Ex) Article Title, Author, Keywords
Ex) Article Title, Author, Keywords
2021; 31(3): 231-255
Published online August 31, 2021 https://doi.org/10.29275/jerm.2021.31.3.231
Copyright © Korea Society of Education Studies in Mathematics.
JeongSuk Pang^{1}, Jin Sunwoo^{2}
Correspondence to:^{†}Jin Sunwoo, camy17@naver.com
ORCID: https://orcid.org/0000-0002-5101-9014
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The purpose of this study was to analyze the changes in pre-service teachers’ noticing through an elementary mathematics methods course along with a practicum. For this purpose, the comments pre-service teachers made on an entire video-based mathematics lesson were collected three times over the semester. Their comments were analyzed in terms of topic, actor, stance, evidence, and alternative strategy. The results showed that the pre-service teachers’ noticing abilities were slightly changed after learning mathematical content and pedagogy related to teaching elementary mathematics. Substantial changes in their noticing occurred after a four-week practicum and subsequent discussions on their own lesson planning, implementation, and reflections. This study has implications for designing a mathematics methods course to develop teacher expertise and refining a teacher noticing analytic framework.
Keywordspre-service teachers, teacher noticing, elementary mathematics methods course, teacher preparation program
Given the significant role that a teacher plays in students’ meaningful mathematics learning, pre-service teachers [PSTs] must develop a variety of classroom expertise through teacher preparation programs. Some studies have explored what kinds of knowledge, beliefs, and identities related to mathematics and mathematics teaching are needed, specifically for teachers (Potari & Chapman, 2020). Others have focused on how to foster PSTs’ actual instructional practices using various tools and processes, such as lesson videos, curriculum resources, technology, or debrief conversations for mathematics lessons (Llinares & Chapman, 2020; van Es & Sherin, 2008). Over the past few decades, teacher educators have made an effort to provide PSTs with more opportunities to develop their knowledge and orientation of mathematics teaching, and to foster actual teaching practice through a teacher preparation program.
Despite the common approach of emphasizing both theoretical and practical aspects in a teacher preparation program, research is lacking on how effective such programs are for enhancing teacher expertise. Among many other indicators, a teacher’s ability to notice what is mathematically significant in a complex classroom situation has been identified as a key indicator of teacher expertise (Mason, 2002; Mitchell & Marin, 2015; van Es & Sherin, 2008). This is mainly because teachers’ instructional decision-making is based on their identification of and reasoning about the salient features of students’ mathematics learning (Jacobs, Lamb, & Philipp, 2010; Mason, 2002). In fact, according to the Association of Mathematics Teacher Educators [AMTE] (2017), well-prepared beginning mathematics teachers are required to “attend to students’ thinking about mathematics content” and “recognize students’ engagement in mathematical practices” (pp. 18-19). In this regard,
In order to develop the ability to notice in a teacher preparation program, teacher educators often provide PSTs with specific pedagogical approaches in which they have opportunities to observe and analyze others’ videos (e.g., Ulusoy, 2020; van Es et al., 2017) or their own lessons (e.g., Roller, 2016; Santagata & Yeh, 2014). These approaches have been reported as effective in developing teacher noticing. However, studies usually do not analyze how PSTs develop their noticing skills over time via the knowledge and experience they gain in a teacher preparation program.
This paper analyzes the
The research questions were as follows: (a) How does PSTs’ noticing in a mathematics lesson change after learning theoretical aspects during a university course compared to their noticing at the beginning of the course? (b) How does PSTs’ noticing in a mathematics lesson change after discussing their own mathematics lessons implemented during the practicum period? Two common pedagogical approaches for educating PSTs on how to teach elementary mathematics are the following: increase the PSTs’ knowledge of mathematics and mathematics teaching and allow the PSTs to implement lessons in an elementary classroom, with subsequent discussion. This paper analyzes what provokes such changes in teacher noticing and describes implications for designing a mathematics methods course.
Expert teachers can identify and understand mathematically meaningful situations in a lesson and make appropriate decisions that foster students’ mathematics learning (Jacobs et al., 2010; Mason, 2002; van Es & Sherin, 2008). For instance, expert teachers can attend to students’ mathematical thinking and provide interpretive comments, whereas novice teachers attend to the whole class environment and provide primarily descriptive comments (van Es, 2011). In a similar vein, expert teachers can present robust evidence in interpreting students’ mathematical thinking, but novice teachers provide limited or no evidence (Jacobs et al., 2010). Given the premise that noticing is a learnable practice (Jacobs & Spangler, 2017), considerable attempts have been made to support the development of PSTs’ noticing skills through teacher education programs (Schack, Fisher, & Wilhelm, 2017).
To develop noticing skills, PSTs need to have knowledge (or resources) to make sense of what is identified as noteworthy features of classroom interactions. In this regard, Sherin and van Es (2009) emphasized
Many studies on enhancing PSTs’ noticing skills have involved lesson videos (Mitchell & Marin, 2015; Santagata & Yeh. 2014; van Es et al, 2017) with which they highlight specific events or moments of mathematical thinking and interpret them. Such videos were selected mostly by the researchers, tended to be short (within 10 minutes), and featured students' particular strategies or thinking processes. In addition to using videos, recent studies have emphasized more specific noticing based on the structured framework for analysis. Such studies have provided PSTs with specific prompts, stimulating them to pay closer attention to the important aspects of a mathematics lesson. For example, McDuffie et al. (2014) provided a specific prompt called
These studies identified the characteristics of PSTs’ noticing and further explored how to foster such a noticing ability through a university-based course. However, the studies were conducted in deliberately designed contexts or interventions to enhance PSTs’ noticing ability. For example, they used short video excerpts, like those already mentioned, discussions of salient interactions within the videos, and specific prompts or protocols to guide noticing. It is plausible that the targeted interventions lead to the successful development of PSTs’ noticing skills (van Es et al, 2017). However, observing a salient part of an entire lesson is different from understanding the overall situation of the lesson. Given that a mathematics lesson involves complex situations in which multiple factors are intertwined (Lampert, 2001), PSTs need to be able to observe the lesson as a whole to fully understand the interactions among the teacher, students, and content (Pang, 2011). This study examined how and what aspects PSTs pay attention to in an elementary mathematics lesson as a whole.
Another aspect that has been overlooked in previous studies is the influence of the PSTs’ practicum and its related learning opportunities on their noticing skills. During a practicum period, PSTs have an opportunity to apply what they have learned through a university course to actual practices in mathematics lessons. They can analyze and reflect on their own lessons instead of others’ videos. Taken together, we as education researchers need to explore the nature and quality of the noticing ability emergent in a rather naturalistic teacher education program which consists of both university lecture and practicum.
Some studies have reported positive or improved outcomes of PSTs’ noticing through a teacher preparation program’s intervention (e.g., Mitchell & Marin, 2015; Star & Strickland, 2008; van Es et al., 2017). But others have reported declines in PSTs’ noticing (e.g., Castro Superfine et al., 2015). In fact, after reviewing 27 studies on PSTs’ noticing, Amador, Bragelman, and Castro Superfine (2021) concluded that a majority of the studies showed “stagnant, mixed level, or inconsistent results” (p. 11). The researchers also highlighted the divergent and ambiguous data collection and analytic methods used to study PSTs’ noticing. These findings imply that researchers on PSTs’ noticing need to be clear about what theoretical framework is used and what methodological approaches are taken, which are elaborated upon in the next section.
Teacher noticing is mostly analyzed in terms of what to notice and how to notice (Amador et al., 2021). This study is based on the Framework for Learning to Notice Student Mathematical Thinking by van Es (2011). The framework classifies four levels of noticing: Level 1 (Baseline), Level 2 (Mixed), Level 3 (Focused), and Level 4 (Extended). For instance, teachers at Level 1 attend to the whole class environment, behavior, and teacher pedagogy, providing descriptive and evaluative comments with little evidence to support their analysis. In contrast, teachers at Level 4 attend to the relationship between particular students’ mathematical thinking and teaching strategies, providing interpretive comments by referring to specific events and interactions as evidence. This study did not employ the levels of noticing per se but used the analytic elements in the framework to present detailed analyses of what and how PSTs notice.
Firstly, regarding what to notice, both
Topics are subtly different from study to study. For instance, Star and Strickland (2008) classified the topics that PSTs noticed from an entire video: the classroom environment, classroom management, mathematical content, tasks, and communication. In another case, van Es et al. (2017) classified the topics that PSTs noticed from a short video clip, specifying the mathematical content and learning goals, student thinking, pedagogies for making thinking visible, and classroom discourse norms. These studies analyzed the frequency of such topics and showed which topics were most often noticed by PSTs. However, such studies were not specific regarding what would be noticed in detail per topic.
Our study expanded the trend of previous studies by further specifying each topic. For instance, the topic of Mathematical Tasks can be further classified depending on whether the main focus is to align with the lesson objective, to adjust students’ interest or motivation to learn, or to promote students’ mathematical thinking. Specifying each topic can reveal more precisely not only what PSTs notice but also what they fail to or struggle to notice (Spizer & Phelps-Gregory, 2017). As this study analyzes changes in PSTs’ noticing through a mathematics methods course for one semester, knowing exactly what PSTs notice and what they miss can help teacher educators better design a methods course.
Secondly, how to notice is related to the ways teachers describe and interpret what they notice. Noticing ability requires teachers to interpret and reason about what they notice. In this sense, van Es and Sherin (2008) analyzed how the teachers participating in a video club made inferences about the classroom events they observed on the basis of their
Teachers are also expected to provide reasonable
Another aspect regarding how to notice is whether teachers can present a detailed description of an alternative teacher action or justification for using a different instructional move. For instance, Santagata, Zannoni, and Stigler (2007) used the criteria of
The study’s research participants were a total of 56 PSTs (15 males and 41 females) enrolled in a three-credit Elementary Mathematics Education Methods course at a public university in Korea^{1)}. The PSTs were in the second semester of the third year out of a four-year teacher preparation program. Due to the number of PSTs, the course was divided into two sections and was taught by the same instructor (i.e., the first author of this paper). The course was compulsory for all elementary PSTs in the university. Previously, the PSTs had taken one basic course dealing with the national mathematics curriculum plus the content and pedagogy of the Number and Operation Strand. The methods course lasted 15 weeks, and the PSTs had their first four-week practicum period in the second half of the semester. As such, they did not have any formal teaching experience at the beginning of the semester. To complete 45 credit hours over the semester, the class met about four hours per week except during the practicum period.
