Developing Epistemic Agency as Pre-service Elementary ...home.oise.utoronto.ca/~cbrett/personal/AERADivIAgency02.pdf · importance of shared discourse as a way to make meaning has

Brett, C. Session 6.10 AERA 2002 03/27/02

1

Developing Epistemic Agency as Pre-service ElementaryMathematics Teachers Supported through Online

Community

Clare Brett, OISE/UT

In Session 6.10, Division IThe value and validity of knowledge building pedagogy and

technology to foster professional development

Paper presented at the American Educational Research Association, NewOrleans, April 2002


2

1Developing Epistemic Agency as Pre-service Elementary Mathematics Teachers:supported through online community.

Clare Brett, OISE/UT

1. ObjectivesOne significant challenge of pre-service elementary education in mathematics is toenable participants with diverse academic backgrounds and experience to acquire theknowledge and pedagogical skill to teach in ways that may either diverge widely fromtheir 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, selectedfrom a cohort of 57 enrolled in a two year certification course at OISE/UT, developidentity as mathematics teachers and learners that they could continue in the field. Thespecific goal of the study was to facilitate the creation of a community that includedthese math anxious pre-service teachers by supporting them in productive engagementthat mapped directly onto three core dimensions of math teaching: 1) Content. Advanceof their mathematics knowledge of three, specific, fundamental concepts relevant toelementary math teaching: (a) place value and renaming, (b) operations with numbers,and (c) patterns/rule-finding; 2) Discourse. Engage in a discourse communitythemselves as part of their own math learning and teaching, and through theseexperiences gain greater understanding of the role of discourse in enriching mathunderstanding; and 3) Pedagogy. Acquire a conceptually-based rather than analgorithmic approach to math pedagogy.

To facilitate these goals, an online environment was used, in addition to small groupdiscussion and open-ended mathematics investigations. The goal of the onlinecommunity was to provide extended, multi-year, time and place independent access toteacher and peer networks of support.

Recent development in pedagogy and conceptions of mathematics on reasoning and theimportance of shared discourse as a way to make meaning has shifted the emphasisaway from traditional algorithmic approaches of mathematics instruction (e.g. Lampert,Rittenhouse & Crumbaugh, 1995). Communications technology offers a potentialsupport for this approach because of the significant role that discourse and reflectionplay in text based discussion environments. Research on collaborative learningenvironments suggests that in certain contexts, such electronic supports can increase thedepth of student learning (Scardamalia & Bereiter, 1995), and evidence from a study ofundergraduates using structured online discussions as an adjunct support to a face-to-face course suggests the electronic component encouraged discussion of issues andfostered interactivity among participants (Hara, Bonk & Angeli, 1998). However,similar research has not been carried out with a group of pre-service teachers with suchprior negative experiences in the domain.

1 Paper accepted for presentation at the American Educational Research Association, New Orleans, April2002.


3

Additionally, research is clear that math anxiety is widespread among both in-serviceand pre-service elementary teachers (Kelly and Tomhave, 1985; Hembree, 1990; Sloan,Vinson, Haynes & Gresham, 1997), and that math anxiety leads to math avoidance,subsequently affecting levels of performance (Trice & Ogden, 1986/7). Identifyingoneself as a math learner is a critical step in developing a positive perspective on mathlearning and teaching. To support this process Year 1 math content work was focusedon a significant subset of concepts, foundational to the elementary math curriculum,and framed within a supportive, discourse-rich learning environment, using smalllearning groups and an online discussion environment available all the time throughoutboth years.

2. Theoretic Framework

This study combines sociocultural and cognitive perspectives to understand thedifferent elements that either facilitated or restricted participants’ engagement as activemembers in the online mathematics environment. The role of epistemic agency(Scardamalia, 2000) is explored as being one of two critical components ofengagement. Scardamalia defines epistemic agency as taking responsibility for personalunderstanding demonstrated through iterative cycles of revising internal and externalideas to a resolution. Developing epistemic agency includes taking responsibility forlearning elements such as goals, motivation and evaluation of understanding that areoften orchestrated by teachers in traditional academic settings (Scardamalia, in press).In this study reported here, epistemic agency may be demonstrated through participant'sefforts to deepen understanding and to work with mathematical ideas towardsdeveloping a conceptually based approach to math knowledge, discourse and pedagogy.However, this particular group of learners may be hampered in developing epistemicagency in respect to mathematics learning because their prior experience in the area hasbeen traditional (and therefore not something that they have had experience in"directing") and that have not left them with feelings of competency, leading to theircurrent levels of anxiety and concern about teaching and learning mathematics.

In Wenger’s (1998) communities of practice model, identity is a critical component ofcommunity membership. Identity in this model is viewed as not a state, but asexperience developed through activity and involvement. As such he suggests that alearning community will become a place of identity if it offers members a place toincorporate their histories (that is their past experience) and provides an experience thatmakes community engagement a significant element in a personal future. Potentially, ashared electronic context would provide a forum for engaging in a community ofpedagogy of mathematics through legitimate peripheral participation (Lave & Wenger,1991). As Barab and Duffy (2000) explain the role of the community in educationalcontexts: “The goal of participation in a community is to develop a sense of self inrelation to society—a society outside of the classroom” (p43). Evidence of communityidentity is demonstrated through participants' reactions to and commentary on thecommunity-focussed elements of the program, and in their attitude shift towardsmathematics.


4

In this paper, epistemic agency and community identity are conceptualized as mutuallyconstituting engagement in community discourse. Epistemic agency represents thefocus on idea development, and identity the connection to the community. In effectidentity (consisting of connectedness in the community, and a basic level of perceivedcompetence in the context) may enable the conditions for epistemic agency to flourish.Through the process of directing learning (epistemic agency), participants activelyengage in community discourse, and through the engagement and feedback, in turnincrease their sense of identity. It is the unpacking of the elements in these twoconcepts of agency and identity that will be addressed in this paper.

