Mar 28, 2002 - by. Clare Brett, Earl Woodruff, Ontario Institute for Studies in Education of the University of Toronto; Rodney Nason, Queensland University of.
C. Brett, OISE/UT
Session 13.81 AERA, 2002
Developing Identity as Pre-service Elementary Mathematics Teachers: the contribution of online community
by Clare Brett, Earl Woodruff, Ontario Institute for Studies in Education of the University of Toronto; Rodney Nason, Queensland University of Technology
In Session 13.81, Division K Paper presented at the American Educational Research Association, New Orleans, April 2002.
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Developing Identity as Pre-service Elementary Mathematics Teachers: the contribution of online community Clare Brett, OISE/UT 1. Objectives One significant challenge of pre-service elementary education in mathematics is to enable participants with diverse academic backgrounds and experience to acquire the knowledge and pedagogical skill to teach in ways that may either diverge widely from their own school experiences or that involve a subject area in which they did not excel. The challenge in this study was to help 20 math-anxious pre-service teachers, selected from a cohort of 57 enrolled in a two year certification course at OISE/UT, develop an identity as mathematics teachers and learners that they could continue in the field. The specific goal of the study was to facilitate the creation of a community that included these math anxious pre-service teachers by supporting them in productive engagement that mapped directly onto three core dimensions of math teaching: 1) Content. Advance of their mathematics knowledge of three, specific, fundamental concepts relevant to elementary math teaching: (a) place value and renaming, (b) operations with numbers, and (c) patterns/rule-finding; 2) Discourse. Engage in a discourse community themselves as part of their own math learning and teaching, and through these experiences gain greater understanding of the role of discourse in enriching math understanding; and 3) Pedagogy. Acquire a conceptually-based rather than an algorithmic approach to math pedagogy. To facilitate these goals, an online environment was used, in addition to small group discussion and open-ended mathematics investigations. The goal of the online community was to provide extended, multi-year, time and place independent access to teacher and peer networks of support. Recent developments in pedagogy and conceptions of mathematics on reasoning and the importance of shared discourse as a way to make meaning has shifted the emphasis away from traditional algorithmic approaches of mathematics instruction (e.g. Lampert, Rittenhouse & Crumbaugh, 1995). Communications technology offers a potential support for this approach because of the significant role that discourse and reflection play in text based discussion environments. Research on collaborative learning environments suggests that in certain contexts, such electronic supports can increase the depth of student learning (Scardamalia & Bereiter, 1995), and evidence from a study of undergraduates using structured online discussions as an adjunct support to a face-toface course suggests the electronic component encouraged discussion of issues and fostered interactivity among participants (Hara, Bonk & Angeli, 1998). However, similar research has not been carried out with a group of pre-service teachers with such prior negative experiences in the domain. Additionally, research is clear that math anxiety is widespread among both in-service and pre-service elementary teachers (Kelly and Tomhave, 1985; Hembree, 1990; Sloan, 1
Paper accepted for presentation at the American Educational Research Association, New Orleans, April 2002.
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Vinson, Haynes & Gresham, 1997), and that math anxiety leads to math avoidance, subsequently affecting levels of performance (Trice & Ogden, 1986/7). Identifying oneself as a math learner is a critical step in developing a positive perspective on math learning and teaching. To support this process math content work was focused on a significant subset of concepts, foundational to the elementary math curriculum, and framed within a supportive, discourse-rich learning environment, using small learning groups and an online discussion environment available all the time throughout both years of the two year program. 2. Theoretic Framework This study combines sociocultural and cognitive perspectives to understand the different elements that either facilitated or restricted participants’ identity as members in the online mathematics environment. In Wenger’s (1998) communities of practice model, identity is viewed as not a state, but as experience developed through activity and involvement. As such he suggests that a learning community will become a place of identity if it offers members a place to incorporate their histories (that is their past experience) and provides an experience that makes community engagement a significant element in a personal future. Potentially, a shared electronic context would provide a forum for engaging in a community of pedagogy of mathematics through legitimate peripheral participation (Lave & Wenger, 1991). Barab and Duffy (2000) explain the role of the community in educational contexts: “The goal of participation in a community is to develop a sense of self in relation to society—a society outside of the classroom” (p43). Evidence of community identity is being conceived of in this study as evidenced through participants' reactions to and commentary on the communityfocussed elements of the program, and in their attitude shift towards mathematics. Also, the role of epistemic agency (Scardamalia, 2000) is explored in the development of participants’ identities as teachers and learners of mathematics. Scardamalia defines epistemic agency as taking responsibility for personal understanding demonstrated through iterative cycles of revising internal and external ideas to a resolution. Epistemic agency is demonstrated through participants' efforts to deepen understanding and to work with mathematical ideas towards developing a conceptually based approach to math pedagogy. In this paper, epistemic agency and community identity are conceptualized as mutually constituting engagement in community discourse. Epistemic agency represents the focus on idea development, and identity the connection to the community. In effect identity (consisting of connectedness in the community, and a basic level of perceived competence in the context) may enable the conditions for epistemic agency to flourish. Through the process of directing learning (epistemic agency), participants actively engage in community discourse, and through the engagement and feedback, in turn increase their sense of identity. It is the unpacking of the elements in these two concepts of agency and identity that will be addressed in this paper.