To analyze how elementary PSTs’ noticing changed while taking the semester-long methods course in conjunction with a practicum, we collected the PSTs’ written comments on an elementary mathematics lesson three times throughout the semester: (a) at the beginning of the course, (b) in the middle of the course after introducing the content and pedagogy of elementary mathematics teaching, and (c) at the end of the course after completing a four-week practicum and subsequent discussions on the mathematics lessons implemented during the practicum (see
Table 1
Instructional phase | Main activities | Time (min) |
---|---|---|
Motivation for the lesson and identification of the lesson goal | • Recall the characteristics of a rectangular prism by showing a cookie box | 5 |
• Think about the method of how to use the least amount of paper when wrapping the cookie box • Present the lesson goal: Let’s explore how to figure out the surface area of a rectangular prism | ||
• Introduce the definition of surface area and write it in a notebook | 2 | |
Exploration of multiple methods to find the surface area of a rectangular prism | • Predict how to find the surface area of the given rectangular prism • Discuss methods in groups | 6 |
Presentation of various solution methods and whole-group discussion | • Whole group discussion regarding how to figure out the surface area of the rectangular prism Method 1: Adding the area of each of the prism’s six faces Method 2: Adding the areas of three different faces and then doubling it Method 3: (Area of the rectangular base)×2 + (Area of the side face) | 7 |
• Calculate the surface area of the rectangular prism with the three methods above • Review each method | 11 | |
Generalization of the formula for finding the surface area of a rectangular prism | • Explore the most effective method and formalize how to figure out the surface area of a rectangular prism | 5 |
Application | • Solve the review exercises in the textbook | 2 |
Closing | • Talk about what you have learned during the lesson • Assignment: Calculate the surface area of a rectangular prism you would find in everyday life | 2 |
After watching the entire lesson for 40 minutes, the PSTs were encouraged to write their comments using the following prompt: Think about the students’ thinking or understanding and the teacher’s instructional strategies in the video and write down whatever you found interesting. As the lesson video was publicly available, the PSTs could watch the video as often as they liked. At the end of the first week of the course, the PSTs were asked to submit, at maximum, two-pages of comments about the video.
After the practicum, the PSTs returned to the university and resumed the course. During the remaining three weeks of the semester, each PST team had an opportunity to debrief on their lesson planning, implementation, and reflection. In debriefing for about 30 minutes, the PSTs were asked to show significant portions of their video-recorded mathematics lesson. The instructor then orchestrated a whole-class discussion on the lesson. The PSTs actively provided feedback to peers, often asking questions about the lesson. Building on the PSTs’ various responses, the instructor raised further questions and aspects to consider in planning and implementing a mathematics lesson. In the course’ final week and with the same guidelines, the PSTs submitted their final comments about the same lesson video concerning the surface area of a rectangular prism (i.e., Data 3).
In summary, the PSTs watched the same video throughout the course and wrote comments. At the beginning of the course, the main purpose of using a lesson video was to identify what and how the PSTs would notice in the video before they had learned any specific content or pedagogy related to teaching measurement. In the middle of the course, the purpose was to explore how the PSTs’ comments might change after the lecture. At the end of the course, the purpose was to analyze how the PSTs’ comments might further change after their practicum in elementary schools and whole-class discussions on their enacted lessons. This data collection timescale, tailored to the PSTs’ main learning experiences throughout one semester, was expected to reveal some changes of their noticing skills related to the impact of their learning experiences.
The PSTs’ comments were divided into semantic units dealing with the same topic. Each unit was coded according to the following five elements resulting from the review of the teacher noticing framework:
The topic element was used to analyze which pedagogical issues the PSTs focused on in the comments. We employed four main topics (i.e., Mathematical Tasks, Instructional Strategies, Learning Environment, and Mathematical Discourse), building on Pang (2011). We then subdivided each topic to capture various aspects in the PSTs’ comments more precisely (see
Table 2
Topic | Subtopic | Code |
---|---|---|
Mathematical Tasks | Using tasks aligned with the lesson objective | MT1 |
Using tasks tailored to students’ prior knowledge and motivation | MT2 | |
Using tasks to promote students’ mathematical thinking and communication | MT3 | |
Instructional Strategies | Using instructional strategies appropriate for the content to be taught | IS1 |
Using instructional strategies appropriate to students’ characteristics | IS2 | |
Using manipulatives and materials to help students’ learning | IS3 | |
Effective implementation of the written lesson plan, including improvisation | IS4 | |
Assessment | IS5 | |
Learning Environment | Establishing a learning environment through appropriate social norms | LE1 |
Physical arrangement and equipment to support learning | LE2 | |
Mathematical Discourse | Mathematical communication between the teacher and students, and among students | MD1 |
Meaningful questions and appropriate feedback | MD2 |
The actor element was used to analyze whom the PSTs mainly focused on in the comments. We divided it into three categories: the teacher, students, and both (see
Table 3
Actor | Description | Code |
---|---|---|
Teacher | The comment mainly focused on the teacher in the video | T |
Students | The comment mainly focused on a particular student, a group of students, or all the students in the video as a whole | S |
Teacher and Students | The comment equally paid attention to both the teacher and the students | T/S |
The stance element was used to analyze how the PSTs mentioned the events they noticed in the video. Building on van Es and Sherin (2008),
Table 4
Stance | Description | Code |
---|---|---|
Describe | Negative statement | D– |
Positive statement | D+ | |
Evaluate | Negative statement | E– |
Positive statement | E+ | |
Interpret | Negative statement | I– |
Positive statement | I+ |
The evidence element was used to analyze whether or not the statements that the PSTs commented on were based on evidence. We intended to analyze not only whether the PSTs referred to specific events’ evidence but also what the main sources of such evidence were. In fact, PSTs’ comments were based on various sources of evidence, partly because they watched the entire lesson instead of lesson segments. These results led us to subdivide the sources of evidence, as shown in
Table 5
Evidence | Code | Evidence | Code |
---|---|---|---|
Students’ responses in the lesson video | S | Mathematical (or theoretical) reason | M |
Personal experience or opinion | E | Affective reason | A |
Connections across the lesson flow | F | Learning environment | L |
Decision of the teacher in the video | T | Comparison with other strategies | C |
No evidence | N |
Finally, the alternative strategy element was used to analyze whether or not the PSTs proposed any alternative approaches to the event they commented on. As described in the literature review, teachers at higher levels of noticing can present alternative pedagogical solutions. In this study, the alternative strategy of the PSTs’ comments was further classified as superficial or specific, and by how many alternative strategies were suggested, as shown in
Table 6
Alternative strategy | Description | Code |
---|---|---|
Level 1 | No alternative strategy proposed | 1 |
Level 2 | Vague or superficial alternative strategy proposed | 2 |
Level 3 | One specific alternative strategy proposed | 3 |
Level 4 | Two or more specific alternative strategies proposed | 4 |
All the PSTs’ comments (i.e., Data 1, 2, and 3) were coded according to the five analytic elements above. The coding was conducted by one researcher (i.e., the co-author of this paper) and three research assistants majoring in elementary mathematics education. Before coding, the coders were told how to code the data, with detailed explanations for each analytical element. They practiced coding with three samples from the PSTs’ comments. Note that there were 168 comments (i.e., three comments from each of 56 PSTs). To check reliability among the coders, we randomly selected 45 comments from 15 PSTs to be coded independently. The results of this coding showed more than 80% agreement for each analytic element: specifically, 90.5% in topic; 85% in actor; 80.8% in stance; 81.9% in evidence; and 95.8% in alternative strategy. After coding all the semantic units in the PSTs’ comments, as illustrated in
Table 7
Excerpt from a PST’s comment (a semantic unit) | Topic | Actor | Stance | Evidence | Alternative strategy |
---|---|---|---|---|---|
It was good that the teacher didn’t mention the most effective method immediately after the students found the three solution methods. Because of that, the students had the opportunity to think of and present why the specific method was the most effective to calculate the surface area of a rectangular prism. | IS1 | T/S | E+ | S | 1 |
Table 8
Topic | Frequency (%)* | |||
---|---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | ||
Mathematical Tasks | MT1** | 13 (2.92) | 11 (2.26) | 26 (4.30) |
MT2 | 21 (4.72) | 24 (4.94) | 30 (4.96) | |
MT3 | 1 (0.22) | 4 (0.82) | 5 (0.83) | |
Total | 35 (7.87) | 39 (8.02) | 61 (10.08) | |
Instructional Strategies | IS1 | 133 (29.89) | 157 (32.30) | 123 (20.33) |
IS2 | 31 (6.97) | 24 (4.94) | 26 (4.30) | |
IS3 | 66 (14.83) | 51 (10.49) | 73 (12.07) | |
IS4 | 12 (2.70) | 25 (5.14) | 29 (4.79) | |
IS5 | 2 (0.45) | 10 (2.06) | 10 (1.65) | |
Total | 244 (54.83) | 267 (54.94) | 261 (43.14) | |
Learning Environment | LE1 | 26 (5.84) | 33 (6.79) | 14 (2.31) |
LE2 | 9 (2.02) | 4 (0.82) | 5 (0.83) | |
Total | 35 (7.87) | 37 (7.61) | 19 (3.14) | |
Mathematical Discourse | MD1 | 70 (15.73) | 65 (13.37) | 119 (19.67) |
MD2 | 61 (13.71) | 78 (16.05) | 145 (23.97) | |
Total | 131 (29.44) | 143 (29.42) | 264 (43.64) | |
Total | 445 (100.00) | 486 (100.00) | 605 (100.00) |
*Frequency refers to the number of semantic units from comments. The percentage (%) is rounded from the third decimal place.
**MT1: Using tasks aligned with the lesson objective. MT2: Using tasks tailored to students’ prior knowledge and motivation. MT3: Using tasks to promote students’ mathematical thinking and communication. IS1: Using instructional strategies appropriate for the content to be taught. IS2: Using instructional strategies appropriate to students’ characteristics. IS3: Using manipulatives and materials to help students’ learning. IS4: Effective implementation of the written lesson plan, including improvisation. IS5: Assessment. LE1: Establishing a learning environment through appropriate social norms. LE2: Physical arrangement and equipment to support learning. MD1: Mathematical communication between the teacher and students, and among students. MD2: Meaningful questions and appropriate feedback.
