EMAT 9640: Spring 2008

John Olive


(706) 542-4557


Meeting Time and Room: Tues/Thurs. 3:30-4:45 p.m. Room 112


Office Hours:   By appointment


Goals of the Course :  The main goal of the course is for the students to develop research questions and a first draft of their prospectus for their dissertation research.  These questions should emerge from an analysis of research designs and theories of research in mathematics education, together with discussions with the students’ major professors and doctoral committees.   Emerging questions will be shared with the class and critiqued.   Students will be expected to perform a literature review for their respective questions.   Opportunities to conduct pilot studies based on these questions will be investigated and carried out where possible.   Presentations of a draft prospectus to the department by each student will be a final goal of the course.

First Article : Cobb, P. (2007). Putting philosophy to work: Coping with multiple theoretical perspectives. In F. Lester (Ed.), The Second Handbook of Research in Mathematics Education, pp. 3-38, Charlotte, NC: NCTM and Information Age Publishing.

Second Article: Click here to download Chapter 2 of Section C of the ICMI Study 17 on Integrating Technology in Mathematics Education. [NO LONGER AVAILABLE. PUBLISHED VERSION COMING SOON.] This is a draft only and cannot yet be cited.  Feedback to the authors is greatly encouraged!  We shall discuss the theoretical constructs introduced in this chapter on Tuesday, February 26, 2008.

Third Article:  Click here to download Chapter 3 of Section C of the ICMI Study 17
on Integrating Technology in Mathematics Education. [NO LONGER AVAILABLE. PUBLISHED VERSION COMING SOON.] This is a draft only and cannot yet be cited.  Feedback to the authors is greatly encouraged!  This Chapter is on Mathematical knowledge and practices resulting from access to digital technologies.

Fourth Article: SimCalc Technical Report on Scaling-Up

For the second week of class, each student should find two articles that relate to the theoretical framework for their research question. Please email the citations for these articles to Dr. Olive.

Links to students' citations: Sam Bennett, Rachael Brown, Shadreck Chitsonga, Kathy Daymude, Hulya Kilic, Hyeonmi Lee, Soo Jin Lee, Margaret Morgan, Ajay Ramful, Susan Sexton, Jaehong Shin

Link to 2003 Student Discussions
and to Jeong Lim Chae's drafts of her prospectus.

Email Discussion Threads: On Schemes

Students' Methodologies

Doctoral Students
Sam Bennett Data mining, Surveys & Interviews (individuals)
Inservice Teachers
Rachael Brown Classroom observations & written work
Inservice Middle School Teachers
Shadreck Chitsonga Questionnaires, classroom observations & individual interviews
Inservice Statistics Teachers
Kathy Daymude Mixed methods: Test-Error analysis, questionnaires, testing data & interviews(?)
High School students, teachers and parents
Hulya Kilic Classroom observations, questionnaires & individual interviews
Pre-service teachers
Hyeonmi Lee Classroom observations, document analysis & individual interviews First-year high school teachers
Soo Jin Lee Teaching Experiment with pairs of students--focus on both teacher/researcher and students
Middle school students and doctoral students
Margaret Morgan Surveys and interviews with individuals
Inservice teachers
Ajay Ramful Clinical interviews with pairs of students
Middle school students
Susan Sexton Clinical interviews with individuals or pairs of students
Pre-service teachers
Jaehong Shin Teaching Experiment with pairs of students--focus on students' thinking
Middle school students

From Sam Bennett:
Koehler, M.S and Grouws, D.A. (1992).  Mathematics teaching practices and their effects. In D.A. Grouws(Eds), Handbook of research on mathematics teaching and learning (pp. 115-126).  New York:  Macmillan.

It is spelled out in this article that the purpose of studying teaching practices is to affect the attitude and achievement of students.  Koehler and Grouws point out the growth in studying teaching practices.  Research began simply with Level 1 studies of teacher personality and characteristics (ex. years of teaching experience, number of math courses taken, the personality trait of enthusiasm, time allocation in the classroom. etc.), and how these affected student outcomes. Level 2 studies were done next, and in these, classroom processes were observed (interactions between students and teachers, classroom management techniques), where teachers influenced students and students influenced teachers.  Level 3 studies followed, in which pupil characteristics (gender differences, iq differences) were included in studies, as well as the understanding that student attitudes, not just achievement, were important.  Level 4 studies were done, in which there was an attempt to pair teaching and learning processes (students’ beliefs, teacher beliefs, etc).  The observations of these two researchers help me to understand the complications of the classroom microcosm.