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3. Methods This research involves a two-year exploratory study that draws upon a number of data sources to better understand the patterns of engagement and disengagement in the development of a mathematical community of twenty preservice teachers enrolled in a two-year pilot program. Such an exploratory structure allows the study of teaching and learning in real world environments, with all the complexity that such contexts entail. Temporal analyses of database activity, categorical ratings of portfolio and database content as well as of interview responses were used to support interpretation. Some of the analyses take a sociocultural perspective, where the focus of understanding is on participation itself and learning is evidenced as contributions to the practice of one’s communities. Other analyses come from a more psychological perspective, including attitudes and beliefs about mathematics and about participant’s views of learning, ability and teaching. Together, these perspectives allow a more complete understanding of the issues affecting engagement in online community because cultural, interpersonal and individual factors can be considered together. In Dewey's words, "the psychological and the social sides are organically related, and that education cannot be regarded as a compromise between the two, or the superimposition of one upon the other" (p444, in McDermott, 1973). 3.1 The program: This study was conducted during the course of a four-term preservice teacher education program. Term 1 Based on whether they had indicated mathematics was a subject they felt anxious about, the cohort of fifty-seven preservice teachers were assigned to eleven small groups of 4-6 members for small group math investigations. Two of these groups consisted only of mathematically anxious females. The other nine groups were heterogeneous in terms of their members’ perceptions about their abilities to do and teach mathematics. From the eleven workshop groups, a sample of four workshop groups (the two mathematically anxious and two heterogeneous workshop groups) formed the twenty participants in the focus group for this study. The whole cohort then participated in a series of 8 two-hour workshops conducted during the fall term in 1995 during which they carried out a number of open-ended mathematical investigations. The goal of these investigations was to overcome many of the preservice teachers’ misconceptions about the nature of mathematics, to make mathematics accessible by demystifying its subject-matter and to provide situations where mathematical discourse would arise naturally. Whilst the mathematical investigation workshops were being conducted, the preservice teachers were given computer accounts and access to a number of computer laboratories on the OISE/University of Toronto network. They also were given training sessions and support in using this technology. To provide ongoing support for reflecting upon and extending the small group class discussions, students were given access to an electronic conference (a First Class conferencing
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system) called Math Inquiry, where they could ask questions, share ideas with peers and faculty, and discuss practicum experiences and share resources. First Class conferencing allows messages to be viewed as threads, and messages can be new or replies, or replies to replies. The title of the message and author information are visible from the header, and red flags denote the presence of new, unread entries. Because there was so much variability in prior computer experience as well as level of computer access—especially home access—considerable attention was given to providing equipment (Alex terminals in the first year and laptops in the second year) and in class training and practice so that people could get connected. Participation in the database was not part of their formal evaluation but they were expected to participate in the same way they were required to attend class The Math Inquiry conferences yielded about 550 notes (varying from a few lines to multiple pages) over the 2 years (they also had opportunity to contribute to as many as 24 other program related conferences on the Tednet system). Term 2 Math activities included a sequence of tutorial/workshops (6 two-hour sessions) in which they investigated how the Jasper Woodbury videodisc and support materials (Learning and Technology Center at Vanderbilt University, 1996) could be utilized to establish and maintain communities of mathematics practice in elementary school classrooms. As in Term 1, the issues and topics discussed in the shared electronic database conferences were in the main decided by the preservice teachers. Term 3 Preservice teachers in the primary/junior division who had chosen the Math/Science and Technology option, attended a sequence of formal mathematics education lecture/tutorials for approximately another 16 hours, but divided among math, science and technology issues. A major component of this term was a six-week block classroom placement. Thus, the major focus of their electronic database conferences during this term was on issues related to preparation for their block classroom placement and/or issues and topics brought up during the formal mathematics education tutorials. Nine out of the thirteen participants in the Primary/Junior/Intermediate option from the Focus group chose the Math/Science/Technology specialization. Another seven of the focus group of twenty were part of the Junior/Intermediate division and therefore were required to take as their teaching subject their background degree specialization, and therefore were not eligible to select MST as a specialization. These were mostly English/Language Arts majors. Term 4 Those in the M/S/T option had approximately 8-10 hours of formal Math Science and Technology teaching, and the main emphasis at this point in the program was on science and technology. The other members of the cohort had no formal math instruction. All the cohort were encouraged to continue their reading and written contributions to the other mathematical conferences on the shared electronic database.
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4. Data Sources For this paper, we will focus on how database content and activity, with interpretations supported by evidence from portfolios and interview ratings can inform how participants engaged with issues of math content, discourse and pedagogy. Type of data Database notes: reading & writing patterns Portfolios Interviews Math content test
Time of collection: Year 1 Math inquiry conference: ongoing
Year 2 Math inquiry plus other program conferences: ongoing
Ongoing throughout Ongoing throughout program program End of year 1 End of year 2 Beginning of Term 1 Year 1 Post program, while teaching in the field
4.1 Database Notes. The number of contributions made by each of the participants to the shared electronic database conferences were recorded and frequencies tabulated of notes written and notes read in the math inquiry as well as the other conferences. The content of the notes was rated using the same categories as the portfolio data (see next paragraph). These were: • Articulating and/or reconceptualizing math knowledge or concepts. • The development of community and discourse about the learning and teaching of elementary mathematics. • Changing views of mathematics pedagogy. Fifty percent of the notes were rated by two raters, with an interrater reliability of r = .90, and differences were resolved. 4.2 Portfolios In their portfolios, the preservice teachers were asked to record significant learning episodes of personal learning occurring during their programme. In each entry to the portfolio, the preservice teachers used the following questions to guide their reflection: (a) What is this entry about? (b) Why did you choose this as an entry? (c) What did you learn or how did you grow? Each entry was approximately 1-3 pages in length with supporting documentation such as an article or lesson plan. The 20 participants in the focus group were asked near the end of the program for permission to photocopy these bodies of work for analyses. They all agreed to do so. The math-related portfolio entries were separated out from the others for more detailed analysis. Next, the portfolio entries for each member of the focus group that were relevant to mathematics were assigned to one of three categories (described above) by two independent raters. Each entry was further rated, within each category, on a scale from 1-3 as to the degree of reflection on issues of understanding (either for the participants/teacher or the future students) evident in the entry. 41% of the portfolios
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were also rated by another rater, and the overall interrater reliability was found to be r = .82. Differences were discussed and agreed upon. 4.3 Interviews At the end of the first year, participants were asked a number of questions in relation to the program. Of particular interest in this paper were questions relating to reactions to their small group experiences, and how they attributed the causes of their math anxiety, specifically: • •
Do you feel you were able to contribute to your group in small group discussions? and has your group experience helped your learning and confidence? How do you account for your feelings about yourself as a math learner based on your experiences as a child and adult?
The first question addressed an issue that could influence their sense of community identity and their subsequent online engagement. The second question touched on cultural effects of early experiences brought to the current learning context. 4.4 Math Content Test. This was developed by the math educator in the program and based on tests by Baturo and Nason (1995). It was focussed on three foundational areas that would be addressed in the content of the small group math investigation problems; place value and renaming, operations with numbers and patterns/rule-finding. The test was administered early in the first year of the program, and then again with the participants who agreed to do so, during their second year of teaching in the field. It was scored by the math educator who developed it.