The PSTs’ noticing did not show major changes between the first and second comments. The total number of PSTs’ comments by the semantic units increased slightly from their first to second comments (i.e., from 445 to 486). However, both sets of the PSTs’ comments tended to focus primarily on Instructional Strategies (approximately 55%) and Mathematical Discourse (approximately 29%), regardless of whether they had learned the mathematical content and pedagogy related to teaching elementary mathematics. Only slight changes occurred regarding the subtopics of Instructional Strategies. Specifically, the frequency of comments about the following subtopics increased: “using instructional strategies appropriate for the content to be taught” (IS1) and “effective implementation of the written lesson plan including improvisation” (IS4). On the other hand, “using instructional strategies appropriate to students’ characteristics” (IS2) and “using manipulatives and materials to help students’ learning” (IS3) decreased. The PSTs’ comments regarding Mathematical Discourse also slightly changed. Specifically, PSTs tended to notice the teacher’s questions and feedback more than the mathematical communication between the teacher and students, and among students.
Given these findings, at best, the university lecture seemed to lead the PSTs to focus more on the teacher’s lesson implementation and the mathematical content than on the students’ characteristics or learning. To emphasize, “using instructional strategies appropriate for the content to be taught” (IS1) was the most frequent subtopic from the first comments and this trend was reinforced in the second comments.
Whereas the changes between the first and second comments were minimal, those between the second and third were considerable. First of all, the total number of semantic units in the third set of comments was quantitatively greater (increasing from 486 in the second comments to 605 in the third). The most frequent topic was Mathematical Discourse, which increased from 29.42% in the second comments to 43.64% in the third comments. Specifically, the percentages of the two subtopics of Mathematical Discourse increased, and the subtopic “meaningful questions and appropriate feedback” (MD2) came to be the most frequently commented upon subtopic.
Along with Mathematical Discourse, the frequency of comments about the topic Mathematical Tasks slightly increased, specifically regarding “using tasks aligned with the lesson objective” (MT1). In contrast, comments on the topics Instructional Strategies and Learning Environment decreased (i.e., the former declined from 54.94% to 43.14%, and the latter from 7.61% to 3.14%). The percentages for the subtopics “using instructional strategies appropriate for the content to be taught” (IS1) and “establishing a learning environment through appropriate social norms” (LE1) notably decreased.
In summary, the main topic that the PSTs noticed was Instructional Strategies, specifically “using instructional strategies appropriate for the content to be taught.” This trend was reinforced through the lecture on the content and pedagogy of teaching elementary mathematics. However, the major change in topic occurred after the PSTs completed their practicum and discussions on their lesson planning, implementation, and reflection. The PSTs noticed more about mathematical communication, questions, and feedback as well as the content to be taught.
Table 9
Actor | Frequency (%) | ||
---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | |
Teacher | 358 (80.45) | 404 (83.13) | 269 (44.46) |
Students | 80 (17.98) | 62 (12.76) | 86 (14.21) |
Teacher and Students | 7 (1.57) | 20 (4.12) | 250 (41.32) |
Total | 445 (100) | 486 (100) | 605 (100) |
The following excerpts illustrate how the actor element changed from one PST’s first comment to the third.
Although the student looked for another method that the teacher had not anticipated and that was difficult to display with the materials the teacher had prepared, the teacher’s skillfulness was outstanding in that she was not embarrassed at all and responded appropriately to explain the method. (PST 25, first comment)
One student presented an answer unexpected from the teacher during the lesson. The student found the surface area of the given rectangular prism by unfolding it into the planar figure and calculating the area of several rectangles shown in the planar figure. This is an application of the method for finding the area of various figures that the student had learned in fourth grade. As I understand it, one of the reasons for this student to use the specific method was because the teacher provided the students with the rectangular prism in a form of its nets. The teacher praised the method she had not anticipated, but at the same time mentioned the limitation of the method by saying “you always must unfold the rectangular prism and check it in order to use this method.” (PST 25, third comment)
In the first comment, PST 25 described the teacher’s skillful response to a student’s unexpected answer. PST 25 did not mention this event in the second comment. In the third comment, PST 25 attended to the student’s answer, inferred how it was derived, and connected the student’s answer to the teacher’s instructional strategies. This change is meaningful in that the PST attended to the relationship between teacher pedagogy and student thinking.
Table 10
Stance | Frequency (%) | |||
---|---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | ||
Describe | Negative | 39 (8.76) | 41 (8.44) | 20 (3.31) |
Positive | 131 (29.44) | 88 (18.11) | 70 (11.57) | |
Total | 170 (38.20) | 129 (26.54) | 90 (14.88) | |
Evaluate | Negative | 45 (10.11) | 76 (15.64) | 164 (27.11) |
Positive | 146 (32.81) | 144 (29.63) | 215 (35.54) | |
Total | 191 (42.92) | 220 (45.27) | 379 (62.64) | |
Interpret | Negative | 20 (4.49) | 42 (8.64) | 52 (8.60) |
Positive | 64 (14.38) | 95 (19.55) | 84 (13.88) | |
Total | 84 (18.88) | 137 (28.19) | 136 (22.48) | |
Total | 445 (100) | 486 (100) | 605 (100) |
The following excerpts illustrate how the stance element changed from one PST’s first comment to the second comment.
At the end of the lesson, the teacher gave an assignment to the students to find the surface area of a rectangular prism in everyday life and to write it down in a math journal.
This assignment seems to be an original task that students could review what they had learned using their math journals (PST 34,italics in origina l, first comment)This lesson is closely linked to real life. The teacher connected real life not only to the lesson activities for motivation and wrap-up but also to the assignment to find the surface area of a rectangular prism that can be found in real life and write it down in a math journal.
This assignment allows students to recognize that the concept of measurement is used in their lives beyond the classroom. This is related to the mathematization by Freudenthal. Through these activities and the assignment, students can better understand what they have learned on that day and develop a sense of volume by actually measuring the surrounding diverse rectangular prisms . (PST 34,italics in original , second comment)
In the first comment, PST 34 attended to the homework given by the teacher at the end of the lesson and evaluated it as an original task to review what was learned. In the second comment PST 34 attended to the same homework, finding a rectangular prism in daily lives and calculating its surface area in the mathematics journal. More importantly, the PST interpreted the homework from various perspectives with mathematical terms, such as “measurement,” “sense of volume,” and “mathematization,” which were learned from the lecture. Similarly, after learning the content and pedagogy of teaching elementary mathematics, some PSTs attempted to interpret the tasks or events of the lesson video based on the theories they had learned in the methods course.
Table 11
Evidence | Frequency (%) | ||
---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | |
Students’ responses in the lesson video | 69 (15.51) | 59 (12.14) | 80 (13.22) |
Personal experience or opinion | 118 (26.52) | 127 (26.13) | 145 (23.97) |
Connections across the lesson flow | 14 (3.15) | 19 (3.91) | 22 (3.64) |
Decision of the teacher in the video | 83 (18.65) | 85 (17.49) | 156 (25.79) |
Mathematical (or theoretical) reason | 8 (1.80) | 29 (5.97) | 67 (11.07) |
Affective reason | 15 (3.37) | 13 (2.68) | 22 (3.64) |
Learning environment | 40 (8.99) | 23 (4.73) | 27 (4.46) |
Comparison with other strategies | 0 (0.00) | 16 (3.29) | 24 (3.97) |
No evidence | 98 (22.02) | 115 (23.66) | 62 (10.25) |
Total | 445 (100) | 486 (100) | 605 (100) |
Regarding the evidence of analysis, the second comments were remarkably similar to the first comments: The PSTs’ second comments were grounded in personal experience or opinion (26.13%), the decision of the teacher in the video (17.49%), and students’ responses in the lesson video (12.14%). The percentage of PST offering no evidence was still high (23.66%). Nevertheless, a few minor changes occurred. There were increases in the percentages of PSTs using mathematical (or theoretical) reason (1.80%→5.97%) and comparison with other strategies (0.00%→3.29%). Also, there was a decrease in the percentage of PSTs referring to the learning environment (8.99%→4.73%). We can infer that the PSTs increased their efforts to interpret classroom events based on the theoretical reasoning or instructional strategies they had learned in the methods course. For instance, as shown below, PST 21 at first evaluated the teacher’s monitoring activity during students’ small-group discussions by stating that it was good. But PST 21 did not specify why such an activity would be effective. In contrast, PST 21 later provided details to explain her thinking by citing the monitoring practice among the five practices she had learned in the methods course.
It was good to see that the teacher actively asked about students’ activities and summarized them while walking around during small group discussions. (PST 21, first comment)
When the small group discussions did not go on well, the teacher asked students a few questions to facilitate the discussion. This situation reminds me of the monitoring practice of 5 practices I learned in the lecture. However, among the groups caught on the camera, there was one group of students who did not discuss actively, and it was regrettable that the teacher only said “little bit more discussion …” and passed by without any further feedback. It would have been better if the teacher could have given the group more specific feedback on how to proceed with the discussion. Considering the 5 practices, selecting in advance whom to present while monitoring students’ activities would be helpful for the teacher to lead the lesson smoothly and to facilitate an effective discussion. By doing so, the students’ understanding could have been enhanced. (PST 21, second comment)
The PSTs’ third comments were grounded in the decision of the teacher in the video (25.79%), personal experience or opinion (23.97%), and students’ responses in the lesson video (13.22%). Note that the percentage referring to the teacher’s decision notably increased. In a similar vein, the percentage centered on mathematical reason increased (5.97%→11.07%). In addition, the percentage of comments that offered no evidence for analysis considerably decreased compared to the percentages in the first and the second sets of comments (22.02%→23.66%→10.25%). It is meaningful that more PSTs could provide evidence to support their analyses at the end of the semester.
Table 12
Alternative strategy | Frequency (%) | ||
---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | |
No alternative strategy | 317 (71.24) | 325 (66.87) | 376 (62.15) |
Superficial alternative strategy | 56 (12.58) | 72 (14.81) | 69 (11.40) |
One specific alternative strategy | 64 (14.38) | 72 (14.81) | 139 (22.98) |
Two or more specific alternative strategies | 8 (1.80) | 17 (3.50) | 21 (3.47) |
Total | 445 (100) | 486 (100) | 605 (100) |
Despite the minor changes, it is encouraging to see the PSTs’ tendency to propose specific alternatives for the events they noticed in the lesson video while taking the methods course (see below for illustrative excerpts).
