Mathematics Teacher Educators’ Decisions in a Time of Crisis: Self-Reflections as a Basis for Community Inquiry

Abstract

A time of crisis is a time of uncertainty, when many decisions need to be made. This study combines self-reflection, along with community inquiry, as three mathematics teacher educators recount a lesson that they taught in the past and how it was changed due to the COVID-19 crisis. Decisions were analyzed in terms of goals, orientations, and resources. The findings showed that the key issue was the immediate requirement to change one’s regular routine. For some, resources were replaced. For others, dominant orientations receded to the background, and new goals were set. A final reflection conducted after returning to the classroom revealed how challenges during the crisis led to change and the adoption of new goals both during and after the crisis, clarifying our values and leading to the use of additional resources today.

1. Introduction

The spring semester began for us on 3 March 2020. The three of us, the authors of this paper, teach mathematics education courses at Tel Aviv University, a major university in a major city in Israel. Our student body is mixed. It includes prospective secondary mathematics teachers studying towards a teaching certificate, as well as practicing elementary and secondary mathematics teachers studying in the graduate mathematics education program. During the first week of the semester, the COVID-19 virus had already made its appearance in Israel, but schools and universities were still open. By the second week, schools and universities were closed, and we were teaching online. This paper is written from our viewpoint as mathematics teacher educators (MTEs).

Schoenfeld [1] called the practice of teaching a “well-practiced” domain “in which individuals have had enough time to develop a corpus of knowledge and routines that shape much of what they do” (p. 457). Routines may be individual or social [2]. On an individual level, a routine may be considered a skill that contributes to an individual’s functioning. On a social level, routines are acknowledged modes of action that make up the practice of teaching mathematics, based on collective knowledge. The COVID-19 pandemic has challenged this knowledge, as well as our beliefs, and has forced us to break with routines. It has also forced us to make decisions quickly. On what basis were these decisions made? Reflecting on these decisions affords us the opportunity to critically analyze our practice and grow. This paper continues in the tradition of previous MTEs who have reflected on their own professional development (e.g., [3,4]). Using Schoenfeld’s [5] theory of decision making, each author chose one lesson and analyzed the differences between the lesson as she had planned and carried it out in the past, and the changes that had to be made the moment crisis hit. We examine the aims, resources, and orientations that influenced those decisions.

A time of crisis is a time of high uncertainty [6]. While teaching inherently involves a certain amount of uncertainty (does one ever really know what to expect when walking into a classroom?), there are different ways of dealing with uncertainty. In her review of prior research, Helsing [7] found that some teachers reacted negatively to uncertainty by compromising standards, relying on familiar routines at the expense of encouraging student participation, and by ignoring or denying the uncertainties they face. On the other hand, a positive stance to uncertainty can lead to increased knowledge, and new expertise through investigation, experimentation, and recognition of problems that arise. One way to deal constructively with uncertainty is by acknowledging its presence, collaborating with others, and seeking additional information [8].

The three authors of this paper have been researching and working together to educate preschool, primary, and secondary teachers to teach mathematics for over ten years. As such, we consider ourselves members of a community of practice, and thus it was natural for us to share our concerns and inquire, together, into our practice. We ask ourselves how awareness of our shared aims, resources, and beliefs as a community of MTEs can help us prepare for future times of uncertainty and how it can help us prepare prospective and practicing teachers for teaching mathematics in times of uncertainty.

2. Decision Making

Decision making is a crucial skill for all professions. For teachers, decisions made while planning and implementing a lesson can impact on a multitude of teachers’ actions, such as how a topic is introduced [9] and the examples presented to students [10], as well as on student outcomes, such as their progression towards learning goals [11]. Studying what lies behind teachers’ decisions may help us analyze essential components of teacher practice to be addressed and developed [11]. In this study, we adopt Schoenfeld’s [1] framework for characterizing decision making in the moment. While this study does not analyze the decisions we made during our lessons, because of the COVID-19 crisis situation and the speed with which we had to make difficult decisions, we consider our decisions very similar to those made in the moment.

