VIEWS OF THE NATURE & JUSTIFICATION OF THREE DOMAINS OF KNOWLEDGE: MATHEMATICS, MATHEMATICS TEACHING, & MATHEMATICS LEARNING

Jennifer B. Chauvot

San Diego State University

This study examined 2 preservice secondary mathematics teachers’ views of sources, evidence, and certainty of knowledge in three domains. Data were collected throughout a year-long program using multiple open-ended data sources. Sources of and evidence for mathematical knowledge were predominantly external, based on the words of an authority. In the context of open-ended mathematical activities, they acknowledged that their peers’ approaches were unique, yet mathematically legitimate. This raised concerns about mathematics learning; however, there was little indication that they searched for knowledge in this domain. For knowledge of mathematics teaching, Brenda exhibited an internal orientation and relied on experience, herself, and intuition whereas Liz continued to rely on perceived authorities for this knowledge. Corresponding implications for teacher education are discussed.

An individual’s epistemology, or philosophy about the nature and justification of knowledge, influences learning in that how an individual perceives a learning experience is contingent on who or what the individual views as sources of and evidence for knowledge, and to what extent the individual perceives that knowledge as certain. Mathematics teacher education programs are environments for learning about domain-specific knowledge such as mathematical knowledge, knowledge of mathematics teaching, and knowledge of mathematics learning. This study examined two preservice secondary mathematics teachers’ views of the nature and justification of knowledge in these three domains in the context of a reform-oriented secondary mathematics teacher education program.

There is no research that concurrently investigates views of knowledge in all three domains. The studies of Arvold (1996), Etchberger and Shaw (1992), and Eggleton (1995) were specific to views of the nature and justification of mathematical knowledge, whereas Mewborn (1999) identified changes in sources of knowledge of mathematics teaching and learning. Arvold and Albright (1995), Cooney and Wilson (1995), and Cooney, Shealy, and Arvold (1998) addressed orientations toward authority (sources and evidence) primarily in the domains of mathematical knowledge and mathematics teaching. This study asked: For each participant, who or what constitutes sources of and evidence for mathematical knowledge, knowledge of mathematics teaching, and knowledge of mathematics learning, and to what extent is that knowledge certain? An understanding of views of knowledge in these three domains has implications for mathematics teacher education.

Similarities across theoretical perspectives about epistemological development (Baxter Magolda, 1992; Belenky, Clinchy, Goldberger, & Tarule, 1986; Perry, 1970) provided a framework for characterizing sources, evidence, and certainty of knowledge (see Figure 1). For example, all

EXTERNAL SOURCE	INTERNAL SOURCE
KNOWLEDGE AS CERTAIN
· Dualism & Multiplistic pre-legitimate (Perry), Silence & Received knowing (Belenky et al), Absolute knowing (Baxter Magolda) · Multiple perspectives are not epistemologically legitimate · EVIDENCE: One right answer based on words of the authority
KNOWLEDGE AS EVENTUALLY CERTAIN
· Multiplistic subordinate (Perry) · Multiple perspectives are epistemologically legitimate · EVIDENCE: One right answer based on words of the authority when authorities figure it out
KNOWLEDGE AS PARTIALLY CERTAIN
· Relativism subordinate (Perry), Impersonal Transitional knowing (Baxter Magolda) · EVIDENCE: Multiple right answers based on logical reasoning processes that use evidence based in context. · Reliance on authorities for when to use reasoning processes	· Multiplistic correlate (Perry), Interpersonal Transitional knowing (Baxter Magolda), Subjective knowing (Belenky et al) · EVIDENCE: Multiple right answers based on personal experiences · Reliance on intuition
EXTERNAL/INTERNAL SOURCES	INTERNAL/EXTERNAL SOURCES
KNOWLEDGE AS UNCERTAIN
· Relativism (Perry), Separate Procedural knowing (Belenky et al), Individual Independent knowing (Baxter Magolda) · EVIDENCE: Multiple right answers based on logical reasoning processes that use evidence based in context. · Recognition of self’s abilities to apply reasoning processes in new contexts.	· Connected Procedural knowing (Belenky et al), Interindividual Independent knowing (Baxter Magolda) · EVIDENCE Multiple right answers based on reasoning processes that strive to understand the personal experiences of others. · Authorities are sources and partners.
INTERNAL-EXTERNAL SOURCES; KNOWLEDGE AS RELATIVE Commitment within relativism (Perry), Constructed knowing (Belenky et al), Contextual knowing (Baxter Magolda) EVIDENCE : Knowledge is relative, based in context, and subject to the values of those involved

Figure 1. Characterizations of Source, Evidence, and Certainty of Knowledge

three perspectives have positions of development in which the individual views sources of knowledge as external, internal, or a combination of both. All three perspectives have positions that characterize evidence to be based on what an authority says, personal opinions and experiences, or reasoned judgment. Finally, all three perspectives highlight positions in which the extent of certainty of knowledge ranges from absolute certainty, free of context and human values to relative certainty, based on context, and subject to the values of those involved. A detailed description of this framework can be found in Chauvot (2000).