It seemed that the number of students presenting in the lesson was limited. I could see some students continuing to present, so I thought that only the students who were presenting were doing it. Of course, it is impossible for all students to present in one lesson, but I think it is good for the teacher to give many students a chance to present. (PST 19, first comment)
When the students were writing on the board how to find the surface area of the cookie box, it was regrettable that the teacher asked the student who had finished writing to present his method first, because other students were still writing. It would be helpful for the student who solved the problem with the first method to listen to how to solve it with the second, third, and fourth methods. As the teacher asked the student who first solved to present, however, the students who solved the problem late could not listen to other students’ presentations. (PST 19, second comment)
It would be better to revise the order of students’ presentations on how to find the surface area of the cookie box. The teacher led the students to present in order of (a) adding the areas of the three pairs of congruent faces, (b) adding the side area and the base area, (c) adding the areas of each face, and (d) dividing the net of the box into parts and adding their areas. I think it is right to make presentations in the order of (c), (a), (b), and (d), because adding the areas of each face requires only a simple computation, but the two methods (i.e., adding the side area and the base area, and adding the areas of the three pairs of congruent faces) require more advanced thinking processes in that they are connected to the properties of a rectangular prism， such as “two faces facing each other are congruent” or “the total width of the side and the sum of edges of the base are the same”. To this end, it would have been better if the teacher had made a note of students' solution methods while walking around (
monitoring ) and led students to present on how to find the surface area of the cookie box in pre-selected order (selecting ). (PST 19,italics in original , third comment)
In the first and second comments, PST 19 only mentioned the limited number of students making presentations in the video and the order of presentations on how to find the surface area of a rectangular prism. In contrast, in the third comment, PST 19 mentioned specific alternatives for how to present students’ ideas. In particular, PST 19 considered students’ thinking process in calculating the surface area of a rectangular prism and suggested an alternative strategy based on monitoring and selecting among the five practices she had learned during the methods course.
The PSTs improved their noticing skills while taking an elementary mathematics methods course that combined both theoretical and practical aspects of teaching mathematics. On the one hand, this result aligns with prior studies that reported PSTs’ enhanced ability to notice (e.g., Mitchell & Marin, 2015; Schack et al., 2013; Star & Strickland, 2008; van Es et al., 2017). On the other hand, our study’s result can be differentiated from prior studies using a popular pre-post measure in analyzing the development of noticing (Amador et al., 2021). Note that this study traced the changes in the PSTs’ noticing in a way that was tailored to their learning experience for one semester. This methodological approach, with the support of a detailed analytical framework, is helpful for unpacking changes in what PSTs notice and how they notice.
Compared to the first comments that the PSTs wrote at the beginning of the methods course, their second comments about the same lesson video indicate that their noticing ability was enhanced. Specifically, the PSTs tended to focus more on mathematical content and the teacher’s lesson implementation. Note that the most frequent subtopic was the use of instructional strategies appropriate to the content to be taught. The PSTs also tended to analyze the lesson in an interpretative stance based on mathematical or theoretical reasons or teaching strategies they had learned in the methods course.
Note that this study’s methods course was not designed specifically to improve teacher noticing. Even though the course covered the Measurement Strand of the elementary mathematics curriculum, it did not elaborate on how to teach the surface area of a rectangular prism. However, the results indicate positive changes in teacher noticing through a lecture on the content and pedagogy related to teaching elementary mathematics in general, even though such changes were minimal in this study.
Compared to the first and the second comments, the third comments the PSTs made at the end of the semester illustrated substantial changes in their noticing abilities across all analytic elements: (a) they focused more on Mathematical Discourse beyond Instructional Strategies (
It is common that a mathematics teacher preparation program includes at least one methods course and the practicum. Note that the methods course of this study connected the lecture-oriented university program to the practicum at elementary schools through a group assignment of lesson planning, implementation, and reflection. The practicum itself may have provided a meaningful experience for PSTs, but just participating in the fieldwork experience does not increase teacher noticing ability (Santagata & Yeh, 2014). In contrast, the group assignment during the practicum period and subsequent discussions on it in the methods course seem to have positively impacted the PSTs’ noticing ability.
Discussing a video-taped lesson among teachers, usually including a teacher educator or facilitator, is a common pedagogical approach that teacher education programs employ to improve teacher noticing (McDuffie et al., 2014; Mitchell & Marin, 2015; van Es & Sherin, 2008). However, unlike many other studies in which discussions centered on video clips selected by researchers, the PSTs in this study discussed in small groups how they planned and enacted their mathematics lessons. They then had an opportunity to discuss further with the instructor and other PSTs from multiple perspectives. The PSTs’ actual experience of teaching mathematics and sharing diverse interpretations seem to have improved their noticing ability (Fernández, Llinares, & Rojas, 2020; Lee & Choy, 2017).
This study shows that PSTs’ noticing ability can be significantly enhanced through a mathematics methods course combining theory and practice. Our study found that the degree and aspects of positive changes in teacher noticing were different depending on when the PSTs’ comments were made. Our findings suggest that there is much room for further development of teacher noticing because even the third round of participant comments provided more evaluative comments than interpretive comments, frequently without alternative strategies. Nevertheless, it is encouraging that one semester-long methods course led PSTs to improve their ability to notice salient features of a mathematics lesson.
In this study, we used the five elements as an analytic framework—
Regarding the analytic element
The detailed analytic framework using subtopics also helped us explore which features need to be emphasized in a teacher education program. For instance, it is encouraging that the PSTs focused more on mathematical tasks while taking the methods course. However, their focus was centered either on using tasks aligned with the lesson objective or on using tasks tailored to students’ prior knowledge and motivation. On the contrary, little focus was on using tasks to promote students’ mathematical thinking throughout the semester. Knowing exactly what the PSTs noticed and what they missed is helpful for re-designing the methods course (Spizer & Phelps-Gregory, 2017).
Regarding the analytic element
In fact, it was encouraging that the PSTs’ comments in this study were grounded in the decision of the teacher in the video as well as their personal experiences or opinions. Their comments were also based on more mathematical reasons as the semester went on. Given that teachers are expected to discuss their lessons throughout their teaching career, any teacher preparation program needs to foster the ability to evaluate or interpret classroom events based on specific and reasonable source.
To summarize, this study devised a more detailed framework to analyze PSTs’ noticing ability, building on prior studies, specifically regarding the actor and evidence elements of analysis. Such a framework broadens our understanding of the changes in teacher noticing throughout a mathematics methods course localized in the Korean context. It is hoped that this study’s analysis of teacher noticing will provoke discussions both on how to design a mathematics methods course in a teacher preparation program and on how to measure noticing as an indicator of teacher expertise.
No potential conflict of interest relevant to this article was reported.
2021; 31(3): 231-255
Published online August 31, 2021 https://doi.org/10.29275/jerm.2021.31.3.231
Copyright © Korea Society of Education Studies in Mathematics.
JeongSuk Pang^{1}, Jin Sunwoo^{2}
^{1}Professor, Korea National University of Education, ^{2}Teacher, Jojong Elementary School, South Korea
Correspondence to:^{†}Jin Sunwoo, camy17@naver.com
ORCID: https://orcid.org/0000-0002-5101-9014
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The purpose of this study was to analyze the changes in pre-service teachers’ noticing through an elementary mathematics methods course along with a practicum. For this purpose, the comments pre-service teachers made on an entire video-based mathematics lesson were collected three times over the semester. Their comments were analyzed in terms of topic, actor, stance, evidence, and alternative strategy. The results showed that the pre-service teachers’ noticing abilities were slightly changed after learning mathematical content and pedagogy related to teaching elementary mathematics. Substantial changes in their noticing occurred after a four-week practicum and subsequent discussions on their own lesson planning, implementation, and reflections. This study has implications for designing a mathematics methods course to develop teacher expertise and refining a teacher noticing analytic framework.
Keywords: pre-service teachers, teacher noticing, elementary mathematics methods course, teacher preparation program
Given the significant role that a teacher plays in students’ meaningful mathematics learning, pre-service teachers [PSTs] must develop a variety of classroom expertise through teacher preparation programs. Some studies have explored what kinds of knowledge, beliefs, and identities related to mathematics and mathematics teaching are needed, specifically for teachers (Potari & Chapman, 2020). Others have focused on how to foster PSTs’ actual instructional practices using various tools and processes, such as lesson videos, curriculum resources, technology, or debrief conversations for mathematics lessons (Llinares & Chapman, 2020; van Es & Sherin, 2008). Over the past few decades, teacher educators have made an effort to provide PSTs with more opportunities to develop their knowledge and orientation of mathematics teaching, and to foster actual teaching practice through a teacher preparation program.
Despite the common approach of emphasizing both theoretical and practical aspects in a teacher preparation program, research is lacking on how effective such programs are for enhancing teacher expertise. Among many other indicators, a teacher’s ability to notice what is mathematically significant in a complex classroom situation has been identified as a key indicator of teacher expertise (Mason, 2002; Mitchell & Marin, 2015; van Es & Sherin, 2008). This is mainly because teachers’ instructional decision-making is based on their identification of and reasoning about the salient features of students’ mathematics learning (Jacobs, Lamb, & Philipp, 2010; Mason, 2002). In fact, according to the Association of Mathematics Teacher Educators [AMTE] (2017), well-prepared beginning mathematics teachers are required to “attend to students’ thinking about mathematics content” and “recognize students’ engagement in mathematical practices” (pp. 18-19). In this regard,
In order to develop the ability to notice in a teacher preparation program, teacher educators often provide PSTs with specific pedagogical approaches in which they have opportunities to observe and analyze others’ videos (e.g., Ulusoy, 2020; van Es et al., 2017) or their own lessons (e.g., Roller, 2016; Santagata & Yeh, 2014). These approaches have been reported as effective in developing teacher noticing. However, studies usually do not analyze how PSTs develop their noticing skills over time via the knowledge and experience they gain in a teacher preparation program.
This paper analyzes the
The research questions were as follows: (a) How does PSTs’ noticing in a mathematics lesson change after learning theoretical aspects during a university course compared to their noticing at the beginning of the course? (b) How does PSTs’ noticing in a mathematics lesson change after discussing their own mathematics lessons implemented during the practicum period? Two common pedagogical approaches for educating PSTs on how to teach elementary mathematics are the following: increase the PSTs’ knowledge of mathematics and mathematics teaching and allow the PSTs to implement lessons in an elementary classroom, with subsequent discussion. This paper analyzes what provokes such changes in teacher noticing and describes implications for designing a mathematics methods course.
Expert teachers can identify and understand mathematically meaningful situations in a lesson and make appropriate decisions that foster students’ mathematics learning (Jacobs et al., 2010; Mason, 2002; van Es & Sherin, 2008). For instance, expert teachers can attend to students’ mathematical thinking and provide interpretive comments, whereas novice teachers attend to the whole class environment and provide primarily descriptive comments (van Es, 2011). In a similar vein, expert teachers can present robust evidence in interpreting students’ mathematical thinking, but novice teachers provide limited or no evidence (Jacobs et al., 2010). Given the premise that noticing is a learnable practice (Jacobs & Spangler, 2017), considerable attempts have been made to support the development of PSTs’ noticing skills through teacher education programs (Schack, Fisher, & Wilhelm, 2017).
