## Pre-Conference

### Teacher’s Approach to Teaching Math

The teacher demonstrated that she is not indifferent to the results of her work and her students’ performance. She said that her aim is to explain the students that mathematics is “not all about numbers on the paper, but all about the world around us”. She stated that she always emphasizes the real-life setting of each notion or operation that they discuss at the lessons.

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### Teacher about the Lesson

At that moment, students were learning division. The teacher planned to devote the lesson to using division in the real-life setting. She said that she wanted to give her students opportunity to better understand where and how they can use division and thus motivate them to studying it with diligence and attention. She planned to offer her students interesting assignments and include the elements of team work.

### Teacher about the Class

The teacher characterized the class as not homogeneous in terms of capability to learning mathematics. She said that the students are strong, well-prepared and motivated to studying, but not all of them are “mathematicians”: some of the students perform better in reading and language arts. She explained that the most difficult task for her when working with this class is to find the balance between the needs of “mathematicians” and “not mathematicians”.

Nevertheless, the teacher considers that it is her duty to give students opportunity to get many-sided, deep comprehension of math. She wants them to get ready to studying algebra and geometry at school and to doing high-level research at college and university.

## Observation and Analysis of Teaching

### Identifying Information

**Setting:**(enter the name)**Gender:**Female**Number of years teaching in current assignment:**6**Grade level:**4**Subject area of lesson:**Math

## Lesson Overview

The topic of the lesson was applying division skills in the real-life settings. The focus was on division cases when a dividend is 2-digit, 3-digit or 4-digit, and a divisor is 1-digit.

The teacher explained that at the previous lessons, they had practiced division skills and were ready to apply them to solve more unexpected assignments. Thus, during the lesson, the students were expected to:

- Practice in applying division skills to solve real-life problems.
- Master division skills.

The lesson included the following stages:

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- The teacher conducted the introductive part of the lesson
- The teacher and the students had the collective work: they fulfilled several assignments aimed at mastering division skills so that students could refresh division.
- The students competed in teams: they had to solve a complex assignment devoted to applying division skills and to present the result.

The final complex assignment implied fulfilling a set of division operations connected by the common “plot”.

## Analysis of Teaching Skills

The analysis has been made on the basis of the scripts taken by me during the lesson. The lesson has been evaluated according to 10 domains offered in (Danielson, 1996).

### Domain 2a – Creating an environment of respect and rapport

This domain implies evaluation of environment at the lesson, relationship between a teacher and the students and between the students (Danielson).

From the first minutes of the lesson, the environment at class was very positive and favorable for work. The teacher began with enthusiasm, and the students also seemed to be ready to intensive work. During the lesson, the teacher was energetic and friendly; the students were involved into the work and actively participated in the lesson.

### Domain 2b – Establishing a culture for learning

This domain focuses on evaluation of students’ understanding of the purpose of the lesson and the importance of the topic, their motivation to learn the new material etc. (Danielson).

When the teacher announced the topic, the students seemed a bit disappointed, as they had learned division for several lessons. However, when a teacher promised that the assignments will be interesting and fun, they cheered up. The class demonstrated that they are not indifferent to what they learn; they work with enthusiasm when an assignment is not ordinary and implies some real-life setting.

### Domain 2c – Managing classroom procedures

This domain implies evaluation of a teacher’s ability to manage materials and supplies, create necessary conditions for conducting a lesson. (Danielson).

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The students were prepared to the lesson appropriately. They all had paper and pens. At the end of the lesson, the teacher provided assignment sheets to the students; despite they were expected to work in teams, each student had a copy.

### Domain 2d – Managing student behavior

This domain refers to evaluation of a teacher’s ability to maintain environment appropriate for learning, notice misbehavior and manage it (Danielson).

When the students fulfilled the team assignment, the teacher successfully maintained silence and discipline. When one “spy” tried to “steal” another team’s answers, she noticed it very quickly and stopped him with a polite jocular remark.

### Domain 2e – Organizing physical space

This domain implies evaluation of a teacher’s ability to make the class space safe and convenient for students (Danielson).

The lesson plan included team work and presentation of its results. The teams had enough space to work with convenience and did not disturb each other. The teacher had opportunity to approach each team when they wanted to ask her something. The students presented results near the blackboard.

### Domain 3a – Communicating Clearly and accurately

This domain implies evaluation of a teacher’s ability to choose appropriate level of complexity when delivering the material, use appropriate vocabulary, and make students understand the material and the assignments. (Danielson).

The teacher used effective wording when she introduced the topic and conducted the collective work. At the same time, when she tried to discuss the material on a more general level, the students seemed somewhat bored.

The teacher provided the detailed explanation of the final assignment so that students did not have difficulties with fulfilling it.

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### Domain 3b – Using questioning and discussion techniques

This domain evaluates how a teacher communicates with the students by means of asking and answering questions (Danielson).

Asking and answering questions is one of the key techniques that the teacher uses to communicate with her students. During the lesson, the questions were addressed in both directions: the teacher answered the students’ questions and asked them “diagnostic” questions to see whether they understood her.

### Domain 3c – Engaging students in learning

The domain implies evaluation of a teacher’s ability to involve students into discussion, make all students equally participate in the class activities etc (Danielson).

