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 5th
and 6th 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
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.
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