5. Results Engagement was assessed through a number of measures. First, we look at depth of understanding as demonstrated through the ratings of the portfolio data. Beyond the actual content analyses of the portfolios and database notes these data show the depth of engagement of the various participants in the focus group. These results will be compared to data in the second section that describes the level of activity for all the participants in the shared electronic database. These data will show the range of participation activity for all the members of the focus group and provide another vantage point from which to consider engagement. 5.1 Engagement viewed as depth of understanding Looking at different models of community in the literature, we find that a distinguishing feature of Knowledge Building Communities (e.g. Scardamalia & Bereiter, 1994) and Communities of Learners (Brown & Campione, 1994) is their focus on increasing depth of understanding. To be effective in accomplishing the three goals of the program (improving content, math discourse and pedagogy), participants in the current study would have to demonstrate some depth of understanding of the three dimensions. Additionally, I would claim that deep understanding in this context
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demonstrates a significant form of engagement, specifically that it indicates a level of epistemic agency (taking charge of one’s own learning). This is because portfolios were self-selected so the math focus reflects efforts to expand learning in this area. As well, reflective elaboration of ideas is a significant aspect of taking agency over one's own learning. Accordingly, we will look first at the Portfolio results for the three program goals. These portfolios, were self-selected examples of learning episodes, and as such provide a measure of participants’ perceptions of their “high points” in learning. Portfolios also provide a content measure completed by everyone and so provide information on all participants independent of technology-related effects. First, the average portfolio ratings, ranked from high to low, are shown for each participant (see Table 1) across the dimensions of math knowledge, discourse and pedagogy. The portfolios were rated for the knowledge dimension on a scale from a low of 1 (an unelaborated mention), to a high of 4 (reflective elaboration of important concepts). To arrive at a single "depth" rating for each participant the combined ratings (from the two raters) were averaged over the number of math portfolios generated for both years. In Table 1, looking at the higher levels of depth scores, we see 9 people scoring 3 or higher for the Discourse and Pedagogy measures, but only 6 scored 3 or higher on the Math Content measure. To score high on the math knowledge depth measure participants had to be dealing with significant conceptual issues relating to mathematical concepts, something which only certain people entered into at that level. Overall, the scores were slightly higher for the Discourse and Pedagogy measures also, suggesting that these areas were more easily accessed by a wider group of people. There is considerable overlap in the individual rankings for each of the measures. The same six people appear within the top six ranks for all three measures: Judy, Wendy, Nora, Sarah, Marissa, and Eileen. There is more variability with the other rankings, although Joan and Stephen do rank low on all three measures of depth. The participants in the middle however, do show higher scores on one or two measures if not all three, suggesting that they are engaging deeply with ideas, but not perhaps as consistently as the group of six identified above. The following examples of portfolio entries with ratings of 3 or higher highlight the reflective nature of the responses by these participants. The first entry is based on a combination of her own learning, reflecting on prior experience and recent professional reading. She is also focused on having students develop their own reflective understanding: I decided to explore the use of math journals as a tool for improving reading skills in math…A large part of my decision to introduce math journals came from my personal experiences with math…I would become very frustrated when I was never given enough time to record my rationale or strategy. I had also done some professional reading about the topic..and am convinced that math journals are an essential part of the math curriculum. I would argue that writing in math forces students to “look inside their own minds” and examine their own understanding of the concept. (Marissa, 11/96).
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This next example also connects early experience with current understanding and then takes it further in applying the solution to a new context. One of the most powerful experiences I had of how explaining a mathematical solution can lead to deeper understanding came when I was doing the math test…Gnomons were described in the problem as l-shaped numbers…and odd numbers. We had to generate a rule for adding consecutive gnomon numbers and justify the rule…I could see the pattern that was developing—when you added consecutive gnomon numbers (beginning with 1) the total would be the square of the number of gnomons you were adding…I was feeling pretty pleased with myself for discovering the pattern and I decided to show the problem to my 15 year old son. He could see that pattern by he immediately asked why. It was when I started to explain to him that I realised I needed to take into account that these gnomon numbers were not just odd numbers but were L-shaped. When you added the L-shapes together you actually created squares. If he hadn’t asked why, and I hadn’t attempted to answer it, I would never have discovered the relationship. (Wendy, 10/96). The summary (Table 2) confirms that the same 6 participants scored above an average of 3 across all areas suggesting that these participants were consistently engaged in depth of understanding of these ideas, across all three areas, at least in the portfolio entries. Next we will look at database activity and see if the portfolio patterns for these individuals are similar to the database engagement patterns. 5.2 Engagement as Online Participation This section describes the range of writing and reading activity among the participants in the database. Activity in the shared database was analysed in two ways. The first was a frequency summary of database participation, in which (a) the number of entries contributed to the database and (b) the proportion of the database read were combined. This summary revealed a general trend during the four terms towards greater participation both in terms of number of entries to the database and the proportion of the database read. For example, the proportion of the computer-mediated math conference read by the focus group (as compared to the whole cohort of 57) increased from 29.9% in Year 1 to 40.5% in Year 2. However, there was much variance among individuals in the focus group. If we order the participants according to how much they participated through reading and writing in the database, we can see four approximate sub-groupings summarized in this table and defined below:
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Group
N
Participation Patterns
Engaged
5
Higher levels of writing and reading in both years
Emergent
6
Reading and writing increase in Yr 2
6
Reading and Writing decrease in Yr 2
3
Low reading and writing rates in both years.
Disengaged • •
•
•
Engaged (n=5) This sub-group started out with the highest levels of reading (>60% of others’ notes) and writing (>10 notes per year), and increased or maintained involvement over the course of the four terms. Emergent (n=6) This sub-group changed most, particularly in relation to the number of the database conferences they read. Their levels of reading started out low but increased steadily over the course of the four terms (>20% increase between Year 1 and Year 2). Written contributions to the math inquiry conference, however, were still low but did increase slightly (>2 notes) between Year 1 and Year 2. Withdrawing (n= 6) This sub-group’s levels of written contributions to the math inquiry conference decreased (or remained very low and unchanged) over the course of the four terms ( a drop of < or = 1) Most of their participation was confined to the reading the contributions of others, although this too decreased between year 1 and year 2 except for two people (a drop of 10%-60%) Disengaged (n=3) This sub-group engaged minimally in the math inquiry conference in both years, either in reading (< 1%) or writing < or =1 note).