Hatfield, L. L., & Ohio Univ, ACCLAIM. (2003). "Up the Back Holler, Down the Dusty Road, Cross the Windy Prairie": Issues, Perspectives, and Strategies for Research in the Crisis of Improving Mathematical Education of Rural Youth. Working Paper Series.  Paper presented at the ACCLAIM Research Symposium, McArthur, Ohio.

This paper is one of Dr Hatfield’s in which he approaches the problems of rural education from a sociocultural theoretical framework.  In this speech, he cites a pervasive “culture of failure”, and encourages the study and understanding of this rural culture as a first step to initiating a transformation in rural mathematics education.  This view of the rural culture helps me to frame my research considerations to include not only the microcosm of specific classrooms, but also the attitudes of individuals and the macrocosms of the community, the community economy--and accepted community norms.
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From Rachael Brown:
Cobb, P., McClain, K., Lamberg, T. d. S., & Dean, C. (2003). Situating Teachers' Instructional Practices in the Institutional Setting of the School and District. Educational Researcher, 32(6), 13-24.

Cobb, McClain, Lamberg, and Dean (2003) share a framework for viewing teaching and teachers. The focus is “to view teaching as a distributed activity and to situate teachers’ instructional practices within the institutional settings of the schools and school districts in which they work” (p. 13). An example is provided with middle school mathematics teachers. The interaction of various communities is analyzed. The idea of viewing the school system as a lived organization versus a designed organization is essential in this framework.

Kazemi, E., & Franke, M. (2004). Teacher Learning in Mathematics: Using Student Work to Promote Collective Inquiry. Journal of Mathematics Teacher Education, 7, 203-235.

Kazemi and Franke (2004) report on a professional development study of elementary teachers where they looked at the learning of the teachers as a whole group. A transformation of participation perspective was used to answer their research questions. This framework analyzed the discussion and identified norms and changes in the teachers’ conversations. One of their research questions was “How is teacher learning evident in shifts in participation in discussions centered on student work?” (p. 204). Looking at learning with this perspective in an Intermath course will be helpful in answering my research question.<>     

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From Shadreck Chitsonga:

Book from which my theoretical framework is drawn:

Crotty, M. (1998). The foundations of social research: Meaning and perspective in the
research process. Thousand Oaks, CA: Sage.

Based on Crotty’s description of epistemology I align myself with constructionism. Crotty has said, “ What constructionism claims is that meanings are constructed by humans as they engage with the world they are interpreting” (p. 43).
According to Crotty, constructionism is “the view that all knowledge, and therefore all meaningful reality as such, is contingent upon human practices, being constructed in and out between human beings and their world, and developed and transmitted within an essentially social context ” (p. 42).
    The particular theoretical framework that I am drawn to is phenomenology. According to Crotty, phenomenology is “ geared towards collecting and analyzing data in ways that do not prejudice their subjective character ” (p. 83) Crotty further states that phenomenology is an “effort to identify, understand, describe, and maintain the subjective experiences of the respondents. It calls into question of what is taken for granted. Further more phenomenology requires us to engage with phenomena in our world and make sense of them directly and immediately.


Garfield, J. (1995). How students learn statistics. International Statistical Review, Vol.63, No. 1, Collection of papers on Statistics education. (Apr., 1995), pp. 25-34.

In this article Garfield reviews some research from the areas of psychology, statistical education, and mathematics education. The article discusses the theories of learning, more especially the emergence of “constructivism”. “Constructivists view students learning a bringing into the classroom their own ideas, material. Rather than “receiving’ material in class as it is given, students restructure the new information to fit into their own cognitive frameworks” (p. 26). Based on what Garfield calls constructivist principles Garfield outlines some principles of learning statistics, which include 1) students learn by constructing knowledge. 2) Students learn by active involvement in learning activities. 3) Students learn to do well only what they practice doing. (Etc)
    This article is important for my research topic because it will enable me to develop a list of attributes on what teachers need to have in order to carry out effective
Classroom assessment.

