LEY Nº 1178 DE LOS SISTEMAS DE ADMINISTRACIÓN Y CONTROL GUBERNAMENTALES

instructors and students included the same questions assessing general beliefs about teaching and learning mathematics. To capitalize on prior research concerning beliefs about teaching and learning mathematics, I took the framework from Principles to

Actions (NCTM, 2014) to describe productive and unproductive beliefs about teaching

and learning mathematics. The ideas within Principles to Actions (see Table 6) use the evolution of standards for mathematics coupled with research studies using data from K- 12 classrooms to articulate a “unified vision of what is needed to realize the potential of educating all students” (NCTM, 2014, p. vii). Even though NCTM focuses on research and evidence of K-12 classrooms, the content of College Algebra is repeated content from middle and high schools. Also, a large percentage of students taking College Algebra are freshmen, and thus only recently were in high school. Therefore, the beliefs outlined in Principles to Actions have at least reasonable if not strong overlap with beliefs about teaching and learning mathematics held by teachers and learners in College

Algebra.

To use these beliefs on the pre- and post-survey, I constructed six separate

questions with each productive and unproductive belief placed on opposite endpoints of a line representing a spectrum. The instructions participants read were, “Below are a list of different beliefs about teaching and learning mathematics. For each question, mark an ‘x’

anywhere on the line indicating where your belief lies in relation to the two beliefs.” See

Table 6

Beliefs about Teaching and Learning Mathematics

Unproductive Beliefs Productive Beliefs

Mathematics learning should focus on practicing procedures and memorizing basic number combinations.

Mathematics learning should focus on developing understanding of concepts and procedures through problem solving, reasoning, and discourse. Students need only to learn and use the same

standard computational algorithms and the same prescribed methods to solve algebraic problems.

All students need to have a range of strategies and approaches from which to choose in solving problems, including, but not limited to, general methods, standard algorithms, and procedures. Students can learn to apply mathematics only after

they have mastered the basic skills.

Students can learn mathematics through exploring and solving contextual and mathematical problems. The role of the teacher is to tell students exactly

what definitions, formulas, and rules they should know and demonstrate how to use this information to solve mathematics problems.

The role of the teacher is to engage students in tasks that promote reasoning and problem solving and facilitate discourse that moves students toward shared understanding of mathematics.

The role of the student is to memorize information that is presented and then use it to solve routine problems on homework, quizzes, and tests.

The role of the student is to be actively involved in making sense of mathematics tasks by using varied strategies and representations, justifying solutions, making connections to prior knowledge or familiar contexts and experiences, and considering the reasoning of others.

An effective teacher makes the mathematics easy for students by guiding them step by step through problem solving to ensure that they are not frustrated or confused.

An effective teacher provides students with appropriate challenge, encourages perseverance in solving problems, and supports productive struggle in learning mathematics.

Note. Taken from Principles to Actions, p. 11, by National Council of Teachers of Mathematics, 2014,

Virginia: The National Council of Teachers of Mathematics Inc.

Figure 3. An example of one of the questions on the pre- and post-survey using NCTM’s

productive and unproductive beliefs.

Students can learn mathematics through exploring and solving contextual and mathematical problems. Students can learn to

apply mathematics only after they have mastered thebasic skills.

Tick marks were placed on the line in order to better quantify where participants indicated their beliefs fell. Using tick marks instead of a typical Likert scale better

represented a spectrum because it steered participants away from choosing a number. The idea behind this choice was this modification allowed participants to pick a place on the line that best reflected their belief in relation to the spectrum. When participants were making their choice for each question, they were not confronted with discrete numbers, instead they had this line with tick marks, which better represented a continuous pathway between the productive and unproductive belief.

Interviews. The instructors and eight students were selected to participate beyond

completing the pre- and post-survey. While eight students were selected, only two students agreed to participate beyond the pre- and post-surveys. More about that part of the data selection will be discussed in the analysis section below. These four participants, whom I refer to as the main participants, were interviewed four times over the course of the semester. Table 7 gives basic information of the four main participants. The first and the last interviews were focused on elaborating the person’s thinking when he or she answered the pre- and post-survey. During both semi-structured interviews, the participant and I always had the participant’s answers to the survey in front of us. Questions for the pre-survey interview mainly came from the pre-survey itself, with additional probing questions asked. For the post-survey interview, I did general

comparisons of the participant’s pre-survey answers with the post-survey answers. One of these comparisons included noting for each question how the answer changed or did not change. I also reexamined and coded the transcript of each participant’s pre-survey interview in order to organize their beliefs into several larger categories for generating

interview questions and prepare myself to ask different probing questions. This process ended with notes for each participant concerning all or most of the following areas: views about mathematics, how they viewed the role of the instructor, how they again viewed the role of students or opinions of students, views of themselves in their role in a math class, and beliefs they admitted to being unsure of at that point of the semester. I created specific questions from this document for each participant for the post interview.

