Math Observation #1
Summary:
The students were given a story problem using decimals and they needed to figure out how to solve it. [Jeff wants to walk a total of 0.23 miles in two days. On Monday, he walked 0.16 miles. How much further will Jeff need to walk on Tuesday to reach his goal?]They were told they needed to solve the problem using the hundredths grid to show their work. The students were allowed to discuss with their table groups how they would use the grids to solve the problem. After that, the teacher brought them back for a whole class discussion. Students came to the board to show how they used the grids to solve the problem. Once the students grasped that concept, they were asked to solve the problem without using the grids. Again they discussed first within their groups, and then the teacher held a whole class discussion. The students were then given a worksheet to complete that had the same types of problems on them. They were allowed to complete one side with their groups but the other side was individual work. Once they were finished with the worksheet they were allowed to play “close to 1” with a partner. [Each player receives 5 cards with different numbers, hundredths and tenths, on them and you need to use the cards to get as close to 1 as you can. They recorded their score on a worksheet.]
Would you characterize the lesson as attending to procedural or conceptual knowledge?
This lesson attended to conceptual knowledge. The teacher provided a sample story problem for the students. The only instructions they received was to use their hundredths grids to solve it. The students weren’t given a procedure or formula to aid them in solving; they needed to come up with ways to solve the problem on their own. After whole group discussion, the teacher made sure the students understood that there was more than one way to solve the problem and that no one way was the “right” way.
Describe how students were actively involved in the mathematics lesson.
After the teacher showed the problem they were going to work on, she allowed them to work together within their table groups. The students participated in a discussion to try to come up with a way to use their hundredths grid to solve the problem. Once students had some ideas, the teacher brought the whole class together. The students were asked to explain their answers and to show how they solved it using their hundredths grid. The students were allowed to come up to the board if they wanted to, if it would help them explain. The students were encouraged to talk to each other about the math and only during independent work, were they to remain silent.
Describe how communication was integrated in the mathematics lesson.
The teacher communicated with her students by talking with them. While students were working in groups, she circulated around to the different groups to answer any questions they may have. While students were in their groups, the teacher also had them communicate ideas/solutions to each other and to her. Also, students were asked to explain their answers and the way they came up with those answers to the whole class. Students were encouraged to ask questions to each other if they needed something clarified. The only time students were asked to not communicate with others was during independent work and when someone was explaining their answer.
Describe how the teacher utilized questioning during the mathematics lesson.
The teacher posed an initial problem/question about math and had the students try to figure it out. While the students were in their groups, she walked around to the students to ask: What do you think? Why do you think that? Could you explain your thinking? If the students were confused or didn’t know what to do, she would ask leading questions so they would come up with the answers on their own. She never game them the correct answer but asked strategic questions to guide them to it. Also, when students were explaining their answers to the whole class, she made sure they were specific. She asked them questions that allowed them to clarify their thinking and to clear up any confusion.
Describe how the teacher assessed the students’ knowledge during the lesson.
The teacher walked around the room throughout the lesson to check for understanding. She asked questions and had the students explain their thinking. She only moved on to the next part once the majority (more than 85%) of the students could use the grid and explain their answers. The students used a worksheet at the end of the lesson and the teacher collected that and graded the independent work.
Reflection:
What was effective about this lesson?
The students were able to understand the grids better and how to use them to figure out a problem because they were allowed to figure that out for themselves. The groups worked to the student’s advantage as well. The students talked with their peers to ask and answer questions and some students were more comfortable asking questions within the group. When the teacher visited each group, it allowed her more one-on-one time with the students so she could pinpoint where they needed instruction.
If you were teaching this lesson, what might you do differently?
I’m not sure I would do anything differently. The process of giving the students a problem to work on by themselves, allowing them to discuss within their groups, and coming back as a whole class to talk about, worked extremely well. The only think I would have stressed was that they needed to use the grid to solve the problem (the first time). The students are used to using traditional algorithms to solve these types of problems and several students went right to this method. This is the first year that these students have been asked to figure out problems for themselves (Common Core) and I think the teacher did an outstanding job of translating the “new” way of doing things to her students.
