2.5+Reflections+on+Cathy's+and+Jo's+Comments

What do you take away from each person’s (Cathy and Jo) reflective comments on the fraction division lesson? Do you have any further questions regarding this case?

Due by Mon10/3 by 11pm. Note your reflections for this paper are targeted towards a different question than what you attend to for the Reading Reflections. I also posted in Drop box the interview Jo mentioned in her comments if you want to watch it.

From the first part of the reading involving Cathy’s reflection of her lesson, there were many questions and thoughts that came to mind. First, Cathy mentions this question, “Is knowing that something makes sense the same as knowing why it works?” This question got me thinking about my proofs class where there were many proofs that I knew made sense but I still was unsure of why that just proved what I wanted to prove in the first place. I feel this is along the same lines as her question.

Next Cathy talks about changing the view of mathematics by saying, “mathematics is never supposed to be a situation in which ‘you don’t know what to do,’ If the teacher has done his or her job and the students have done their job, they should always be able to apply a fact, rule, or procedure to obtain an answer quickly.” How do we as math teachers change this misconception, we know that math is not always that easy, nor is that always the point. Also, how can we attempt to change this misconception when many standardized test questions want students to just know how to use a formula or apply a rule and not truly know where it comes from or what it means. After all aren’t we just supposed to teach to the test?

Cathy continues her reflection by discussing the purposes small group discussions serve which according to her include giving more students a chance to explain their thinking and giving the teacher a valuable way to understand what students are thinking, “eavesdropping is a powerful assessment tool that allows me to make better decision about what the thrust of a whole-class discussion should be,” she states. I also agree with Cathy that explaining a concept or idea to someone helps “clarify thinking and solidify learning.” The more you explain something the deeper your understanding of that topic goes and the more connections you have the opportunity of making.

One last comment I thought was interesting was when Cathy talked about calling only on volunteers and how there are disadvantages, however she wants to avoid putting children on the spot. I think this is a fine line to walk, how do we know when it is ok to call on a student who is not volunteering or to simply pick a volunteer. For me, I also think I would avoid putting students on the spot simply because I hate it when teachers do it to me, I’m not volunteering for a reason, is my thinking. I think to avoid putting students on the spot is to ask them privately if they would mind sharing like Cathy did in her lesson. During her rounds to the different groups she asked a student if he would share, I think this is a much less intimidating approach for a student and yet the teacher still gets new voices.

Moving on Jo’s analysis of Cathy’s reflection, the first thing that caught my attention was her analysis of Cathy having students act as skeptics. In this way, “If students do not understand something, they do not have to declare their lack of knowledge; instead they can say that insufficient evidence has been given, requiring the speaker to explain in different or better ways.” Jo goes on to say, “This pedagogical move may be a helpful resource for teachers who want students to offer their ideas without fear of being incorrect and who would like high standards of representation and justification in their classrooms,” and isn’t that what we all want as teachers?

The next point in the analysis I found interesting the mention of classroom norms and how they pertained to the quality of the class discussion. I also think it is important to set and follow classroom norms. For example, I want my students to know that my classroom is a safe space, that everyone’s opinions need to be respected and that it is ok to be wrong because it is part of the learning process.

Finally, the last point I would like to discuss is the part about Cathy asking her students to “hold that thought.” Jo comments by saying, “Asking a student to hold her thought is an important pedagogical move that allows the teacher to respect a student’s thinking and at the same time maintain an important mathematical direction. At this point I am still not quite sure how we as teachers would know when to have a student hold their thoughts or continue speaking; perhaps we learn it from practice. I am also not sure how I see this can be beneficial when we want to hear students thoughts, again perhaps because I have not yet been in that situation.

Overall I thought this reading was filled with a lot of good information and techniques that will help us in our future classroom. Kaitlin Froehlke