3. Methods

This research involves a two-year exploratory study that draws upon a number of datasources to better understand the patterns of engagement and disengagement in thedevelopment of a mathematical community of twenty preservice teachers enrolled in atwo-year pilot program. Such an exploratory structure allows the study of teaching andlearning in real world environments, with all the complexity that such contexts entail.Temporal analyses of database activity, categorical ratings of portfolio and databasecontent as well as of interview responses were used to support interpretation. Some ofthe analyses take a sociocultural perspective, where the focus of understanding is onparticipation itself and learning is evidenced as contributions to the practice of one’scommunities. Other analyses come from a more psychological perspective, includingattitudes and beliefs about mathematics and about participant’s views of learning,ability and teaching. Together, these perspectives allow a more completeunderstanding of the issues affecting engagement in online community becausecultural, interpersonal and individual factors can be considered together. In Dewey'swords, "the psychological and the social sides are organically related, and thateducation cannot be regarded as a compromise between the two, or the superimpositionof one upon the other" (p444, in McDermott, 1973).

3.1 The program:This study was conducted during the course of the four-term preservice teachereducation program.

Term 1 Based on whether they had indicated mathematics was a subject they felt anxiousabout, the cohort of fifty-seven preservice teachers were assigned to eleven smallgroups of 4-6 members for small group math investigations. Two of these groupsconsisted only of mathematically anxious females. The other nine groups wereheterogeneous in terms of their members’ perceptions about their abilities to do andteach mathematics. From the eleven workshop groups, a sample of four workshopgroups (the two mathematically anxious and two heterogeneous workshop groups)formed the twenty participants in the focus group for this study. The whole cohort thenparticipated in a series of 8 two-hour workshops conducted during the fall term in 1995during which they carried out a number of open-ended mathematical investigations.


5

The goal of these investigations was to overcome many of the preservice teachers’misconceptions about the nature of mathematics, to make mathematics accessible bydemystifying its subject-matter and to provide situations where mathematical discoursewould arise naturally. Whilst the mathematical investigation workshops were beingconducted, the preservice teachers were given computer accounts and access to anumber of computer laboratories on the OISE/University of Toronto network. Theyalso were given training sessions and support in using this technology. To provideongoing support for reflecting upon and extending the small group class discussions,students were given access to an electronic conference (a First Class conferencingsystem) called Math Inquiry, where they could ask questions, share ideas with peersand faculty, and discuss practicum experiences and share resources.

First Class conferencing allows messages to be viewed as threads, and messages can benew or replies, or replies to replies. The title of the message and author information isvisible 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 ofcomputer access—especially home access—considerable attention was given toproviding 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 inthe database was not part of their formal evaluation but they were expected toparticipate in the same way they were required to attend class The Math Inquiryconferences yielded about 550 notes (varying from a few lines to multiple pages) overthe 2 years (they also had opportunity to contribute to as many as 24 other programrelated conferences on the Tednet system).

Term 2Math activities included a sequence of tutorial/workshops (6 two-hour sessions) inwhich they investigated how the Jasper Woodbury videodisc and support materials(Learning and Technology Center at Vanderbilt University, 1996) could be utilized toestablish and maintain communities of mathematics practice in elementary schoolclassrooms. As in Term 1, the issues and topics discussed in the shared electronicdatabase conferences were in the main decided by the preservice teachers.

Term 3Preservice teachers in the primary/junior division who had chosen the Math/Scienceand Technology option, attended a sequence of formal mathematics educationlecture/tutorials for approximately another 16 hours, but divided among math, scienceand technology issues. A major component of this term was a six-week blockclassroom placement. Thus, the major focus of their electronic database conferencesduring this term was on issues related to preparation for their block classroomplacement and/or issues and topics brought up during the formal mathematics educationtutorials. Nine out of the thirteen participants in the Primary/Junior/Intermediate optionfrom the Focus group chose the Math/Science/Technology specialization. Anotherseven of the focus group of twenty were part of the Junior/Intermediate division andtherefore were required to take as their teaching subject their background degree


6

specialization, and therefore were not eligible to select MST as a specialization. Thesewere mostly English/Language Arts majors.

Term 4Those in the M/S/T option had approximately 8-10 hours of formal Math Science andTechnology teaching, and the main emphasis at this point in the program was onscience and technology. The other members of the cohort had no formal mathinstruction. All the cohort were encouraged to continue their reading and writtencontributions to the other mathematical conferences on the shared electronic database.

4. Data Sources

For this paper, we will focus on how database content and activity, with interpretationssupported by evidence from portfolios and interview ratings can inform howparticipants engaged with issues of math content, discourse and pedagogy.

Type of data Time of collection: Year 1 Year 2Database notes:reading & writingpatterns

Math inquiry conference:ongoing

Math inquiry plus other programconferences: ongoing

Portfolios Ongoing throughoutprogram

Ongoing throughout program

Interviews End of year 1 End of year 2Math content test 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 electronicdatabase conferences were recorded and frequencies tabulated of notes written andnotes read in the math inquiry as well as the other conferences. The content of thenotes 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.Two raters rated fifty percent of the notes, with an interrater reliability of r = .90, anddifferences were resolved.

4.2 Portfolios:

In their portfolios, the preservice teachers were asked to record significant learningepisodes of personal learning occurring during their programme. In each entry to theportfolio, the preservice teachers used the following questions to guide their reflection:


7

(a) What is this entry about? (b) Why did you choose this as an entry? (c) What did youlearn or how did you grow? Each entry was approximately 1-3 pages in length withsupporting documentation such as an article or lesson plan. The 20 participants in thefocus group were asked near the end of the program for permission to photocopy thesebodies of work for analyses. They all agreed to do so.The math-related portfolio entries were separated out from the others for more detailedanalysis. Next, the portfolio entries for each member of the focus group that wererelevant to mathematics were assigned to one of three categories (described above) bytwo independent raters. Each entry was further rated, within each category, on a scalefrom 1-3 as to the degree of reflection on issues of understanding (either for theparticipants/teacher or the future students) evident in the entry. 41% of the portfolioswere 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 tothe program. Of particular interest in this paper were questions relating to reactions totheir 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? 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 communityidentity and their subsequent online engagement. The second question touched oncultural 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 Baturoand Nason (1995). It was focussed on three foundational areas that would be addressedin the content of the small group math investigation problems; place value andrenaming, operations with numbers and patterns/rule-finding, The test was administeredearly in the first year of the program and then again with the participants who agreed toso, during their second year of teaching in the field. The math educator who developedit scored the test.