Two preservice secondary mathematics teachers (Liz and Brenda) from a larger research project, Research and Development Initiatives Applied to Teacher Education (RADIATE[i]), were informants for this study. Data collected through RADIATE served as the data for this study. The goals of the RADIATE instructional program were consistent with a reform vision of mathematics teaching and learning (NCTM, 2000). The RADIATE research questions addressed the nature and structure of the preservice teachers’ knowledge and beliefs about mathematics, mathematics teaching, and mathematics learning, and the extent to which the preservice teachers were able to be reflective about their teacher education experiences. Data collection occurred throughout a four-quarter sequence of two content/pedagogy courses, a student-teaching quarter, and a post-student-teaching seminar. Data sources included 34 journal entries, 6-13 semi-structured interviews of at least 45 minutes in length, open-ended surveys, course artifacts, and fieldnotes. Journal prompts, interview protocols, surveys, course artifacts, and detailed descriptions of selected mathematical activities of RADIATE can be found in Chauvot (2000).

The data and perspectives provided in the work of Baxter-Magolda, Belenky et al., and Perry provided insight for coding data for each knowledge domain. For example, the authors used information about their participants’ expectations of the learner, peers, and instructors in the learning process as a means to discuss participants’ epistemological views. Therefore, for the domain of mathematical knowledge, I identified indications of each participant’s expectations from herself, peers, and instructors while involved in mathematics courses and any mathematics-related activities of the RADIATE program. For the domains of knowledge of mathematics teaching and mathematics learning, I identified indications of each participant’s expectations of herself, peers, instructors, and cooperating teachers in the program experiences. Inductive analysis strategies (Patton, 1990) were used to identify patterns in the data. Figure 2 summarizes the findings in terms of the framework. In the discussion that follows, the findings are organized by source which then includes explanations related to corresponding assumptions about evidence and certainty.

For the domain of mathematical knowledge, sources were predominantly external; mathematical knowledge was certain with correctness based on the words of the teacher or textbook. This was evident in Liz and Brenda’s recollections of their past learning experiences and characterizations of good mathematics teaching. Mathematics teachers were expected to explain

MK		MT		ML
External	Internal	External	Internal	External	Internal
Knowledge as Certain		Knowledge as Certain		Knowledge as Certain
Liz, Brenda		Liz	Liz, Brenda		Liz, Brenda
Knowledge as Eventually Certain		Knowledge as Eventually Certain		Knowledge as Eventually Certain
					Liz, Brenda
Knowledge as Partially Certain		Knowledge as Partially Certain		Knowledge as Partially Certain
Liz, Brenda			Brenda
External/Internal	Internal/External	External/Internal	Internal/External	External/Internal	Internal/External
Knowledge as Uncertain		Knowledge as Uncertain		Knowledge as Uncertain

Internal-External Sources; Knowledge as Relative

Figure 2. Views of the Nature and Justification of Knowledge in All Three Domains

material clearly, be willing to answer students’ questions, and be willing to re-teach the material when necessary. The student was expected to pay attention in class, study regularly, and get help from the teacher when needed. Also, both participants spoke of learning by doing, learning by seeing, and learning through discovery. They expected students to practice problems, use manipulatives in small groups, and use technology. In practice, students were actively involved, but student contributions to the mathematical discourse was limited. Sources and evidence for knowledge remained with Liz or Brenda or the textbook. The mathematical knowledge existed with certainty, free of context and human values.

The RADIATE mathematical activities sometimes provided a context that indicated an awareness of partial certainty of mathematical knowledge. Both participants were surprised to realize that their peers approached mathematical problems in unique, yet mathematically legitimate ways. The uncertainty of mathematical knowledge was due to different mathematical thinking by peers. Evidence for mathematical knowledge was not necessarily the words of an authority. For example, in response to the Function Card Sort Activity (see Chauvot, 2000), Liz “found it interesting to see how students with similar backgrounds in math could see totally different answers (backed with perfect logic) to one problem” (4/7/94j).