For over two decades, Schoenfeld [1,12,13] has worked towards developing and employing a theory of teaching in context to study how teachers make choices when they are engaged in the act of teaching. His early theory modeled teaching as a function of teachers’ goals, knowledge, and beliefs. Goals are the aims we have and what we set out to achieve, consciously or unconsciously. Schoenfeld [13] described three types of goals. Overarching goals are the “consistent long-term goals that the teacher has for a class. They tend to manifest themselves frequently in instruction” (p. 50). The second type are called major instructional goals. These goals “may be oriented toward content or toward building a classroom community. Content goals tend to be more short-term, reflecting major aspects of the teacher’s agenda for the day or unit” (p. 50). Finally, there are local goals related to specific circumstances. Schoenfeld [1] claimed that while a teacher’s choices may be made in the moment, they may still be considered rational in that they represent choices consistent with that teacher’s beliefs, knowledge, and high-priority goals.

More recently, Schoenfeld [1,5] modified and extended his original theory. Knowledge was amended to resources, which include not only knowledge in its various forms, such as subject matter and pedagogical content knowledge (SMK and PCK) and its derivatives (e.g., [14,15]), but also “the social and material resources that are available to the teacher when teaching” (Schoenfeld, [1], p. 459). For mathematics teacher educators (MTEs), resources include theories and previous research studies [16], as well as case studies [17]. In one study [18], theories of mathematical knowledge were introduced to prospective and practicing teachers, who in turn used those theories to analyze cases. Cases may be authentic [18] or they may be hypothetical [19]. Like the use of theories, the use of cases is dependent on the goals of the MTE. Pang [20], for example, used video cases to help prospective teachers focus on mathematical accuracy and mathematical communication, rather than focusing on the classroom atmosphere and general teaching strategies. Tirosh et al. [18] used cases to bridge theory and practice, specifically focusing on the use of examples in mathematics classes.

In Schoenfeld’s [1,5] current goal-oriented decision-making theory, beliefs were amended to orientations and included “beliefs, values, preferences, and tastes” (p. 460). While beliefs involved the attribution of some external truth to a set of propositions, values refer to “personal truths or commitments cherished by individuals. They help motivate long-term choices and shorter-term priorities” [21] p. 135). Thus, choosing which resource to draw on in a specific situation may be dependent on associated orientations. In one study [22], for example, MTEs engaged kindergarten teachers with mathematical tasks that were beyond the level of early childhood mathematics because they believed that teachers should be aware of how mathematics learned in kindergarten is connected to mathematics learned during school years. Thomas and Yoon [10] employed fine-grained analysis of the interactions between a secondary school teacher’s goals, resources, and orientations in a calculus class. They found that at times, the teacher produced a new goal that was an amendment of one of the original goals, but which blended in with the other goals. The authors further claimed that the new goal was determined mostly by the teacher’s orientations, taking into consideration current resources, thus influencing the teacher’s decisions.

How does the process of decision making work? According to Schoenfeld [1], decisions made in the moment take into consideration the specific context and the specific resources and orientations that come into play at that particular point. Thus, the first step is that an individual enters a context and orients themself to the situation. At this point, “Certain pieces of information and knowledge become salient and are activated” (p. 461). In other words, we enter a situation with certain previous orientations and then, depending on the context, some of those orientations rise to the surface, while others retreat. These orientations then guide us in choosing which resources to use. Based on orientations and resources, we establish goals, which may be new goals, adjusted goals, or even the same goals that we had originally, which at this point are acknowledged anew. Decisions are based on these goals.