Verkoeijen, P., Imbos, T., van de Wiel, M. W. J., Berger, M. P. F., & Schmidt, H. G. (2002) "Assessing Knowledge Structures in a Constructive Statistical Learning Environment" Journal of Statistics Education [Online], 10(2)

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From Kathy Daymude:
Creswell, J. W. & Plano Clark, V. L. (2007). Designing and Conducting Mixed Methods Research. Thousand Oaks, CA: Sage.
Click here for Kathy's synopsis of Chapter 1 and Dr. Olive's feedback.
Click here for Kathy's synopsis of Chapter 2 and Dr. Olive's feedback.

This is a test error analysis summary sheet.  Students fill it out after each chapter test.  The matrix itself can be tailored to a given course and semester.  Feel free to e-mail me if you'd like more details on the process or the research:  kdaymude@uga.edu

Cobb, P. (1994).  Where is the Mind?  Constructivist and Sociocultural
Perspectives on Mathematical Development.  Educational Researcher, 23 (7) 13-20. PDF copy for download.

This article compares and contrasts sociocultural and constructivist theoretical perspectives and encourages a blend of these two perspectives rather than choosing between them.  Cobb argues that “neither an individual student’s mathematical activity nor the classroom microculture can be adequately accounted for without considering the other.”  They provide the background for each other and together they give a complete picture of mathematical learning.

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From Hulya Kilic:
Gage, N. L. (1989). The paradigm wars and their aftermath:  A “historical” sketch of research on teaching since 1989. Educational Researcher, 18(7), 4-10

The paper was based on discussions about three different research paradigms, namely, objective-quantitative, interpretative-qualitative, and critical-theoretical. First, Gage stated how interpretative-qualitative and critical-theoretical paradigms had emerged in the 1980s. He stated critiques of interpretivists and critical theorists to current research paradigms at the beginning of the 1980s. Then he described how such critiques had influenced the researchers’, teachers’, and publishers’ views as well as methodology of researches in social sciences.

He stated that interpretivists focus on the specifics of an action and its meaning. He explained that interpretivists “[They] rejected the conception of cause as mechanical or chemical or biological…” (p.5). Furthermore, he stated that interpretivists rejected the assumption of uniformity in nature and also rejected “the use of linear causal models applied to behavioral variables as a basis for inferring causal relations among the variable…” (p.5). Instead, interpretivists emphasize the phenomenological perspective of the persons behaving. Interpretivists “…regard individuals as able to construct their own social reality rather than having reality always be the determiner of the individual’s perceptions” (p.5).

Howe, K. R. (1998). The interpretative turn and the new debate in education. Educational Researcher, 27, 12-30.

In this paper, Howe presented the debate between quantitative and qualitative paradigms in terms of their perspectives on human nature and the relationship between theory and practice. He supported ‘critical educational research model’ since he claimed that “positivism is untenable and interpretivism is incomplete” (p.243).
Howe identified interpretivism as “…insiders’ perspective regarding the interpretations of the meaning and implications of social events and arrangements.” (p.238) However, he stated that “…the interpretivist conception construes humans as so radically different from other things in the natural world that they are totally inexplicable in terms of such methods [methods of natural sciences]” (p.243).

Crotty, M. (1998). The foundations of social research: Meaning and perspective in the
research process. Thousand Oaks, CA: Sage.

Crotty defined interpretivism as “emerged in contradistinction positivism in attempts to understand and explain human and social reality” (p.5-6). He stated that interpretivist approach looks for “culturally derived and historically situated interpretations of the social life-world” (p.6).
He briefly explained the roots of interpretivism and compared it with other research paradigms.

Click here for definitions of Pedagogical Content Knowledge

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From Hyeonmi Lee:
Goos, M. (2005). A sociocltural analysis of the development of pre-service and beginning teachers' pedagogical identities as users of technology. Journal of Mathematical Teacher Education, 8, 35-59.

This first one is a case study that investigated the pedagogical practices and beliefs of prservice teachers and beginning teachers in integrating technology into the teaching of secondary school mathematics. It was conducted by Merrilyn Goos, in the University of Queensland in Australia. Goos used Valsiner's concepts as a theoretical framework to theorize dynamic teachers' learning as increasing participation in sociocultural practices. She documented how one teacher's modes of working with technology changed over time and across different school contexts and identified relationships between a range of personal and contextual factors which influenced the teacher's identity development in this case study.