To develop noticing skills, PSTs need to have knowledge (or resources) to make sense of what is identified as noteworthy features of classroom interactions. In this regard, Sherin and van Es (2009) emphasized
Many studies on enhancing PSTs’ noticing skills have involved lesson videos (Mitchell & Marin, 2015; Santagata & Yeh. 2014; van Es et al, 2017) with which they highlight specific events or moments of mathematical thinking and interpret them. Such videos were selected mostly by the researchers, tended to be short (within 10 minutes), and featured students' particular strategies or thinking processes. In addition to using videos, recent studies have emphasized more specific noticing based on the structured framework for analysis. Such studies have provided PSTs with specific prompts, stimulating them to pay closer attention to the important aspects of a mathematics lesson. For example, McDuffie et al. (2014) provided a specific prompt called
These studies identified the characteristics of PSTs’ noticing and further explored how to foster such a noticing ability through a university-based course. However, the studies were conducted in deliberately designed contexts or interventions to enhance PSTs’ noticing ability. For example, they used short video excerpts, like those already mentioned, discussions of salient interactions within the videos, and specific prompts or protocols to guide noticing. It is plausible that the targeted interventions lead to the successful development of PSTs’ noticing skills (van Es et al, 2017). However, observing a salient part of an entire lesson is different from understanding the overall situation of the lesson. Given that a mathematics lesson involves complex situations in which multiple factors are intertwined (Lampert, 2001), PSTs need to be able to observe the lesson as a whole to fully understand the interactions among the teacher, students, and content (Pang, 2011). This study examined how and what aspects PSTs pay attention to in an elementary mathematics lesson as a whole.
Another aspect that has been overlooked in previous studies is the influence of the PSTs’ practicum and its related learning opportunities on their noticing skills. During a practicum period, PSTs have an opportunity to apply what they have learned through a university course to actual practices in mathematics lessons. They can analyze and reflect on their own lessons instead of others’ videos. Taken together, we as education researchers need to explore the nature and quality of the noticing ability emergent in a rather naturalistic teacher education program which consists of both university lecture and practicum.
Some studies have reported positive or improved outcomes of PSTs’ noticing through a teacher preparation program’s intervention (e.g., Mitchell & Marin, 2015; Star & Strickland, 2008; van Es et al., 2017). But others have reported declines in PSTs’ noticing (e.g., Castro Superfine et al., 2015). In fact, after reviewing 27 studies on PSTs’ noticing, Amador, Bragelman, and Castro Superfine (2021) concluded that a majority of the studies showed “stagnant, mixed level, or inconsistent results” (p. 11). The researchers also highlighted the divergent and ambiguous data collection and analytic methods used to study PSTs’ noticing. These findings imply that researchers on PSTs’ noticing need to be clear about what theoretical framework is used and what methodological approaches are taken, which are elaborated upon in the next section.
Teacher noticing is mostly analyzed in terms of what to notice and how to notice (Amador et al., 2021). This study is based on the Framework for Learning to Notice Student Mathematical Thinking by van Es (2011). The framework classifies four levels of noticing: Level 1 (Baseline), Level 2 (Mixed), Level 3 (Focused), and Level 4 (Extended). For instance, teachers at Level 1 attend to the whole class environment, behavior, and teacher pedagogy, providing descriptive and evaluative comments with little evidence to support their analysis. In contrast, teachers at Level 4 attend to the relationship between particular students’ mathematical thinking and teaching strategies, providing interpretive comments by referring to specific events and interactions as evidence. This study did not employ the levels of noticing per se but used the analytic elements in the framework to present detailed analyses of what and how PSTs notice.
Firstly, regarding what to notice, both
Topics are subtly different from study to study. For instance, Star and Strickland (2008) classified the topics that PSTs noticed from an entire video: the classroom environment, classroom management, mathematical content, tasks, and communication. In another case, van Es et al. (2017) classified the topics that PSTs noticed from a short video clip, specifying the mathematical content and learning goals, student thinking, pedagogies for making thinking visible, and classroom discourse norms. These studies analyzed the frequency of such topics and showed which topics were most often noticed by PSTs. However, such studies were not specific regarding what would be noticed in detail per topic.
Our study expanded the trend of previous studies by further specifying each topic. For instance, the topic of Mathematical Tasks can be further classified depending on whether the main focus is to align with the lesson objective, to adjust students’ interest or motivation to learn, or to promote students’ mathematical thinking. Specifying each topic can reveal more precisely not only what PSTs notice but also what they fail to or struggle to notice (Spizer & Phelps-Gregory, 2017). As this study analyzes changes in PSTs’ noticing through a mathematics methods course for one semester, knowing exactly what PSTs notice and what they miss can help teacher educators better design a methods course.
Secondly, how to notice is related to the ways teachers describe and interpret what they notice. Noticing ability requires teachers to interpret and reason about what they notice. In this sense, van Es and Sherin (2008) analyzed how the teachers participating in a video club made inferences about the classroom events they observed on the basis of their
Teachers are also expected to provide reasonable
Another aspect regarding how to notice is whether teachers can present a detailed description of an alternative teacher action or justification for using a different instructional move. For instance, Santagata, Zannoni, and Stigler (2007) used the criteria of
The study’s research participants were a total of 56 PSTs (15 males and 41 females) enrolled in a three-credit Elementary Mathematics Education Methods course at a public university in Korea^{1)}. The PSTs were in the second semester of the third year out of a four-year teacher preparation program. Due to the number of PSTs, the course was divided into two sections and was taught by the same instructor (i.e., the first author of this paper). The course was compulsory for all elementary PSTs in the university. Previously, the PSTs had taken one basic course dealing with the national mathematics curriculum plus the content and pedagogy of the Number and Operation Strand. The methods course lasted 15 weeks, and the PSTs had their first four-week practicum period in the second half of the semester. As such, they did not have any formal teaching experience at the beginning of the semester. To complete 45 credit hours over the semester, the class met about four hours per week except during the practicum period.
To analyze how elementary PSTs’ noticing changed while taking the semester-long methods course in conjunction with a practicum, we collected the PSTs’ written comments on an elementary mathematics lesson three times throughout the semester: (a) at the beginning of the course, (b) in the middle of the course after introducing the content and pedagogy of elementary mathematics teaching, and (c) at the end of the course after completing a four-week practicum and subsequent discussions on the mathematics lessons implemented during the practicum (see
Table 1 .
Instructional phase | Main activities | Time (min) |
---|---|---|
Motivation for the lesson and identification of the lesson goal | • Recall the characteristics of a rectangular prism by showing a cookie box | 5 |
• Think about the method of how to use the least amount of paper when wrapping the cookie box • Present the lesson goal: Let’s explore how to figure out the surface area of a rectangular prism | ||
• Introduce the definition of surface area and write it in a notebook | 2 | |
Exploration of multiple methods to find the surface area of a rectangular prism | • Predict how to find the surface area of the given rectangular prism • Discuss methods in groups | 6 |
Presentation of various solution methods and whole-group discussion | • Whole group discussion regarding how to figure out the surface area of the rectangular prism Method 1: Adding the area of each of the prism’s six faces Method 2: Adding the areas of three different faces and then doubling it Method 3: (Area of the rectangular base)×2 + (Area of the side face) | 7 |
• Calculate the surface area of the rectangular prism with the three methods above • Review each method | 11 | |
Generalization of the formula for finding the surface area of a rectangular prism | • Explore the most effective method and formalize how to figure out the surface area of a rectangular prism | 5 |
Application | • Solve the review exercises in the textbook | 2 |
Closing | • Talk about what you have learned during the lesson • Assignment: Calculate the surface area of a rectangular prism you would find in everyday life | 2 |
After watching the entire lesson for 40 minutes, the PSTs were encouraged to write their comments using the following prompt: Think about the students’ thinking or understanding and the teacher’s instructional strategies in the video and write down whatever you found interesting. As the lesson video was publicly available, the PSTs could watch the video as often as they liked. At the end of the first week of the course, the PSTs were asked to submit, at maximum, two-pages of comments about the video.
After the practicum, the PSTs returned to the university and resumed the course. During the remaining three weeks of the semester, each PST team had an opportunity to debrief on their lesson planning, implementation, and reflection. In debriefing for about 30 minutes, the PSTs were asked to show significant portions of their video-recorded mathematics lesson. The instructor then orchestrated a whole-class discussion on the lesson. The PSTs actively provided feedback to peers, often asking questions about the lesson. Building on the PSTs’ various responses, the instructor raised further questions and aspects to consider in planning and implementing a mathematics lesson. In the course’ final week and with the same guidelines, the PSTs submitted their final comments about the same lesson video concerning the surface area of a rectangular prism (i.e., Data 3).
In summary, the PSTs watched the same video throughout the course and wrote comments. At the beginning of the course, the main purpose of using a lesson video was to identify what and how the PSTs would notice in the video before they had learned any specific content or pedagogy related to teaching measurement. In the middle of the course, the purpose was to explore how the PSTs’ comments might change after the lecture. At the end of the course, the purpose was to analyze how the PSTs’ comments might further change after their practicum in elementary schools and whole-class discussions on their enacted lessons. This data collection timescale, tailored to the PSTs’ main learning experiences throughout one semester, was expected to reveal some changes of their noticing skills related to the impact of their learning experiences.
The PSTs’ comments were divided into semantic units dealing with the same topic. Each unit was coded according to the following five elements resulting from the review of the teacher noticing framework:
The topic element was used to analyze which pedagogical issues the PSTs focused on in the comments. We employed four main topics (i.e., Mathematical Tasks, Instructional Strategies, Learning Environment, and Mathematical Discourse), building on Pang (2011). We then subdivided each topic to capture various aspects in the PSTs’ comments more precisely (see
Table 2 .
Topic | Subtopic | Code |
---|---|---|
Mathematical Tasks | Using tasks aligned with the lesson objective | MT1 |
Using tasks tailored to students’ prior knowledge and motivation | MT2 | |
Using tasks to promote students’ mathematical thinking and communication | MT3 | |
Instructional Strategies | Using instructional strategies appropriate for the content to be taught | IS1 |
Using instructional strategies appropriate to students’ characteristics | IS2 | |
Using manipulatives and materials to help students’ learning | IS3 | |
Effective implementation of the written lesson plan, including improvisation | IS4 | |
Assessment | IS5 | |
Learning Environment | Establishing a learning environment through appropriate social norms | LE1 |
Physical arrangement and equipment to support learning | LE2 | |
Mathematical Discourse | Mathematical communication between the teacher and students, and among students | MD1 |
Meaningful questions and appropriate feedback | MD2 |
The actor element was used to analyze whom the PSTs mainly focused on in the comments. We divided it into three categories: the teacher, students, and both (see
Table 3 .