Of importance to this study is Schoenfeld’s differentiation between two possible scenarios. The first scenario, in which the actors are familiar with the situation and the decision-making process, is relatively automatic, often carried out on an unconscious level. Implementing actions towards reaching some goals usually means relying on routines and schema and enacting well-practiced scripts. The second scenario is when one finds themselves in an unfamiliar situation. In that case, the individual’s decision-making process is more conscious and involves an assessment of the use of available resources, in accordance with orientations. In both cases, what follows is implementation and monitoring. Instead of seeing orientations and resources as leading to goals, Thomas and Yoon [10] state that orientations shape the way we see the world, giving rise to goals, which are then attained by seeking out available and pertinent resources. This might be the case in a crisis when what were reliable sources become unattainable.

In the current study, we use Schoenfeld’s decision-making theory to analyze decisions we made due to the COVID-19 crisis. We begin with multiple case studies and continue by reporting on community inquiry.

3. Communities of Inquiry

As mentioned previously, the three authors of this paper have been teaching and researching at the same university for several years. However, what makes us a community of practice is not our place of work, but that we “are actively involved in communicating with each other about important issues and working together toward common goals” (Forman, [23], p. 104). Wenger [24] described three dimensions of communities of practice: mutual engagement, a joint enterprise, and a shared repertoire of resources. As Goos [25] pointed out, mutual engagement need not require homogeneity, but may involve disagreements and tensions. Negotiating a joint enterprise entails coordinating complementary expertise. Finally, participants share and reconstruct their repertoire of resources, including terms and expressions, shared artefacts, routines, and experiences that facilitate the attainment of shared goals [26].

Belonging to a community of practice not only entails engaging with the practice, but also alignment with norms, conditions, characteristics, and expectations of the practice [24]. Jaworski [27] suggested a form of critical alignment, where participants align “with aspects of practice while critically questioning roles and purposes as a part of their participation for ongoing regeneration of the practice” (p. 188). When participants adopt a critical stance and when the community’s way of being is to reflect on, research, and critique its current practice, it may be said to be a community of inquiry. Participating in a community of inquiry requires participants to “look critically at their practices as they engage with them, to question what they do as they do it, and to explore new elements of practice” ([28], p. 103).

Our research questions are as follows:

1. According to the MTEs’ reports of two versions of the same lesson, before and during the crisis, what were the goals, resources, and orientations that accompanied the decision-making process for each lesson?

a. Did the goals change? Did we lean on different resources? Which orientations came to the fore, and which moved to the background?

b. How did inquiring as a community contribute to each person’s reflection?

2. As a community of inquiry reflecting on our individual instructions at a time of uncertainty, what have we learned that may inform future practice?

4. Methodology

There were three parts to this study: personal self-reflection, self-reflection as a basis for community inquiry, and community inquiry. The type of self-reflection carried out in the first part may be termed reflection-on-action [29]. This type of reflection “can be thought of as a person’s posterior analysis of his/her own actions. In such reflection, the individual, free of the restrictions and demands of the specific situation, can systematically apply his/her conceptual tools and analytical strategies to understanding and assessing his/her past actions” (García et al., [29], p. 2). The actions we chose to reflect on are those we took when the university obliged us, overnight, to teach online. Prior to writing down our self-reflections, we agreed to use Schoenfeld’s theory of decision making as the conceptual tool to analyze our actions. As such, each of us chose one lesson that we revised because of the immediate physical closing of the university and wrote down our account. The account included a description of how that lesson was conducted in the past, followed by an analysis of the goals, resources, and orientations that led to the implementation of that lesson. The account continued with a description of the revised lesson, accompanied by an analysis of the goals, resources, and orientations that led to the revised lesson. This concluded the first step of personal self-reflection.

The second step was emailing each other our stories and analysis so that each could read and write comments (using the “track changes” tool on Word) on what the others wrote. The commented reports were then sent back to the author, who read the comments and replied to the comments made by the others. The double-commented written reports are our first data source.