If we look at which participants fall into these groups (Table 3 shows participation rates for individuals within groups), we find that in the Engaged group, four of the participants receiving the highest rankings in portfolio ratings, Judy, Wendy, Nora and Eileen are members. Sarah’s online participation patterns locate her in the Emergent group and Marissa in the Withdrawing group. This pattern suggests that a core group were able to develop a sense of community identity in the online environment sufficient to let them consistently engage in discourse about mathematics during the program. We can also look at the amount that participants read each other’s notes both in the Math Inquiry conference and across conferences on other topics, and see how the database is valued as a shared discourse environment. In Table 4 we can see both in the Math Inquiry conference and across other conferences, the patterns of reading in each of the four groups. The shaded cells indicate an increase between years for the proportion read of other people’s math notes, and the clear cells indicate an increase between years for the proportion of others’ notes read in conferences apart from math. First, we notice an increase across years in both math and other subjects for the Engaged and Emergent groups, and in conferences other than Math, for the Withdrawing and the Disengaged group. This suggests that overall, the database community support was found to be useful by most of the focus group participants, because all of these participants increased their use of the online conferences overall, if
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not for the math inquiry conference itself. However, the math conference was used most by the Engaged and Emergent group members. The role of knowledge by itself as measured by the incoming math test does not explain the level of activity as measured by this engagement metric. One might argue that people who are more knowledgeable would feel comfortable in contributing to the database. However, the correlation between the number of contributions written in the math conference and the average math test score is not significant in either Year 1 (r=0.28) or Year 2 (r= 0.30), suggesting that factors other than incoming knowledge level are critical for database involvement. Another view of database participation was a summary temporal analysis by month across the two years indicating a) when each participant contributed notes to the database; b) the times of vacation and practicum placements; and c) the duration of four (one from each term) of the most sustained database discussions. These data allowed the following observations: First, most members of the focus group of 20, from the Engaged, Emergent and Withdrawing subgroups contributed to all four discussions, supporting the idea that even math anxious members found some role within this emerging community. Second, contributions from the focus group to the discussions occurred during both class and during practicum time, and increased in the second year, suggesting that the database was seen as a community support even when students were in geographically distant locations during their practica. Third, the temporal representation of these data indicated that contributions were distributed most widely among the focus group participants during December of Year 1. This timing coincides with a number of programme events. First, by this time everyone who was going to use a computer consistently was doing so. Second this was the most intense period of the small group math investigations series, and therefore database activity was closely supported by temporally contiguous classroom activity. For a number of participants, particularly those in the Withdrawing group, this close connection was important. Without that support, their online participation dropped off considerably in the second year. These second year discussions also drew more widely from people across the programme community, including faculty from other parts of the program, and associate teachers. In such extended contexts, some may have felt less confident in being able to contribute substantively and so shifted their attention to other, non-math related online conferences, as suggested by earlier analyses of the reading data patterns comparing the math inquiry participation to other online conferences (Brett, Woodruff & Nason 1999). For participants in the Emergent group however, the generally positive experiences in the small group math investigations seemed to allow them to increase their online activity during the second year, although the actual level of mathematical analysis undertaken by participants in this group did not consistently reach the level of the Engaged group members, as shown by lower reflectivity ratings of portfolio and database entries. 5.3 Epistemic Agency: Working with ideas The content of the database discourse was categorized to examine commentary on mathematics content, discourse and pedagogy. Many of these comments demonstrate
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shifts in epistemic agency, as we see the participants setting their own agendas for learning within the social context of the database. 5.3.1 a Mathematics Content. These included participant's conceptions of what mathematics was as well as how they worked with substantive mathematical ideas. In Table 5, the amount of content related comments from each year are tabulated. It is clear that the largest amount of commentary came from the Engaged group participants, followed by the Emergent group. The database comments about views of mathematics can be summarized as falling into three main categories: 1) the significance of contextual assumptions in problem-solving, 2) mathematical reasoning extending beyond algorithms and 3) math as a way of thinking. Often using the problems discussed in the small group math investigations, participants noted shifts in their own thinking. In the first example is a reflection on how assumptions in a problem affect the answer: We could have spent hours examining different assumptions that lead us to different answers but the most valuable lesson for me was learning that every answer you get is based on a series of assumptions (some more valid than others). These assumptions need to be explored before the answer can be fully understood. So now I know that there is more to an answer than meets the eye! (Megan, Emergent).
