2.4+Reactions+to+Cathy's+class

Read Cathy’s reflection on her teaching and Jo’s analysis. Watch the video of the interview with the children. What do you take away from all of this? Do you have any further questions? Post this on the wiki. Due Thursday by 11pm.

The “Division of Fractions” classroom study brought a number of pedagogical practices to my attention that I will want to expand in my classrooms.

Teachers should expect students who come with different mathematical backgrounds to solve problems in different ways. They should also be prepared for students solving problems with incorrect logic that is based on misconceptions or a misapplication of rules they have previously been taught.

Utilizing small groups encourages intimate and safe discussion between students initially. The students interviewed expressed that they learned mathematics from explaining their logic to someone else. They also learned the skill of articulation and debate. The strategy to “convince yourself, convince a friend, convince a skeptic” expands the growth of our students from mere “number-crunchers” to high-level logical thinkers of mathematics.

The move to the large group discussion expands the entire classroom to a safe environment of knowledge sharing and debate. The students noted that it helped them to see how other students were solving the problem. The large group discussion also highlights for the teacher the different thinking strategies employed by students and the most common mistakes made. As Cathy stated, and as I have experienced, having a number of students make a common mistake is wonderful for the learning environment. It is rewarding to dispel myths and to make students really think about what they are doing.

As Jo noted in her summary, the teacher’s role is dynamic and changes from lesson to lesson. The teacher utilizes a moment-to-moment decision making process when determining what path to take students down. Teachers will make mistakes, but I feel that teaching is an art. Learning good techniques is vital, but implementing them takes practice, intuition, and talent. Even after teaching for a number of years, an experienced teacher can get surprises in the classroom! **//(Susan Copeland)//**

I took away the concept of getting multiple answers into the open before saying which one is right, and allowing students to discuss their ideas and justifications. Listening to the whole class and using what you learn to guide the class is a good assessment tool. “Convince yourself, convince a friend, and convince a skeptic” is a way to keep the class focused on justification and reasoning, rather than having students argue about who's right and who's wrong. It is important not to just correct students and tell them they are wrong, but allow them to explore ideas and discover on their own and with peers what answer is correct and why. The wrong answer given to a question is useful in a classroom, because it allows the teacher to have students explore the reasoning for math, and gives them a deeper understanding.

Asking students to be skeptical is useful, because it helps eliminate the fear of being wrong for students. It also raises the standards for students and their explanations. This also makes the other students in the class have to pay attention to detail, so they can pick up on small mistakes of the presenting student. Showing interest as the teacher will get students interested in the topic as well, and this will cause them to be more thoughtful and involved in the discussion.

It is important to pay attention to the needs of individuals, not just the class. Teachers need to make sure that the each individual understands the material, as well as the class as a whole understanding it. I want to see a hand up on each group is a good tool to test for understanding, because if a group does not have a hand raised, maybe none of them understand it, or they did not have enough time to reason out the problem.

Allowing students help each other while one is working at the board allows each students to work on something at the board without the risk of feeling dumb, because they have their classmates to help notice or solve mistakes.

Hold that thought is when you ask a student to wait a few more minutes while you unpack or work to help the class understand the current idea being discussed. Also, asking students to put their thoughts and pictures on the board allows interaction between that student and the class. This allows different representations and showing their ideas. **Josh Kaylor**

As I have learned from this classroom study, it is very important to really utilize the students and their thought process in order to create a better learning environment. Letting them explain their own thoughts and going through the same steps they did, can really help them realize what they did wrong or right. Also, having the students explain their logic to each other is greatly beneficial because it helps them learn to articulate the way they solve the problem, and they might be able to explain it to each other in a way that the teacher can't. Allowing the class to discover their own answers helps them remember it more, and it helps them fully understand what they did wrong or right since it is thoroughly explained. Making mistakes is a vital part of the learning process since they often have common misconceptions that they come across, so it is important to address them and make sure the students understand the reasoning behind the mathematics. If they never learn the mechanics behind the problems, and why their mistakes are actually wrong, then they'll never really grasp the concepts fully. One of the difficulties, though, with teaching middle school mathematics is that the teacher needs to be able to make decisions on the fly and decide which paths to take when conducting a classroom. There might be something interesting that one student does, so you want to direct the class' attention to their method and get their thoughts or reactions to it. You can never completely know ahead of time what the kids in the classroom will come up with, so teachers need to be able to adjust the lesson appropriately with each class. A difficulty with that, though, is trying to adjust the lesson also to accommodate the needs of all different levels of students. Not everybody will be able to learn the same way, or understand the material with the same amount of effort. Overall, I learned from this classroom study to let the students really express their thoughts and ideas, and to be flexible with teaching. When you step back and let them really have a voice in the classroom, they are more interested in the material and thus, become better mathematics students. **-Shanna Thorn**

After watching the videos of Cathy's instruction, what her students said about her, and what Cathy and Jo stated, I have found much success in Cathy's methods that have impacted how I view a successful mathematics classroom.