Three virtual meetings followed (via the Zoom platform), each devoted to one account, where that person’s self-reflection and everyone’s comments served as the basis for further community inquiry. The first aim of those meetings was to clarify and challenge a person’s account and analysis. Inquiring led to fine tuning and furthering self-reflection as others continued to ask probing questions. The second aim of each meeting was to discuss together what we think that person might adapt from the revised lesson and use in the future. Our guiding questions were as follows: (1) Next year, when you go back to the classroom and plan to give this lesson, would you revert to your original lesson, or are there elements of the revised lesson that you would implement in the future? (2) Knowing that crises unfortunately occur and without warning, what could you do to plan ahead? All virtual conversations were recorded, and these recordings served as a second set of data.

The final step of our study took place as another virtual meeting, this time focused on what we learned from inquiring into our practice that might help mathematics teachers prepare for a future crisis.

5. Findings

5.1. Ruthi’s Account: Connecting Theory and Practice

The following are excerpts of Ruthi’s written description of the original lesson she chose to present:

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.

Ruthi’s original analysis for this lesson may be seen in

Table 1. Ruthi’s analysis of her goals, resources, and orientations.

	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.

* Points added as a result of the written community inquiry; ** Points added during the virtual community inquiry.

in statements that appear without an asterisk. An example of Michal’s and Esther’s written comments to Ruthi’s analysis of her resources, as well as Ruthi’s written replies, are given below and are indicated in Table 1 with one asterisk.

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.

In the second part of the self-reflection, Ruthi described her new lesson due to the crisis:

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.

In Table 1, two asterisks indicate a change that was made due to the virtual community inquiry. For Ruthi, there was quite a shift in goals, and this was a focus of the inquiry as follows:

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.

After further discussion, the MTEs discussed resources and then orientations. During the discussion of O2 (we learn by doing), the discussion returned to goals.

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.

Inquiring together into Ruthi’s practice brought to light for Ruthi another goal (G2) that she had not considered at first.

Summarizing Ruthi

Ruthi revised her entire lesson because she lost her main resource, which was authentic situations brought by the students. Before the collective inquiry, Ruthi listed the students’ authentic situations as her only resource in the original lesson and her self-authored hypothetical situation as her only resource in her revised lesson. For Ruthi, changing resources when required to revise her lesson was the basis for her decision making. Each of these resources was accompanied by various orientations, some the same and some different. However, when her major resource was removed, Ruthi found a different resource, one with which she was familiar, to plan her revised lesson. This resource led to new goals and brought to light additional beliefs.

5.2. Michal’s Account—Focusing on Student Interactions

The following are excerpts from Michal’s description of the lesson she chose to present:

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.

Michal’s original analysis for this lesson may be seen in

Table 2. Michal’s analysis of her goals, resources, and orientations.

	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.

** Points added during the virtual community inquiry.

in statements that appear without an asterisk. The second part of Michal’s self-reflection was as 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.

When commenting on Michal’s written report, Ruthi noted that the question posed to students during the revised lesson (what may be considered good research in mathematics education) differed from the question posed to students in the original (how can research in mathematics education be characterized). Ruthi wrote to Michal: “This question is different from the one you wrote in the original lesson. It seems that your goals changed a bit—no?” Michal replied, “You are correct. The terminology (of the question) is a bit different… because of the papers I gave the students to read”.

This was brought up and discussed during the virtual community inquiry (see Figure 1 which illustrates how the community inquiry was actualized).

Michal:

Ruthi is right. I didn’t realize the change in my goals until she pointed it out. There is a difference.

Later, the issue of goals came up again:

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.

Although G2 was a goal for the original lesson, G5 became an acknowledged goal of the revised lesson due to the community inquiry. Regarding decision making, we see from both Ruthi and Michal that decisions might have been taken consciously, but goals were not always set up front and only came to light when reflecting together as a community.