Another shift was starting to see math knowledge as based in understanding and reasoning that was constructed rather than transmitted, a view of learning that was being conveyed throughout their program. I think it’s of vital importance that every learner discover THEIR own personal pattern or else it’s meaningless. My father-in-law has told me many times how he remembers his phone number but I can’t for the life of me remember because his pattern is meaningless to me. I didn’t discover it, I can’t remember it. Now extend this to math. Sure I memorized the operations, the theorems the teacher presented but I didn’t understand the basic principles and couldn’t make them mine. How does math taste? It’s becoming more palatable by the minute. For years I didn’t want to take a bite. I feel I’ve discovered a whole new field. (Eileen, Engaged 29/09/96)
The final viewpoint about mathematics that emerged most clearly during Year 2 was that mathematics is not just a tool but a way of thinking. This viewpoint was only found amongst five of the focus group participants. It perhaps was most effectively articulated by Wendy when she wrote: Mathematics is not only just a useful tool that helps us make change or buy hamburger buns or whatever. It is a way of thinking and a way of organizing thought. It is a way of interpreting and explaining what we see and do. The only way we can begin to really understand Math is by doing it. We really don’t begin to understand numeracy unless we “handle” numbers - play with them - move them around. You can do neat things with calculators - but you only really understand how they neat they are if you understand what’s behind it. (Wendy, Engaged)
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5.3 1.b. Mathematical idea development. A sequence of notes generated during one of the small group investigations illustrates how participants engaged in principled mathematical understanding. The preservice teachers were attempting to justify and/or prove their solutions to the investigation’s set of problems. Understanding the proofs of the solutions to these problems required principled understanding (Leinhardt, 1988) of place value. When engaged in this process one participant went to great pains to lay out her thinking as clearly as she could in words—taking on the challenge of communicating her mathematical thinking. What helped me understand this proof was when I realized that in using multiple representations of numbers you don’t always have to describe the number in a fixed order of hundreds, tens, and ones. That you can say in this case 3 hundreds, 9 tens, and 5 ones OR 5 ones, 9 tens, 3 hundreds or possibly even 9 tens, 3 hundreds or 5 ones - it’s still the same number. Therefore, if I was working with a group of young students I think I would want to get across not just the notion that you can say 85 is 85 or 8 tens and 5 ones or 6 tens and 25 ones but also 5 ones and 8 tens etc. That when you describe a number this way, the actual order of the numerals does affect the value the way it does when you are representing it the usual way. I think a lot of us, while we get the idea that you can break the number up in different ways, when you are describing it as hundreds, tens and ones it’s still a leap to actually changing the “order” around, i.e., putting the ones first instead of last. I don’t know if this will help anyone else. It seems to me using multiple representations of numbers actually makes them more flexible or manipulable. (Wendy, Engaged)
Another participant was helped to understand the same idea, from Wendy's description of her own thinking process: Wendy, you helped me understand why we could suddenly turn the number backwards, once grouped, and it would still be the same number. Of course it would!!! How did I ever get to high school math without realizing this? (Eileen, Engaged)
In a collaborative response to the first note above, another two participants supported the idea from Wendy’s note and added a context where the idea could be applied: G. and I both agree with you 100%. We too have been taught that 395 is 3 hundreds 9 tens and 5 ones. If you ask people after the test (referring to the math pretest), most got the question on place values wrong as a result of this. This helps me and I think it will help students as well. However, they first have to understand that they have to make and can make other groups from the same number and transfer it. This will help kids understand the concept and value of money. It’s always the same amount but shown differently. Try re-arranging picture patterns in order for students to understand regrouping, while the number or picture stays the same. (Anna, Withdrawing)
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Two participants in response, in another collaborative note, elaborated on why they felt Wendy’s note was so good: Wendy, it was interesting to see how you broke down your numbers (hundreds/tens/ones) It is great to see you applying what you have learned in class to your own teaching situations. We (Elaine and I) would have never thought of showing the students that they can rearrange the numbers and they will still remain the same. It seems like common sense, but it is something we would probably never have thought of. (Elaine, Disengaged and Janice (a non-focus group participant)
As one reads these notes, it can be seen that the first participant’s entry2 is a reflective analysis of her understanding, what had helped her to gain that understanding and how that shift in understanding could be integrated into teaching. Furthermore, her note indicates that a serious effort was being made to understand some of the underlying principles of the place value numeration system. The knowledge-building intent of her participation in this discourse thus was apparent. The second participant’s response extends the discourse by giving support and extending the idea to another context (using money). However, this participant’s response does not add anything to the discourse about the characteristics and properties of the numeration system. The final participants’ entry is a reflection about their own learning and also a supportive comment. It adds socially to the discourse, but does not elaborate the ideas substantively. The contributions of the second and third participants therefore did not really contribute much towards the further development of the focus group’s collective principled understanding of the numeration system. Thus, although all of the responses demonstrated engagement, they varied as to whether they were focused on both the mathematics and the social elements, in a principled way, or were simply reflective of their own learning. That difference would be a defining feature of knowledge building, apart from simply collaborative discourse (Scardamalia and Bereiter, 1994). This example is representative of the level of knowledge building discourse in the database across topics. Certain people were able to engage at this level, showing epistemic agency in pursuing understanding and opening their reasoning up to the community, but it was only a small group of people who were able to do this at the level of knowledge-building, and most of these people were in the Engaged group. The others certainly made progress--they became aware of their thinking processes, advanced in their understanding, but did not necessarily contribute to the discourse in the same way. 5.3.2 Discourse: In Table 6, we can see how commentary on the discourse and community subcategories (Value of Discourse, Metacomments on Learning, Discourse as Assisting Learning and Sharing of classroom experiences) was distributed among the participants. The Engaged and Emergent group members expressed their sense of the value of discourse This entry was part of an ongoing attempt to write up a proof of how the individual digits of multiples of 9 always add up to 9
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and working together as a community. However, only the Engaged group consistently made metacomments on their own learning through the database notes (Table 6, notes falling in the 2B Metacommentary category in Year 1). Encouragingly, the Engaged, Emergent and Withdrawing group members expressed an understanding of the importance of using discourse and community to assist learning. Finally, the Emergent group did the most unelaborated sharing of classroom experiences. In this way they were using the database as a social or communicative space, but taking a less analytical or reflective stance on their activities. 5.3.3. Pedagogy: Table 9 displays the categories related to pedagogy in the database notes. The Emergent group members along with the Engaged group, share rationales and ideas for instruction, take the learners' understanding into account, reflect on how their own learning development will impact their pedagogy, and consider different issues that impact pedagogy, such as motivation. In Year 1, we also see a slightly less, but still similar pattern of activity for the Withdrawing group, except that they reflect less on the different issues that impact pedagogy. Overall we see the greatest distribution of comments across participants in all groups in the database notes, on issues of pedagogy (as compared to discourse or math content). This is one context in which more people were able to actively contribute. Also, all the engaged participants and half of each of the emergent and withdrawing participants made direct connections between their own learning experiences and its impact on their teaching. This finding is encouraging in that it suggests participants are broadly trying to reframe their approach to math pedagogy. “I always had a math phobia..and I do not want to pass that on to my own students. After having witnessed and participated in…activities at the faculty that directly related to math, I found myself interested in the strategies and wanting to apply them in my own classroom”. (Nora, Engaged)
There was reflection in the database and portfolios about applying pedagogical strategies learned in other aspects of the program to the mathematics context (such as using journals in mathematics class to encourage reflection and articulation of ideas). In addition, they commented on how activity centres and collaborative work could be applied in a mathematics context. I think the difficulty is not with the fact that it is a real life problem but the pre-existing knowledge they should have. I consider the fact that as a class at FEUT, we all had some basic knowledge of the fundamentals of math and it still took us all day and we still had problems. This is with pre-existing knowledge. (Jessica, Engaged. 9/4/96)
Further they are demonstrating a level of epistemic agency, as shown in the example above, in actively debating the potential value of different teaching strategies that would avoid recreating the problems they had experienced during their own schooling.