Lee, H. S. (2005). Facilitating students' problem solving in a technological context: Prospective teachers' learning trajectory. Journal of Mathematics Teacher Education, 8, 223-254.

This second study is also a case study conducted by Hollylynne Stohl Lee, in North Carolina State University. In this study, Lee closely looked at how three prospective teachers interpreted and developed their role of facilitating students' mathematical problem solving with a technology tool. She used case study methods to identify and critical events that occurred throughout the different phases of the study and to analyze how these events may have influenced the prospective teachers' work with students. Click here to download a more detailed summary.

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From Soo-Jin Lee:
Izsák, A. (in press). Mathematical knowledge for teaching fraction multiplication. Cognition and Instruction.

Mathematical knowledge for teaching fraction multiplication.

This (from Dr.Izsak) is a study which used theoretical framework that I want to adopt for my study.  most of my research questions are compatible to his questions except that I will add another situation to teachers which is the professional development program. Dr.Izsak conducted case studies with two middle school mathematics teachers to examine knowledge of fraction multiplication that they used while teaching under various drawing representations. Specifically he focused on teachers’ use of three levels of unit structures with an aid of the distributive property in adapting their mathematical knowledge when teachers interact with kids with drawn representation problems in the class. Thus he formulated his theoretical framework into two parts. The first category was unit structures and knowledge of multiplication; the second was uses for drawn representation. It turned out that teacher who could work with three levels of units was more flexible in adapting her knowledge with kids’ various drawing representations than another teacher who showed only two levels of units in working fraction multiplication problems. Emphasizing individual learning and reasoning with three levels of unit in fraction multiplication problems suggests that the author might share some aspects of cognitive (Piagetian) psychology. Moreover from the use of drawn representation as a cognitive tool for teachers to understand students’ reasoning, and placing teacher and learner in the context of immediate classroom phenomenon, I could verify some aspects of distributed or situated  psychology. Thus it seems that he used interpretive framework for the study although he did not explicitly stated the term on the paper.

Putnam, R. T., & Borko, H. (2000). What do new views of knowledge and thinking have to say about research on teacher learning? Educational Researcher, 29(1), 4-15.

What do new views of knowledge and thinking have to say about research on teacher learning?

This article (from Putnam & Borko) is useful for my research as it described situative perspective which I think is pretty compatible to distributed psychology in much detail focusing more on teacher learning aspect. As Cobb described Putnam as pragmatic realist in his paper “putting philosophy to work”, this school of people, as John Dewey, rejected the dualism between external reality originated from Plato and reality which is composed of people. How they view of learning and knowing framed the situative perspective, and suggested three conceptual perspective of cognition: cognition as situative, cognition as social, and cognition as distributed. For my study, as I will use interpretive framework, I will add one more category “cognition as individual” on to their conceptual perspective of cognition. Learning is not only the process self-organization but also the process of social enculturation.
Putnam and Borko also stated an importance of emerging teachers into various discourse communities which ranged from professional disciplines, to groups of people sharing common interest, to particular classrooms. Moreover, they provided several examples of actual professional development programs conducted in this field sharing similar perspective. They also emphasized that those discourse communities gave rise to the cognitive tools so that teachers adopted while they tried to make sense of new experience. As cognition distributed across persons and tools for them, they described how cognitive tools (in the paper, specifically talked about technology) could enhance learning and knowing in school context. 

Tzur, R. (1999). an integrated study of children's construction of improper fractions and the teacher's role in promoting that learning. Journal for Research in Mathematics Education, 30(4), 390-416.

Possible research questions for Soo-Jin

If I collect my data from Does it Work project:

1. How do teachers situated in the professional development program - where
the focus is on introducing new ideas or technologies to teach students fractions and
discussing students’ errors or difficulties with fractions - adapt their knowledge in
their own classroom while they interact with their students?

2. How do diverse groups of teachers with different types of knowledge and
expertize come together in the program (a type of discourse community
suggested by Putnam and Borko) and create rich conversations and new insights
into teaching and learning of fractions? Can teachers do collective reflection?

