(REFLECTION: Part 1)
(Representation and Analysis of My LessOn)
My lesson got off on the wrong foot from the very beginning. My partner student teacher and I both scheduled our last Term III lessons within fifteen minutes of each other on the Wednesday before Thanksgiving. We agreed to video record each other's lessons. Since her lesson went over by fifteen minutes and I was recording her lesson, I did not have very much time to set up my materials. More importantly, I did not have time to get myself together mentally. I felt very flustered and rushed from the very beginning because I knew that my students had to go to lunch at 12:15. Instead of altering the amount of material that I wanted to cover in the lesson, I tried to fit the same amount of material into a shorter time frame. My Penn Mentor noted the fast pace of my lesson in her observation notes. In hindsight, trying to enact my full lesson plan was not a very wise in the moment decision.
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"Students work with partners to create their own word problems- Problems are shared with the group (rushed again)." - Penn Mentor |
This is the poster that I made to go with the first word problem that we modeled. I had to alter the problem to reflect our new scenario.
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I also did not think about the effect of teaching my small group lesson on the day before the long Thanksgiving weekend. Although it is hard to say whether the timing of this lesson significantly affected my students’ behavior, it did affect our attendance. One-third of our class was absent on the day of my math lesson. This was problematic for me because two of my six students were absent. Since I specifically selected my group of six students based on what I knew about their understanding of subtraction, I did not want to pull two other students into my group. I thought that change unknown subtraction word problems might be a challenge for the small group of students that I selected, which led me to think that it could have brought other students to a level of frustration.
Although teaching a lesson with four students instead of six students might seem like it would have a minimal little impact, I think that it affected my lesson. I brought posters with my word problems, because I have learned that students benefit from visual aids (as opposed to just reading the problem out loud). Since Janine and I talked about having direct modeling in the lesson, I planned to have the students model the word problem. However, I had to alter the word problem from 6 - 2 = ? to 4 - 2 = ?, so that it would represent our group of four students. Many of the students kept talking about how they knew the answer was two. I had difficulty discerning whether they were getting the number "two" from the word problem or from solving the problem and giving the answer. I think the new problem made it difficult for the students to see the difference between the change and the end. |
After we went over the two types of problems- end unknown and change unknown, I had the students write their own word problems with partners. I hoped that this would give me a better understanding of what they thought the difference was between the end and change unknown problems. Unfortunately, as I mentioned earlier, I was only able to give the students about four minutes to write their word problems. This was definitely not enough time. I also realized that this task was not just assessing their understanding of the different contexts for subtraction situations, but it also relied heavily upon the students' ability to record their thinking with words. Since we ran out of time toward the end of the lesson, I was unable to have the students solve and discuss the word problems that their classmates created. Writing, solving, and discussing the word problems were my main forms of assessment. Fortunately, I conferenced with individual students as they tried to find the unknown in the second word problem. Had I not done that, I would have been left with very little information about what my students understood.
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This was the poster that I made to go with the second word problem. I wanted to have the students apply what we did in the first problem (start, change, and end table) to this change unknown problem.
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(TASKS)
My enacted Term III Math lesson was comprised of group, individual, and partner tasks:
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As Hiebert (1997) discussed, "students form their perception of what a subject is all about from the kinds of tasks that they do" (p. 17). Knowing this, I knew that I did not want all of the tasks to involve filling in worksheets and solving problems. Although I did include worksheets and writing number sentences, I wanted to make sure that I also included opportunities for the students to share their thinking and discuss their strategies. I also thought that having them write their own word problems might give them a different understanding of mathematics because usually students are given word problems that may not mean anything to them.