5. Results

Engagement was assessed through a number of measures. First, we look at depth ofunderstanding as demonstrated through the ratings of the portfolio data. Beyond theactual content analyses of the portfolios and database notes these data show the depthof engagement of the various participants in the focus group. These results will becompared to data in the second section that describes the level of activity for all theparticipants in the shared electronic database. These data will show the range of


8

participation activity for all the members of the focus group and provide anothervantage 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 adistinguishing feature of Knowledge Building Communities (e.g. Scardamalia &Bereiter, 1994) and Communities of Learners (Brown & Campione, 1994) is their focuson increasing depth of understanding. To be effective in accomplishing the three goalsof the program (improving content, math discourse and pedagogy), participants in thecurrent study would have to demonstrate some depth of understanding of the threedimensions. Additionally, I would claim that deep understanding in this contextdemonstrates a significant form of engagement, specifically that it indicates a level ofepistemic agency (taking charge of one’s own learning). This is because portfolios wereself-selected so the math focus itself reflects efforts to expand learning in this area. Aswell, reflective elaboration of ideas is a significant aspect of taking agency over one'sown learning. Accordingly, we will look first at the Portfolio results for the threeprogram goals. These portfolios, were self-selected examples of learning episodes, andas such provide a measure of participants’ perceptions of their “high points” inlearning. Portfolios also provide a content measure completed by everyone and soprovide information on all participants independent of technology-related effects. First,the average portfolio ratings ranked from high to low, for each of the three dimensionsof math knowledge, discourse and pedagogy are shown in Table 1 for each participant,identified by a pseudonym.

The portfolios were rated for the knowledge dimension on a scale from a low of 1 (anunelaborated mention), to a high of 4 (reflective elaboration of important concepts). Toarrive at a single "depth" rating for each participant the combined ratings (from the tworaters) were averaged over the number of math portfolios generated for both years. InTable 1, looking at the higher levels of depth scores, we see 9 people scoring 3 orhigher for the Discourse and Pedagogy measures, but only 6 scored 3 or higher on theMath Content measure. To score high on the math knowledge depth measureparticipants had to be dealing with significant conceptual issues relating tomathematical 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. Thesame 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. Theparticipants in the middle however, do show higher scores on one or two measures ifnot all three, suggesting that they are engaging deeply with ideas, but not perhaps asconsistently as the group of six identified above.

The following examples of portfolio entries with ratings of 3 or higher highlight thereflective nature of the responses by these participants. The first entry is based on acombination of her own learning, reflecting on prior experience and recent professional


9

reading. She is also focused on having students develop their own reflectiveunderstanding:

I decided to explore the use of math journals as a tool for improvingreading skills in math…A large part of my decision to introduce mathjournals came from my personal experiences with math…I wouldbecome very frustrated when I was never given enough time to recordmy rationale or strategy. I had also done some professional readingabout the topic..and am convinced that math journals are an essentialpart of the math curriculum. I would argue that writing in mathforces students to “look inside their own minds” and examine theirown understanding of the concept. (Marissa, 11/96).

This next example also connects early experience with current understanding and thentakes it further in applying the solution to a new context.

One of the most powerful experiences I had of how explaining amathematical solution can lead to deeper understanding came when Iwas doing the math test…Gnomons were described in the problem asl-shaped numbers…and odd numbers. We had to generate a rule foradding consecutive gnomon numbers and justify the rule…I could seethe pattern that was developing—when you added consecutivegnomon numbers (beginning with 1) the total would be the square ofthe number of gnomons you were adding…I was feeling prettypleased with myself for discovering the pattern and I decided to showthe problem to my 15 year old son. He could see that pattern by heimmediately asked why. It was when I started to explain to him that Irealised I needed to take into account that these gnomon numberswere not just odd numbers but were L-shaped. When you added theL-shapes together you actually created squares. If he hadn’t askedwhy, and I hadn’t attempted to answer it, I would never havediscovered the relationship. (Wendy, 10/96).

The summary able (Table 2) confirms the same 6 participants scored above an averageof 3 across all areas suggesting that these participants were consistently engaged indepth of understanding of these ideas, across all three areas, at least in the portfolioentries.Next we will look at database activity and see if the portfolio patterns for theseindividuals 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 participantsin the database. Activity in the shared database was analysed in two ways. The firstwas a frequency summary of database participation, in which (a) the number of entries


10

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 greaterparticipation both in terms of number of entries to the database and the proportion ofthe database read. For example, the proportion of the computer-mediated mathconference read by the focus group (as compared to the whole cohort of 57) increasedfrom 29.9% in Year 1 to 40.5% in Year 2. However, there was much variance amongindividuals in the focus group. If we order the participants according to how much theyparticipated through reading and writing in the database, we can see four approximatesub-groupings summarized in this table and defined below:

Group N Participation Patterns

Engaged 5 Higher levels of writing and reading in bothyears

Emergent 6 Reading and writing increase in Yr. 2

Withdrawing 6 Reading and Writing decrease in Yr. 2

Disengaged 3 Low reading and writing rates in both years.

• 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 maintainedinvolvement over the course of the four terms.