During the virtual community inquiry, we asked Michal if in quieter times, she would revert to her original resources and lesson plan. Her answer was as follows:

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

Like Ruthi, Michal also revised her entire lesson, but for a different reason. For Michal, the main issue was not the resources, but her strong belief in learning as increased participation in a community [24], in this case, the community of mathematics education research. Her orientations that accompanied both the original and revised lessons were the same. These orientations transformed into goals, which guided her decision to break up students into rooms where they could discuss the papers. Previous studies have also pointed out the strong connection between beliefs and goals. For example, Aguirre and Speer [36] demonstrated how shifts in goals during a lesson may be explained by examining that teacher’s beliefs. However, for Michal, her beliefs were not merely the background for her goals; they were her goals. Her belief that talking with peers in a small group situation can enhance learning, transformed into the goal of having students recognize the difference between thinking on one’s own and thinking with others. This was an overarching goal [13] for Michal, not just a way of implementing a lesson.

5.3. Esther’s Account—Must I Really Revise the Lesson?

The following is an excerpt from Esther’s description of the lesson she chose to present:

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.

Esther’s original analysis for this lesson may be seen in

Table 3. Esther’s analysis of her goals, resources, and orientations.

	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.)

* Points added as a result of the written community inquiry.

in statements that appear without an asterisk. The second part of Esther’s self-reflection was as follows:

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.

In Table 3, note that G1–G4 are written as goals for both lessons. However, Esther had written those goals only for her original lesson, which Ruthi then commented on.

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.

During the virtual community inquiry discussion, these goals came up again.

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.

During the virtual community inquiry, another point of discussion was how to divide students into groups as our classes are multi-cultural and include students with different mother tongues. Michal stated that she always makes sure to form multi-lingual and multi-cultural groups, while Esther believed in letting the students choose with whom they work. When thinking about the future and returning to classrooms, Esther stated, “Maybe I would try to mix up the groups. I wonder how it might affect their learning, working with students from different cultural backgrounds. Now, during the crisis, is not a time to judge if it works, because things are so strange. But in a regular year… maybe”.

Summarizing Esther

Unlike Ruthi and Michal, Esther decided to retain her original resources (not linked to the physical classroom) and most of her orientations. Unlike Michal, she did not specify the attributes of the virtual classroom she would use. Unlike Ruthi, who changed her lesson goals because her resources changed, Esther did not change her goals. Hanging on to familiar routines when terrain is unfamiliar is usually thought of as a negative reaction to crisis [7]. However, Esther was not necessarily running away from the crisis. Instead, her reaction was in line with her belief of flexible lesson plans.

6. Discussion

As we pointed out in the beginning, a time of crisis is a time of uncertainty. To untangle decision making in times of crisis, we turn to the first aim of this study. What were the goals, resources, and orientations that accompanied the decision-making process, as a particular lesson plan implemented in the past was revised due to the crisis?

For Ruthi and Michal, one of their first decisions was to search for new resources as they deemed their previous resources either unattainable (Ruthi) or unusable (Michal). That Ruthi and Michal were quickly able to find new resources exemplifies that it is not merely academic knowledge that interplays with orientation and goals, but “knowledge inventory” [1], that is, knowledge, information, and materials. In Ruthi’s case, this inventory included previously written hypothetical situations. In Michal’s case, it included having current information regarding research studies.

Esther made the decision to retain her resources, exchanging only her physical classroom with a virtual classroom. Furthermore, her overarching orientations regarding the importance of raising teachers’ awareness of their mathematics beliefs, as well as their beliefs regarding teaching and learning mathematics, remained steadfast. Ruthi’s account, however, exemplifies how different situations bring different beliefs to the fore [9]. In her original lesson, Ruthi stressed her belief in the importance of students learning from their field experiences. This belief was challenged during the pandemic. Ruthi’s solution to bring to class hypothetical situations that she had previously authored still allowed students to discuss together situations that arise in school. Furthermore, it went along with another belief, that of using cases and mathematical situations to bridge academic courses with field work. In her revised lesson, she stressed her belief in learning from mathematical errors. These beliefs lived side by side without any conflict until as a community, we confronted them both and, with Ruthi, we inquired which, in the future, she would adopt. Her reply that she would revert to using situations brought by the students shows that having students share their field experiences is more than a passing belief, but a value that she cherishes. As noted by DeBellis and Goldin [21], values motivate our long-term choices.

