TOOLS FOR INVESTIGATING HIGH SCHOOL STUDENTS’
UNDERSTANDING OF GEOMETRIC PROOF
|
Sharon Soucy McCrone Illinois State University |
Tami S. Martin Illinois State University tsmartin@math.ilstu.edu |
Abstract: Although proof and
reasoning are seen as fundamental components of learning mathematics, research
shows that many students continue to struggle with understanding geometric
proofs. In a preliminary study to a
larger research project, we investigated two components of students’
understanding of proof – their beliefs about what constitutes a proof and their
ability to construct proofs. This
preliminary study focused on the development of two instruments to gather
information regarding these two components of students’ understanding of
geometric proofs. Results from the
study have aided us in revising the instruments for the larger study, and have
given us some information that has been useful for focusing our research
questions in the current larger research project.
Introduction
Proof is fundamental to the discipline of mathematics because it is the convention that mathematicians use to establish the validity of mathematical statements within a given axiomatic system. In addition, the teaching of proof as a sense-making activity is important for developing student understanding in geometry and other areas of mathematics. Despite the fact that student difficulty with proof has been well established in the literature, existing empirical research on pedagogical methods associated with the teaching and learning of geometric proof is insufficient (Chazan, 1993; Hart, 1994; Martin & Harel, 1989). Our work in this area has begun to address the need for research into the pedagogy of geometric proof instruction. In a preliminary study we developed and piloted research instruments for measuring students’ understanding of geometric proof. We are currently undertaking research funded by the National Science Foundation (NSF) in which revised versions of these instruments are being used to help us develop an empirically grounded theoretical model that relates pedagogy to student understanding of proof.
The objectives of the preliminary study were (1) to construct, administer, and refine two instruments for measuring students’ understanding of geometric proof; and (2) to conduct preliminary analyses of the data collected from students’ responses to the instruments in order to inform the current NSF project. In developing the research instruments, we focused on two components of student understanding of proof, namely, students’ beliefs about what constitutes a proof and students’ proof-construction ability.
Existing research documents students’ poor
performance on proof items and identifies common, fundamental misunderstandings
about the nature of proof and generalization in a number of mathematical
content areas (Chazan, 1993; Hart, 1994; Martin & Harel,
1989; Senk, 1985). In trying to make sense of students’
difficulties with geometric proof, Dreyfus and Hadas (1987) articulate six
principles that form a basis for understanding geometric proof. These principles address many of the student
misunderstandings of proof cited in the literature (Chazan, 1993;
Hart, 1994; Martin & Harel, 1989; Senk, 1985). A revised version of the six principles
guided the development of the research questionnaire to assess students’
beliefs about what constitutes a proof.
Other perspectives on students' reasoning abilities that we considered
when developing our instruments can be found in Harel and Sowder (1998), Hoyles
(1997), and Simon and Blume (1996). The findings from
our three-year study will be used to make connections between pedagogy and
various levels of student understanding of proof.
In order to measure beliefs about what constitutes a proof, we constructed a questionnaire that assessed students’ agreement with a revised set of six principles. It was necessary to add more detail to Dreyfus and Hadas’ (1987) principles in order to develop items that could be reliably identified with particular principles. Questionnaire items consist of items modified from instruments used by Chazan (1993), Healy and Hoyles (1998), and Williams (1979), as well as some original items. Part I of the questionnaire, requires students to indicate whether they agree, disagree, or are unsure about statements that correspond to the revised six principles. Part II of the questionnaire includes open-response items related to the same principles.
The instrument designed to assess proof construction ability includes items in which students must construct partial or entire proofs, as well as generate conditional statements and local deductions. In addition to some original items, the instrument includes items modified from Healy and Hoyles (1998), Senk (1985), and the Third International Mathematics and Science Study [TIMSS] (1995).
In the pilot study, the instruments were given to first and second semester geometry students in a local high school summer program. The same version of the questionnaire was given to all students early in the semester. There were four versions of the performance assessment: before instruction and after instruction versions for the first semester students as well as before instruction and after instruction versions for the second semester students. Students received about two weeks of instruction on proof between the two administrations of the performance assessments.