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In addition, there were shifts towards a much more positive view of mathematics pedagogy. Specifically, in the interview data at the end of Year 1 about how the program had impacted on their teaching experiences, fifteen of the participants mentioned incorporating collaborative groups and discourse to their math pedagogy. One indicator of their increased positive attitudes towards math generally was that 8 out of the 13 participants in the Primary/Junior division (who were the only ones who could choose a specialization other than their degree program) chose Math, Science and Technology in Year 2. They explicitly stated that they now had the confidence to tackle these areas in which they felt they had less prior experience and incoming confidence. 5.4 Identity: The language of collaboration and inclusion in the database. Identity in the online community can also be seen in the actual patterns of discourse in there. Inviting people to respond to your ideas suggests that you are comfortable in that context and prepared to interact. Encouragement to other participants to join in and shown in statements such as. “.please ask me questions and perhaps it will help me find a better way to explain it.” This invitation suggests that other’s feedback will help both the asker of the question, who will hopefully get their question answered, and the originator of the explanation. Such a focus could encourage cycles of interaction in which questions do not simply get taken up and responded to, but worked with iteratively. Statements with a similar stance were found frequently throughout the database. Some of these statements were specifically focused on feedback for an idea, for example: “Is this an incredibly simplistic attitude? What do you think?” (Eileen, Engaged 22/11/95)
Others requested specific help, often requesting assistance with teaching specific lessons: If you have any ideas about teaching a Kindergarten lesson on patterns, please respond, I would really appreciate it!! (Janet, Withdrawing 8/1/96)
and: ...But I felt that that wasn’t really the best way to do it. I would say the math take up was pretty much a disaster! Any comments? I know this is like dumping stuff on you, but I really need some guidance. Thanks, R (an instructor) (Jessica, Engaged. 4/12/95)
the following requests a response to her interpretation of an issue: Anyone have experience with this? Am I right off the wall with my theory? Signed: Help me out here! (Eileen.Engaged 20/3/96)
Some were more general however, such as Write back if you’ve had a similar experience (Marissa, Withdrawing 4/12/95)
Others gave a more indirect request for people’s feedback such as in the next example:
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Well I think I have written enough to be going on with. I look forward to discussing this some more. (Judy, Engaged 15/3/96)
Frequently, reciprocity was directly encouraged and elicited through asking questions, as the next example and its response illustrate: Question: I wonder...if I had continued the weekly estimation, would their answers have gotten more accurate (your “practice’ principle) (Eileen, Engaged 10/3/97)
6. Social and Contextual Factors Contributing to Engagement and Identity Data sources which detail contextual factors influencing engagement such as responses to interview questions about the usefulness of the database and the origins of their mathematical anxiety, give a perspective on what most influenced the likelihood of participant's engagement in the database.
6.1 Reactions to the database Responses to interview questions about database usefulness suggest that participants are more likely to make written contributions to a conference to which they feel able to substantively contribute. However, reading substantial portions of the database in a timely manner as participants in the Engaged group (in both years) and Emergent group (in year 2) and the Withdrawing group (in year 1) did suggests that the content was perceived as useful. If, on the other hand, participants felt a need for help in math, this did not predict a greater degree of asking for help in the database—something that might be a logical prediction from such information. Instead, feelings of vulnerability and lack of confidence may have prevented people from asking so publicly for assistance. It was noticeable that the people who either asked for help or who openly discussed problems were predominantly Engaged and Emergent group participants, suggesting that they had a more secure sense of their identity in this online context. 6.2 Attributions of causes of math anxiety. Another important factor seemed to be how they attributed the origins of their mathematical anxiety. In interviews, Disengaged participants blamed themselves for being unable to understand the mathematics they were taught in school. They also did not report any successful experiences with math at any time in their own schooling. By contrast, Engaged participants gave clear and detailed accounts of the external factors such as teachers’ attitudes and specific experiences that had caused them to feel inadequate mathematically. They all appeared to have reflected on this in interviews and in the database, analyzing the causes rather than simply internalizing the negative experiences. Withdrawing and Emergent participants tended to give accounts that mixed internal and external factors. The categories and results are shown in Table 8. One interpretation of these data is that those who are self-blaming, also view math learning as based on innate ability. With such a theory of learning, it is hard to develop a sense of identity as a potentially successful math learner, engaging in a discourse
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community about mathematics. While many of the participants did comment on the usefulness of discourse and reflection in math pedagogy, there was still a reluctance to do this publicly themselves. 6.3 Small Group Math Investigations. Interview data also revealed that the face to face mathematical investigations conducted in Semester 1 played a critical role in the establishment of an identity within a mathematics education community of practice for the Engaged, Emergent and Withdrawing participants. These interactions provided a structure for their math discourse and mediated the development of a support network that encouraged participants to share their ideas and concerns. Their responses were categorized and the results are shown in Table 9. While there is variability among the Emergent and Withdrawing groups, there is consistency in the positive reactions among the Engaged group, and consistency in the mixed reactions among the Disengaged group. Only the Engaged group reported consistently positive experiences from the beginning, and there were most reports of problems (scores of 1) in the Withdrawing and Disengaged groups. Even so, no-one reported entirely negative experiences. The main problem reported by those who found the experience less satisfying was feeling diffident about their own ability to contribute, although they found the situation a very useful one for their own learning. This diffidence was also reported as a reason for not making written online contributions. In sum, it appears that while for most people, the online environment supported an emerging identity as members of a mathematical community, self-blame for prior negative mathematics experiences, and technical fears reduced the effectiveness for some. A separate indicator of the overall increased confidence was that 9 out of the 13 participants in the Primary/Junior division (who could choose a specialization other than their degree program) chose Math, Science and Technology in Year 2, stating that they now had the confidence to tackle these areas that they had previously avoided. 7. Scientific Importance This study offers insights on the role of electronic environments for supporting participants’ emerging identities as mathematics teachers. First, online environments can provide an opportunity for some pre-service teachers to develop their ideas in a framework of social discourse. Second, there is considerable variability in how people benefit from such contexts. Even the technological developments (simpler interfaces; more reliable connectivity) since these data were collected would likely have reduced the technological frustration expressed by the Disengaged and some Withdrawing participants. While this may have increased overall participation it may not have changed the level of written contributions made by these participants. Their activity appeared determined more by factors outside the electronic context. These included a sense of their own agency as math learners (affected by how they attribute the origins of their math anxiety) and the degree to which they felt able to substantively contribute
