.” In contrast to Jasmine, Kathleen quickly offered that her initial reaction to the problems was that, though they are different, she liked them because she felt free: “I think, cool, I can do this. I just think and think to draw pictures or do this instead. It gives me the chance to think about what I’m doing and helps me to note things that I’m doing and apply it to the future.”

Throughout our conversation, Kathleen regularly referred to her frustration in math classes where repetition and speed was the focus. She felt that open-ended problems helped to build her confidence and allowed her to use various methods of problem solving. I know from working with her that Kathleen often has outside-the-box ways of looking at things, and even she stated that her problem solving strategies often make other students go “huh?” It became evident through our interview that open-ended problems have been an outlet for her creative thinking: “If I have enough time, I can figure almost anything out!” I would love to have all of my students voice this kind of confidence about math. I’m hoping that open-ended problems can be one means to this end.

The two students I have interviewed so far about open-ended problems say that they have grown as mathematicians through working on these problems and that the overall experience has been positive. Though they describe different approaches and feelings, both feel that the problems challenged them and built their confidence to work on future problems. By taking closer looks at their journals and conducting additional interviews and focus groups over the next few months, I want to look at the different strategies students employ in solving these problems and see if there are trends in their feelings about these problems, particularly as related to their feelings of success in math. Only by understanding how students experience math in my classroom can I design open-ended math problems that are useful, enjoyable, and effective in helping students improve their problem solving skills, make connections in math, and therein see a glimpse of the beauty that this subject can hold for everyone. I hope that through my continued research I can help math teachers work together to ensure that fewer students walk out of our classrooms saying, “Oh, I’m just not a math person,” and that more feel empowered to take on any problem that comes their way.

Tips for Teachers

  • 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 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.

Resources for Teachers

Books

Shuster, L. and Anderson, N. (2005). Good questions for math teaching: Why ask them and what to ask? (grades 5-8). Sausalito, CA: Math Solutions Publications.
This book presents open-ended problems in a clear progression, so you can either use all of the problems as the backbone of your curriculum, or pull problems out and intersperse them with what you already have planned for your class.

Swan, P. (2003) Messy math: A collection of open-ended math investigations (grades 4-7). Rowley, MA: Didax.
This book offers a wide range of open-ended problems with reproducible handouts, answer keys, and teachers’ notes.

Becker, J. and Shimada, S. (1997). The open-ended approach: A new proposal for teaching mathematics. Reston, VA: National Council of Teachers of Mathematics.
After reviewing the theoretical underpinnings and early research on open-ended problems, the authors offer sample lesson plans and examples of teaching across the elementary and secondary grades.

Kabiri, M. and Smith, N. (2003). Turning traditional textbook problems into open-ended problems. Mathematics Teaching in the Middle School, 9(4), 186-192.
This article offers practical strategies for turning traditional problems into open-ended problems. It gives examples at various levels to model the changes.

Websites

http://math.exeter.edu/dept/materials/
Phillips Exeter Academy is a boarding school in Exeter, New Hampshire that has developed wonderful materials, including many open-ended problems (primarily grades 9-12).

http://sln.fi.edu/school/math2/
This website offers a nice selection of open-ended math problems broken down by month. They are primarily for grades 6-7 but could be adapted for any grade.

http://nrich.maths.org/public/
This website is sponsored by NRICH and the University of Cambridge and has a great selection of open-ended problems for both middle and high school, including possible problems of the week.

http://mathcounts.org/NETCOMMUNITY/Page.aspx?pid=355&srcid=195
MATHcounts is an organization aimed at improving math at the middle school level. This site offers a selection of fantastic open-ended problems, including archives of problems and solutions from the past eight years.

References

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, 42(4), 212-227.

Jarrett, D. (2000). Open-ended problem solving: Weaving a web of ideas. Northwest Education Quarterly, 1(1), 1-7.

McKenzie, J. (1997, September). Telling questions and the search for insight. From Now On: The Educational Technology Journal, 7(1). Retrieved from: http://fno.org/sept97/telling.html.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation: Standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Passow, H.A. (1982). Differentiated curricula for the gifted/talented. Ventura, CA: Ventura County Superintendent of Schools Office.

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.

Steen, L. (2007). How mathematics counts. Educational Leadership Journal, 65(3), 8-15.

Trei, L. (2005, February 2). How urban high schoolers got math. The Stanford Report. Retrieved from: http://news-service.stanford.edu/news/2005/february2/boaler-020205.html