Actor | Description | Code |
---|---|---|
Teacher | The comment mainly focused on the teacher in the video | T |
Students | The comment mainly focused on a particular student, a group of students, or all the students in the video as a whole | S |
Teacher and Students | The comment equally paid attention to both the teacher and the students | T/S |
The stance element was used to analyze how the PSTs mentioned the events they noticed in the video. Building on van Es and Sherin (2008),
Table 4 .
Stance | Description | Code |
---|---|---|
Describe | Negative statement | D– |
Positive statement | D+ | |
Evaluate | Negative statement | E– |
Positive statement | E+ | |
Interpret | Negative statement | I– |
Positive statement | I+ |
The evidence element was used to analyze whether or not the statements that the PSTs commented on were based on evidence. We intended to analyze not only whether the PSTs referred to specific events’ evidence but also what the main sources of such evidence were. In fact, PSTs’ comments were based on various sources of evidence, partly because they watched the entire lesson instead of lesson segments. These results led us to subdivide the sources of evidence, as shown in
Table 5 .
Evidence | Code | Evidence | Code |
---|---|---|---|
Students’ responses in the lesson video | S | Mathematical (or theoretical) reason | M |
Personal experience or opinion | E | Affective reason | A |
Connections across the lesson flow | F | Learning environment | L |
Decision of the teacher in the video | T | Comparison with other strategies | C |
No evidence | N |
Finally, the alternative strategy element was used to analyze whether or not the PSTs proposed any alternative approaches to the event they commented on. As described in the literature review, teachers at higher levels of noticing can present alternative pedagogical solutions. In this study, the alternative strategy of the PSTs’ comments was further classified as superficial or specific, and by how many alternative strategies were suggested, as shown in
Table 6 .
Alternative strategy | Description | Code |
---|---|---|
Level 1 | No alternative strategy proposed | 1 |
Level 2 | Vague or superficial alternative strategy proposed | 2 |
Level 3 | One specific alternative strategy proposed | 3 |
Level 4 | Two or more specific alternative strategies proposed | 4 |
All the PSTs’ comments (i.e., Data 1, 2, and 3) were coded according to the five analytic elements above. The coding was conducted by one researcher (i.e., the co-author of this paper) and three research assistants majoring in elementary mathematics education. Before coding, the coders were told how to code the data, with detailed explanations for each analytical element. They practiced coding with three samples from the PSTs’ comments. Note that there were 168 comments (i.e., three comments from each of 56 PSTs). To check reliability among the coders, we randomly selected 45 comments from 15 PSTs to be coded independently. The results of this coding showed more than 80% agreement for each analytic element: specifically, 90.5% in topic; 85% in actor; 80.8% in stance; 81.9% in evidence; and 95.8% in alternative strategy. After coding all the semantic units in the PSTs’ comments, as illustrated in
Table 7 .
Excerpt from a PST’s comment (a semantic unit) | Topic | Actor | Stance | Evidence | Alternative strategy |
---|---|---|---|---|---|
It was good that the teacher didn’t mention the most effective method immediately after the students found the three solution methods. Because of that, the students had the opportunity to think of and present why the specific method was the most effective to calculate the surface area of a rectangular prism. | IS1 | T/S | E+ | S | 1 |
Table 8 .
Topic | Frequency (%)* | |||
---|---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | ||
Mathematical Tasks | MT1** | 13 (2.92) | 11 (2.26) | 26 (4.30) |
MT2 | 21 (4.72) | 24 (4.94) | 30 (4.96) | |
MT3 | 1 (0.22) | 4 (0.82) | 5 (0.83) | |
Total | 35 (7.87) | 39 (8.02) | 61 (10.08) | |
Instructional Strategies | IS1 | 133 (29.89) | 157 (32.30) | 123 (20.33) |
IS2 | 31 (6.97) | 24 (4.94) | 26 (4.30) | |
IS3 | 66 (14.83) | 51 (10.49) | 73 (12.07) | |
IS4 | 12 (2.70) | 25 (5.14) | 29 (4.79) | |
IS5 | 2 (0.45) | 10 (2.06) | 10 (1.65) | |
Total | 244 (54.83) | 267 (54.94) | 261 (43.14) | |
Learning Environment | LE1 | 26 (5.84) | 33 (6.79) | 14 (2.31) |
LE2 | 9 (2.02) | 4 (0.82) | 5 (0.83) | |
Total | 35 (7.87) | 37 (7.61) | 19 (3.14) | |
Mathematical Discourse | MD1 | 70 (15.73) | 65 (13.37) | 119 (19.67) |
MD2 | 61 (13.71) | 78 (16.05) | 145 (23.97) | |
Total | 131 (29.44) | 143 (29.42) | 264 (43.64) | |
Total | 445 (100.00) | 486 (100.00) | 605 (100.00) |
*Frequency refers to the number of semantic units from comments. The percentage (%) is rounded from the third decimal place..
**MT1: Using tasks aligned with the lesson objective. MT2: Using tasks tailored to students’ prior knowledge and motivation. MT3: Using tasks to promote students’ mathematical thinking and communication. IS1: Using instructional strategies appropriate for the content to be taught. IS2: Using instructional strategies appropriate to students’ characteristics. IS3: Using manipulatives and materials to help students’ learning. IS4: Effective implementation of the written lesson plan, including improvisation. IS5: Assessment. LE1: Establishing a learning environment through appropriate social norms. LE2: Physical arrangement and equipment to support learning. MD1: Mathematical communication between the teacher and students, and among students. MD2: Meaningful questions and appropriate feedback..
The PSTs’ noticing did not show major changes between the first and second comments. The total number of PSTs’ comments by the semantic units increased slightly from their first to second comments (i.e., from 445 to 486). However, both sets of the PSTs’ comments tended to focus primarily on Instructional Strategies (approximately 55%) and Mathematical Discourse (approximately 29%), regardless of whether they had learned the mathematical content and pedagogy related to teaching elementary mathematics. Only slight changes occurred regarding the subtopics of Instructional Strategies. Specifically, the frequency of comments about the following subtopics increased: “using instructional strategies appropriate for the content to be taught” (IS1) and “effective implementation of the written lesson plan including improvisation” (IS4). On the other hand, “using instructional strategies appropriate to students’ characteristics” (IS2) and “using manipulatives and materials to help students’ learning” (IS3) decreased. The PSTs’ comments regarding Mathematical Discourse also slightly changed. Specifically, PSTs tended to notice the teacher’s questions and feedback more than the mathematical communication between the teacher and students, and among students.
Given these findings, at best, the university lecture seemed to lead the PSTs to focus more on the teacher’s lesson implementation and the mathematical content than on the students’ characteristics or learning. To emphasize, “using instructional strategies appropriate for the content to be taught” (IS1) was the most frequent subtopic from the first comments and this trend was reinforced in the second comments.
Whereas the changes between the first and second comments were minimal, those between the second and third were considerable. First of all, the total number of semantic units in the third set of comments was quantitatively greater (increasing from 486 in the second comments to 605 in the third). The most frequent topic was Mathematical Discourse, which increased from 29.42% in the second comments to 43.64% in the third comments. Specifically, the percentages of the two subtopics of Mathematical Discourse increased, and the subtopic “meaningful questions and appropriate feedback” (MD2) came to be the most frequently commented upon subtopic.
Along with Mathematical Discourse, the frequency of comments about the topic Mathematical Tasks slightly increased, specifically regarding “using tasks aligned with the lesson objective” (MT1). In contrast, comments on the topics Instructional Strategies and Learning Environment decreased (i.e., the former declined from 54.94% to 43.14%, and the latter from 7.61% to 3.14%). The percentages for the subtopics “using instructional strategies appropriate for the content to be taught” (IS1) and “establishing a learning environment through appropriate social norms” (LE1) notably decreased.
In summary, the main topic that the PSTs noticed was Instructional Strategies, specifically “using instructional strategies appropriate for the content to be taught.” This trend was reinforced through the lecture on the content and pedagogy of teaching elementary mathematics. However, the major change in topic occurred after the PSTs completed their practicum and discussions on their lesson planning, implementation, and reflection. The PSTs noticed more about mathematical communication, questions, and feedback as well as the content to be taught.
Table 9 .
Actor | Frequency (%) | ||
---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | |
Teacher | 358 (80.45) | 404 (83.13) | 269 (44.46) |
Students | 80 (17.98) | 62 (12.76) | 86 (14.21) |
Teacher and Students | 7 (1.57) | 20 (4.12) | 250 (41.32) |
Total | 445 (100) | 486 (100) | 605 (100) |
The following excerpts illustrate how the actor element changed from one PST’s first comment to the third.
Although the student looked for another method that the teacher had not anticipated and that was difficult to display with the materials the teacher had prepared, the teacher’s skillfulness was outstanding in that she was not embarrassed at all and responded appropriately to explain the method. (PST 25, first comment)
One student presented an answer unexpected from the teacher during the lesson. The student found the surface area of the given rectangular prism by unfolding it into the planar figure and calculating the area of several rectangles shown in the planar figure. This is an application of the method for finding the area of various figures that the student had learned in fourth grade. As I understand it, one of the reasons for this student to use the specific method was because the teacher provided the students with the rectangular prism in a form of its nets. The teacher praised the method she had not anticipated, but at the same time mentioned the limitation of the method by saying “you always must unfold the rectangular prism and check it in order to use this method.” (PST 25, third comment)
In the first comment, PST 25 described the teacher’s skillful response to a student’s unexpected answer. PST 25 did not mention this event in the second comment. In the third comment, PST 25 attended to the student’s answer, inferred how it was derived, and connected the student’s answer to the teacher’s instructional strategies. This change is meaningful in that the PST attended to the relationship between teacher pedagogy and student thinking.
Table 10 .
Stance | Frequency (%) | |||
---|---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | ||
Describe | Negative | 39 (8.76) | 41 (8.44) | 20 (3.31) |
Positive | 131 (29.44) | 88 (18.11) | 70 (11.57) | |
Total | 170 (38.20) | 129 (26.54) | 90 (14.88) | |
Evaluate | Negative | 45 (10.11) | 76 (15.64) | 164 (27.11) |
Positive | 146 (32.81) | 144 (29.63) | 215 (35.54) | |
Total | 191 (42.92) | 220 (45.27) | 379 (62.64) | |
Interpret | Negative | 20 (4.49) | 42 (8.64) | 52 (8.60) |
Positive | 64 (14.38) | 95 (19.55) | 84 (13.88) | |
Total | 84 (18.88) | 137 (28.19) | 136 (22.48) | |
Total | 445 (100) | 486 (100) | 605 (100) |
The following excerpts illustrate how the stance element changed from one PST’s first comment to the second comment.