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to the discussion through answering peers questions, which together seemed to shape identity among these participants. In conclusion, participants in the Engaged Emergent and Withdrawing groups found the shared database socially supportive, and a source of assistance and ideas, but it was mainly the Engaged participants who used it as a context for the building of knowledge about mathematics and mathematics education. The small group math investigations seem to provide these participants with the levels of mathematical confidence necessary to “jump-start” them into wanting to become involved deeply in discourse about mathematical ideas as well as the teaching of mathematics and self-reflection, at which point the electronic environment became a useful tool. Members of the other groups may have benefited from an extended series of the small group investigation workshops. This would have provided greater opportunity to become immersed in the construction of mathematical content knowledge and gain a stronger identity as developing mathematicians. In turn, this may have made the reflective advantages of the online environment a reality for a broader group of participants. Despite the potential for democratizing the classroom, using online environments to create a culture of discourse in which engagement is widespread fundamentally requires that participants have identity within that community. This in turn allows them to start to develop enough of a sense of epistemic agency to engage in the community knowledge building discourse. In the data from this study we see a sense of community identity in the confidence they felt during the small group discussions; the language of collaboration and inclusion in the database; and the extent to which they defined the origins of their mathematical anxiety as stemming from factors outside (such as the teacher or teaching practices) rather than their own internal deficiency. Through community identity we see ourselves in the discourse as legitimate participants, and as contributors to the discourse we are able to shape and then share in the community discourse. This was the case most strongly with the engaged group who were involved in the communal discourse from the beginning. This extended participation may have enabled the development of a history and hence (Barab & Duffy, 2000) would argue, these members were able to develop a sense of self in that community. To the extent that we identify ourselves as deficient in relevant knowledge or ability, we experience less identity, engage less and thereby fail to increase our own level of knowledge. Such a conclusion has implications for all community discourse in that novices are not necessarily drawn in as apprentices to the centre over time, unless they can develop a sense of identity based on their own perceived value and power in that community.
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References Barab, S. & Duffy, T. M. (2000) From practice fields to communities of practice. In Jonassen, D. and Land, S. (Eds.) Theoretical Foundations of Learning Environments. Lawrence Erlbaum Associates. NJ. Brett, C. Woodruff, E. & Nason, R. (1999) Online Community: What can reading and writing patterns tell us about participation? Paper presented at the Annual Meeting of the American Educational Research Association, Montreal. Brown, A. L. & Campione, J. (1994). Guided discovery in a community of learners. In Kate McGilly (Ed.). Classroom lessons: Integrating cognitive theory (pp. 229-270). Cambridge, MA: MIT Press. Hara, N. Bonk, C. J. & Angeli, C. (1998) Content analysis of online discussion in an educational psychology course. CRLT Technical Report No. 2-98. Hembree, R. (1990). The nature, effects and relief of mathematics anxiety. Journal for Research in Mathematics Education, 21, 33-46. Kelly W.P.& Tomhave, W. (1985). A study of math anxiety and math avoidance in preservice elementary teachers. Arithemetic Teacher, 32. 51-53. Lampert, M., Rittenhouse, P., & Crumbaugh, C. (1995). Agreeing to disagree: developing sociable mathematical discourse. In D. Olson & N. Torrance (Eds.), Handbook of Education and Human Development: New Models of Learning, Teaching and Schooling (pp. 731-764). Oxford: Basil Blackwell Lave, J. & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press. Scardamalia, M. & Bereiter, C. (1995). Adaptation and understanding: A case for new cultures of schooling. In S. Vosniadou, E. De Corte, R. Glaser, & H. Mandel (Eds.), International perspectives on the psychological foundations of technology-based learning environments (pp. 149-163). Mahwah, NJ: Lawrence Erlbaum Associates. Scardamalia, M., & Bereiter, C. (1994). Computer support for knowledge building communities. The Journal of the Learning Sciences, Vol.3, 265-283. Scardamalia, M., & Bereiter, C. (in press). Knowledge Building. In Encyclopedia of Education, Second Edition. New York: Macmillan Reference, USA. Sloan, T. R., Vinson, B., Haynes, J. & Gresham, R. (1997). A comparison of preand post- levels of mathematics anxiety among preservice teacher candidates enrolled in
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a mathematical methods course. Paper presented November 12-14, 1997 at the annual meeting of the MidSouth Educational Research Association in Memphis, TN. Trice, A.D. & Ogden, E.D. (1986/7). Correlates of mathematics anxiety in first year elementary school teachers. Educational Research Quarterly, 11 (3) 3-4. Wenger, E. (1998). Communities of practice: Learning, meaning and identity. Cambridge University Press.
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Table 1 Average ratings of Portfolios according to level of reflective elaboration for each of three dimensions of Math content, discourse and pedagogy. Average Portfolio Ratings Ranked for each of 3 Dimensions Math Name Judy Nora Marissa Wendy Eileen Sarah Alicia Katherine Amy Megan Kristina Elaine Anna Jessica
Discourse
Content
Name 4 4 3.7 3.6 3.2 3
Judy Nora Sarah Marissa Eileen Wendy
2.7 Nadine 2.65 2.5 2.25 2 2 2 1.8
Janet Kristina Katherine Maia Alicia Amy Maureen
Maia
1.7 Elaine
Maureen Janet Stephen Nadine Joan
1.5 1.3 1.3 1.3 0.7
Anna Stephen Joan Jessica Megan
Math
& Community Name 4 4 3.75 3.7 3.6 3.4
Judy Nora Sarah Marissa Wendy Eileen
Pedagogy 4 3.8 3.75 3.7 3.6 3.6
3.3 Katherine
3.3
3 3 2.8 2.8 2.8 2.5 2
3.3 3 2.8 2.8 2.5 2.5 2.5
Nadine Maureen Maia Jessica Amy Kristina Megan
2 Alicia 2 2 2 1.8 1.8
Janet Elaine Anna Stephen Joan
2.45 2.3 2 2 1.8 1.8
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Table 2. Average ratings of Portfolios according to level of reflective elaboration for combined dimensions of Math knowledge, discourse and pedagogy
Name Judy Nora Marissa Wendy Eileen Sarah
Average Portfolio Depth Rating 4 3.9 3.7 3.53 3.5 3.5
Katherine
2.9
Nadine Alicia Amy Kristina Maia Janet Maureen
2.5 2.5 2.5 2.5 2.4 2.2 2.2
Jessica
2.1
Megan Elaine Anna Stephen Joan
2.1 2 2 1.7 1.5
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Table 3 Grouping of participants according to reading and writing participation in the Math Inquiry database, by year and group.
