Mathematics Teacher Educators’ Decisions in a Time of Crisis: Self-Reflections as a Basis for Community Inquiry
Abstract
:1. Introduction
2. Decision Making
3. Communities of Inquiry
4. Methodology
5. Findings
5.1. Ruthi’s Account: Connecting Theory and Practice
The course I describe here is entitled “Student-teaching workshop” and accompanies the eight prospective secondary school mathematics teachers (PTs) currently doing their student teaching. In ‘regular’ times, the course meets once a week at the university to discuss issues that arise from the PTs’ field experience. Each PT is required to hand in a report of a situation that occurred in one of the observed lessons that raised a dilemma for that PT. This dilemma might be mathematical (e.g., the PT has a mathematical question), pedagogical (e.g., the PT would like to discuss how to respond to students’ mistakes), affective (e.g., the students do not seem motivated), social norms, or related to classroom management. Each PT (with permission from the class teacher and students) audio-records a mathematics lesson, taking pictures of the whiteboard as the lesson proceeds. The PTs then transcribe and analyze the recording in terms of the mathematical ideas discussed in the lesson and the pedagogical moves of the teacher. During the course lesson, each PT hands out the transcription of the situation to the other PTs, who first discuss it in pairs, followed by a whole group discussion led by me.
- Michal:
- You did not present your own resources. I am sure that you have resources which you bring to this lesson.
- Esther:
- That’s it? You have additional resources such as your own analysis of case studies. Also, in the past you wrote situations for use in professional development.
- Ruthi:
- Esther, you are right. I have a lot of experience analyzing situations that students bring, and inventing situations that can be used to enhance PTs’ mathematical and pedagogical content knowledge. Michal, Esther said the same thing and I replied to her.
The university and schools closed without warning before any PT had a chance to report on any situation or dilemma. The first decision I needed to make was how to proceed without students being able to report on their student teaching. I chose for this lesson to send them a homework assignment to submit within a week. The assignment included a ‘transcribed’ hypothetical situation that involved a teacher and three students discussing a problem in plane trigonometry. PTs were required to read the transcription and point out two correct and two incorrect mathematical ideas found in the transcription.
- Esther:
- In your analysis of your revised lesson, you wrote that the first aim of your original lesson was no longer relevant. That’s quite a strong statement.
- Ruthi:
- Because they bring authentic situations… Now, it’s more… They read a transcript of a story. Sometimes, what they bring has less mathematics, but I can always find a way to raise mathematical issues.
- Ruthi:
- I believe that students learn from their experiences in class.
- Esther:
- What do you mean?
- Ruthi:
- They observe, then transcribe, then analyze what they saw. They complain about having to transcribe. But then, they realize how much they learn from that. They tell me, “When I wrote the transcription, I said to myself, wow, I didn’t even notice that happened.”
- Esther:
- So, maybe in the original lesson, another goal was for them to reflect.
- Michal:
- But it is a reflection on what someone else is doing (meaning the classroom teacher and students).
- Esther:
- Or maybe the aim is to teach them how to learn more from their observations.
- Ruthi:
- Yes. Exactly.
Summarizing Ruthi
5.2. Michal’s Account—Focusing on Student Interactions
The course I focus on is entitled “Mathematics and science education research methodologies.” This year, 45 students are participating in the course, some studying towards a secondary school teaching license and some who are graduate students. In the past, this lesson was the first lesson of the semester, but turned into the first and second lessons during the crisis this year. In the past, the general aim of the lesson was to characterize what counts as research in mathematics education. I implement a variation of a “think-pair-share” strategy [30]. Students write down their own thoughts regarding what counts as research, and then share what they have written with 4–5 other students, comparing and contrasting their answers, coming up with agreed upon characterizations. I then hand out to each student in the group a written quote from different mathematics education researchers (e.g., [31]), characterizing research in mathematics education. After sharing their quotes, discussing in the group if their former characterizations should be changed, they put up on the board their answers and a whole class discussion follows.
Due to the COVID-19 virus, I had to teach the first lesson of the semester online. Because I could not imagine how I would implement the think-pair-share strategy in a virtual classroom, or how I would hand out quotes to different students in real time online, I had to decide on another way to introduce my students to what it means to conduct research in mathematics education. The first decision I made was to upload to the course website four different papers [32,33,34,35] and assign each student one paper to read and summarize. Thus, the first lesson was asynchronized. The next lesson was synchronized, and I decided to implement a “jigsaw” lesson format. I broke up the students into rooms based on the paper they read, where they discussed the paper and summed up what they wished to present to those who had not read that paper. Next, I mixed up the students such that in each group there was one representative of each paper. Each group was tasked to answer the following two questions (reporting their answers on a google slide that I had prepared and shared with students prior to the lesson): Are the four papers coherent? What is good research in mathematics education? I visited the separate rooms to check on progress. A final whole class discussion was not conducted.