In the process of piloting the instruments, we used a number of techniques to determine the quality of the instruments. For example, four experts examined all questionnaire items in order to assess the content validity. In order to determine the construct validity of the questionnaire, we compared responses to open-ended items with responses on parallel multiple choice items and assessed the consistency of those responses. Consistency issues are discussed in the results section. In order to estimate the questionnaire’s reliability, Part I of the questionnaire was divided into two 14-item half tests, matched on content. The split-half reliability estimate, with a Spearman-Brown correction, was 0.67. To assist us in interpreting these measures, we also collected student feedback regarding the clarity of the questionnaire items.
The quality of
the performance assessment instrument (proof quiz) was also assessed using
several measures. Construct and
criterion-related validity of the proof quiz were measured by a simple
correlation between students’ scores on the pilot instruments and their final
examination grade. The correlation
coefficient of 0.66 is fairly high and indicates that the proof quiz did a good
job of differentiating between high- and low-ability students. The experts also confirmed that before and
after versions of the proof quiz were roughly equivalent. The coefficients of stability
and equivalence for the versions given to first and second year students,
respectively, were 0.78 and 0.91, indicating that the two forms were reasonable
equivalents and that we could expect student performance on these test
instruments
to be stable over time.
Cronbach’s a coefficients, which
estimate reliability, were 0.66 and 0.86 for the before and after versions of
the quiz given to first year students and, likewise, 0.57 and 0.68 for the
second year versions. The researchers
and the classroom teacher used a common written rubric to score students’
responses to the performance assessment items.
Rater agreements for the four versions of the performance assessment
were 96%, 93%, 89%, and 90%.
There are two
types of results for this study. The
first type of result is the collection set of
instruments for assessing student understanding about geometric proof. Based on student responses as well as
validity and reliability estimates from the pilot study, we have revised the
instruments. We are currently using
these revised instruments in our ongoing work and will continue to assess their
quality and make further revisions as necessary. We recognize that creating appropriate and useful instruments is
problematic because “understanding of proof” is difficult to define and more
difficult to measure.
A second type of result from the study is evidence of students’ beliefs about proof and abilities to construct proof from administration of the instruments. For example, by their responses to related items on the questionnaire -- which assesses students' agreement with the six principles of understanding proof -- most students (89-100%) showed evidence of agreeing with the principle that proof has a dual purpose, to convince and to explain. In contrast, few students (22%) indicated agreement with the principle that proofs have internal logic requirements. Both as a group and individually, many students gave inconsistent responses to different items that addressed the same principle. An example of this relates to the generality requirements of a proof. On one item, students claimed to believe that a proof must be general (78% agreement on the item), but on another item, 78% of students accepted as valid a "proof" based on a few particular examples. In some cases, students who provided consistently correct responses to items related to a particular principle in the multiple choice section of the questionnaire were unable to produce correct, coherent reasons for their conclusions to items corresponding to the same principle in the open response component of the questionnaire. For this reason, we are developing an interview protocol so that we may learn more about inconsistencies in students' beliefs. In our pilot study, many of the inconsistencies were due to wording of the instrument or students' inability to communicate ideas in writing. However, by questioning the students, we hope to be able to identify instances in which their beliefs about a particular principle are incomplete or contradictory.
The performance assessment instruments revealed results that were not surprising since they echoed the findings of Chazan (1993) and Senk (1985). For example, student scores were lowest (ranging from 0% to 33%) on quiz items that required them to construct a proof, even when provided with an outline of the proof. Student scores were a bit higher (ranging from 29% to 44%) on an item that showed a proof with missing statements and reasons, for which students were asked to fill in the missing steps. Student scores were highest (ranging from 29% to 60%) on an item that asked them to rewrite a conjecture in “if-then” form and determine the given information as well as what was to be proved.
In summary, the instruments we have created appear promising for
collecting evidence about students’ beliefs about proofs and their ability to
construct proofs. We realize that
results from the initial administration of the instruments may not be readily
generalizable due to the fact that summer students may not be representative of
the general high school population.
However, initial findings identify several specific weaknesses and
inconsistencies in proof understanding for further consideration.
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