Engaged Wendy Judy Eileen Jessica Nora Average Emergent Alicia Sarah Maia Nadine Megan Stephen Average Withdrawing Kristina Janet Joan Anna Marissa Katherine Average Disengaged Maureen Amy Elaine
Year 1 Year 2 % read # written % read # written 87 33 84 19 46 18 58 15 76 9 79 12 92 13 100 40 96 10 91 11 79.4
16.6
82.4
19.4
15 12 47 56 51 4.9 31.0
2 0 1 3 9 4 3.2
82 44 92 97 80 34 71.5
9 4 0 10 10 7 9.0
72 9 10 7 11 20 21.5
1 2 2 5 2 5 2.8
10 3 8 37 21 15 15.7
1 0 1 1 1 3 1.5
0 0 2 0.6
0 0 2 0.6
0 0 0 0
0 0 0 0
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Table 4 Proportion of others' conference entries read, in math compared to all other conference topics combined and averaged for year 1 and year 2.
ENGAGED
62.72
Reading Others' All other Conf. Yr 1 % 29.72
EMERGENT
30.95
13.65
71.48
68.1
WITHDRAWING
21.35
18.75
15.78
35.92
DISENGAGED
0.6
2.5
0
3.63
O = > All Other O = > Math
Reading Others' Math Yr 1 %
Reading Others' Math Yr 2 %
Reading Others' All Other Conf. Yr 2 %
82.66
79.2
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Table 5 Math Content in Database Notes, by category for years 1 and 2.
# Notes Engaged Wendy Judy Eileen Jessica Nora
20 18 9 13 10
Average 14 Emergent Alicia 2 Sarah 0 Maia 1 Nadine 3 Megan 9 Stephen 4 3.2 Average Withdrawing Kristina 1 Janet 2 Joan 2 Anna 5 Marissa 2 Katherine 5 2.8 Average
Year 1 Math Content 1a Math 1b Nature Concepts 11 2 0 7 5 5
1 1 3 1 1.5
1 1 2
of math 5 2 1 1 1 2
3 1 2
Year 2 Math Content 1a Math 1b Nature
1c Connection to real world Concepts 0 6 2 4 0 2 2 1 1 4 1
1 0 0.5
of math 6
1c Connection to real world 3
3 3
3
3.4
3
3
1
1 1
1
1
1
1
1
1 1 1 1
1
1 1
1 1
2 2
Dusengaged Maureen 0 Amy 0 Elaine 2
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Table 6 Discourse and community comments in Database Notes, by category for years 1 and 2
Name
Engaged Wendy Judy Eileen Jessica Nora
Year 1 Discourse and Community 2A Value 2B metaconments of discourse on Learning 3 1 0 2 2
7 7 3 1 6
Year 2 Discourse and Community 2C 2D Sharing 2A Value 2B 2C Assists 2D Sharing Assists Metacomments Learning Class exp. of On Learning Learning Class exp. discourse 10 2 1 3 6 2 1 4 7 1 1 4 1 1 4 3 4
Emergent Alicia Sarah Maia Nadine Megan Stephen Withdrawing Kristina Janet Joan Anna Marissa Katherine Dusengaged Maureen Amy Elaine
2
1 1
3 1
1 2 3
1
0 0
6 2
2
1 1 1 3
2 0 1
2
1
1 2
1 2
1
1
1
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Table 7 Math Pedagogy comments in Database Notes, by category for Years 1 and 2
Name Engaged Wendy Judy Eileen Jessica Nora
Year 1 Year 2 Rethinking Math Pedagogy Rethinking Math Pedagogy 3A Sharing 3B learners 3C own learning 3A Sharing 3B learners 3C own learning Ideas mind impacts teaching Ideas mind impacts teaching 5 2 4 11 1 0 2 2 7 5 1 0 2 11 2 3 5 2 0 1 4 5 2 1 3 1 3
Emergent Alicia Sarah Maia Nadine Megan Stephen
2
1 2
4 1
1 2 2
4 2 2
1 1
1 1 2
1
1 1 2 0
2 0
Withdrawing Kristina Janet Joan 1 Anna Marissa 2 Katherine 2
1
Dusengaged Maureen Amy Elaine
1
2
3 2 2
1
1
2 1
1
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Table 8 Location of attributions of origins of mathematical anxiety
Engaged Emergent Withdrawing Disengaged
External* 80 % (4)** 14 %(1)
Mixed 20 % (1) 28 %(2) 60 %(3) 33 % (1)
Internal 57 % (4) 40 %(2) 66% (2)
1. Internal: problems attributed to a lack of ability or aptitude located within oneself which still is in effect (e.g. “I am still afraid of math”), or an internalising of teacher attitudes (e.g. “girls can’t do math”), 2. Mixed: problems attributed to a combination of ability lacks and unhelpful teacher attitudes. 3. External: problems due to inadequate teaching or support or an unsympathetic school structure, and evidence that they have overcome these experiences (e.g. “I can do math now”) **Numbers in brackets indicate the actual number of participants whose answers fell into each category.
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Table 9 Small Group Experiences
Name
Small Group Exp*
Engaged Wendy
3
Judy
3
Eileen
3
Jessica
3
Nora
3
Average
3.0
Emergent Alicia
1
Sarah
3
Maia
2
Nadine
2
Megan
2
Stephen
3
Average
2.2
Withdrawing Kristina
3
Janet
3
Joan
1
Anna
3
Marissa
1
Katherine
2
Average
2.2
Disengaged Maureen
1
Amy
2
Elaine
1
Average
1.3
* These were categorized as follows: 1= Participants reported significant problems either due to the math content or the collaborative experience, which detracted from their sense of confidence. 2= Participants reported the group experience was something that they needed to adjust to, and came to enjoy and feel confident in. 3= Participants reported the group experience as a positive, confidence-building one right from the beginning.
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31