After watching the video, I was amazed at how productive and well oiled Cathy’s classroom was. I thought there was little if nothing that I would have wanted to improve upon. This is why, when I read Cathy’s own notes and reflections about her lesson, I was amazed at how well she could critique and analyze her thoughts and actions. One statement which stood out to me was when Cathy said she has a hard time grasping what she means by asking students what makes sense (2-2). She then goes on to question whether or not knowing why something works is the same as knowing if it makes sense. This leads some students to thinking if the answer is right then this must make sense. All of Cathy’s questioning made me think about how the young mind works and even more, how are we to interpret it? Cathy admitted in her reflection she was excited to get the wrong answer of six (2-3). She was able to see how many of her other students thought this was a right answer and then she could build upon this to facilitate the discussion. If I were Cathy, I don’t know if I would have been this excited. However, after reading her reflection and reasoning, I can see by allowing this “wrong” answer to become a discussion point, she is allowing the student to grow and develop their own reasoning. I believe the main concepts I gained out of Cathy’s reflection is there is always room for improvement. I thought her lesson was very well put together, however Cathy reflected that she had other things she could have said and done and wondered if that would have changed how the discussion went Jo’s reflection really shined a new light, for me, on the before aspect of a lesson like this. The idea of the skeptic is great within a classroom. I remember being in math classes and afraid to say anything because I thought the answers were wrong and I was not convinced. Being able to be a skeptic allows the student to move from confusion to the person asking for the justification and reasoning which is no prevalent in a solution (6-2). From Jo’s reflection, I was able to take away the idea of preparation and careful attention to the classroom. Having the student know what the role of the skeptic is, seeming interested in their answers, knowing which solutions to expand on, and so much more seem to be fundamental parts of the discussion process. As Jo said, it may seem in the teacher doesn’t do much in a discussion but that is not the truth. I now see it as a teacher has the most important and difficult job in the whole discussion. I would say my only question to this entire lesson and the reflection of it is how? I see this lesson being taught so well with all the students communicating and playing a role. In Jo’s interview with some of the students who thought the answer was six later told her they weren’t embarrassed and it was a learning opportunity (10-1). Most children the age of those in the classroom would not think this way. I want to know how they were able to get them to think this way and how they were able to get them to play their roles. It seems like it would take a long time and a lot of hard work to break down the construction of previous math classes. Katey Cook

After watching the video clip of Cathy’s lesson, it was intriguing to see how she evaluated her own teaching and the students’ learning process. Until she highlighted it, I never realized the importance of the wrong answer as potential for future discussion (44-3). From observing the class, I realized that without the wrong answer being supported, the entire discussion would have no basis to it. Interestingly, students’ incorrect answers are often developed using logic; however, this logic is based on misapplying certain rules or properties they have learned before (44-3). Cathy attempts, through her probing questions, to delve deeply into these misunderstandings and allow students to think about what really makes sense (45-1).

I really liked how Cathy shared with us her thinking process during the discussion. It showed me that teachers are constantly making adjustments and improvements as the lesson progresses; we cannot plan every aspect of the class period ahead of time, no matter how much we prepare. Additionally, I thought an ingenious idea for finding out what and how students are thinking is to have them write about what they learned that day, as Cathy brought up at the end (48-2). This would allow me to understand what they actually took away from one of my lessons and give me a framework for improvement in the future.

Cathy clearly did not think her teaching was perfect. In fact, she criticized herself rather harshly for using the lumber example, where she decided to bring context into the discussion (48-1). The first time I watched the video, I could not figure out how dividing by two thirds made sense until the moment she presented that example. Additionally, Jo mentioned that a student brought up this example in an interview much later. Thus, Cathy’s example, although she didn’t particularly like it, proved to be effective.

In her analysis, Jo acknowledges that coordinating a productive class discussion is very complex and incorporates a number of different teaching strategies (48-3). A few of the techniques Jo underscores in Cathy’s lesson were particularly interesting for me. Giving students the role of skeptics shifts the focus to explanation and justification rather than right or wrong answers (49-1). Also, multiple answers are legitimate for their role in learning, not necessarily for their mathematical correctness (50-3). One aspect I had never considered before was balancing the needs of individuals and the needs of the whole class (51-1); I will definitely need to figure out how to accomplish this goal in my classroom.

This article, although very insightful, left me with a few questions. First, how do we resolve misconceptions without creating more confusion for our students? A faulty explanation could result in further misunderstanding. Also, I really liked Cathy’s insight that “The trouble with teaching is that there are so many paths to take, each with different results!” (48-2). How do we, as future teachers, determine which results we want and which path to choose in order to achieve these? This appears to be an overwhelming dilemma. Finally, Jo mentioned that Cathy ended the discussion to give it closure. How can we know when a discussion has progressed far enough? When do we cut it off? And how do we reconcile limited time to accomplish certain goals with our students with our desire to have deep, interactive whole-class discussions? Clearly, leading a discussion is an activity that requires extensive practice and much skill.