• Emergent (n=6) This sub-group changed most, particularly in relation to the numberof the database conferences they read. Their levels of reading started out low butincreased steadily over the course of the four terms (>20% increase between Year 1and Year 2). Written contributions to the math inquiry conference, however, werestill 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 mathinquiry conference decreased (or remained very low and unchanged) over thecourse of the four terms ( a drop of < or = 1) Most of their participation wasconfined to the reading the contributions of others, although this too decreasedbetween 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 inquiryconference 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 ratesfor individuals within groups), we find that in the Engaged group, four of theparticipants receiving the highest rankings in portfolio ratings, Judy, Wendy, Nora andEileen are members. Sarah’s online participation patterns locate her in the Emergentgroup and Marissa in the Withdrawing group. This pattern suggests that a core groupwere able to develop a sense of community identity in the online environment sufficientto 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 theMath Inquiry conference and across conferences on other topics, and see how thedatabase is valued as a shared discourse environment. In Table 4 we can see both in the


11

Math Inquiry conference and across other conferences, the patterns of reading in eachof the four groups. The shaded cells indicate an increase between years for theproportion read of other people’s math notes, and the clear cells indicate an increasebetween 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 theEngaged and Emergent groups, and in conferences other than Math, for theWithdrawing and the Disengaged group. This suggests that overall, the databasecommunity 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, ifnot for the math inquiry conference itself. However, the math conference was usedmost by the Engaged and Emergent group members.

The role of knowledge by itself as measured by the incoming math test does not explainthe level of activity as measured by this engagement metric. One might argue thatpeople who are more knowledgeable would feel comfortable in contributing to thedatabase. However, the correlation between the number of contributions written in themath 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 knowledgelevel are critical for database involvement.

Another view of database participation was a summary temporal analysis by monthacross the two years indicating a) when each participant contributed notes to thedatabase; 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 groupof 20, from the Engaged, Emergent and Withdrawing subgroups contributed to all fourdiscussions, supporting the idea that even math anxious members found some rolewithin this emerging community. Second, contributions from the focus group to thediscussions occurred during both class and during practicum time, and increased in thesecond year, suggesting that the database was seen as a community support even whenstudents were in geographically distant locations during their practica. Third, thetemporal representation of these data indicated that contributions were distributed mostwidely among the focus group participants during December of Year 1. This timingcoincides with a number of programme events. First, by this time everyone who wasgoing to use a computer consistently was doing so. Second this was the most intenseperiod of the small group math investigation series, and therefore database activity wasclosely supported by temporally contiguous classroom activity. For a number ofparticipants, particularly those in the Withdrawing group, this close connection wasimportant. Without that support, their online participation dropped off considerably inthe second year. These second year discussions also drew more widely from peopleacross the programme community, including faculty from other parts of the program,and associate teachers. In such extended contexts, some may have felt less confident inbeing able to contribute substantively and so shifted their attention to other, non-mathrelated online conferences, as suggested by earlier analyses of the reading data patternscomparing the math inquiry participation to other online conferences (Brett, Woodruff& Nason 1999). For participants in the Emergent group however, the generally positiveexperiences in the small group math investigations seemed to allow them to increase


12

their online activity during the second year, although the actual level of mathematicalanalysis undertaken by participants in this group did not consistently reach the level ofthe Engaged group members, as shown by lower reflectivity ratings of portfolio anddatabase entries.

5.3 Epistemic Agency: Working with ideas

The content of the database discourse was categorized to examine commentary onmathematics content, discourse and pedagogy. Many of these comments demonstrateshifts in epistemic agency, as we see the participants setting their own agendas forlearning 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 theyworked with substantive mathematical ideas. In Table 5, the amounts of content relatedcomments from each year are tabulated. It is clear that the largest amount ofcommentary came from the Engaged group participants, followed by the Emergentgroup. The database comments about views of mathematics can be summarized asfalling into three main categories: 1) the significance of contextual assumptions inproblem-solving, 2) mathematical reasoning extending beyond algorithms and 3) mathas a way of thinking. Often using the problems discussed in the small group mathinvestigations, participants noted shifts in their own thinking. In the first example is areflection on how assumptions in a problem affect the answer:

We could have spent hours examining different assumptions that lead us todifferent answers but the most valuable lesson for me was learning thatevery answer you get is based on a series of assumptions (some more validthan others). These assumptions need to be explored before the answer canbe fully understood. So now I know that there is more to an answer thanmeets the eye! (Megan, Emergent).

Another shift was starting to see math knowledge as based in understanding andreasoning that was constructed rather than transmitted, a view of learning that wasbeing conveyed throughout their program.

I think it’s of vital importance that every learner discover THEIR ownpersonal pattern or else it’s meaningless. My father-in-law has told memany times how he remembers his phone number but I can’t for the life ofme 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 theoperations, the theorems the teacher presented but I didn’t understand thebasic principles and couldn’t make them mine.

How does math taste? It’s becoming more palatable by the minute. Foryears 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 wasthat mathematics is not just a tool but a way of thinking. This viewpoint was only


13

found amongst five of the focus group participants. Wendy perhaps most effectivelyarticulated it when she wrote:

Mathematics is not only just a useful tool that helps us make change or buyhamburger buns or whatever. It is a way of thinking and a way oforganizing thought. It is a way of interpreting and explaining what we seeand do. The only way we can begin to really understand Math is by doingit. We really don’t begin to understand numeracy unless we “handle”numbers - play with them - move them around. You can do neat things withcalculators - but you only really understand how they neat they are if youunderstand what’s behind it. (Wendy, Engaged)

5.3.1.b Mathematical ideas.

A sequence of notes generated during the period of one of the small groupinvestigations illustrates how participants engaged in principled mathematicalunderstanding. The preservice teachers were attempting to justify and/or prove theirsolutions to the investigation’s set of problems. Understanding the proofs of thesolutions to these problems required principled understanding (Leinhardt, 1988) ofplace value.

When engaged in this process one participant went to great pains to lay out her thinkingas clearly as she could in words—taking on the challenge of communicating hermathematical thinking.