Collective Reflection: Cobb (1997) used this term to correlate social and
psychological notion of reflective abstraction. In particular, he discussed how
teacher and student contributions brought forth shifts of focus in the discourse,
from generating various results of a problem to making those results an explicit
object of collective reflection and later to directly operate on those results.

3. What knowledge do teachers use as they respond to students’ thinking when
they are coordinated one by one?

If I collect my data from OAK project:

1. What is the role of teacher in providing problematic situations so that students
can transform partitive into the iterative fraction scheme? In general, what is the
role of teacher in giving them problem situations so that they can promote students
to self-regulate their knowledge? How can we make problem situations
meaningful so that students can learn and study?

Three types of tasks need to be presented to learners (Balacheff, 1990; Cobb, 1997)
• Initial tasks are used to promote the creation and recognition of a certain
experience and activity.
• Reflective tasks are used to promote an awareness of activity–result
• Anticipatory tasks are used to promote an abstraction of such relationships in
the absence of the activity.

2. How students reorganize their knowledge of fractions based on their
prior experience and accommodate to construct an improper fraction scheme? What
are the operations, units, and schemes necessary in an improper fraction scheme?

3. How does a teacher accommodate her knowledge of fractions as she tries to fit
the constraints she experiences with her students to understand her students’ reasoning
as well as to motivate students’ learning?

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From Margaret Morgan:
The theoretical frameworks described below are applicable to my research because they focus on adult populations similar to those at ATC and seek to explore why these adults are or are not successful in learning mathematics.  ATC has identified that many of its adult learners are not successful in the college algebra course and I hope to explore why they are unsuccessful and/or how to help them become successful in learning the mathematics in this course.

In my search, I found a group of researchers with an interest in adults and mathematics education.  Below is a link to the groups website:

The following websites describe the efforts of some members to develop a theoretical framework for researching :

Wedege, T. To know or not to know—mathematics, that is a question of context.     Educational Studies in Mathematics v. 39 no. 1-3 (1999) p. 205-27

This first article I chose is “To know or not know—mathematics that is question of context” by Tine Wedege.  She gives a detail description of her theoretical framework.  The first aspect of her framework that I like is that she focuses on adults similar to the ones I will study.  She says, “The ‘adults’ I am speaking of are those with brief schooling whose perspective with regard to education is about training themselves for the job on hand or for skilled work.” (207)  She further describes this group as having mathematical competence in the real world that does not apply in the classroom and explains that mathematics education is an economic issue for this group.  In discussing the learning of mathematics by this group, she recognizes both emotional and cognitive aspects.  Her theory draws on the theory of situated learning as described by Jave and Wegner and habitus as described by Bourdieu.
Burton, L. From failure to success: changing the experience of adult learners of     mathematics. Educational Studies in Mathematics v. 18 (1987) p. 305-316.

This second article I chose was one cited in the Wedege article—“From failure to success: Changing the experience of adult learners of mathematics” by Leone Burton. This author describes the theoretical context in less detail than Wedege, but shares with her a recognition of the emotional element of learning for adult learners.  This author’s theoretical context focused on the idea that an adult’s image of mathematics and the adult’s relationship to mathematics affect the adult’s success or failure in learning mathematics.

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From Ajay Ramful:

Vergnaud, G. (1994). Multiplicative conceptual field: Why and what? In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 41–59 ). Albany: State University of New York Press.

Vergnaud, G. (1996). The theory of conceptual fields. In L. P. Steffe & P. Nesher (Eds.), Theories of mathematical learning (pp. 219-239). Mahwah, New Jersey: Lawrence Erlbaum Associates.

Essentially, the two papers explain Vergnaud’s theory of conceptual fields and the importance of such a structure in analyzing additive and multiplicative problems. The different constituents of the theory like, ‘schemes’, ‘operational invariants’,  ‘concepts-in-action’ and ‘theorems-in-action’ are defined. Further, the concept of multiplicative conceptual field (MCF) is explained through a range of examples. The second paper Vergnaud (1996) also includes a discussion on the relevance of conceptual fields in the domains of number, space and elementary algebra.
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From Susan Sexton:
Lapp, D.A. (1997). A theoretical model for student perception of technological authority: Implementing technology in the classroom. Paper presented at the Third International Conference on Technology in Mathematics Teaching, Koblenz, Germany.