Mandi Mills

In Part Deux of the “Division of Fractions” case we are given a reflection by Cathy. I can understand why Cathy was thankful for a wrong answer (1-3), but what would she have done if the entire class got the same wrong answer? It is nice for a teacher to see a wrong answer as long as the teacher knows one of the students got the right answer with a convincing argument that can persuade the rest of the students to change their answer. But how do you approach a class that sees more logic in finding 6 rather than 1½ for an answer. I really liked the distinction Cathy made between “making sense” and knowing “why it works” (2-2). As a student I can remember saying I understood something to get onto the next problem and finish my homework. I feel like a majority of students look for it to make sense procedurally and ignoring why it works that way.

Cathy mentioned something that I question a lot. She mentioned that she prefers to choose volunteers rather than putting children on the spot” (4-3). Maybe it would be established in the classroom norms that all students would participate, but in many of the classes I have taken only a handful of people really volunteer. How do you gain everyone’s participation by only taking volunteers? Surely there will be a few students who refuse to contribute to the discussion.

It was good that Cathy had a variety of students show their methods to solving the problem put before them. As a teacher I think it will take a while for me to allow students to attack problems differently than how I would. I would be very tempted to influence how students do certain problems. I will need to be accepting of more than one way (if there is more than one way) to do different problems, and model these different solutions for the students.

I thought it was good that Cathy could take a step back and recognize that her lumber example wasn’t ideal for the lesson (5-1). I think I will be pretty good at recognizing my examples are bad, hopefully before I teach the same lesson later in the day.

I kind of see myself wanting to lead and steer discussions in the classroom like Jo mentions (6-1). It will take a lot of class periods for me to learn how not to steer the discussion and still reach the wanted results. Trusting my students will be a struggle, especially at the beginning of the school year when I don’t always know how strong their math background is. How do we develop these discussion management skills before entering a classroom?

I like the idea of having students play the role of the skeptic (6-2). This forces students to students to explain clearly their thoughts on mathematical concepts and persuade other students to see things the way that they do. How do you develop a classroom that teaches students not to be afraid of being wrong? I also liked what Jo had to say about having students support each other at the board (9-3). I think it would help to employ a support system that includes having a buddy at the board to instill confidence in students who present in front of the class. One final question I would like answered is how long did those students have Cathy as their teacher before this lesson? It seems that it would take some time to instill the norms that were present in the classroom, and I’m just wondering how long it takes form the classroom norms?

Mike Freeland

In Cathy’s reflection on her own teaching there were a few things that I either identified with or would like to try and test in my own future classroom. Cathy says “small-group discussions are a valuable way for me to understand what students are thinking. “Eavesdropping” on their conversations is a powerful assessment tool that allows me to make better decisions about what the thrust of a whole-class discussion should be,” (45-3). I //completely// agree with her on this one. So far in my pre-internship, I like that my mentor teacher uses small-group discussions often because I’ve gotten a chance to do some of the “eavesdropping” she mentions and notice who is explaining totally false ideas with other group members and who is on track with a possible path to a correct solution.

In Jo’s analysis of Cathy’s classroom, I liked that she mentioned the importance of requiring students to be skeptical (49-2). Having a skeptic be a required role in the classroom discussion allows students who normally don’t like to answer in fear of being wrong to stick up for their reasoning against the skeptic’s inquiries. Students shouldn’t be afraid to try and answer publically.

In all honesty, I don’t have any further questions. I understood some of the common mistakes the students in the video did. This could, again, be a result of the timing of my pre-internship and I’ve seen various ways of trying to have small-group discussions that include a skeptic turn it into a free flowing whole-group discussion. Tori Ward

CATHY:
 * Children's errors frequently have a logic based on misconceptions or misapplication of rules.
 * Pushing students to be skeptics as they listen to their peers' answers totally changes the classroom atmosphere.

JO:
 * Students like Christine knew it was alright to be wrong because it played a productive role in the classroom.
 * Teaching is a constant act of navigation between attending to the needs of individuals and attending to those of the whole class

-kyle d.

In Cathy’s reflection of her own teaching during the division of fractions lesson, she explains her reasoning behind the numerous moment-to-moment decisions made. It was enlightening to read about the smaller decisions that a classroom discussion requires of the teacher, as this is where the difference between effective and ineffective teaching lies, according to Jo (10-2).