Considering that the students demonstrated different level of capability to understand the material, it was difficult for the teacher to keep her focus broad and address the needs of each student at once. Sometimes when the “mathematicians” worked actively and asked questions, the “not mathematicians” seemed to be a bit “out of the game”.

### Domain 3d – Providing feedback to students

This domain evaluates a teacher’s ability to communicate with the students, answer their questions, react on their participation in the lesson (Danielson).

The teacher found good balance between delivering the material and giving students opportunity to actively participate in the lesson. She encouraged them to ask and answer questions and supported each student when he/she answered or shared his/her thoughts.

### Domain 3e – Demonstrating flexibility

This domain implies evaluation of how a teacher grasps students’ feedback (verbal and non-verbal) and adapts the way he/she conducts the lesson (Danielson).

During the first part of the lesson students sometimes seem bored. First, the “mathematicians” were bored when the class worked on reviewing division skills. Then, when a teacher tried to discuss division on a more general level operating the terms “quotient”, “dividend”, “divisor”, all students seemed somewhat bored. At the moment, they seem to prefer to limit themselves to fulfilling operations rather than conduct a discussion using terms. At the same time, the “mathematicians” started an active conversation about the remainder in the real life and seemed interested in it. Sometimes the teacher failed to take these changes of the students’ mood into account and did not react quickly.

## Summary Statement

It was very interesting to observe this lesson. The teacher worked with enthusiasm and encouraged students to actively participate in the lesson. She had managed to awake the students’ interest to mathematics and had trained them to aspire for deep understanding of the topics. Students asked questions, which demonstrated that they want to understand the essence of the issues they learn.

Despite involving the students into active work, the teacher successfully maintained environment that was favorable for learning. She answered the students’ questions and nevertheless followed the course of the lesson. The discipline in the class was of the high level, even during the team work. The material of the lesson corresponded to the New Jersey curriculum framework for Grades 3-4 (State of New Jersey Department of Education).

At the same time, the class demonstrated differentiation in capability to deep understanding of the material. While some students grasped the teacher’s explanation and asked questions, the others seemed to be somewhat “out of he game”. The teacher understands this differentiation, as she mentioned it at the pre-observation stage, but at the moment, she has not find the way to manage this situation and address it during her lessons.

Besides, the students stayed indifferent to the teacher’s attempt to discuss division on the more general level. The teacher demonstrated her intent to gradually introduce the elements of “arithmetic-algebra” transition, but at the moment, the students seem to be not prepared enough.

After the team work, the students presented the results. The teacher asked not about the answers only, but also about the way the students thought, that is why the presentation was quite interesting.

Despite the students did not demonstrate their individual division skills, I think the lesson was nevertheless useful: the students had to work quickly and thus mastered their skills. Besides, they understood how important learning division is.

The overall impression about the lesson is very positive. The teacher has created good environment, and the math lessons are conducted with enthusiasm and pleasure.

## Post-Conference

I talked with the teacher in one hour. She said that she was quite satisfied with the results of the students’ team work. Only one team did not get the right answers.

I explained my concerns about the differentiation in the class and the complexity of discussion of division on a general level. The teacher said that at that moment, she had been looking for the solution of the situation with the students’ different capacity and had not found the answer yet.

## Professional Growth Plan

Having observed the lessons, I have defined two points that I would recommend the teacher to work on:

### Point 1

The teacher needs to develop the optimal approach to teaching the class with the explicit differentiation in capacity to learning mathematics.

### Objectives

The teacher should improve her ability to identify the needs of the “not mathematicians” and conduct her lessons so that they will not feel themselves left “out of the game”.

### Point 2

The teacher should improve her approach to realization of her intent to introduce the elements of transition from arithmetic to algebra.

### Objectives

The teacher should understand the level of the students’ cognitive readiness to advanced learning of math and develop the teaching strategy that will correspond to it.

Recommended activities: I recommend the teacher to work with the sources devoted to cognitive aspects of learning mathematics and cognitive readiness to algebra learning. She may get familiarized with the researches and recommendations of Kieran, Brumbaugh and Rock, Carraher and Schliemann, Chazan, Davis et al devoted to identifying the level of children’s cognitive development, considering cognitive aspects of preparing them to algebra et al. She will understand her students better and address the needs of “not mathematicians” more precisely.

The teacher may also conduct the testing to assess the level of the students’ cognitive development. She will be able to find the assignments in the sources devoted to this topic.

Next meeting: in 1 month.

When observing the lesson, I tried to make my observation many-sided and to pay attention to different aspects.

After having read my scripts, I noticed that I focused on such issues as communication between the teacher and the students, their feedback to each other, the students’ behavior et al. At the same time, I did not manage to script the minor details of how exactly the teacher delivered the material, though it was very interesting: she found effective wording to communicate with the students.

Nevertheless, the material that I scripted was enough to complete the Observation and Analysis part. Next time, I will try to pay attention to both the way the material is delivered and the way the students respond.

## Bibliography

Danielson, Charlotte*. Enhancing Professional Practice: A Framework for Teaching.* Umatilla, 1996. Web.

*NJ Mathematics Curriculum Framework: Standard 6 – Number Sense – Grades 3-4.* State of New Jersey Department of Education. Web.