Theoretical Framework

            Problem solving in mathematics, specifically non-routine problems, has been one focus of mathematical reforms throughout the past 20 years. A non-routine problem has no standard algorithm for representing or solving it, and involves high cognitive demand (Schoenfeld, 1992). Smith and Stein (1998) describe levels of cognitive demand as lower level, such as memorization and procedures without connections, and higher level, such as procedures with connections and “doing mathematics.” 

Students often struggle with fractions and their representations when solving problems. It is crucial for teachers to help students build flexible understandings of the meanings of fractions via multiple representations.  We define representations as mathematical models, tools or products that may include, images, words and/or symbols, that illustrate a student's mathematical thinking and understanding (Goldin, 2003; Monk, 2003; NCTM, 2000; Smith, 2003). The concept of unit is foundational to fractional understanding and to the common meaning of fractions: part-whole relationship, measure, quotient, ratio/rate, and operator (Barnett-Clark, Fisher, Marks, & Ross, 2010; Lamon, 2005; Petit, Laird, & Marsden, 2010). Each problem structure elicits particular student understandings and representations.

            Formative assessment helps teachers better understand student thinking. Teachers gather and interpret evidence of student understanding and use that evidence to plan subsequent instruction and improve learning (Bell & Cowie, 2001; Black & Wiliam, 2009; Wiliam, 2007). Formative assessment comes in many forms, including discussion, interviews, written work, and non-routine problems.

            A framework is a systematic way of thinking about learning, based on educational research. Frameworks are tools that allows teachers to view their students' learning and make distinctions among levels of understanding (Panasuk, 2010).

Methods

The questions guiding our research were:

How can middle level math teachers use student work to better understand student mathematical reasoning?

·         How do teachers articulate the use of non-routine problems in their classroom?

·         What strategies and representation do students use to solve non-routine fraction problems?

·         How do teachers use a fraction framework to analyze student work generated from non-routine problems to inform instruction?

We analyzed the data qualitatively using three data sources: student written work, teacher focus group interviews, and field notes from classroom observations.

Our research subjects were 5^th and 6^th grade teachers in a Midwestern state involved in a two-year professional development program between 2005-2010, and their students. As part of the professional development program, teachers gave their students non-routine problems, adapted from reform curricula, each spring and fall. The student work was coded independently by researchers, using the Ongoing Assessment Project, OGAP, (Petit & Vermont Institutes, 2009) framework to analyze the strategies students used when solving a non-routine fraction problem. The main categories of the framework are fractional, transitional, fractional strategy with an error, non-fractional reasoning, and unable to determine strategy (see Table 1).

Table 1. Coding Categories for Fraction Problem #1

Fractional Strategies	Number Sense
	Equivalence
	Estimation
	Efficient Algorithm
Transitional Strategies	Student Generated Model
Fractional strategy with an error	Student generated model that is appropriate but solution contains an error
	Appropriate operation or strategy, but solution contains an error
Non-Fractional Reasoning	Whole number reasoning
	Inappropriate model, operation or strategy
	Rule based reasoning, not linked to understanding
Unable to determine strategy	Not enough evidence to understand student thinking
	Blank or no mathematical value

Note. The codes were derived from OGAP fraction frameworks. 

Twelve teachers involved in the professional development program were invited to participate in a focus group interview, designed to help teachers understand a fractional framework and examine student work. Seven teachers participated in the recorded and transcribed focus groups, and of those, the two most willing to learn more about a fraction framework became the subjects of further study. Those teachers administered one non-routine problem, then the teacher and a researcher coded the student work, and together designed a subsequent lesson.

Results

The research subjects had participated in a high-quality professional development program, and therefore understood the potential for student growth from using non-routine problems. The teachers believed non-routine problems made their students think deeply about mathematics and stated they tried to use non-routine problems in their classroom, but time was the main barrier. 

            The two teachers involved in the follow-up study worked with a researcher to code their students' fraction work and then design a subsequent lesson. The teachers and researcher chose to focus on equivalence as a strategy for problem solving. Teachers used modeling and discussed the meaning of fractions, emphasizing that models need to be the same size and divided into equal pieces. This emphasis addressed the needs of students using transitional and non-fractional strategies because they were able to see how the models connected to the equivalence strategy.

            After the follow up lessons, both teachers thought the OGAP framework helped them understand student thinking better than other methods used. Heather[1] specifically said she thought it was more beneficial to look at what strategies the students used rather than simply answers.

            When asked whether they could see themselves using a framework for formative assessments, the teachers said they would use it with additional practice. Heather explained that frameworks would be particularly valuable with difficult topics. Eric said, “What it does is it breaks down the problem into specific skills and it helps you individualize the education of each student. You are looking at each student to see where they need instruction.” 

Educational Importance

            We learned that teachers viewed using non-routine problems in their classroom as important. Teachers understand that engagement in mathematical thinking allows students to build conceptual understanding, and gives teachers access to gain knowledge about what each student understands. When teachers use knowledge about student understanding to plan subsequent instruction, they are able to see the impact of formative assessment. Teachers also recognized the usefulness of a framework as a formative assessment tool. Teachers realized it was essential to use multiple representations to help students understand fractions. Fractions remain a topic that students struggle to understand, however, the OGAP framework gives teachers a tool to help them understand student work via formative assessment.

References

Barnett-Clarke, C., Fisher, W., Marks, R., & Ross, S. (2010). Developing essential understanding of rational number: Grades 3 – 5. Reston, VA: National Council of Teachers of Mathematics.

Bell, B. & Cowie, B. (2001). Formative assessment and science education. Boston, MA: Kluwer Academic Publishers.

Black, P. & Wiliam, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability, 21(1), 5-31. Doi: 10.1007/s11092-008-9068-5

Goldin, G. A. (2003). Representation in school mathematics: A unifying research perspective. In J. Kilpatrick, W. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Lamon, S. (2005). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum Associates, Publishers.

Monk, S. (2003). Representation in school mathematics: Learning to graph and graphing to learn. In J. Kilpatrick, W. Martin, & D. Schifter (Eds.), A research companion to principles and 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.

Panasuk, R. M. (2010). Three phase ranking framework for assessing conceptual understanding in algebra using multiple representations. Education, 131(2), 235-257.

Petit, M. M., Laird, R. E., & Marsden, E. L. (2010). A focus on fractions: Bringing research to the classroom. New York, NY: Routledge.

Petit, M. M. & Vermont Institutes. (2009, June). Vermont Mathematics Project Ongoing Assessment Project: Fraction Framework. Retrieved from http://www.margepetit.com

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning. New York, NY: Macmillan Publishing Company.

Smith, M. S. & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3, 344-350.

Smith, S. P. (2003). Representation in school mathematics: Children's representations of problems. In J. Kilpatrick, W. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Wiliam, D. (2007). Keeping learning on track: Classroom assessment and the regulation of learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning. Charlotte, NC: Information Age Publishing.