At the end of the lesson, the teacher gave an assignment to the students to find the surface area of a rectangular prism in everyday life and to write it down in a math journal.
This assignment seems to be an original task that students could review what they had learned using their math journals (PST 34,italics in origina l, first comment)This lesson is closely linked to real life. The teacher connected real life not only to the lesson activities for motivation and wrap-up but also to the assignment to find the surface area of a rectangular prism that can be found in real life and write it down in a math journal.
This assignment allows students to recognize that the concept of measurement is used in their lives beyond the classroom. This is related to the mathematization by Freudenthal. Through these activities and the assignment, students can better understand what they have learned on that day and develop a sense of volume by actually measuring the surrounding diverse rectangular prisms . (PST 34,italics in original , second comment)
In the first comment, PST 34 attended to the homework given by the teacher at the end of the lesson and evaluated it as an original task to review what was learned. In the second comment PST 34 attended to the same homework, finding a rectangular prism in daily lives and calculating its surface area in the mathematics journal. More importantly, the PST interpreted the homework from various perspectives with mathematical terms, such as “measurement,” “sense of volume,” and “mathematization,” which were learned from the lecture. Similarly, after learning the content and pedagogy of teaching elementary mathematics, some PSTs attempted to interpret the tasks or events of the lesson video based on the theories they had learned in the methods course.
Table 11 .
Evidence | Frequency (%) | ||
---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | |
Students’ responses in the lesson video | 69 (15.51) | 59 (12.14) | 80 (13.22) |
Personal experience or opinion | 118 (26.52) | 127 (26.13) | 145 (23.97) |
Connections across the lesson flow | 14 (3.15) | 19 (3.91) | 22 (3.64) |
Decision of the teacher in the video | 83 (18.65) | 85 (17.49) | 156 (25.79) |
Mathematical (or theoretical) reason | 8 (1.80) | 29 (5.97) | 67 (11.07) |
Affective reason | 15 (3.37) | 13 (2.68) | 22 (3.64) |
Learning environment | 40 (8.99) | 23 (4.73) | 27 (4.46) |
Comparison with other strategies | 0 (0.00) | 16 (3.29) | 24 (3.97) |
No evidence | 98 (22.02) | 115 (23.66) | 62 (10.25) |
Total | 445 (100) | 486 (100) | 605 (100) |
Regarding the evidence of analysis, the second comments were remarkably similar to the first comments: The PSTs’ second comments were grounded in personal experience or opinion (26.13%), the decision of the teacher in the video (17.49%), and students’ responses in the lesson video (12.14%). The percentage of PST offering no evidence was still high (23.66%). Nevertheless, a few minor changes occurred. There were increases in the percentages of PSTs using mathematical (or theoretical) reason (1.80%→5.97%) and comparison with other strategies (0.00%→3.29%). Also, there was a decrease in the percentage of PSTs referring to the learning environment (8.99%→4.73%). We can infer that the PSTs increased their efforts to interpret classroom events based on the theoretical reasoning or instructional strategies they had learned in the methods course. For instance, as shown below, PST 21 at first evaluated the teacher’s monitoring activity during students’ small-group discussions by stating that it was good. But PST 21 did not specify why such an activity would be effective. In contrast, PST 21 later provided details to explain her thinking by citing the monitoring practice among the five practices she had learned in the methods course.
It was good to see that the teacher actively asked about students’ activities and summarized them while walking around during small group discussions. (PST 21, first comment)
When the small group discussions did not go on well, the teacher asked students a few questions to facilitate the discussion. This situation reminds me of the monitoring practice of 5 practices I learned in the lecture. However, among the groups caught on the camera, there was one group of students who did not discuss actively, and it was regrettable that the teacher only said “little bit more discussion …” and passed by without any further feedback. It would have been better if the teacher could have given the group more specific feedback on how to proceed with the discussion. Considering the 5 practices, selecting in advance whom to present while monitoring students’ activities would be helpful for the teacher to lead the lesson smoothly and to facilitate an effective discussion. By doing so, the students’ understanding could have been enhanced. (PST 21, second comment)
The PSTs’ third comments were grounded in the decision of the teacher in the video (25.79%), personal experience or opinion (23.97%), and students’ responses in the lesson video (13.22%). Note that the percentage referring to the teacher’s decision notably increased. In a similar vein, the percentage centered on mathematical reason increased (5.97%→11.07%). In addition, the percentage of comments that offered no evidence for analysis considerably decreased compared to the percentages in the first and the second sets of comments (22.02%→23.66%→10.25%). It is meaningful that more PSTs could provide evidence to support their analyses at the end of the semester.
Table 12 .
Alternative strategy | Frequency (%) | ||
---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | |
No alternative strategy | 317 (71.24) | 325 (66.87) | 376 (62.15) |
Superficial alternative strategy | 56 (12.58) | 72 (14.81) | 69 (11.40) |
One specific alternative strategy | 64 (14.38) | 72 (14.81) | 139 (22.98) |
Two or more specific alternative strategies | 8 (1.80) | 17 (3.50) | 21 (3.47) |
Total | 445 (100) | 486 (100) | 605 (100) |
Despite the minor changes, it is encouraging to see the PSTs’ tendency to propose specific alternatives for the events they noticed in the lesson video while taking the methods course (see below for illustrative excerpts).
It seemed that the number of students presenting in the lesson was limited. I could see some students continuing to present, so I thought that only the students who were presenting were doing it. Of course, it is impossible for all students to present in one lesson, but I think it is good for the teacher to give many students a chance to present. (PST 19, first comment)
When the students were writing on the board how to find the surface area of the cookie box, it was regrettable that the teacher asked the student who had finished writing to present his method first, because other students were still writing. It would be helpful for the student who solved the problem with the first method to listen to how to solve it with the second, third, and fourth methods. As the teacher asked the student who first solved to present, however, the students who solved the problem late could not listen to other students’ presentations. (PST 19, second comment)
It would be better to revise the order of students’ presentations on how to find the surface area of the cookie box. The teacher led the students to present in order of (a) adding the areas of the three pairs of congruent faces, (b) adding the side area and the base area, (c) adding the areas of each face, and (d) dividing the net of the box into parts and adding their areas. I think it is right to make presentations in the order of (c), (a), (b), and (d), because adding the areas of each face requires only a simple computation, but the two methods (i.e., adding the side area and the base area, and adding the areas of the three pairs of congruent faces) require more advanced thinking processes in that they are connected to the properties of a rectangular prism， such as “two faces facing each other are congruent” or “the total width of the side and the sum of edges of the base are the same”. To this end, it would have been better if the teacher had made a note of students' solution methods while walking around (
monitoring ) and led students to present on how to find the surface area of the cookie box in pre-selected order (selecting ). (PST 19,italics in original , third comment)
In the first and second comments, PST 19 only mentioned the limited number of students making presentations in the video and the order of presentations on how to find the surface area of a rectangular prism. In contrast, in the third comment, PST 19 mentioned specific alternatives for how to present students’ ideas. In particular, PST 19 considered students’ thinking process in calculating the surface area of a rectangular prism and suggested an alternative strategy based on monitoring and selecting among the five practices she had learned during the methods course.
The PSTs improved their noticing skills while taking an elementary mathematics methods course that combined both theoretical and practical aspects of teaching mathematics. On the one hand, this result aligns with prior studies that reported PSTs’ enhanced ability to notice (e.g., Mitchell & Marin, 2015; Schack et al., 2013; Star & Strickland, 2008; van Es et al., 2017). On the other hand, our study’s result can be differentiated from prior studies using a popular pre-post measure in analyzing the development of noticing (Amador et al., 2021). Note that this study traced the changes in the PSTs’ noticing in a way that was tailored to their learning experience for one semester. This methodological approach, with the support of a detailed analytical framework, is helpful for unpacking changes in what PSTs notice and how they notice.
Compared to the first comments that the PSTs wrote at the beginning of the methods course, their second comments about the same lesson video indicate that their noticing ability was enhanced. Specifically, the PSTs tended to focus more on mathematical content and the teacher’s lesson implementation. Note that the most frequent subtopic was the use of instructional strategies appropriate to the content to be taught. The PSTs also tended to analyze the lesson in an interpretative stance based on mathematical or theoretical reasons or teaching strategies they had learned in the methods course.
Note that this study’s methods course was not designed specifically to improve teacher noticing. Even though the course covered the Measurement Strand of the elementary mathematics curriculum, it did not elaborate on how to teach the surface area of a rectangular prism. However, the results indicate positive changes in teacher noticing through a lecture on the content and pedagogy related to teaching elementary mathematics in general, even though such changes were minimal in this study.
Compared to the first and the second comments, the third comments the PSTs made at the end of the semester illustrated substantial changes in their noticing abilities across all analytic elements: (a) they focused more on Mathematical Discourse beyond Instructional Strategies (
It is common that a mathematics teacher preparation program includes at least one methods course and the practicum. Note that the methods course of this study connected the lecture-oriented university program to the practicum at elementary schools through a group assignment of lesson planning, implementation, and reflection. The practicum itself may have provided a meaningful experience for PSTs, but just participating in the fieldwork experience does not increase teacher noticing ability (Santagata & Yeh, 2014). In contrast, the group assignment during the practicum period and subsequent discussions on it in the methods course seem to have positively impacted the PSTs’ noticing ability.
Discussing a video-taped lesson among teachers, usually including a teacher educator or facilitator, is a common pedagogical approach that teacher education programs employ to improve teacher noticing (McDuffie et al., 2014; Mitchell & Marin, 2015; van Es & Sherin, 2008). However, unlike many other studies in which discussions centered on video clips selected by researchers, the PSTs in this study discussed in small groups how they planned and enacted their mathematics lessons. They then had an opportunity to discuss further with the instructor and other PSTs from multiple perspectives. The PSTs’ actual experience of teaching mathematics and sharing diverse interpretations seem to have improved their noticing ability (Fernández, Llinares, & Rojas, 2020; Lee & Choy, 2017).
This study shows that PSTs’ noticing ability can be significantly enhanced through a mathematics methods course combining theory and practice. Our study found that the degree and aspects of positive changes in teacher noticing were different depending on when the PSTs’ comments were made. Our findings suggest that there is much room for further development of teacher noticing because even the third round of participant comments provided more evaluative comments than interpretive comments, frequently without alternative strategies. Nevertheless, it is encouraging that one semester-long methods course led PSTs to improve their ability to notice salient features of a mathematics lesson.