Table 7

Basic Information for Four Main Participants

Name Level of School Role in College Algebra Prior Teaching/Learning Experiences Sally 2nd _{year PhD student Main Instructor (GTA) * Instructor of College Algebra prior}

semester

* Calculus Recitations

* Tutoring in undergraduate school Cara Sophomore

undergraduate

Support Instructor (LA) * LA of College Algebra prior semester * tutoring in high school

Brad Freshman other undergraduate

Student N/A

Michael Freshman other undergraduate

Student * Took Intermediate Algebra at UNL previous semester

For the two interviews in the middle of the semester, I chose to focus the interview on specific instances of teaching and learning that occurred in their College Algebra course. To do this, I videotaped class sessions, then edited those recordings to short two- to three-minute video clips. The intention of using video clips as the center of these interviews was to focus the conversations on common interactions within the classroom (Speer, 2008), rather than letting participants openly reflect. Using video clips

for teacher development is a common way to help teachers reflect more deeply on their own practices (Borko, Jacobs, Eiteljorg, & Pittman, 2008; Sherin, 2002; Sherin & van Es, 2009). Hartman’s (2010) dissertation study involved teachers using videos of themselves when they were students in a mathematics course to support reflection on themselves as learners. Thus, by using video clips for these interviews, I sought to help both the instructors and students reflect more deeply on their practices as an instructor or as a student.

For each interview in the middle of the semester, each participant and I watched three video clips and then using a semi-structure interview technique (Creswell, 2013). I gathered more in-depth data from the participant about their beliefs. Watching each clip several times, I then picked certain points during the video, created video clips, to direct the conversation towards and created questions for these semi-structured interviews using these video clips. Once a point in the video was chosen, I would create correlating

questions for each participant. For example, in one video clip, a point was identified in which a student presented a solution and the GTA asked the student to further explain her solution. For this point in the video, I asked the GTA why she chose to ask the student to explain further, then I also asked the LA and two students why they thought the GTA chose to push the student for more of an explanation. At the end of this process I had a list of questions for the GTA, the LA, and one list for the students (see Appendix B). Figure 4 denotes the general timeline in which these four interviews occurred for each main participant.

Figure 4. General timeline of interview data collection for each main participant.

Table 8 gives a brief summary of each video clip used and why it was picked. There were a variety of clips used in the two interviews. Two of the six video clips focus on Sally talking directly about beliefs and reasons behind teaching decisions.

Participant selection.

Choosing one section of college algebra. The only criteria for picking this

section of College Algebra to study was having a main instructor who was a GTA with less than one year in teaching experience. I met with the Director of First Year

Mathematics Programs to ask for his suggestions of which section to contact first. He gave me two initial sections to contact because he believed both of the GTAs would be willing to be open and thoughtful about their ideas. The first GTA I contacted indicated she would be out of town for about three weeks during the semester, but was willing to let me study her section. I decided to choose the second section whose GTA was also willing to participate and did not intend to be gone for a substantial period of time. Her section met three consecutive days of the week, each for a period of 1 hour and 15 minutes, giving a total contact time of 3 hours and 45 minutes per week. The classroom

Interview #1: Focus on pre- survey answers Interview # 2: Focus on post- survey answers Interview #2: Focus on 3 video clips Interview #3: Focus on 3 video clips

80 Table 8

Summary of Video Clips Used in the Middle Interviews

Mathematical

Focus Brief Summary of Clip Why I chose this Clip Interview #2: Focus on 3 Video Clips The Bridge Problem Problem solving using the The Seven Bridges of Königsberg problem

Sally presented a solution to The Bridge Problem, a problem students had been working on the previous class session. During her

presentation, Sally used a specific representation of The Bridge Problem and asked students questions. Afterwards, Sally got on a “soap box” where she explained her reasoning behind giving this type of problem.

* It was a challenging problem with no solution

* I was not sure whether the students had come up with the representation Sally used

* Sally explained to the whole class her reasons behind this teaching decision Cara Helping A Group of Students Graphing and interpreting asymptotes

Cara initially helped two students with a calculator issue then helped a different group of students working on a question about asymptotes.

* Cara directly intervened when helping the students with the calculator * Cara was obviously

trying to lead the 2nd group of students by making leading statements Composi tion of Function s with Pizza Composition of Functions

After giving a brief overall picture of what composition of functions mean, Sally worked through a specific example of composition of functions using the price a pizza parlor charges and that cost that is required to make a pizza. While Sally worked through this example, the students seemed lost and there was a moment where Sally tried to clear up why people might be lost.

* Sally gave a

conceptual picture of the content first * I really wanted to see

what each participant saw as reasons for confusion

* This was a moment where Sally, Cara, and many students were engaged in one conversation

81 Mathematical

Focus Brief Summary of Clip Why I chose this Clip Interview #3: Focus on 3 Video Clips The Magnitu de Discussi on The meaning of exponents

A student presented her solution to the whole class. Sally asked her to explain more details about the solution, but the student got hung up. Other students and Cara tried to help explain more details of the solution.

