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See Section 3. Significance We next describe the benefits of the supplement to students, the materials to be developed during the project, and the anticipated research results of the project. After that, we discuss the intellectual merit and broader impacts of the project. Even if transition-to-proof courses are offered, introductory real analysis and abstract algebra courses must continue the transition from lower-level problem-based courses to upper- level proof-based courses. This entire transition process takes time and can be a potential stumbling block for many students.
These courses are typically populated by three groups of students: preservice secondary education mathematics majors, undergraduate mathematics majors, and incoming mathematics graduate students requiring remediation.
All three groups have the potential to benefit from their participation in the supplement. First constructing a proof framework enables students to begin and provides a structure around which students can organize their thoughts, thereby revealing the problems for which key ideas are required to complete the proof.
Students are then able to focus their attention on finding such key ideas. This process gives students a sense of empowerment and self-efficacy Bandura, which is particularly valuable for mathematics majors and remediating mathematics graduate students who will require such proving skills in subsequent courses We have informal evidence indicating that our students write and organize their proofs more clearly than those who do not participate in the supplement.
See Section 2. As a result, it easier for an instructor to pinpoint problems in the students' thinking, so that more focused feedback can be provided. This will translate into fewer students failing and needing to repeat the course. The graduate student facilitators will expand their repertoire of teaching techniques as they will be exposed to a more interactive way of teaching and will gain upper-level teaching experience, thus strengthening their resumes. The preservice secondary mathematics teachers taking the supplement will also benefit from being exposed to a more interactive way of teaching, since people tend to teach the way they were taught.
We have found that preservice teachers who enter the course fearing the proving process actually come to enjoy it. And as a result, their secondary students will be better prepared to continue their studies in the STEM disciplines and fewer students will be lost from mathematics. However, the supplement is designed to run independently of any particular real analysis course.
Instructors can teach in their usual manner. The course itself does not need to be restructured; the number of credits can remain the same, and the supplement can be facilitated by an advanced graduate student using our materials.
Finally, the idea of the supplement is generalizable and could be developed as a compliment to other proof-based courses e. For an example, see Figure 1, p. The proof of one of these is to be assigned as homework while the proof of the other is to be co-constructed in the supplement. For each proof problem, we will produce a description of the proving process that includes not only a final written proof but also a detailed description of the actions needed to construct it.
Descriptions will be written for each proof problem in a pair so that the role of homework and supplement problems can be interchanged as deemed appropriate by the teacher. Those descriptions will be handed out at the end of the supplement classes. While the descriptions might not accurately reflect the proof produced during the supplement, they will nonetheless aid students in reflecting on what happened.
The descriptions will also aid the graduate student facilitators. While the mathematics education research literature on proving has been globally useful in designing our pilot supplement classes, it has not been very helpful in guiding our moment-to-moment teaching practices in those supplement classes.
For example, knowing that undergraduate students often have difficulty with mixed quantifiers e. However, our theoretical perspectives Section 3 have been very useful in helping us help students to overcome their proving difficulties and to succeed in producing well-organized and well-written proofs.
Our intent is to collect and analyze qualitative data in order to investigate the effectiveness of the supplement. We expect that many of our findings will be in the form of qualitative descriptions of the way students benefit from the supplement over time, that is, how they improve their proof constructing and proof writing skills.
We will concentrate on the following three research areas. We will analyse and describe the effect of the supplement on the proving skills of secondary mathematics education majors, mathematics majors, and graduate students taking the course as remediation.
We will also interview students about their attitudes towards proofs and mathematics. Case studies. We expect to produce in-depth case studies of the proving progress of at least two students one who is participating in the supplement and one who is not over the course of a semester.
We will describe and analyze how students prove theorems individually. We will recruit students who have previously participated in the supplement to prove theorems outside of their regular classes on tablet computers using screen capture software such as Camtasia.
We hope to understand what students think or do when they are working on proving alone, in a naturalistic setting. Another feature of the research that will be particularly novel is to study moment-to- moment proving done individually recorded on tablet computers.
To the best of our knowledge, there have been little or no results of this latter kind to date in the mathematics education research literature, so that any results in this direction would be a significant breakthrough. Development plans include: 1 an extensive set of paired proof problems, along with detailed descriptions of the actions involved in proving them; 2 a video for use in training graduate student facilitators that will give a sense of how these interactive supplement classes are to be conducted.
We expect that a department's adoption of this or a similar supplement would be relatively easy because of its cost-effective design. It would not require altering an existing introductory real analysis course.