I feel like all of the tasks in this lesson are vital in seeing if students are able to identify the unknowns in different types of subtraction word problems. However, I think the order of some of the tasks may have hindered rather than helped students better understand different types of word problems. I wanted the students to use the start, change, and end table to represent the problem and help generate the number sentence. In the first problem, I had the students generate the number sentence before the table, because I had not introduced the table yet. I think my rationale behind this was to have the students write the number sentence independently to get a sense of what they knew about identifying parts of the problem before I introduced the start, change, and end table. It would be very difficult to tell where individual students were in their understanding of what the word problem was asking if they worked collaboratively. The sequence made sense for my needs, but I do not think it was necessarily appropriate for my students' needs.
I think modeling how I would approach a word problem and filling in a larger copy of the worksheet together was one of the strengths of this lesson. However, after enacting my lesson, I realized that I sequenced my tasks in a way involved modeling the more familiar end unknown problem and having the students independently identify the pieces of the problem type that I was new to them. I think modeling both problem types and having a discussion about the differences between the word problems would have been more helpful for the students. My Penn Mentor noted that some of my students "appear to be confused". This suggests to me that the sequence of the tasks was more detrimental than helpful to the students and that more time spent on discussion was needed.
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"Most students are focused and engaged, but some appear to be confused." - Penn Mentor |
"I just put the question mark because you're supposed to." |
Although I did not explicitly state this as a goal or objective, I wanted the students to stop thinking about math as a time to focus on finding the "right answer" and moving on to solve the next problem. I now realize that by making the start, change, and end table into a worksheet, the tool that was supposed to be a representation intended to make math meaningful actually just became a new way for the students to "find the right answer". When I asked Matthew about the end unknown problem, he said, "I just put the question mark because you're supposed to." (20:51) I asked him to explain why he put it at the end and he said, "Because that's what we're supposed to do." (20:58) Looking at the rubric, Matthew also wrote "-2" for the change and did not include a question mark anywhere. This suggests to me that he was more focused on solving the problem and finding the "right answer" than he was in using the representation as a way to deconstruct the word problem to better understand what you are trying to solve for.
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Looking at the Rubric:
S= What did the student say was the start from the change unknown problem?
C= What did the student say was the change from the change unknown problem?
E= What did the student say was the end from the change unknown problem?
NS= Did the number sentence represent the change unknown problem?
W.P. End= Did the student create an end unknown word problem?
W.P. Change= Did the student create a change unknown word problem?
Solve W.P. 1= Was the student able to solve/identify the parts in the first word problem?
Solve W.P. 2= Was the student able to solve/identify the parts in the second word problem?
Check Understanding= At the end of the lesson/during the discussion, did the student seem to have an understanding of the difference between change and end unknown word problems?
Other Notes:
C= What did the student say was the change from the change unknown problem?
E= What did the student say was the end from the change unknown problem?
NS= Did the number sentence represent the change unknown problem?
W.P. End= Did the student create an end unknown word problem?
W.P. Change= Did the student create a change unknown word problem?
Solve W.P. 1= Was the student able to solve/identify the parts in the first word problem?
Solve W.P. 2= Was the student able to solve/identify the parts in the second word problem?
Check Understanding= At the end of the lesson/during the discussion, did the student seem to have an understanding of the difference between change and end unknown word problems?
Other Notes:
(TOOLS)
There were four main tools involved in this lesson: the start, change, and end table, the question mark symbol, the poster with questions to think about when looking at word problems, and the posters that were copies of the students' worksheets. The start, change, and end table was connected to my goal because it was a way for the students to see the differences in word problems visually. I hoped that they would see that sometimes word problems provide the start and change and other times they provide the start and end. (I did not mention start unknown problems during this lesson.)