What helped me understand this proof was when I realized that in usingmultiple representations of numbers you don’t always have to describe thenumber in a fixed order of hundreds, tens, and ones. That you can say inthis case 3 hundreds, 9 tens, and 5 ones OR 5 ones, 9 tens, 3 hundreds orpossibly 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 wouldwant to get across not just the notion that you can say 85 is 85 or 8 tens and5 ones or 6 tens and 25 ones but also 5 ones and 8 tens etc. That when youdescribe a number this way, the actual order of the numerals does affect thevalue 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 upin different ways, when you are describing it as hundreds, tens and ones it’sstill a leap to actually changing the “order” around, i.e., putting the onesfirst instead of last. I don’t know if this will help anyone else. It seems tome using multiple representations of numbers actually makes them moreflexible or manipulable. (Wendy, Engaged)

Another participant was helped to understand the same idea, from Wendy's descriptionof her own thinking process:

Wendy, you helped me understand why we could suddenly turn the numberbackwards, once grouped, and it would still be the same number. Of courseit would!!! How did I ever get to high school math without realizing this?(Eileen, Engaged)


14

In a collaborative response to the first note above, another two participants supportedthe 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 3hundreds 9 tens and 5 ones. If you ask people after the test (referring to themath pretest), most got the question on place values wrong as a result ofthis. This helps me and I think it will help students as well. However, theyfirst have to understand that they have to make and can make other groupsfrom the same number and transfer it.

This will help kids understand the concept and value of money. It’s alwaysthe same amount but shown differently. Try re-arranging picture patternsin order for students to understand regrouping, while the number or picturestays the same. (Anna, Withdrawing)

Two participants in response, in another collaborative note, elaborated on why they feltWendy’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 learnedin class to your own teaching situations. We (Elaine and I) would havenever thought of showing the students that they can rearrange the numbersand they will still remain the same. It seems like common sense, but it issomething we would probably never have thought of. (Elaine, Disengagedand Janice (a non-focus group participant)

As one reads these notes, it can be seen that the first participant’s entry2 is a reflectiveanalysis of her understanding, what had helped her to gain that understanding and howthat shift in understanding could be integrated into teaching. Furthermore, her noteindicates that a serious effort was being made to understand some of the underlyingprinciples of the place value numeration system. The knowledge-building intent of herparticipation in this discourse thus was apparent. The second participant’s responseextends the discourse by giving support and extending the idea to another context(using money) However, this participant’s response does not add anything to thediscourse about the characteristics and properties of the numeration system. The finalparticipants’ entry is a reflection about their own learning and also a supportivecomment. It adds socially to the discourse, but does not elaborate the ideassubstantively. The contributions of the second and third participants therefore did notreally contribute much towards the further development of the focus group’s collectiveprincipled understanding of the numeration system.

Thus, although all of the responses demonstrated engagement, they varied as to whetherthey were focused on both the mathematics and the social elements, in a principledway, or were simply reflective of their own learning. That difference would be adefining feature of knowledge building, apart from simply collaborative discourse(Scardamalia and Bereiter, 1994). This example is representative of the level ofknowledge building discourse in the database across topics. Certain people were ableto engage at this level, showing epistemic agency in pursuing understanding and

This entry was part of an ongoing attempt to write up a proof of how the individual digits of multiples of9 always add up to 9


15

opening their reasoning up to the community, but it was only a small group of peoplewho were able to do this at the level of knowledge-building, and most of these peoplewere in the Engaged group. The others certainly made progress--they became aware oftheir thinking processes, advanced in their understanding, but did not necessarilycontribute 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 andSharing of classroom experiences) was distributed among the participants. TheEngaged and Emergent group members expressed their sense of the value of discourseand working together as a community. However, only the Engaged group consistentlymade metacomments on their own learning through the database notes (Table 6, notesfalling in the 2B Metacommentary category in Year 1). Encouragingly, the Engaged,Emergent and Withdrawing group members expressed an understanding of theimportance of using discourse and community to assist learning. Finally, the Emergentgroup did the most unelaborated sharing of classroom experiences. In this way theywere using the database as a social or communicative space, but taking a less analyticalor reflective stance on their activities.

5.3.3. Pedagogy:

In Table 9 are the categories related to pedagogy in the database notes. The Emergentgroup members along with the Engaged group, share rationales and ideas forinstruction, take the learners' understanding into account, reflect on how their ownlearning development will impact their pedagogy, and consider different issues thatimpact pedagogy, such as motivation. In Year 1, we also see a slightly less, but stillsimilar pattern of activity for the Withdrawing group, except that they reflect less on thedifferent issues that impact pedagogy.

Overall we see the greatest distribution of comments across participants in all groups inthe 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 theengaged participants and half of each of the emergent and withdrawing participantsmade direct connections between their own learning experiences and its impact on theirteaching. This finding is encouraging in that it suggests participants are broadly tryingto reframe their approach to math pedagogy.

“I always had a math phobia..and I do not want to pass that on to my ownstudents. After having witnessed and participated in…activities at thefaculty that directly related to math, I found myself interested in thestrategies and wanting to apply them in my own classroom”. (Nora,Engaged)

There was reflection in the database and portfolios about applying pedagogicalstrategies learned in other aspects of the program to the mathematics context (such asusing journals in mathematics class to encourage reflection and articulation of ideas).


16

In addition, they commented on how activity centres and collaborative work could beapplied in a mathematics context.

I think the difficulty is not with the fact that it is a real life problem but thepre-existing knowledge they should have. I consider the fact that as a classat FEUT, we all had some basic knowledge of the fundamentals of math andit still took us all day and we still had problems. This is with pre-existingknowledge. (Jessica, Engaged. 9/4/96)

Further they are demonstrating a level of epistemic agency, as shown in the exampleabove, in actively debating the potential value of different teaching strategies thatwould avoid recreating the problems they had experienced during their own schooling.