Though Cathy elaborates on many of her decisions, one of her over-arcing goals throughout the lesson seemed to be encouraging dissenters to voice their opinions so that skeptics may be convinced to a more reasonable method of solving the problem. This way, those who are skeptics play a role in the “greater understanding,” (3-3) by demanding further justification of the class’s collective thinking. I understand why she encourages students to be skeptical of those who provide hasty justifications, because it shifts the focus of the lesson from being solution-oriented to being process and understanding-oriented. In Jo’s analysis she mentions another positive aspect of encouraging skepticism in the classroom that had not occurred to me. She notes that students who are playing the role of dissenter do not have to publicly voice their confusion, which could cause students to feel uncomfortable by standing out, and instead they only need to claim “insufficient evidence” (6-2). This will force those who already understand the concept to explain it in a different and more clarifying way. After the second explanation, not only will the dissenter have a better understanding, but the student who had to re-explain it will also have a better understanding because he or she had to think of the problem in a new way.

Despite the positive outcomes of having skeptics in the classroom voice their opinions, there is one instance where Christine claims that she is still unconvinced of the answer 1.5, but Cathy chooses to Christine’s dissent on hold. I thought that this was an interesting move. In Jo’s analysis, she says that Cathy’s decision was the mark of a practiced teacher, as novice teachers would have been more likely to allow Christine to continue with her dissenting opinion (9-2). This worried me because I felt that were I in that situation, I would have allowed Christine to continue in the hopes that the class would have seen her faulty reasoning and consequently be more convinced of the reasoning behind the answer 1.5. In the same section, Jo explains that Cathy must have put Christine’s comment on hold because she wanted to further the discussion of Cheryl’s idea. My question is that, knowing that I would have most likely made the opposite decision, how are we to know when to silence the opinions of skeptics and when to let them enter into the conversation? They have an obvious value for generating discussion, but when do skeptics’ ideas become distracting from the correct reasoning?

Cathy’s quick decision making with putting Christine’s comment on the backburner is an example of the kind of snap judgment that we, as teachers, will need to be able to make. These small decisions can alter the outcome of the classroom discussion, by either keeping the discussion on its’ current track or deciding when it is appropriate to veer off-course with a dissenting opinion.

Valerie Gipper

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Something I thought was extremely interesting about these two takes was the different feel of time coming from the teacher's perspective and the observer's When reading Cathy's take on the situation, everything seemed hectic and split-second and like she needed to do five-hundred things at once. Meanwhile, from Jo's end everything Cathy did comes off as calculated brilliance.

Throughout Cathy's self-assessment of the situation she went through a step-by-step of why she made the moment to moment decisions she did. She mentioned that children often make errors based on misconceptions and misuse of rules they've been taught previously. (44-3) It's very interesting that she likes to try to exploit these errors in logic to help lead discussion. I very much agree that failure is a huge catalyst towards understanding anything, because nothing will stick with someone quite like a mistake, especially if it's discussed. Her decision to allow the students to beak off into groups when she heard the two answers, one of which came as a great surprise to her, was also extremely interesting. Instead of just discrediting the one and focusing on the right answer or moving to a student who said the correct answer and why it was the correct answer, she allowed the kids to talk it out with one another and explain why. This enabled her to get to where the students with the incorrect answer were coming up with it, and more aptly correct the problem.

Another interesting thing was that as the groups worked together she walked around to see what the various groups were working on and talking about and allowed for various students to share ideas she thought were interesting. One of the students idea to create two pie charts to represent 2/3 and 1 respectively and show that 1 and 1/2 of those 2/3 could fit into 1 was particularly brilliant. (46-2) I was extremely surprised at that kind of insight from a middle school child.

From the entire thing I was also rather surprised at how many decisions needed to be made in such an extremely small time-frame. There were a lot more nuances and on-the-spot stuff than I would expect from a math course. The way she went about letting the students lead the discussion seems almost overwhelming, but from the response it seems to have done a much better job than simple repetition ever could have.

From Jo's side, it was interesting to see how much different all the interaction came off to an outsider. Jo talked about allowing the students to be skeptics and as she wrote 'This simple act changes the entire nature of the classroom environment, as students no longer have to fear the shadow that hangs over many mathematics classrooms---that of being wrong." (49-2) Which is something I completely agree with. It's very hard to ask questions in class, and especially so with that fear of being wrong. But by encouraging the students to ask questions about any and every answer presented it allows a great deal more freedom to them to be wrong and not have to feel embarrassed.