In this study, we used the five elements as an analytic framework—
Regarding the analytic element
The detailed analytic framework using subtopics also helped us explore which features need to be emphasized in a teacher education program. For instance, it is encouraging that the PSTs focused more on mathematical tasks while taking the methods course. However, their focus was centered either on using tasks aligned with the lesson objective or on using tasks tailored to students’ prior knowledge and motivation. On the contrary, little focus was on using tasks to promote students’ mathematical thinking throughout the semester. Knowing exactly what the PSTs noticed and what they missed is helpful for re-designing the methods course (Spizer & Phelps-Gregory, 2017).
Regarding the analytic element
In fact, it was encouraging that the PSTs’ comments in this study were grounded in the decision of the teacher in the video as well as their personal experiences or opinions. Their comments were also based on more mathematical reasons as the semester went on. Given that teachers are expected to discuss their lessons throughout their teaching career, any teacher preparation program needs to foster the ability to evaluate or interpret classroom events based on specific and reasonable source.
To summarize, this study devised a more detailed framework to analyze PSTs’ noticing ability, building on prior studies, specifically regarding the actor and evidence elements of analysis. Such a framework broadens our understanding of the changes in teacher noticing throughout a mathematics methods course localized in the Korean context. It is hoped that this study’s analysis of teacher noticing will provoke discussions both on how to design a mathematics methods course in a teacher preparation program and on how to measure noticing as an indicator of teacher expertise.
No potential conflict of interest relevant to this article was reported.
Table 1
Instructional phase | Main activities | Time (min) |
---|---|---|
Motivation for the lesson and identification of the lesson goal | • Recall the characteristics of a rectangular prism by showing a cookie box | 5 |
• Think about the method of how to use the least amount of paper when wrapping the cookie box • Present the lesson goal: Let’s explore how to figure out the surface area of a rectangular prism | ||
• Introduce the definition of surface area and write it in a notebook | 2 | |
Exploration of multiple methods to find the surface area of a rectangular prism | • Predict how to find the surface area of the given rectangular prism • Discuss methods in groups | 6 |
Presentation of various solution methods and whole-group discussion | • Whole group discussion regarding how to figure out the surface area of the rectangular prism Method 1: Adding the area of each of the prism’s six faces Method 2: Adding the areas of three different faces and then doubling it Method 3: (Area of the rectangular base)×2 + (Area of the side face) | 7 |
• Calculate the surface area of the rectangular prism with the three methods above • Review each method | 11 | |
Generalization of the formula for finding the surface area of a rectangular prism | • Explore the most effective method and formalize how to figure out the surface area of a rectangular prism | 5 |
Application | • Solve the review exercises in the textbook | 2 |
Closing | • Talk about what you have learned during the lesson • Assignment: Calculate the surface area of a rectangular prism you would find in everyday life | 2 |
Table 2
Topic | Subtopic | Code |
---|---|---|
Mathematical Tasks | Using tasks aligned with the lesson objective | MT1 |
Using tasks tailored to students’ prior knowledge and motivation | MT2 | |
Using tasks to promote students’ mathematical thinking and communication | MT3 | |
Instructional Strategies | Using instructional strategies appropriate for the content to be taught | IS1 |
Using instructional strategies appropriate to students’ characteristics | IS2 | |
Using manipulatives and materials to help students’ learning | IS3 | |
Effective implementation of the written lesson plan, including improvisation | IS4 | |
Assessment | IS5 | |
Learning Environment | Establishing a learning environment through appropriate social norms | LE1 |
Physical arrangement and equipment to support learning | LE2 | |
Mathematical Discourse | Mathematical communication between the teacher and students, and among students | MD1 |
Meaningful questions and appropriate feedback | MD2 |
Table 3
Actor | Description | Code |
---|---|---|
Teacher | The comment mainly focused on the teacher in the video | T |
Students | The comment mainly focused on a particular student, a group of students, or all the students in the video as a whole | S |
Teacher and Students | The comment equally paid attention to both the teacher and the students | T/S |
Table 4
Stance | Description | Code |
---|---|---|
Describe | Negative statement | D– |
Positive statement | D+ | |
Evaluate | Negative statement | E– |
Positive statement | E+ | |
Interpret | Negative statement | I– |
Positive statement | I+ |
Table 5
Evidence | Code | Evidence | Code |
---|---|---|---|
Students’ responses in the lesson video | S | Mathematical (or theoretical) reason | M |
Personal experience or opinion | E | Affective reason | A |
Connections across the lesson flow | F | Learning environment | L |
Decision of the teacher in the video | T | Comparison with other strategies | C |
No evidence | N |
Table 6
Alternative strategy | Description | Code |
---|---|---|
Level 1 | No alternative strategy proposed | 1 |
Level 2 | Vague or superficial alternative strategy proposed | 2 |
Level 3 | One specific alternative strategy proposed | 3 |
Level 4 | Two or more specific alternative strategies proposed | 4 |
Table 7
Excerpt from a PST’s comment (a semantic unit) | Topic | Actor | Stance | Evidence | Alternative strategy |
---|---|---|---|---|---|
It was good that the teacher didn’t mention the most effective method immediately after the students found the three solution methods. Because of that, the students had the opportunity to think of and present why the specific method was the most effective to calculate the surface area of a rectangular prism. | IS1 | T/S | E+ | S | 1 |
Table 8
Topic | Frequency (%)* | |||
---|---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | ||
Mathematical Tasks | MT1** | 13 (2.92) | 11 (2.26) | 26 (4.30) |
MT2 | 21 (4.72) | 24 (4.94) | 30 (4.96) | |
MT3 | 1 (0.22) | 4 (0.82) | 5 (0.83) | |
Total | 35 (7.87) | 39 (8.02) | 61 (10.08) | |
Instructional Strategies | IS1 | 133 (29.89) | 157 (32.30) | 123 (20.33) |
IS2 | 31 (6.97) | 24 (4.94) | 26 (4.30) | |
IS3 | 66 (14.83) | 51 (10.49) | 73 (12.07) | |
IS4 | 12 (2.70) | 25 (5.14) | 29 (4.79) | |
IS5 | 2 (0.45) | 10 (2.06) | 10 (1.65) | |
Total | 244 (54.83) | 267 (54.94) | 261 (43.14) | |
Learning Environment | LE1 | 26 (5.84) | 33 (6.79) | 14 (2.31) |
LE2 | 9 (2.02) | 4 (0.82) | 5 (0.83) | |
Total | 35 (7.87) | 37 (7.61) | 19 (3.14) | |
Mathematical Discourse | MD1 | 70 (15.73) | 65 (13.37) | 119 (19.67) |
MD2 | 61 (13.71) | 78 (16.05) | 145 (23.97) | |
Total | 131 (29.44) | 143 (29.42) | 264 (43.64) | |
Total | 445 (100.00) | 486 (100.00) | 605 (100.00) |
*Frequency refers to the number of semantic units from comments. The percentage (%) is rounded from the third decimal place.
**MT1: Using tasks aligned with the lesson objective. MT2: Using tasks tailored to students’ prior knowledge and motivation. MT3: Using tasks to promote students’ mathematical thinking and communication. IS1: Using instructional strategies appropriate for the content to be taught. IS2: Using instructional strategies appropriate to students’ characteristics. IS3: Using manipulatives and materials to help students’ learning. IS4: Effective implementation of the written lesson plan, including improvisation. IS5: Assessment. LE1: Establishing a learning environment through appropriate social norms. LE2: Physical arrangement and equipment to support learning. MD1: Mathematical communication between the teacher and students, and among students. MD2: Meaningful questions and appropriate feedback.
Table 9
Actor | Frequency (%) | ||
---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | |
Teacher | 358 (80.45) | 404 (83.13) | 269 (44.46) |
Students | 80 (17.98) | 62 (12.76) | 86 (14.21) |
Teacher and Students | 7 (1.57) | 20 (4.12) | 250 (41.32) |
Total | 445 (100) | 486 (100) | 605 (100) |
Table 10
Stance | Frequency (%) | |||
---|---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | ||
Describe | Negative | 39 (8.76) | 41 (8.44) | 20 (3.31) |
Positive | 131 (29.44) | 88 (18.11) | 70 (11.57) | |
Total | 170 (38.20) | 129 (26.54) | 90 (14.88) | |
Evaluate | Negative | 45 (10.11) | 76 (15.64) | 164 (27.11) |
Positive | 146 (32.81) | 144 (29.63) | 215 (35.54) | |
Total | 191 (42.92) | 220 (45.27) | 379 (62.64) | |
Interpret | Negative | 20 (4.49) | 42 (8.64) | 52 (8.60) |
Positive | 64 (14.38) | 95 (19.55) | 84 (13.88) | |
Total | 84 (18.88) | 137 (28.19) | 136 (22.48) | |
Total | 445 (100) | 486 (100) | 605 (100) |
Table 11
Evidence | Frequency (%) | ||
---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | |
Students’ responses in the lesson video | 69 (15.51) | 59 (12.14) | 80 (13.22) |
Personal experience or opinion | 118 (26.52) | 127 (26.13) | 145 (23.97) |
Connections across the lesson flow | 14 (3.15) | 19 (3.91) | 22 (3.64) |
Decision of the teacher in the video | 83 (18.65) | 85 (17.49) | 156 (25.79) |
Mathematical (or theoretical) reason | 8 (1.80) | 29 (5.97) | 67 (11.07) |
Affective reason | 15 (3.37) | 13 (2.68) | 22 (3.64) |
Learning environment | 40 (8.99) | 23 (4.73) | 27 (4.46) |
Comparison with other strategies | 0 (0.00) | 16 (3.29) | 24 (3.97) |
No evidence | 98 (22.02) | 115 (23.66) | 62 (10.25) |
Total | 445 (100) | 486 (100) | 605 (100) |
Table 12
Alternative strategy | Frequency (%) | ||
---|---|---|---|
First comment (data 1) | Second comment (data 2) | Third comment (data 3) | |
No alternative strategy | 317 (71.24) | 325 (66.87) | 376 (62.15) |
Superficial alternative strategy | 56 (12.58) | 72 (14.81) | 69 (11.40) |
One specific alternative strategy | 64 (14.38) | 72 (14.81) | 139 (22.98) |
Two or more specific alternative strategies | 8 (1.80) | 17 (3.50) | 21 (3.47) |
Total | 445 (100) | 486 (100) | 605 (100) |
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