* A student presenting was something relatively new in this class

* The student’s struggle is a typical kind of error seen in College Algebra

* The problem and Sally were pushing students to think beyond the procedures

* Cara interjected in the group conversation Explorin g Even and Odd Function s Examining properties of even and odd functions

In a whole group discussion, Sally started leading the class through the example of what happens at x=0 when you reflect an even function across the y axis. The class came to the consensus that x could be anything when you reflect an even function across the y- axis. Then the class examined what happens when an odd function is reflected across the y-axis. Unlike the even function, the class sees that x had to be 0 when the odd function is reflected.

* The students appeared to be really engaged in this discussion * Sally set up the

discussion so that the students really had to think about what happened at x=0 for both problems Sharing Mid- Semester Feedback

Not Applicable Sally shared the feedback she received on mid-semester evaluations she previously gave. The feedback was positive but also contradictory at points where students described what was helping them learn and other things they wanted to help them learn.

* Another instance where Sally is being pubic about her teaching decisions * Insight into student

in which they met was a recently redone room with a white board covering most of the walls, which were a light grey color all around. The room was filled with seven brand new tables;e each table had between four to seven new rolling chairs around them. It was a recently renovated classroom, unlike older classrooms with chalkboards and rows of individual desks.

Initial information of the main participants. The main participants in this study

included two instructors and two students. The main instructor, Sally, was a second year doctoral student in the mathematics department. Before this Spring 2015 semester, she had only been the main instructor of College Algebra in the previous semester; the rest of her teaching experiences included being a recitation instructor for Calculus and being engaged in tutoring during her undergraduate years. Cara, the Learning Assistant (LA), was a sophomore undergraduate student majoring in mathematics who was the LA with the main instructor in the previous semester. The LA had also been a tutor previously, during her later years in high school. Both instructors were young females who were outwardly enthusiastic about mathematics and approachable. While both instructors had great intentions of being helpful to others, they differed slightly on how they enacted these intentions. If a person were to ask Sally for help with a mathematics problem, she would kneel down next to that person and first ask him or her what he or she has already tried. Then she would wait for the person to respond and follow up with more questions like the first one she asked. If a person were to ask Cara for help with the same

mathematics question, she would ask the person what he or she has tried but then almost immediately ask him or her more questions or make more statements that hint more strongly at the answer. I quickly found out from both participants after choosing this

section for this study that Cara and Sally taught together in College Algebra the prior semester, Fall 2014. Throughout the semester, both of them often referred to this semester together and believed that it had significant impact on this semester of Spring 2015.

Using the two dimensions from the pre-survey, the following four categories were created as a way for picking different types of students: low self-efficacy and traditional beliefs, high self-efficacy and traditional beliefs, low self-efficacy and progressive beliefs, and high self-efficacy and progressive beliefs. Eight students, two from each of the four categories, were initially contacted to be part of the study, but only two agreed to further participate beyond the pre-survey. The two students scored high on the self- efficacy part of the pre-test, but differed in terms of how progressive their views were on the beliefs portion of the pre-test. Table 9 summarizes these four categories and how the students were selected. Also, both students self-identified as the “freshman other” level of undergraduate education meaning they were not within one year of being out of high school and only possessed enough credit hours to be considered a freshman.

Table 9

Summarizing Categories Used to Select Student Participants

High self-efficacy, traditional beliefs High self-efficacy, progressive beliefs Low self-efficacy, traditional beliefs Low self-efficacy, progressive beliefs Student A Brad Student B Michael Student C Student D Student E Student F

From his pre-survey responses, Michael indicated high confidence in his

mathematical abilities and identified more progressive beliefs about teaching and learning mathematics. Brad also identified high confidence in his mathematics abilities, but

differed from Michael: Brad held more traditional beliefs about teaching and learning mathematics. Both students were fairly shy upon first meeting but indicated that they wanted to help me out by participating. Because of this willingness, it became more important to me to spend more effort keeping them involved in the study than to try to fulfill the original plan of getting several students with different levels of self-efficacy and beliefs about teaching and learning mathematics. Not having students from the other categories in Table 9 is a limitation for this study.

Data Analysis

Analysis of the data occurred in three separate phases. Phase 1 involved analyzing the pre- and post-surveys primarily for the purpose of collecting richer qualitative data, which will be explained in further detail below. Phase 2 is where the interviews,

observation notes, and video clips were analyzed for understanding what happened to the instructors’ and students’ beliefs about teaching and learning mathematics. Interviews from the beginning and end of the semester, as well results from the first phase were used to gather a sense of what happened to the participants’ beliefs. Then, data from the

middle of the semester is used to support what was declared to have happened with each participants’ beliefs. After Phase 2, I realized the analysis had only been considering participants’ beliefs individually, which meant all four participants’ beliefs had not been considered together. From this realization emerged Phase 3, which involved zooming in on three video clips, and the participants’ responses to specific questions within these

video clips to understand the second research question about how participants’ beliefs