In addition, there were shifts towards a much more positive view of mathematicspedagogy. Specifically, in the interview data at the end of Year 1 about how theprogram had impacted on their teaching experiences, fifteen of the participantsmentioned incorporating collaborative groups and discourse to their math pedagogy.One indicator of their increased positive attitudes towards math generally was that 8 outof the 13 participants in the Primary/Junior division (who were the only ones who couldchoose a specialization other than their degree program) chose Math, Science andTechnology in Year 2. They explicitly stated that they now had the confidence totackle these areas in which they felt they had less prior experience and incomingconfidence.

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 inthere. Inviting people to respond to your ideas suggests that you are comfortable in thatcontext and prepared to interact. Encouragement to other participants to join in andshown in statements such as. “.please ask me questions and perhaps it will help me finda better way to explain it.” This invitation suggests that other’s feedback will help boththe asker of the question, who will hopefully get their question answered, and theoriginator of the explanation. Such a focus could encourage cycles of interaction inwhich questions do not simply get taken up and responded to, but worked withiteratively. Statements with a similar stance were found frequently throughout thedatabase. 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 specificlessons:

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 themath take up was pretty much a disaster! Any comments? I know this is


17

like dumping stuff on you, but I really need some guidance. Thanks, R (aninstructor) (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 asWrite back if you’ve had a similar experience (Marissa, Withdrawing4/12/95)

Others gave a more indirect request for people’s feedback such as in the next example:Well I think I have written enough to be going on with. I look forward todiscussing 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 theiranswers 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 responsesto interview questions about the usefulness of the database and the origins of theirmathematical anxiety, give a perspective on what most influenced the likelihood ofparticipant's engagement in the database.

6.1 Reactions to the database

Responses to interview questions about database usefulness suggest that participantsare more likely to make written contributions to a conference to which they feel able tosubstantively contribute. However, reading substantial portions of the database in atimely manner (as participants in the Engaged group (in both years) and Emergentgroup (in year 2) and the Withdrawing group (in year 1) did suggests that the contentwas 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 thatmight be a logical prediction from such information. Instead, feelings of vulnerabilityand lack of confidence may have prevented people from asking so publicly forassistance. It was noticeable that the people who either asked for help or who openlydiscussed problems were predominantly Engaged and Emergent group participants,suggesting that they had a more secure sense of their identity in this online context.


18

6.2 Attributions of causes of math anxiety.

Another important factor seemed to be how they attributed the origins of theirmathematical anxiety. In interviews, Disengaged participants blamed themselves forbeing unable to understand the mathematics they were taught in school. They also didnot report any successful experiences with math at any time in their own schooling. Bycontrast, Engaged participants gave clear and detailed accounts of the external factorssuch as teachers’ attitudes and specific experiences that had caused them to feelinadequate mathematically. They all appeared to have reflected on this in interviewsand in the database, analyzing the causes rather than simply internalizing the negativeexperiences. Withdrawing and Emergent participants tended to give accounts thatmixed internal and external factors. The categories and results are shown in Table 8.One interpretation is that those who are self-blaming, also tend to have views of mathlearning based on innate ability. With such a theory of learning, it is hard to develop asense of identity as a potentially successful math learners, engaging in a discoursecommunity about mathematics. While many of the participants did comment on theusefulness of discourse and reflection in math pedagogy, there was still a reluctance todo this publicly themselves.

6.3 Small Group Math Investigations.

Interview data also revealed that the face to face mathematical investigations conductedin Semester 1 played a critical role in the establishment of an identity within amathematics education community of practice for the Engaged, Emergent andWithdrawing participants. These interactions provided a structure for their mathdiscourse and mediated the development of a support network that encouragedparticipants to share their ideas and concerns. Their responses were categorized and thecategories and results are shown in Table 9. While there is variability among theEmergent and Withdrawing groups, there is consistency in the positive reactions amongthe Engaged group, and consistency in the mixed reactions among the Disengagedgroup. Only the Engaged group reported consistently positive experiences from thebeginning, and there were most reports of problems (scores of 1) in the Withdrawingand Disengaged groups. Even so, no one reported entirely negative experiences. Themain problem reported by those who found the experience less satisfying was feelingdiffident about their own ability to contribute, although they found the situation a veryuseful one for their own learning. This diffidence was also reported as a reason for notmaking written online contributions.

In sum, it appears that while for the Engaged and to a lesser extent, the Emergent groupmembers, the online environment supported the emergence of epistemic agency aroundteaching and learning mathematics, as well as supporting an emerging identity asmembers of a mathematical community. The pervasiveness of these effects werelimited in the Withdrawing and Disengaged group members by self-blame for priornegative mathematics experiences, and technical fears that reduced the effectiveness ofthe online community for these participants. A separate indicator of the overallincreased confidence was that 9 out of the 13 participants in the Primary/Junior division


19

(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 tacklethese areas that they had previously avoided.

7. Scientific Importance

This study offers insights on the role of electronic environments for supportingparticipants’ emerging identities as mathematics teachers. First, online environmentscan provide an opportunity for some pre-service teachers to develop their ideas in aframework of social discourse, and thereby increasing their epistemic agency as mathlearners and teachers. Second, there is considerable variability in how people benefitfrom such contexts. Even the technological developments (simpler interfaces; morereliable connectivity) since these data were collected would likely have reduced thetechnological frustration expressed by the Disengaged and some Withdrawingparticipants. However, while this may have increased overall participation it may nothave changed the level of written contributions made by these participants. Theiractivity appeared determined more by factors outside the electronic context. Theseincluded internal attributions for the origins of their math anxiety, which suggests thatthey see math ability as an attribute, and not something open to change throughlearning. Such a belief may have reduced their confidence in answering peersquestions or substantively contributing in other ways to the discussion. These factorstogether seemed to reduce a sense of community identity among these participants, andthus they seemed not to demonstrate the same level of agency as math learners asmembers of the Engaged and Emergent groups.