One of the most memorable experiences from Cathy's teaching was that her students were confident enough to make mistakes in front of each other. Personally, I do not like to make mistakes in front of other people and I especially did not like to be seen making mistakes as a middle school student. Yet, Cathy's students were eager to show what they did on the problem with the chance that they may be wrong. Furthermore, Cathy's students actually found mistakes important as discussed in the video. The students' said that mistakes were "important to learning" and saw them as an opportunity to learn. The source of this confidence is also discussed by Cathy's students.

The students said that Cathy encouraged them to make mistakes by celebrating mistakes. Instead of Cathy correcting student mistakes she used them as a mode of discussion and embraced them. One of the the students said that Cathy got very excited when new mistakes were made and liked seeing different ways of reasoning, especially if they were incorrect because other students could discover why it was incorrect. Cathy herself said that it is just as important to see the right answer as it is to see common misconceptions.

The most influential observation I made was that students actually enjoyed Cathy's class and developed an interest in mathematics. The students said that they enjoyed being able to discuss what they were thinking because in most classes they were not able to talk. They also liked the discussions when they didn't have to raise their hands. I think that allowing students to do what they like to do (TALK) as a learner tool is a major reason why Cathy's students enjoyed her class.

Thus, creating discussion and encouraging students to take risks in the classroom works for Cathy because students are becoming mathematical thinkers and are developing an interest in mathematics. --- Katie

====The Division of Fractions study made me thing of a few things I would want in my future classroom. I really like that most of the students respected each other. That no one was made fun of if they got the wrong answer. Also that the students felt comfortable with giving answers they think are right and going up to the board in front of their peers.==== ====I also really liked the convince yourself, a friend, and a skeptic. First you start out by convincing yourself, because in order to convince someone else, you need to know how you got your answer. By convincing a friend and skeptic, if the student is right, they can get a better understanding of how they came up with the answer and convince someone who had it wrong. If the student was wrong, by trying to explain it to someone, they might realize that they were wrong and realize the mistake they made.==== ====I like how Cathy got multiple answers out of the students and had them discuss it rather than just telling them the answer. That the students went up to the board the tell the class what they thought or talked about it in smaller groups. **(Nicole Parry)**====

First off, I thought that it was really awesome how the whole class was involved in the problem of dividing one by two-thirds. Even though most everyone came up with their own way of figuring out the problem, they were still all correct in their own sense. Some of them were actually the correct answer that the teacher was looking for and the others were correct for what they were actually set up to do. The video was much more helpful because I could watch and see how the students were acting, whereas the script of the interview was harder to comprehend or really get the feel of the environment in the classroom.

I was impressed with how Cathy wanted to convince yourself, a friend, and then a skeptic. It is all to easy for you to make sense of something yourself and at times it is all too easy for you to convince one of your friends as well. Convincing a skeptic is one of the important parts of Cathy’s strategy because you must fully understand what it is you are trying to convince them with and if the skeptic can convince you they are right then maybe you really weren’t convinced yourself. All too many times I found myself not wanting to go up to the board to show my answer because I was not fully convinced myself or maybe I was just afraid of being wrong when up at the board.

The time that the students had to themselves was a good time for them to discuss what was really being asked from Cathy. By having the students demonstrate what they thought the answer would be and how they got to their answer, Cathy was able to show the class with only guiding the student up at the board with proper and mind-triggering questions that would allow other students to find understanding.

Being able to react with the live classroom interactions is a key that Jo presents in her analysis. A problem may not go as you plan it to go, but a teacher must be able to go with it and react to how the students are reacting to the problem. Overall, I was impressed with Cathy’s control of the class, the environment she had created in the class, and how well she was able to guide the students through a through-provoking activity. **(Ryan Sherman)**