For the domain of knowledge of mathematics teaching, one source was personal experience as learners. Both participants admitted that their own learning of mathematics in the context of the RADIATE mathematical activities served as evidence for knowledge of mathematics teaching. Liz was quite adamant about this. Brenda was more reserved. The evidence for knowledge focused on the perception of an increased understanding of mathematics. Also, these activities fulfilled the participants’ desires to have their students more actively engaged in the learning process. Consequently, beliefs about learning mathematics (e.g., students learn by doing) also served as evidence for knowledge of mathematics teaching.

A second identified source of knowledge of mathematics teaching was practical teaching experience. This was especially important to Brenda:

The field experience I think is really important [for my professional development] because experience is something you can learn from. And no matter how much you talk about teaching geometry, you’re not going to know what it’s really like until you get up there and actually do it. I think that [experience] is probably the most important thing there is (Interview 4, 10/94).

A third source of knowledge of mathematics teaching was best expressed as a combination of opinions of self, RADIATE instructors and peers, and cooperating teachers. What became significant in this analysis was the weighting system each participant used to decide the veracity of the knowledge she heard from the different sources. Liz’s focus was external and authority-oriented. She placed more value on the opinions and experiences of the RADIATE instructors and her cooperating teachers as compared to the views of her peers.

I get so sick of everyone. We'll spend like three days in class doing something, and then the first comment is "Well you can't do it in a classroom." And I'm like "Give it a chance." I mean we're not here for no reason. I just get kinda upset with people just immediately saying they can't do it in a classroom (Interview 4, 10/94).

In terms of self, she frequently assumed she was wrong. “I might be totally wrong cause I’ve found out that I’m totally wrong about a lot of things recently” (Interview 2, 5/94).

Brenda focused on herself, experience, and intuition. The opinions of her cooperating teachers and RADIATE instructors and peers were valuable, but secondary. She did not expect the RADIATE instructors or her cooperating teachers to transmit knowledge of mathematics teaching to her. Instead, she intended to rely on experience and herself as the primary source.

I realize that many of her [the cooperating teacher] teaching skills come from experience. I hope to be able to determine what amount of class discussion, lecture, and guided practice is appropriate for each of my classes as she seems to have done (Paper, 5/26/94).

Also specific to Brenda was an unwillingness to pass judgment on the opinions of her peers, her cooperating teachers, and the RADIATE instructors.

I think [my cooperating teacher] did a real good job with the class that she had. I don’t know how I would have done anything different. … Because of the students that were in there. I mean, different students in the same class may have responded differently and students in a different class may have responded the same way. I don’t know (Interview 3, 6/94).

Brenda seemed to feel that everyone had a right to his or her own opinion about mathematics teaching. Opinions were based on practical teaching experience. Brenda was aware that individual experiences varied; consequently, she seemed reluctant to pass judgment.

There was little indication that the participants searched for knowledge of mathematics learning. Consistently, each participant assumed that students learn mathematics in the same manner that she had learned mathematics. However, the participants were concerned about using “it worked for me” as evidence for this knowledge. They were particularly concerned about how lower-level students learn mathematics. Both concluded that such students simply needed more time and patience: Learning occurred the same way, but at a slower pace. In addition, the awareness that their RADIATE peers, who were assumed to have “similar backgrounds in math,” approached mathematical problems in different, yet legitimate ways, created concern. Consequently, they turned toward practical teaching experience as a source of knowledge of mathematics learning. Both hoped that the field experiences and beginning years of working with students would inform them about student thinking. This conclusion seemed to discontinue the search for any other sources of knowledge of mathematics learning. Interestingly, there was no indication that they turned toward the RADIATE instructors or their cooperating teachers for this knowledge.

Findings from this examination of teachers’ views of knowledge in the three domains have implications for teacher education. For example, interactions with peers in the context of open-ended mathematical activities stimulated new thinking about the certainty of mathematical knowledge (see Borasi, Fonzi, Smith, & Rose (1999) for similar results). This in turn led to concern about mathematics learning. These are promising results, particularly because there was little evidence to indicate that the participants were even searching for knowledge of mathematics learning. This limited indication in itself is a significant finding for teacher educators. Perhaps the program was not as explicit in this domain of knowledge. Or perhaps the participants were too naïve at this point in their development to reflect on student thinking. Finally, regarding the domain of knowledge of mathematics teaching, teacher educators must be cautious of teachers (e.g., Liz) who seem to eagerly and readily accept new ideas about mathematics teaching without careful deliberation. At the same time, almost total reliance on experience and self (e.g., Brenda), also without careful deliberation, has limitations as well. Teacher educators need to consider program experiences that both honor teachers’ views as well as promote further professional growth.

References

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