Another thing Jo's writing showed was how things Cathy did might appeal to students. Such as the way she went from group to group and showed an active interest in what they were doing. She encouraged her students to share what they had and talk to the class about them to help clear various things up, and from the view point of a student this can be a huge deal for their self-confidence because it's not forced and they were able to get some feedback from other members of the class first.

Finally, I thought the two perspectives on the finishing of the class was rather fascinating. Cathy was embarrassed by her lumber yard example, yet two of the students who originally thought the answer was six said in an interview later cited it specifically as very useful in helping them understand why six was incorrect and one and a half was the correct answer. (50-3) Jo also thought that bringing it all together at the end was a sound decision and the way in which she did it was well-founded too. These things made the strong self-criticism over the lumber problem all the more interesting. Sometimes, it seems, it's harder to get a grasp of what works and what doesn't when you're the one making the decisions, but from the outside looking in everything also seems to be more controlled and pre-planned than it might actually be.

- Doug Wills

Cathy’s reflection gave me a few insights into running a classroom. The main thing I took away was the fact that her students felt comfortable and safe enough to discuss and argue their points freely and honestly. The fact that Christine felt safe enough defend her answer although it was wrong means that Cathy has created an environment that is conducive to learning. (46-3) Cathy also reflected on her use of the “convince yourself, convince a friend, and convince a skeptic.” I liked that although Ben had the wrong idea initially because he was playing the role of the skeptic he was able to accurately defend the right answer. (46-2) She systematically called on a few more students until she felt there was an adequate argument for the correct answer. Once she obtained this she was able to put the question back into context by “shinning the light” on a partial answer that had shown signs of cognitive reasoning going on. (48-1)

Jo’s analysis of Cathy’s reflection was pretty insightful as well. Obviously the role of the skeptic was a good idea to stimulate discussion.(50-1) During an exit interview a student said they learned more from learning the wrong answer then just to be told the right answer and the procedure that follows. (50-2) Jo also noticed that Cathy was balancing between the needs of the individual and needs of the whole by waiting to further the discussion until all the groups had one person willing to explain their solution. (51-2)

The biggest question this leaves me with is when do you end the conversation for the sake of time? Do I play the numbers and when the majority has it move on?

Christopher Cardon

I would first like to say how interesting it is that we first learned about the preparation that Cathy put into this lesson, and then watched the video of her implementing her lesson. Lastly, we read the article of Cathy reflecting on her teaching method. This assignment has been a realistic example of how the process of planning, teaching, and reflecting occur in a middle school mathematics classroom. By Cathy reflecting, it shows us the importance of evaluating and reflecting on our own lessons in order to make necessary changes to improve our curriculum.

Cathy wanted the students to forget the rule and reason through the problem. In the reflection, Cathy says the transition between these two things can be hard for middle school students (45-2). I find the transition hard for myself. Once the algorithms are presented, they become second-nature, and it makes it difficult to reason why you do something. When only algorithms are taught, students start to think that they will always know what to do on math problems (45-2). But, when you have to put reasoning behind work, there will often be situations when the approach isn’t known right away. Cathy used small-group discussions for the students, which I feel eased the shock of having to reason through a problem and not use the rule. Cathy’s lesson provided students with many ways to make sense of the division problem.

Jo reflected on the teaching strategies that Cathy used in her lesson. Cathy used the convince a skeptic method, which I think is very clever because it gets rid of that fear of being wrong (49-2). Students are more interested and engaged in lessons like Cathy’s, and it makes me wonder why my teachers didn’t teach this way? I found it surprising that some of the students in Cathy’s class enjoyed mathematics for the first time that year (50-3). When I was in middle school, I enjoyed math class as it was, which consisted of answering thirty problems for homework every night. This just goes to show that each student is different and in order to reach all of our students, we sometimes have to step out of our comfort zone and step away from the algorithms.

Hailey McDonell

Cathy's reflection showed me how a teacher thinks as she goes over practice in the real world. First she states how the answer six had was beneficial because of its potential for a useful discussion.44-3 Having wrong answers let students learn how to things correctly in order to come up with the right solution. I like how she put students into small group discussions. She says this gives students a chance to explain their own thinking. 45-3 I think having students explain their own thinking makes them further justify their answer to prove to somebody that their answer is correct. She also says small group discussions are a valuable way to help understand what students are thinking. 45-3 She says "Eavesdropping" on their conversations is a powerful assessment tool.45-3 I believe it is also powerful because you can gain a sense of what students are thinking and how they are coming up with their solutions. I like the strategy "convince yourself, convince a friend, convince a skeptic". 46-1. This strategy is a great way to help students solidify their understand of the problem. Once the students came up with the right solution the teacher gave an explanation on why six was too big then she used the example of lumber to defend her answer.

