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In the figure below (Figure 1), we have included a typical formative assessment. The first questions of our formative assessments were conceptual questions related to the current content.

The two open questions always appear as the last two questions of every formative assessment.

When we analyzed students’ documents, we looked for student errors in the content questions, and checked the homework assignments and test following the formative assessment how persistent the error was. For the open questions, we coded all student questions and comments about what they did not understand that were directed to the instructor as peripheral participation, and considered the written statement in the penultimate question to identify concepts students claimed to be transferring from other areas. After coding the open response questions, we looked at students’ summative homework assignments and exams for further evidence of improvement and transfer.

15TH Annual Conference on Research in Undergraduate Mathematics Education 407 Directions: Answer the following questions to the best of your ability. Responses need not be lengthy, but should answer all parts of the question. Please type your answers into this word document and email it back to [Me] at [Your.instructor@email.edu] by [9 pm tonight].

1. Fill in blanks with the letter(s) from the definition of the derivative to label the quantities marked on the graph of y  f ( x) as illustrated below.

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2. Write a short paragraph that answers the following two questions. What mathematical concepts or phrases used so far this week do you recognize from calculus? From other mathematics courses?

3. What questions do you have about the material we have covered so far in class?

Figure 1 Formative Assessment Two Given the lack of qualitative literature on formative assessment, particularly with American undergraduates, we chose to conduct an exploratory study. The first level of our analysis was the classroom, where we conducted a macro-level analysis of the learning trajectory of the classroom for two introductory calculus classes. At the second level, we analyzed formative assessments, homework assignments, and exams as artifacts of the learning trajectory (Patton, 1990) from four students in each class. The first author also observed the classrooms the day before and the day after the weekly formative assessment was distributed to the students and debriefed the instructors on a weekly basis to obtain their observations of student and classroom learning trajectories. We analyzed the data using an open coding thematic analysis, which was peer checked (Patton, 2002).

The classes we recruited participants from utilized Oehrtman’s (2008) approximation framework as a coherent instructional approach which uses limits to develop the concepts in introductory calculus. This framework is built upon an approximation metaphor for limits Oehrtman (2008, 2009) based on approximating an unknown quantity. For each approximation there is an associated error which one needs to bound in order to have some sense of the accuracy of the approximation. While the actual student usage of approximation metaphors can be highly idiosyncratic (Martin & Oehrtman, 2010), systematic structuring of the elements and relationships among approximations, errors, error bounds reinforce common limit structures within and across different limit contexts. The goal of the instructional framework is for students’ use of the metaphor to become more systematized in ways that reflect the structure of 408 15TH Annual Conference on Research in Undergraduate Mathematics Education formal limit definitions but are intuitively accessible to the students (Oehrtman, 2008); the goal of the formative assessments was to facilitate this systemization. The systematic metaphor can encourage the abstraction of a common structure while engaging in multiple activities within a limit context and the results of such abstractions further support abstractions of common structures across different limit contexts that can provide a more coherent understanding of the role of limit throughout all of calculus and beyond. As a student’s approximation schema becomes well organized these ideas become a cognitive tool that can guide students’ informal investigation into concepts formally defined in terms of limits.

Results The classroom-level learning trajectory outlined by Oehrtman’s (2008) approximation framework was unchanged with the addition of formative assessment; the class still needed to engage in the same cognitive challenges to master the framework. However, since the instructor was able to use formative assessments to provide feedback that immediately addressed misconceptions, students who completed the formative assessments appeared to make fewer mistakes on their unit tests than students who did not do the assignments. While the early questions on each formative assessment allowed instructors to communicate with students what material was important, the final two questions of each formative assessment contained some evidence of student self-monitoring and actor-orientated transfer.

The thematic analysis of the data suggested three factors helped individual students develop more systematic and less idiosyncratic conceptual structures related to the approximation framework. First, the formative assessment provided students a legitimate peripheral participatory role; the open response question allowed students to ask questions of their instructor without any loss of face, and gave students some say over what happened in class. As Max explained after class one day, Everyone at my table is so much smarter than me, and I know they really get it, but when we do the formative assessments, it’s over email, so no one has to see me not get it. I know they [my table] get bored the next day, but it makes all the difference for me to have my questions answered.

Second, by asking students to reflect on what concepts they did and did not understand, the

formative assessment scaffolded student self-monitoring. As Robin explained:

Before the first one [formative assessment] I thought I understood most everything. But them when I had to sit down and write a paragraph about what I didn’t understand I started to realize I really didn’t know how the pieces fit together. Then, when we talked about it [the formative assessment] the next day, I knew I had to pay extra close attention.

Third, by asking students to reflect each week on what concepts they had seen before, together with the improvements in self-monitoring, students improved their incidence of actor-orientated transfer. The responses on the next to last question on each formative assessments that asked students to make connections not only increased in length, but students began to correctly claim that approximation ideas were applicable from week to week. In the figure below (Figure 2), we have provided the responses of a typical student’s responses from the first two formative assessments in the derivative chapter, in the third and fourth weeks of class. While the student is mostly noticing common vocabulary words at this stage, this is a necessary first step to further transfer of concept (Barnett, 2002).