- Michal:
- Ruthi is right. I didn’t realize the change in my goals until she pointed it out. There is a difference.
- Esther:
- You wrote that your second goal (see G2 in Table 2) was not achieved. Was this (G2) an explicit aim of your revised lesson?
- Michal:
- I did not meet this aim.
- Esther:
- But did you want to achieve this aim?
- Michal:
- It seems that less so. It moved to the background. It disappeared.
- Esther:
- Because your new goal was characterizing what is good research in mathematics education?
- Michal:
- Correct.
In the original lesson, I began with what each student knew, added to that other students’ knowledge, and then finally, added experts’ (in this case researchers’) knowledge. In the revised lesson, I began with the researchers. On the one hand, there are not many opportunities in this course for students to articulate what they know; thus, the original lesson has merit. On the other hand, I am not sure when the opportunity would arise for students to read the four, in my opinion, important papers written by Cai et al. [32,33,34,35]. Perhaps in the future, I would begin in the same way as I did in the original lesson and end the course with the four editorial papers, as a summary for the course. Yes, this might be a solution.
Summarizing Michal
5.3. Esther’s Account—Must I Really Revise the Lesson?
The course I teach is entitled “Affective aspects of learning and teaching mathematics”. I have 18 students comprising secondary school PTs, as well as practicing teachers in the mathematics education graduate program. The lesson I describe here was the second lesson of the semester. During the first lesson, participants were introduced to the notions of beliefs, attitudes, emotions, and values, and in particular, beliefs related to the nature of mathematics and learning and teaching mathematics.
In ‘regular’ times, the second lesson focuses on beliefs. Participants divide themselves into groups without interference from me. Usually, students with the same mother tongue work together, perhaps because it easier for them to communicate with each other. Some students know each other from previous courses and choose to work together. Each group must design a mini-study that will investigate beliefs of school students or teachers related to a specific area that interests the group. Some groups focus on students while others focus on teachers. Students may refer to sample questionnaires that I gleaned from previous studies and have made available on the course website. I go from group to group to listen, to ask questions, and to offer suggestions. While allowing the groups’ autonomy, I share ideas from one group with another. The outcome of this lesson is that each group has a short questionnaire that I have reviewed and approved of, and that they will implement with students or teachers, the results of which they will present later in the course.
This year, between the first and second lessons, the university closed, and I decided to implement the same lesson in my new virtual classroom. Having no experience hosting a virtual lesson, my teaching assistant and I practiced using the features of the virtual classroom. For most of my students, this was their first experience working in groups in a virtual classroom.
The lesson began by explaining the task and reminding the students that sample belief questionnaires could be found on the course website. Then, I took the quicker technical path, and had the system randomly assign participants to four different rooms. I first went from room to room to make sure that everyone understood the instructions and how to communicate with each other using the new tool. The second time I visited each room was to make sure that participants were active, to see if anyone had questions, and offer some suggestions. Because I could not visually assess at once that all participants were indeed engaged with the task, I felt the need to jump from group to group, which left me less time to evaluate the products of their collaboration. Thus, at the end of the lesson, it turned out that all groups had decided to focus on students’ beliefs. Furthermore, I was not sure that each group had satisfactorily completed the task of designing a short questionnaire. Thus, I requested each group to email their questionnaire within 2–3 days for my approval.
- Esther wrote:
- In theory, the goals (referring to G1–G4) should have been the same. However, they were not. Honestly, my goal was that the students and I would be able to utilize the new platform and that I would succeed in breaking them into groups without them complaining about who they were with.
- Ruthi wrote:
- Why is there a contradiction?
- Esther replied:
- It’s not really a contradiction, but when I was reflecting [on the revised lesson], this was my first goal. The other goals were less important. I don’t know.
- Esther:
- The goals of the revised lesson are actually the same goals, and in addition (laughs a bit), my goal was to survive the lesson (Michal laughs).
- Michal:
- In an ordinary lesson, you have no doubt that you will be able to divide them into groups. Suddenly, this was an issue.