Jo's reflection was interesting as well. he described that, "Having the students play the role of skeptics changes the entire nature of the classroom environment, as students no longer have to fear the shadow that hangs over many mathematicians classrooms of being wrong." 49-2 With students not having a fear of being wrong they are not scared to share mathematical insight to problems that they are not sure of. Students can then build on each others statements to come to a more powerful conclusion. Jo says that " Teaching is a constant act of navigation between attending to the needs of individuals and attending to those of the whole class." 51-2 This statement I feel is the sum of all Jo's analysis. with this everybody is learning, from the student to the classroom. Teachers should know when to attend to specific person and when to convince a whole class. It is this method which keeps intellectual thought flowing.

As far as questions I have none for this case. For it was all of observing and listening to how things were being run in the classroom. As long as everyone in the class learned how to give the correct answer I see no need as far as to asking more questions.

Fredrick Martin

The most interesting bit of Cathy’s perspective was her focus on classroom norms and the view of wrong answers in the classroom. As she said, the people who admitted they were wrong had their status in the classroom suffer as a result. However, also as she says, the errors of these students can be valuable tools for understanding mathematics because you can pick the brains of those who do procedures incorrectly and also get students to learn why those mistakes do not work. She states this in the beginning of her reflection when talking about how the incorrect answer of 6 appeared. (1-3) She used the situation as a leaping board into discussion.

I also find Cathy’s piece on mathematics making sense an important concept. The question she brings up challenges the view of mathematics merely being procedure. (2-2) The ability to use mathematical reasoning is more valuable to students. This is because if they are faced with a problem they have not seen, they might be able to solve the problem with no procedure, and even if they cannot solve the problem, they will still utilize their knowledge rather than giving up because they do not have that procedure.

Skipping to later in the article, Cathy also talks about the multitude of teaching approaches that she could have taken during the discussion. (4-4) It really emphasizes the importance of having more than one tools in the toolbox so to speak. Also, it shows Cathy’s ability of reflection. Did I see multiple teaching methods when I watched the video? Not especially, I was focused on what the students were saying. She closes her reflection by talking about how many paths she could have taken. (5-2) She also notes two students who took little from her lesson and how she should have approached them.

In Jo’s analysis, she focuses on the difficulties of discussion and one would have to agree with her. (6-1) I have now been convinced that direct instruction leaves a lot to be desired in a student’s development, but I am only getting a rough idea of how to teach through a different method, such as class discussion that was used for this piece.

The skeptics in the classroom are what I have always found uncomfortable. (6-2)I never liked hearing in class how people achieved wrong answers and would instead focus on how “stupid” they were for not being able to get the correct answer. It took me a long time to not feel this way when another student would ask a question. I also think that students do judge someone based on when they ask a question, but when the two girls who had the wrong answer were asked about the lesson, they said that they found importance in their answers. (7-3) I find it amazing that they were comfortable presenting, as it appears to me, such ludicrous examples.

Jo has many important insights, but I am running out of space to discuss them all. Communication, student’s role in the class, Cathy’s ability to mold the discussion to her choosing, and other topics are also worth mentioning and discussing in detail. As Jo states in her conclusion, “The case revealed a selection of different pedagogical moves that were employed in the service of learning.” (10-2) I have definitely found this to be true.

-Marcus Edgette

In the comments about the video I found it particularly telling that they make it a point to restate what Cathy thought or said. I feel like because it was Cathy reflecting on her own work that she could reveal her thoughts about it and admits her dislike for her example (1-1) and her relief at a wrong answer (1-3). I also like how she talks about the usefulness of different actions and acknowledging how the students must feel and relating it to her previous knowledge (4-3). I think Jo did a great job because it was a different perspective than that of Cathy’s. She seemed to point out the things Cathy missed and analyzing her different methods. I feel though that she pointed out what Cathy does with the wrong answers as being productive (6-10). The idea that everyone has been accepted into the classroom atmosphere even if they are wrong has also been a point we have touched on many times in class. In her analyses she talks about the importance of how the teacher communicates with the students and what it helps (8-1). The questions I have of this reading is that why does Cathy feel as this class session was not as productive as having them complete some sort of writing on an individual basis? Does this mean we should think of giving homework in the sense of will it be productive or non productive? Denise Slate