As I mentioned, I think that the start, change, and end table was useful in supporting my students' thinking about the different word problems. However, it became almost like a new way to show that they arrived at the right answer. In this case, the right answer did not mean solving the problem, it meant representing the problem accurately. Matthew was able to fill in the table correctly, but he did not explain why he put the numbers and symbols into those specific boxes. Instead, he kept saying that it was, "because you're supposed to". This suggests to me that the table was not effective at supporting his deeper understanding of the problem. I also think based on one of his comments that he already has an understanding of the relationship between subtraction and addition, which was one of my bigger picture goals. During our discussion, Matthew said, "I don't see it, but if you have two and you do a number line and add two more, you get four. And then if you add four and you take away two, you get another two." (29:05)
The question mark symbol was a universal way for us to represent the part of the story or number sentence that was unknown to us. I wanted the students to be able to have some say in how to represent the unknown, which is why we voted on the symbol. The students unanimously agreed to use a question mark (?) rather than an empty box ( [ ] ). The question mark was intended to help the students see visually what part of the problem they were trying to solve for.
As I mentioned, I think that the start, change, and end table was useful in supporting my students' thinking about the different word problems. However, it became almost like a new way to show that they arrived at the right answer. In this case, the right answer did not mean solving the problem, it meant representing the problem accurately. Matthew was able to fill in the table correctly, but he did not explain why he put the numbers and symbols into those specific boxes. Instead, he kept saying that it was, "because you're supposed to". This suggests to me that the table was not effective at supporting his deeper understanding of the problem. I also think based on one of his comments that he already has an understanding of the relationship between subtraction and addition, which was one of my bigger picture goals. During our discussion, Matthew said, "I don't see it, but if you have two and you do a number line and add two more, you get four. And then if you add four and you take away two, you get another two." (29:05)
The question mark symbol was a universal way for us to represent the part of the story or number sentence that was unknown to us. I wanted the students to be able to have some say in how to represent the unknown, which is why we voted on the symbol. The students unanimously agreed to use a question mark (?) rather than an empty box ( [ ] ). The question mark was intended to help the students see visually what part of the problem they were trying to solve for.
The poster with the word problem questions was intended to support students by providing a structure for them to pick out the important aspects of the situation. Identifying the components of the problem is helpful when trying to represent the situation in a number sentence. I thought this tool would be important because I knew that my classroom mentor had a similar chart in her classroom that the students used to think about word problems. Even though I think that this tool was familiar to my students, I am not sure that they used it when they looked at the word problem. They did not mention it when I asked them to explain how they filled in their tables.
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In addition to having the word problems on posters in the front of the table, I also tried to have larger poster versions of the worksheets that the students were supposed to fill out. My Penn mentor has told me many times that it helps if students have a visual. She taught me that it is important to model exactly what the teacher is looking for when it comes to recording information or completing worksheets. I also learned the importance of introducing new ways of organizing information from my literacy lesson. If students are unfamiliar with the graphic organizer, they need direct instruction.
(DISCOURSE)
One of the things that my Penn mentor commented on was how I encouraged collaboration. Knowing that she noticed how I tried to incorporate student to student talk in different ways made me feel successful. Having spent a lot of time talking about the limitations on relying on patterns of discourse that tend to follow the initiation-response-evaluation pattern, I went in to all of my lessons with a desire to encourage more student to student talk. I wanted the students to turn and talk to each other, but I also wanted them to talk to each other in our group settings (Kazemi and Hintz, 2014, p. 21).
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"Students are encouraged to 'turn and talk'- collaborate- help each other- revise answers if needed - SHARE!!" - Penn Mentor |
Me: "So, we started with how many people, Stephen?" |
Although my Penn mentor noted the student to student talk, when I watched the video of my lesson, I realized that I started my lesson with a lot of teacher to student questions that were more about regurgitating the information from the problem and less about how they identified that particular piece of information as important.
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I noticed that I did not always ask students to repeat what I said or what the problem was asking. Some of the teacher to student questions that I asked were intended to encourage students to repeat what their classmates had said (Kazemi and Hintz, 2014, p. 21). There were a number of times where I asked things like, "So DeSean, can you remember what he just said? Can you tell the group?" (5:43).