15TH Annual Conference on Research in Undergraduate Mathematics Education 409 Formative assessment #1 Formative assessment #2 This week I recognize the phrase average I recognize slope from past math classes and speed. That is when you take the change in the instantaneous rate of change from physics height and divide it by the change in speed. I classes as well as calc. I also recognize error recognize instantaneous speed from physics. bound because we have been discussing it over I’m not exactly sure how you find it, but I do the last few classes.

recognize the word. I recognize the slope of a line and relate it to when I learned about it back in algebra. The approximation value, remind me of last week when we worked on limits.

Figure 2. Sample Formative Assessment Responses.

The analysis of individual students’ artifacts suggested that the opportunity to ask questions and gain specific feedback was crucial in addressing individual misconceptions.

Conclusions While the formative assessments are graded for completion and only worth a few token percent of the students’ final grades, the act of completing the formative assessment help students understand what concepts the instructor values, reflect on what they understand, ask questions without losing face, and ponder connections between topics on weekly, unit and semester scales. This suggests that, for undergraduate mathematics students using asynchronous formative assessment, the peripheral participatory role can be included in Black & Wiliam’s (2009) theoretical framework. Since their framework is based on verbal and whole class formative assessments, students who have questions about the material must feel safe admitting this in front of their peers; participation is not peripheral.

Since formative assessment improved students’ self-monitoring, formative assessments could be designed and implemented for any introductory mathematics course. As we move forward on data

collection, we are interested in obtaining feedback from peers, particularly the following questions:

(1) how else might formative assessment influence the learning process? (2) How can we improve our coding scheme for evidence of transfer?

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Bereiter, C. (1994). Implications of Postmodernism for Science, or, Science as Progressive Discourse. Educational Psychologist, 29, 3-12.

Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation, and Accountability, 21(1), 5-31.

Brown, J. S., Collins, A., & Duguid, P. (1989). Situated Cognition and the Culture of Learning. Education Researcher, 18(1), 32-42.

Clark, I. (2010). Formative assessment: There is nothing so practical as a good theory.

Australian Journal of Education, 54(3), 341-352.

Clark, I. (2011). Formative assessment: Policy, perspectives, and practice. Florida Journal of Educational Administration and Policy, 4(2), 158-180.

410 15TH Annual Conference on Research in Undergraduate Mathematics Education Jordan, B. & Henderson, A. (1995). Interaction analysis: Foundations and practice. Journal of the Learning Sciences, 4(1), 39-103.

Lave, J., and Wenger, E. (1991). Situated Learning. Cambridge: Cambridge University Press.

Lobato, J., & Siebert, B. (2002). Quantitative reasoning as a reconceived view of transfer.

The Journal of Mathematical Behavior, 21(1), 87-116.

Martin, J., Oehrtman, M., Roh, K., Swinyard, C., & Hart-Weber, C. (2011). Students’ reinvention of formal definitions of series and pointwise convergence. Paper presented at the 14th Annual Conference on Research in Undergraduate Mathematics Education, Portland, OR: Portland State University.

Ning, L., & Sun, Z. (2011). An experimental research on the transfer of mathematics skills based on self-monitoring strategy. Journal of Mathematics Education, 4(1), 67-74.

Oehrtman, M. (2008a). “Layers of abstraction: Theory and design for the instruction of limit concepts”. In Making the connection: Research and practice in undergraduate mathematics, MAA Notes Volume, Edited by: Carlson, M. and Rasmussen, C. 65–80.

Washington, DC: Mathematical Association of America. 73.

Oehrtman, M. (2009). Collapsing dimensions, physical limitation, and other student metaphors for limit concepts. Journal for Research in Mathematics Education, 40, 396–426.

Oehrtman, M., Swinyard, C., Martin, J., Roh, K., & Hart-Weber, C. (2011). From intuition to rigor: Calculus students’ reinvention of the definition of sequence convergence. Paper presented at the 14th Annual Conference on Research in Undergraduate Mathematics Education, Portland, OR: Portland State University.

Patton, M. Q. (2002). Qualitative research & evaluation methods (3rd ed.). Thousand Oaks, California: SAGE Publications, Inc.

Salomon, G. & Perkins, D. (1997). Individual and Social Aspects of Learning. In Review of Research in Education, 23, 1-24.

Sfard, A. (1998). On Two Metaphors for Learning and the Dangers of Choosing Just One.

Educational Researcher, 27, 4-13.

Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.

Journal for Research in Mathematics Education, 26(2), 114-145.

Swinyard, C. (2011). Reinventing the Formal Definition of Limit: The Case of Amy & Mike.

Journal of Mathematical Behavior, 30(2), 93-114.

15TH Annual Conference on Research in Undergraduate Mathematics Education 411 Szydlik, J. E. (2000). Mathematical beliefs and conceptual understanding of the limit of a function. Journal for Research in Mathematics Education, 31(3), 258-276

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Abstract In this report we detail linear algebra students’ interpretations of linear transformations. Data for this analysis comes from mid semester, semi-structured problem solving interviews with 13 undergraduate students in linear algebra. We identified two main categories for student reasoning students in completing three tasks: 1) students who used structural reasoning with entries of the matrix, columns of the matrix, and orientation of the shape and 2) students who used operational reasoning through matrix and vector multiplication. We examine the patterns that emerged from student strategies, and discuss possible explanations for these patterns.

Key words: linear algebra, linear transformations, operational and structural reasoning, concept development

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