Community inquiry was especially important when delineating our goals. Michal only recognized that a goal of her lesson had changed due to our inquiry. During crisis situations, people often set new goals, either because new goals take higher priority [37] or because the situation is perceived as an opportunity to set new goals [7]. After acknowledging that she indeed had set a new goal, Michal, along with the community, discussed adopting this goal in the future. Thus, when a community inquires into their practice, they may assist each other in setting new goals.

Some of the MTEs’ goals remained stable. Goals such as promoting PTs’ SMK, and PCK (Ruthi) and raising teachers’ awareness of their own and their students’ mathematics beliefs (Esther) are major goals of mathematics teacher education [38], goals which do not change even in a time of crisis. Michal’s goal of having students share their ideas with their peers in small group settings is in line with current mathematics education research that calls for student collaboration for sensemaking [39]. Deciding what should not change is equally important as deciding what can change.

Community inquiry made Esther reflect on her routines for group work. In recent years, much has been said regarding collaborative mathematics learning, student groupings, and mathematics teacher education (e.g., [40]). By sharing grouping routines and experiences as a community [26], Esther began to consider further goals in mathematics teacher education, such as addressing issues of diversity. In a course aimed at raising teachers’ awareness of beliefs towards mathematics and teaching mathematics, multi-cultural groupings can offer teachers a chance to confront their beliefs regarding diversity and equity in mathematics education [41] and how students’ lives outside the classroom can impact their activity and experiences in the mathematics classroom [42].

The second aim of this study was to reflect on what we learned as a community of inquiry that may inform our future practice and help us, as well as prospective and practicing teachers, make decisions during a crisis. Looking back at each of our coping strategies, we each reverted to some familiar routine to persevere through the moment. Ruthi implemented a familiar routine with a familiar resource. Michal reverted to the familiar “jigsaw” lesson format, and Esther attempted to change as little as possible. Leaning on familiar routines can help us persevere through crises. Thus, one way to prepare for the future is to acknowledge the existence of various routines and examine those routines for flexibility and adaptability. For example, although Ruthi’s routine for her lesson included having students discuss and analyze their field experience, she was able to bend this routine and have students analyze hypothetical situations. We may also say that Ruthi, as well as Michal, flexibly used resources. Instead of focusing on specific resources, we may look at general resources. As MTEs, one of our resources is theories of mathematics education research [16]. Michal changed the specific theories of mathematics education research she introduced to students, but the overall resource remained. For Ruthi, the overall resource was using case studies [17]. Flexibly, she turned from using authentic cases to using hypothetical situations. Thus, instead of “breaking” with routines, we can adapt routines to different circumstances.

Reflecting on the entire process, i.e., self-reflections, community inquiry, and analyzing decision making, two additional results stand out, results that we would recommend MTEs and mathematics teachers to consider. First, we regard the methodology of this study. As Schoenfeld [5] suggested, “having teachers reflect on their decision making—thinking about the options they might have considered and did not, about why they made the choices they did, and about how those choices reflect their beliefs, goals, and knowledge—can serve as a catalyst for professional growth” (p. 153). Yet, as seen in this study, self-reflection is not always enough, nor is community inquiry a substitute for self-reflection. Instead, self-reflection can serve as a basis for community inquiry, one that takes into consideration individual resources, orientations, and goals, as well as the community’s resources, orientations, and goals. We also add that forming a community of inquiry, in this case a community of MTEs, requires all participants to recognize the right of each to express themselves honestly and freely, without fear of repercussions [43]. This involves trust. Such trust is found when teachers show mutual support and encouragement [44], such as when Esther and Michal showed Ruthi that she had many more resources than she at first admitted.