In conclusion, participants in the Engaged Emergent and Withdrawing groups found theshared database socially supportive, and a source of assistance and ideas, but it wasmainly the Engaged participants who used it as a context for the building of knowledgeabout mathematics and mathematics education. The small group math investigationsseem to provide these participants with the levels of mathematical confidence necessaryto “jump-start” them into wanting to become involved deeply in discourse aboutmathematical ideas as well as the teaching of mathematics and self-reflection, at whichpoint the electronic environment became a useful tool. Members of the other groupsmay have benefited from an extended series of the small group investigationworkshops. This would have provided greater opportunity to become immersed in theconstruction of mathematical content knowledge and gain a stronger identity asdeveloping mathematicians. In turn, this may have made the reflective advantages ofthe online environment a reality for a broader group of participants.

Despite the potential for democratizing the classroom, using online environments tocreate a culture of discourse in which engagement is widespread fundamentally requiresthat participants have both agency and identity within that community. Identitysupports the development of epistemic agency that allows them to direct their learningand thereby engage in the community knowledge building discourse. In the data fromthis study we see community identity in data such as the confidence they felt during thesmall group discussions; the language of collaboration and inclusion in the database; in


20

whether they defined the origins of their mathematical anxiety as stemming fromfactors outside (such as the teacher or teaching practices) rather than their own internal“deficiency”. Through community identity we see ourselves in the discourse aslegitimate participants, and as contributors to the discourse we are able to shape andthen share in the community discourse. This was the case most strongly with theengaged group who was involved in the communal discourse from the beginning. Thisextended participation may have enabled the development of a history and hence(Barab & Duffy, 2000) would argue, these members were able to develop a sense ofself in that community.

Epistemic agency (taking charge of their own learning) is demonstrated through thelevel of reflective elaboration in Math Portfolios across the three dimensions; in the useof metacognitive comments in the database about understanding and insights; and inarticulating math knowledge, concepts and strategies in the database discourse.

To the extent that we identify ourselves as deficient in the ability to develop relevantknowledge, we experience less identity, engage less and thereby fail to increaseepistemic agency through increasing our own level of knowledge and control over ourlearning. Such a conclusion has implications for all community environments in thatnovices are not necessarily drawn in as apprentices to central participation over time,unless they can develop a sense of identity based on their own perceived value andpower in that community.


21

References

Barab, S. & Duffy, T. M. (2000) From practice fields to communities ofpractice. In Jonassen, D. and Land, S. (Eds.) Theoretical Foundations of LearningEnvironments.Lawrence Erlbaum Associates. NJ.

Brett, C. Woodruff, E. & Nason, R. (1999) Online Community: What canreading and writing patterns tell us about participation? Paper presented at the AnnualMeeting of the American Educational Research Association, Montreal.

Brown, A. L. & Campione, J. (1994). Guided discovery in a community oflearners. 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 inan 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 avoidancein 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, Teachingand Schooling (pp. 731-764). Oxford: Basil Blackwell

Lave, J. & Wenger, E. (1991). Situated learning: Legitimate peripheralparticipation. Cambridge: Cambridge University Press.

Scardamalia, M. & Bereiter, C. (1995). Adaptation and understanding: A case fornew cultures of schooling. In S. Vosniadou, E. De Corte, R. Glaser, & H. Mandel (Eds.),International perspectives on the psychological foundations of technology-basedlearning environments (pp. 149-163). Mahwah, NJ: Lawrence Erlbaum Associates.

Scardamalia, M., & Bereiter, C. (1994). Computer support for knowledgebuilding communities. The Journal of the Learning Sciences, Vol.3, 265-283.

Scardamalia, M., & Bereiter, C. (in press). Knowledge Building. In Encyclopediaof Education, Second Edition. New York: Macmillan Reference, USA.

Scardamalia, M. (in press) Collective cognitive responsibility. In B. Jones (Ed.)Liberal Education in the Knowledge Age. Chicago. Open Court.


22

Sloan, T. R., Vinson, B., Haynes, J. & Gresham, R. (1997). A comparison of pre-and post- levels of mathematics anxiety among preservice teacher candidates enrolled ina mathematical methods course. Paper presented November 12-14, 1997 at the annualmeeting of the Mid South Educational Research Association in Memphis, TN.

Trice, A.D. & Ogden, E.D. (1986/7). Correlates of mathematics anxiety in firstyear elementary school teachers. Educational Research Quarterly, 11 (3) 3-4.

Wenger, E. (1998). Communities of practice: Learning, meaning and identity.Cambridge University Press.


23

Table 1 Average ratings of Portfolios according to level of reflective elaboration for each of threedimensions of Math content, discourse and pedagogy.

Average Portfolio Ratings Ranked for each of 3 Dimensions

Math Discourse Math

Name Content Name & Community Name Pedagogy

Judy 4 Judy 4 Judy 4

Nora 4 Nora 4 Nora 3.8

Marissa 3.7 Sarah 3.75 Sarah 3.75

Wendy 3.6 Marissa 3.7 Marissa 3.7

Eileen 3.2 Eileen 3.6 Wendy 3.6

Sarah 3 Wendy 3.4 Eileen 3.6

Alicia 2.7 Nadine 3.3 Katherine 3.3

Katherine 2.65 Janet 3 Nadine 3.3

Amy 2.5 Kristina 3 Maureen 3

Megan 2.25 Katherine 2.8 Maia 2.8

Kristina 2 Maia 2.8 Jessica 2.8

Elaine 2 Alicia 2.8 Amy 2.5

Anna 2 Amy 2.5 Kristina 2.5

Jessica 1.8 Maureen 2 Megan 2.5

Maia 1.7 Elaine 2 Alicia 2.45

Maureen 1.5 Anna 2 Janet 2.3

Janet 1.3 Stephen 2 Elaine 2

Stephen 1.3 Joan 2 Anna 2

Nadine 1.3 Jessica 1.8 Stephen 1.8

Joan 0.7 Megan 1.8 Joan 1.8


24

Table 2. Average ratings of Portfolios according to level of reflective elaborationfor combined dimensions of Math knowledge, discourse and pedagogy

Average Portfolio

Name Depth Rating

Judy 4

Nora 3.9

Marissa 3.7

Wendy 3.53

Eileen 3.5

Sarah 3.5

Katherine 2.9

Nadine 2.5

Alicia 2.5

Amy 2.5

Kristina 2.5

Maia 2.4

Janet 2.2

Maureen 2.2

Jessica 2.1

Megan 2.1

Elaine 2

Anna 2

Stephen 1.7

Joan 1.5


25

Table 3

Grouping of participants according to reading and writing participation in theMath Inquiry database, by year and group.