Pea, R.D. (1998). Learning and teaching with educational technologies. In Walberg, H.J. & Haertel, G.D. (Eds.) Psychology and Educational Practice (pp. 274-296). Berkeley, CA: McCutchan.

Click here for descriptions of these two articles

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From Jaehong Shin:
Olive, J., & Steffe, L. P. (2002). Schemes, schemas and director systems-an integration of Piagetian scheme theory with Skemp's model of intelligent learning. In D. Tall & M. Thomas (Eds.), Intelligence, learning and understanding in mathematics. Flaxton, Australia.

The project ‘OAK (Ontogenesis of Algebraic Knowing)’ from which I collect the data for my dissertation could be categorized under cognitive psychology, especially from the actor’s viewpoint by Cobb’s criteria. According to Cobb, cognitive psychology focuses on how the epistemic individual successively reorganizes its activity and comes to act in a mathematical environment. I think one of the most viable ways to build up epistemic individual is Piagetian scheme theory. The above article explains how schemes, in addition to the comparison with Skemp’s schemas and director systems, can be used to build a viable, theoretical model of children’s constructive activity in the context of learning about fractions. The below sentence directly drawn from the article would be sufficient to explain why I chose the ‘Scheme theory’ for my theoretical framework.

Our (my) focus on schemes would be appropriate because we (I) regard children’s fractional (algebraic) knowledge as consisting of schemes of action and operation that are functioning reliably and effectively.

Thompson, P. W. (1990). A theoretical model of quantity-based reasoning in arithmetic and algebra. Progress report to the National Science Foundation. San Diego State University, Center for Research in Mathematics and Science Education.

One of the hardest parts of teaching experiment using scheme theory is that the researcher does not decide a priori how to base children’s construction of an emerging scheme on the already built-up schemes even if established epistemic individual can be referred to usefully for researcher’s modeling a new scheme. Despite the difficulty, Thompson’s quantity-based reasoning might be a good start in order to analyze my collected data in OAK project because many research results imply that quantity-based reasoning might foster the development of the algebraic reasoning. In this article, Thompson presents the cognitive processes and structures that enable the kind of quantitative reasoning

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More Articles on Theoretical Frameworks:

Eisenhart Plenary for PME-NA, 1991

Peressini (2004)  Concept Framework

ICMI  Study17, Theme C, Chapter 2 on Theories for research on integrating technology in mathematics education (Drijvers & Kieran, 2008)

We shall also use this web page as a public forum for  issues raised by students in this class. Following is an example from a student in 2003:

Previous Students' Contributions

Jeong-Lim Chae's first draft of her prospectus

Jeong-Lim Chae's final draft of her prospectus

From Brian Lawler on 1/16/03:

Today, XXXX brought forth the common flawed assumptions regarding a 
cause-effect relationship between curriculum and high scores on tests 
of mathematical success. He offered as valid data the example of his 
success with Saxon -> high math scores.

I offer another example of data that indicates an alternate conclusion, 
less so to trumpet the curriculum but more so to encourage a 
questioning of the assumptions (and presuppositions) that are often 
hidden in statements about the power of an extant curriculum - like 
these above.

The first link below is a simple "Executive Summary" demonstrating a 
curriculum very different from Saxon to be successful. The rest of the 
research report can be found on the 2nd link.


This next report demonstrates, on a more broad scale, the successes 
measured by students of IMP.


Of course, Dr. Jo Boaler, now at Stanford, made an attempt to 
understand what effect curricular (and other classroom characteristics) 
had on student success on "tests".


Finally, a more personal, and "small data set"  example - once again. 
This California HS has been "graded" in the top 3 for every year the 
state has used this "API" index (API stands for Academic Performance 
Index, although many CA teachers refer to it as the Affluent Parent 
Index). Every student in the high school is taking IMP as their math 
curriculum. And every 9th grade student is heterogeneously grouped  in 
Year 1 of the curriculum, 10th grade in Year 2, etc.


And finally, I will leave the last word to our colleague Jeremy 
Kilpatrick, from JRME Vol. 32 (4) pp. 421-427: "Where's the Evidence?"