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"Students are encouraged to repeat the responses of others." |
Me: "Can anyone tell us what Dana just said?" (3:41) |
While I think that it is good that I encouraged repeating, I think I could have emphasized repeating the thinking behind what the students said rather than merely repeating what they said. Repeating can be used to check comprehension of concepts as well as "restate important parts of a complex idea" (Kazemi and Hintz, 2014, p. 21). By making it about what the other student "just said", I put more emphasis on the words than the understanding of others' ideas.
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I wanted to provide opportunities for student to student talk. It was important for me to show the students that when we share our answers, it is not just about sharing their ideas with the teacher and having the teacher evaluate their response. The students in our classroom are encouraged to reference their classmates' ideas by beginning their sentences with "I agree with ______ because..." and "I disagree with ______ because..." but they are not given many opportunities to converse with each other about their ideas in a group setting. I thought that I might have to encourage the students to discuss their thoughts with each other in a way that was more conversational than merely referencing their ideas.
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Me: "Why don't you two talk to each other and everyone else will listen?" (5:28) |
Matthew: "What's a number sentence?" (6:01) |
Although the majority of the discourse in this lesson was teacher-student-student or student-student, there were some instances where a student would ask a question. Unfortunately, I do not think that I succeeded in creating an atmosphere where students asked deep conceptual questions. The questions that were generated by students were more clarifying questions like, "What's a number sentence?"
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One of my main goals for this lesson was to encourage students to talk more about their process than their answers. I tried to ask questions like, "Can you explain to me what you did here?" (22:20) My Penn mentor noted that, "Students are consistently encouraged to explain answers/strategies- focus on thinking- not answers." I was happy that she picked up on my intention, but when I watched the video of my lesson, I found myself saying little things that were counter to my intentions. Even though I wanted my students to think more about the process, it seems like I inadvertently encouraged our conversation to become more "answer-driven" because I said, "Matthew and Dana did you get the same number sentence? Stephen and DeSean did you get the same number sentence?" (8:15) Even though I was trying to see if there would be a debate or a disagreement, it might have come off to the students that there was a "right" answer. One thing that I could do next time is tell the students that I came up with a different number sentence. I could then ask them to explain why they thought their number sentence reflected the word problem and how mine differed from theirs. I recognize that this would have been difficult to do with this particular word problem since the problem was 4 - 2 = 2.
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DeSean: "4 minus 2 equals what?" |
(NORMS)
From the very beginning of my lesson, I tried to establish a norm around mathematics as a collaborative subject. I hoped that my classroom arrangement communicated this idea. Even though I had to use the library space to teach my lesson, I felt like the arrangement was suited for this lesson because the students were grouped in pairs so they could collaborate. Also, the chairs were facing each other which meant that the students can make eye contact during the discussion and direct their comments at each other. I would not have wanted them to be at individual desks facing me, because I did not want to suggest that all of their attention should be directed at me.
As I mentioned in the previous discourse section, one of the norms that I wanted to establish was the importance of talking about our reasoning rather than our answer. After we talked about the word problem, Stephen said, "Where are we supposed to write it?" I replied, "Oh, we're not writing yet." (2:59) I intended to communicate to Stephen that we're not just trying to find the answer and record it. We're going to talk about it first.
Since I am emphasizing the process rather than the answer, it is important to recognize that some students may need additional time to think about their thinking and then figure out how to communicate that to others. One of the things that I could work on is increasing my wait time. Although I tended to give between 5 and 7 seconds of wait time, I could have waited longer before asking other students to share their thinking. I noticed that there were a couple of times when students would begin to share their thinking, but then either I would make a side comment to someone about their behavior or there would be an outside noise that might have distracted them. Instead of realizing that they might have lost their train of thought because of the distraction and given them more wait time, I gave less wait time and then asked other students to share. (3:41) By correcting behavior while the student was thinking, I also may have inadvertently communicated to the students that addressing behavioral issues was more important to me than waiting for the student to collect his thoughts.