Second, a time of uncertainty can be a time of growth. “Routines are mechanisms that intend to make teaching and learning mathematics predictable and safe” ([2], p. 102). Yet, safety that can obstruct change. Sometimes, routines need to be discarded, not only teaching routines, but how we go about making decisions. Thomas and Yoon [10] stated that we seek out available resources to attain goals. However, as seen in this study, in a crisis, attainable resources may come first and goals may come second. During a crisis, we may also seek out new resources from our community. Thus, participating in a community of practice can be critical. In a study of people stranded during a transit crisis, it was found that in order to make decisions, people purposely drew together to pool resources in the construction of knowledge, while integrating information [45]. As a community of inquiry, this is what we do.

7. Epilogue

It has now been over a year since we all returned to our classrooms. We do not necessarily teach the same courses as we did previously. Yet, we continue to reflect and ask ourselves the following: how has our work in the classroom changed after our experience during the crisis?

For Ruthi, it may be said that her belief in having students share their field experience was indeed a stable value. During the crisis, she had no choice, but as soon as the pandemic was over, she went back to having students observe classrooms, transcribe authentic situations, and analyze them. However, Ruthi now also teaches an asynchronous course for prospective mathematics teachers and has adopted the use of hypothetical situations in this course. Reflecting on the difference between authentic and hypothetical situations, Ruthi noted that the hypothetical situations are more focused on the mathematical content and pedagogical content knowledge than situations brought from the field. Incorporating both resources has raised further questions for Ruthi as follows: How may reading and analyzing hypothetical situations raise future teachers’ noticing skills [46] during their observations in the field? How may working with hypothetical situations help the students reflect and analyze authentic situations? For Ruthi, her belief that hypothetical situations are at best a second choice when authentic situations are unavailable was challenged. Instead, it is the combination of both resources that has the potential to increase students’ mathematical and pedagogical mathematical knowledge.

Michal has not taught the same course again since the pandemic. However, the resources that she adopted during that year, including the four papers authored by Cai et al. [32,33,34,35], have become part of her repertoire, especially as she mentors graduate students in mathematics education who are writing a thesis. While Michal read those papers when they were first published and always thought about integrating them into her teaching, it took the discussion of these papers with students during the pandemic to realize that these papers can help students conceptualize what it means to conduct (good) research in mathematics education. Regarding her belief that learning occurs through active participation, Michal acknowledges today that this is a value that withstood the difficulties of implementation during the pandemic; today, she feels as strongly as ever about small group work. Finally, Michal acknowledges that the move to Zoom during the pandemic was a shock, and that she could not fathom teaching online prior to the crisis. That belief has been challenged. Today, if she or her students cannot meet on campus for some reason (e.g., snowstorms blocking the road), she seamlessly switches to teaching online, showing that the online platform has become a handy resource, which also affected her beliefs regarding the positive role of online teaching.

For Esther, addressing issues of diversity and equity in all her courses was something that she continued to reflect upon and continues today to incorporate in her teaching. This issue arose during the collective inquiry stage when discussing how breakout rooms caused Esther to change her ways of grouping students during shared assignments. Her belief in offering students the autonomy to make decisions about group learning was challenged during the pandemic. As a result of her experience with random groupings and breakout rooms during the pandemic, Esther found other ways to offer her students autonomy in their learning and has now added a new goal to her previous goals: that her students will understand how culture and language impact one’s beliefs regarding teaching and learning mathematics. To this end, Esther assigns the students to groups, ensuring diversity. Furthermore, Esther has acquired additional resources in the form of research studies that investigate cultural aspects of mathematics education (e.g., [47]) and has added them as required reading for her students. Students still say that they would rather work with others who have the same mother tongue or live in the same village or city. After all, these are the students’ routines. However, we do not necessarily need to wait for a crisis to change our routines, and that is a lesson worth learning.

As we saw in this study, and in line with Schoenfeld [1], orientations, goals, and resources act in synergy. In times of uncertainty, resources may need to be replaced, some orientations may come to the fore while others take a step back, and we might set new goals. When the crisis has passed, instead of reverting to what was, we may reflect on what we have gained.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.