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Spring 2008 Discussion on Schemes:

1.     On Wed, Feb 27, 2008 at 12:15 AM, soo jin lee <sjjh0314@uga.edu> wrote:
Dr.Olive and all of our classmates,

I think I finally understood, after hours of perturbation:), Vergnaud's definition of scheme is compatible to Piaget's one.The first paragraph is Vergnaud's definition of scheme, and the second paragraph is statements that I pulled out from von Glasersfeld's and Steffe's papers:
the definition of Vergnaud (1996): a scheme is an invariant organization of behavior for a given class of situations. More informally: a scheme is a more or less stable way to deal with specific situations or tasks.

von Glasersfeld says that "all knowledge consists of invariants which the experiencer creates and maintains in fact of changing experience", moreover Piaget defined scheme of action as "repeatable and generalized through application to new object".

HOWEVER, this question is open to all of our classmates!
I am still struggling with this stability. Is it my English problem? I still think this scheme is subject to vary as all cognitive organism tries to pursue equilibrium albeit we can never completely have equilibrium state as von Glasersfeld said. Even Piaget said that it is generalized through application to new object, and wouldn't it mean that it has variability?

Thank you!
Soo-Jin Lee

2.      On Feb 27, 2008, at 1:05 AM, Susan Sexton wrote:

Hi Soo Jin,
I believe that stability means that the scheme provides stability for the experiencer, not that the scheme, itself, is stable. I think that you are correct in your idea that a scheme varies and changes (or accomodates?) depending on the situation (or perturbation?). (I'm still wrapping my mind around the new terminology. :-)

3.  On Feb 27, 2008, at 8:03 PM, Rachael Brown wrote:

I agree with both of you. I think of a scheme as stable to the person who has it. For example, if you were to ask me to add 7 + 5, the way I deal with this (I know, sad) is I think 7 + 3 is 10 and 2 more is 12. I realize that many people have this memorized but for some reason 7 + 5 and 8 + 5 is hard for me. : )  Anyway, this scheme I have is something I do every time and I'm confident it works. This scheme can be changed by a perturbation and by me thinking about other ways to deal with 7 + 5. I don't think something is a scheme if a student isn't consistent applying it. Does that seem to fit in with what you all are thinking?


4.     On Feb 28, 2008, at 3:21 AM, soo jin lee wrote:

Thanks Susan and Rachael,

It is making much more sense to me.
As a scheme is observer’s conceptual tool to analyze children’s action and their language, I have kept thinking of it only in terms of observer's perspective.
Moreover, even as an observer, we may think of “a scheme” as stable in the given situation until an experiencer faces perturbation because once it is modified then it seems like we are talking about a new scheme now. If an experiencer faces with some perturbations depending on the situation, it may split one scheme into two (by generalizing assimilation) or it may constitute a new scheme (by accommodation). Thus “a scheme” becomes “schemes” or another scheme  cited from Glasersfeld, E. v. (1980). The concept of equilibration in a constructivist theory of knowledge. Autopoiesis, communication, and society. F. Benseler, P. M. Hejl and W. K. Koeck. Frankfurt/New York, Campus: 75-85.

Another similar statement from von Glasersfeld “an assimilation is the process of incorporating new experiences into the scheme, while an accommodation is any changes that results in either creation of a new or more elaborated scheme or in the splitting of the scheme into subschemes (von Glasersfeld, p82, the abstraction of counting units, proceeding for PME-NA)”

What was confused me a lot was reorganized or re-interiorized schemes. I did not think of it as a new scheme even though it was accommodated. For example, if I use Steffe’s children’s counting schemes, children reorganized or re-interiorized their previous counting scheme such as ENS to constitute more advanced counting number scheme which is GNS. So if a child is a GNS kid, then we don’t say he discarded all the operations of ENS, TNS, and INS. At that moment, the child is a GNS kid with all aspects of INS, TNS, and ENS embedded in it. Thus, I thought that it is not a new scheme for both observer and a kid. BUT it may not be totally a different scheme but it is a new scheme as kids are using different or more effective operations now!
So now I agree with Vergnaud's definition of scheme.

From various articles from radical constructivists, I knew that cognitive organisms are always trying to make some relatively regular or invariant reality out of their experience to have this equilibrium. However, it was hard for me to conceptualize self-regulatory (capability to maintain invariant) as an important aspect of a scheme. Whenever I read these articles, I felt that I knew all at the time but it seems like I am still missing very important pieces!
Thanks for all your help!!

See you in the class!
Soo-Jin Lee