Moreover, there would be no need to use additional senior faculty at graduate degree-granting institutions because advanced graduate students could be trained to serve as facilitators of the supplement using our materials. Broader impacts. Four groups of students that can be found at most U. We have informally found that many of these students actually come to enjoy the supplement, and so are more likely to implement proving with their future students as recommended by the NCTM Standards and Common Core State Standards The video and accompanying documentation will aid in the implementation of the supplement at other sites.
Also, the idea of the supplement is generalizable and could be developed as a compliment to other proof-based courses e. Schedule and Plans for Development of Materials, Data Collection and Analysis The proposed project is for two years, that is, four semesters and two summers starting in Spring The introductory real analysis course Math will be taught in Spring and Spring Each week in these semesters, the teacher will select at least one proof homework problem.
This will produce a document about four times as long as the proof itself. In Fall and Fall , the project team will produce expanded proofs for a popular real analysis textbook different from that used at NMSU.
All expanded proofs will be checked by at least two project team members for correctness and clarity. Only a few paired proof problems already exist from the pilots Section 2.
She is well qualified, as during the pilots, she observed and videoed all supplement classes and participated in the subsequent debriefs. Three other advanced graduate students will be selected as trainee facilitators. We will meet with them to discuss this project for several hours, watching excerpts from the pilot supplement videos.
After that, each trainee will facilitate at least one supplement class session. After that, the facilitator trainees will meet with the project team for several hours to watch the videos of themselves facilitating the supplement. Finally, they will be asked to provide observations and advice on the training of future facilitators.
In the Summer and Fall , we will write a document drawing on the above experience and our theoretical perspective that explains what a facilitator should do and why Section 5.
We will also produce a video of excerpts from actual facilitations, illustrating the points made in the written document. In Spring , we will select three other advanced graduate students, train them, as if they were at another university, using only the materials produced in Summer and Fall As a test of our training materials, the three graduate student trainees will be given the opportunity to facilitate the supplement. The remaining supplement classes will be facilitated by Kerry McKee, our full-time graduate research assistant.
Finally the three graduate student trainees will together meet with the project team to provide observations and feedback on the training materials. In the Summer and Fall , we will make adjustments to the above training materials as suggested by the results of the Spring facilitation and feedback from the graduate student facilitators.
The data will support three research interests described in Sections 5. Data will include copies of homework, tests, and exams from the Spring and Spring introduction to real analysis courses Math Video recordings and field notes will be made of all supplement class sessions. This will inform the teacher which students might benefit from the supplement and provide a baseline for judging progress.
For case studies, every two weeks we will conduct videoed interviews that are partly task-based and partly attitude-based with a student who is, and one who is not, attending the supplement. These videos will be transcribed by undergraduate student workers and checked by a project team member. Transcriptions will leave adequate space for project team comments and text from speech will be interspersed between relevant blackboard images captured from the videos.
This transcription method has been tried by Milos Savic, our proposed part-time graduate student research assistant, and it works well. Additional videos will be made of the project team debriefs, all interviews, and the training sessions for the graduate student facilitators. When we are not running the supplement in Fall Semesters and , we will collect data on how students construct proofs independently at home on the tablet computers as if they were doing homework.
For this purpose, we will select two volunteers who have taken the real analysis supplement and two who have not. We will collect screen and sound images using screen capture software. This data collection technique seems not to have been used before in mathematics education research.
However, we are sure the hardware works because Milos Savic has tried it himself using his own tablet computer. This Summer he will pilot the method on some mathematics graduate student colleagues and a few mathematics faculty. The variety of data collected will facilitate considerable triangulation Mathison, The research team will meet to review the tablet computer screen captures, videos, and field notes shortly after these are produced, planning ahead and taking note of student difficulties and successes with various proving actions, as well as noting more general patterns of behavior.
In the Summer and Fall Semesters of and , we will undertake a retrospective analysis of our collected observations. Because we will be conducting exploratory research, we cannot know or describe what we will find in advance and cannot know what coding categories will emerge. We will look for evidence that students in the supplement have taken up various helpful behavioral schemas i. Dissemination Plans We expect to distribute a booklet containing, or to provide a webpage containing, a collection of proof problems consisting of pairs having similar actions that could be assigned in an undergraduate introductory real analysis course, complete with a detailed description of the actions involved, and some information for graduate assistant facilitators on how to conduct a supplement using our co-construction technique.
See Section 5. The paired proof-problems, with an explanation of the supplement and its facilitation, could be published in the Journal of Inquiry-Based Learning in Mathematics, which publishes course notes along with a description of their implementation. In addition, we will attempt to interest some publishers of introductory real analysis textbooks to add something about this supplement to their informational materials.
It is likely that we will succeed in this with at least one book publisher because Mary Ballyk, the PI, is currently co-authoring an introductory real analysis textbook.
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