Summarizing Esther
6. Discussion
7. Epilogue
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Original Lesson | Revised Lesson | |
---|---|---|
Goals | G1: To allow PTs the opportunity to present a mathematical situation observed during their field work in which dilemmas arose. G2 **: To learn how to observe lessons. G3: To raise PTs’ awareness of social and affective issues that arise in the mathematics classroom. | G6: To raise PTs’ awareness of various (mis)conceptions held by secondary students learning mathematics, as well as various correct and incorrect ways students solve problems. G7: To discuss with PTs the use of students’ mistakes as a springboard for learning. |
G4: To promote PTs’ SMK and PCK. G5: To discuss ways of increasing mathematical discourse in the classroom. | ||
Resources | R1: Situations that students bring from their field work. | R4: Hypothetical situations that I had previously authored. |
R2 *: Experience as a teacher mentor. R3 *: Experience authoring hypothetical situations for mathematics and methods courses and for professional development. | ||
Orientations | O1: Having PTs present authentic situations from their own field work raises motivation and interest in learning. | O4: Mathematical mistakes may be used as a springboard for promoting mathematical knowledge. |
O2: We learn by doing (e.g., by observing lessons, transcribing, analyzing a situation, and presenting a situation). O3: Using case studies or mathematical situations is a way to bridge academic courses with field work. |
Original Lesson | Revised Lesson | |
---|---|---|
Goals | G1: To present students with different characterizations of research in (mathematics) education. G2: To raise students’ awareness of the complexity for defining research and the benefits of characterizing research. | G5 **: To familiarize students with the way (influential) scholars characterize good research in mathematics education. |
G3: To have students actively participate in the lesson. G4: To have students share ideas with their peers and recognize the difference between thinking on one’s own and thinking with others. | ||
Resources | R1: Past scholarly essays regarding what is considered research in education [31]. | R4: Four editorial papers [32,33,34,35]. R5: The Zoom feature that allows the host to manually split the students into separate rooms. R6: My experience with the jigsaw pedagogical format. |
R2: My experience as a mathematics education researcher and the ways in which I characterize research in mathematics education. R3 **: Knowledge about what students had learned during the first semester. | ||
Orientations | O1: Active learning is better than listening to a lecture. O2: Talking with peers in a small group situation can enhance learning. O3: When working in groups, it is important for each student to have a unique role in the group, giving each person a reason and need for participating. O4: The way I organize lessons can set an example and possibly influence the way teachers can organize their classrooms in school. |
Original Lesson | Revised Lesson | |
---|---|---|
Goals | G5: That the students and I would be able to utilize the online platform and work in breakout rooms without complaints. | |
G1 *: To have students think about the types of beliefs school students might hold regarding learning mathematics. G2 *: To have students think about their own beliefs regarding teaching and learning mathematics. G3 *: To familiarize students with the types of questions to ask when inquiring into others’ beliefs. G4 *: To encourage students to share their experiences, either as learners of mathematics or as teachers of mathematics, related to beliefs. | ||
Resources | R1: A classroom with chairs that can be moved so students can sit and work together; a standard white board; computer and projector. | R4: A virtual classroom. |
R2: My experience as a researcher investigating teachers’ beliefs for teaching mathematics and my knowledge of studies that investigated teachers’ and students’ beliefs. R3: Various questionnaires taken from journal articles related to beliefs when learning and teaching mathematics. | ||
Orientations | O6: I can give the same lesson in a different environment, without much change in planning, and it will still be successful. (Note: If you believe in a flexible lesson plan, then it is not difficult to believe that changing the physical environment will not change the essence of your lesson.) | |
O1: It is important for teachers to be aware of their own mathematics education beliefs. O2: To raise awareness of beliefs, teachers need to share them with others and think of what they would want to know about others’ beliefs. O3: Teachers need to recognize that students also hold beliefs and attitudes regarding mathematics, and how mathematics should be taught. O4: I believe in interweaving theory and practice. O5: I believe in a flexible lesson plan. (My lessons are not rigidly formatted, and students have a great deal of autonomy because I believe that they will invest more effort into a topic that interests them.) |
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Levenson, E.S.; Barkai, R.; Tabach, M. Mathematics Teacher Educators’ Decisions in a Time of Crisis: Self-Reflections as a Basis for Community Inquiry. Educ. Sci. 2023, 13, 453. https://doi.org/10.3390/educsci13050453
Levenson ES, Barkai R, Tabach M. Mathematics Teacher Educators’ Decisions in a Time of Crisis: Self-Reflections as a Basis for Community Inquiry. Education Sciences. 2023; 13(5):453. https://doi.org/10.3390/educsci13050453
Chicago/Turabian StyleLevenson, Esther S., Ruthi Barkai, and Michal Tabach. 2023. "Mathematics Teacher Educators’ Decisions in a Time of Crisis: Self-Reflections as a Basis for Community Inquiry" Education Sciences 13, no. 5: 453. https://doi.org/10.3390/educsci13050453