Year 1 Year 2Engaged % read # written % read # writtenWendy 87 33 84 19Judy 46 18 58 15

Eileen 76 9 79 12Jessica 92 13 100 40Nora 96 10 91 11

Average 79.4 16.6 82.4 19.4Emergent

Alicia 15 2 82 9Sarah 12 0 44 4Maia 47 1 92 0

Nadine 56 3 97 10Megan 51 9 80 10Stephen 4.9 4 34 7

Average 31.0 3.2 71.5 9.0Withdrawing

Kristina 72 1 10 1Janet 9 2 3 0Joan 10 2 8 1Anna 7 5 37 1

Marissa 11 2 21 1Katherine 20 5 15 3

Average 21.5 2.8 15.7 1.5DisengagedMaureen 0 0 0 0

Amy 0 0 0 0Elaine 2 2 0 0

0.6 0.6 0 0


26

Table 4 Proportion of others' conference entries read, in math compared to allother conference topics combined and averaged for year 1 and year 2.

O = > All Other

O = > Math

Reading Others'Math Yr 1 %

ReadingOthers'All other Conf.Yr 1 %

Reading Others'Math Yr 2 %

Reading Others'All Other Conf.Yr 2 %

ENGAGED 62.72 29.72 82.66 79.2

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


27

Table 5 Math Content in Database Notes, by category for years 1 and 2.

Year 1 Year 2

Math Content Math Content

# Notes 1a Math 1b Nature 1cConnection

1a Math 1b Nature 1cConnection

Engaged Concepts of math to real world Concepts of math to real world

Wendy 20 11 5 0 6 6 3Judy 18 2 2 2 4Eileen 9 0 1 0 2Jessica 13 7 1 2 1 3Nora 10 5 1 1 4 3 3

Average 14 5 2 1 3.4 3 3

Emergent

Alicia 2 1 1 1Sarah 0 1Maia 1 1Nadine 3 1 1Megan 9 3 3 1 1 1 1Stephen 4 1 1 0 1

Average 3.2 1.5 2 0.5 1 1 1

WithdrawingKristina 1Janet 2 1 1Joan 2 1Anna 5 2 1 1Marissa 2 1 1Katherine 5 2

Average 2.8 2

DusengagedMaureen 0Amy 0Elaine 2


28

Table 6 Discourse and community comments in Database Notes, by category foryears 1 and 2

Year 1 Year 2

Name Discourse and Community Discourse and Community2A Value 2B metaconments 2C Assists 2D Sharing 2A Value 2B Metacomments 2C Assists 2D Sharing

Engaged of discourse on Learning Learning Class exp. of discourse On Learning Learning Class exp.

Wendy 3 7 10 2 1 3Judy 1 7 6 2 1 4Eileen 0 3 7 1 1Jessica 2 1 4 1 1Nora 2 6 4 3 4

Emergent

Alicia 2 1 3Sarah 1 1Maia 1Nadine 2 3Megan 2 0 6 0 2 1 1 3Stephen 3 0 2 1 2 3

Withdrawing

Kristina 1Janet 1Joan 1Anna 1 2 3Marissa 1Katherine 2 1

DusengagedMaureenAmyElaine 1 1


29

Table 7 Math Pedagogy comments in Database Notes, by category forYears 1 and 2

Year 1 Year 2Name Rethinking Math Pedagogy Rethinking Math Pedagogy

3A Sharing 3B learners 3C own learning 3A Sharing 3B learners 3C own learningEngaged Ideas mind impacts teaching Ideas mind impacts teaching

Wendy 5 2 4 11 1 0Judy 2 2 7 5Eileen 1 0 2 11 2 3Jessica 5 2 0 1 4Nora 5 2 1 3 1 3

Emergent

Alicia 2 1 4 2Sarah 1 2 1MaiaNadine 1 1 4 3Megan 2 2 2 2 1 2Stephen 0 0 2 2 1 2

WithdrawingKristina 1Janet 1Joan 1 1 1Anna 2 2Marissa 2 1 1Katherine 2 1

Dusengaged

MaureenAmyElaine 1 1


30

Table 8 Location of attributions of origins of mathematical anxiety

External* Mixed InternalEngaged 80 % (4)** 20 % (1)Emergent 14 %(1) 28 %(2) 57 % (4)Withdrawing 60 %(3) 40 %(2)Disengaged 33 % (1) 66% (2)

1. Internal: problems attributed to a lack of ability or aptitude located within oneself which stillis in effect (e.g. “I am still afraid of math”), or an internalising of teacher attitudes (e.g. “girlscan’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 schoolstructure, 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 eachcategory.


31

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 collaborativeexperience, which detracted from their sense of confidence.2= Participants reported the group experience was something that they needed to adjust to, and came toenjoy and feel confident in.3= Participants reported the group experience as a positive, confidence-building one right from thebeginning.


32

Documents

Developing Epistemic Agency as Pre-service Elementary ...home.oise.utoronto.ca/~cbrett/personal/AERADivIAgency02.pdf · importance of shared discourse as a way to make meaning has