Although my wait time could be increased and improved, I think that I did a good job of communicating to the students that I wanted to hear what everyone thought. At various points during the lesson, I asked students to put their thumbs up or thumbs down if they agreed with what was being said (5:07). I encouraged both the students who put their thumbs up and down to explain why they agreed or disagreed. I also think that by walking around, observing, and listening to what the students were doing and saying, I was showing them that I was interested in their thinking. I communicated to students that I was focused on them by making comments to the group about things that I noticed. For example, at one point I paused the turn and talks and said, "I heard Matthew use a good word- equation. A number sentence is like an equation." (7:05)
As I mentioned in the previous discourse section, one of the norms that I wanted to establish was the importance of talking about our reasoning rather than our answer. After we talked about the word problem, Stephen said, "Where are we supposed to write it?" I replied, "Oh, we're not writing yet." (2:59) I intended to communicate to Stephen that we're not just trying to find the answer and record it. We're going to talk about it first.
Since I am emphasizing the process rather than the answer, it is important to recognize that some students may need additional time to think about their thinking and then figure out how to communicate that to others. One of the things that I could work on is increasing my wait time. Although I tended to give between 5 and 7 seconds of wait time, I could have waited longer before asking other students to share their thinking. I noticed that there were a couple of times when students would begin to share their thinking, but then either I would make a side comment to someone about their behavior or there would be an outside noise that might have distracted them. Instead of realizing that they might have lost their train of thought because of the distraction and given them more wait time, I gave less wait time and then asked other students to share. (3:41) By correcting behavior while the student was thinking, I also may have inadvertently communicated to the students that addressing behavioral issues was more important to me than waiting for the student to collect his thoughts.
Although my wait time could be increased and improved, I think that I did a good job of communicating to the students that I wanted to hear what everyone thought. At various points during the lesson, I asked students to put their thumbs up or thumbs down if they agreed with what was being said (5:07). I encouraged both the students who put their thumbs up and down to explain why they agreed or disagreed. I also think that by walking around, observing, and listening to what the students were doing and saying, I was showing them that I was interested in their thinking. I communicated to students that I was focused on them by making comments to the group about things that I noticed. For example, at one point I paused the turn and talks and said, "I heard Matthew use a good word- equation. A number sentence is like an equation." (7:05)
Stephen: "Two students went to go find Ms. V." |
I encouraged students to use the resources that were around us such as the poster with the questions to ask yourself and the posters with the word problems. I wanted the students to know that the posters were not up there because I thought they looked nice. I made sure the posters were visible so that the students could refer to them to re-read the problem (12:07), find accurate information (22:38), and make sure that their claims were based on evidence (28:43).
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"You can look up there [at the poster]. It's right up there." (22:38) |
Throughout my lessons, I wanted to encourage students to see the value in revising their thinking. I wanted students to know that nothing is set in stone and if they receive new information, it is okay to change your mind (Kazemi and Hintz, 2014, p. 21). I think that there were moments where I subtly suggested to students that everyone makes mistakes and revisions. For instance, I asked one of the students two show me where he saw the number two in one of the sentences. Sure enough, Stephen pointed to the word "two" that I had written underneath one of the words in that sentence. I had intended for it to be read with the following sentence, but because it was in between the lines he thought that it went with the previous sentence. I crossed it out and wrote "two" over the word that I originally meant for it to replace. (28:49) I also intentionally let students know that revising their thinking is a positive thing. I told two students, "I love how you're revising that, Stephen and DeSean." (13:26) The rubric also shows that one student, Dana, revised her start, change, and end chart. Although I did not record it, I remember that she initially had 4 in the start box, -2 in the change box, and ? in the end box. I encouraged her to refer back to the problem and asked her if she could show me where she found each of those numbers. Looking back on it, I could have asked her to talk to me about the differences between the first (end unknown) and second (change unknown) word problem to help her see that while the questions used the same numbers, they were asking for different things. Therefore, it might have helped her see that the numbers in the boxes could not be the same.
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"I love how you're revising that, Stephen and DeSean." (13:26) |