The students were given a story problem using decimals and they needed to figure out how to solve it. [Jeff wants to walk a total of 0.23 miles in two days. On Monday, he walked 0.16 miles. How much further will Jeff need to walk on Tuesday to reach his goal?]They were told they needed to solve the problem using the hundredths grid to show their work. The students were allowed to discuss with their table groups how they would use the grids to solve the problem. After that, the teacher brought them back for a whole class discussion. Students came to the board to show how they used the grids to solve the problem. Once the students grasped that concept, they were asked to solve the problem without using the grids. Again they discussed first within their groups, and then the teacher held a whole class discussion. The students were then given a worksheet to complete that had the same types of problems on them. They were allowed to complete one side with their groups but the other side was individual work. Once they were finished with the worksheet they were allowed to play “close to 1” with a partner. [Each player receives 5 cards with different numbers, hundredths and tenths, on them and you need to use the cards to get as close to 1 as you can. They recorded their score on a worksheet.]
Would you characterize the lesson as attending to procedural or conceptual knowledge?
This lesson attended to conceptual knowledge. The teacher provided a sample story problem for the students. The only instructions they received was to use their hundredths grids to solve it. The students weren’t given a procedure or formula to aid them in solving; they needed to come up with ways to solve the problem on their own. After whole group discussion, the teacher made sure the students understood that there was more than one way to solve the problem and that no one way was the “right” way.
Describe how students were actively involved in the mathematics lesson.
After the teacher showed the problem they were going to work on, she allowed them to work together within their table groups. The students participated in a discussion to try to come up with a way to use their hundredths grid to solve the problem. Once students had some ideas, the teacher brought the whole class together. The students were asked to explain their answers and to show how they solved it using their hundredths grid. The students were allowed to come up to the board if they wanted to, if it would help them explain. The students were encouraged to talk to each other about the math and only during independent work, were they to remain silent.
Describe how communication was integrated in the mathematics lesson.
The teacher communicated with her students by talking with them. While students were working in groups, she circulated around to the different groups to answer any questions they may have. While students were in their groups, the teacher also had them communicate ideas/solutions to each other and to her. Also, students were asked to explain their answers and the way they came up with those answers to the whole class. Students were encouraged to ask questions to each other if they needed something clarified. The only time students were asked to not communicate with others was during independent work and when someone was explaining their answer.
Describe how the teacher utilized questioning during the mathematics lesson.
The teacher posed an initial problem/question about math and had the students try to figure it out. While the students were in their groups, she walked around to the students to ask: What do you think? Why do you think that? Could you explain your thinking? If the students were confused or didn’t know what to do, she would ask leading questions so they would come up with the answers on their own. She never game them the correct answer but asked strategic questions to guide them to it. Also, when students were explaining their answers to the whole class, she made sure they were specific. She asked them questions that allowed them to clarify their thinking and to clear up any confusion.
Describe how the teacher assessed the students’ knowledge during the lesson.
The teacher walked around the room throughout the lesson to check for understanding. She asked questions and had the students explain their thinking. She only moved on to the next part once the majority (more than 85%) of the students could use the grid and explain their answers. The students used a worksheet at the end of the lesson and the teacher collected that and graded the independent work.
Reflection:
What was effective about this lesson?
The students were able to understand the grids better and how to use them to figure out a problem because they were allowed to figure that out for themselves. The groups worked to the student’s advantage as well. The students talked with their peers to ask and answer questions and some students were more comfortable asking questions within the group. When the teacher visited each group, it allowed her more one-on-one time with the students so she could pinpoint where they needed instruction.
If you were teaching this lesson, what might you do differently?
I’m not sure I would do anything differently. The process of giving the students a problem to work on by themselves, allowing them to discuss within their groups, and coming back as a whole class to talk about, worked extremely well. The only think I would have stressed was that they needed to use the grid to solve the problem (the first time). The students are used to using traditional algorithms to solve these types of problems and several students went right to this method. This is the first year that these students have been asked to figure out problems for themselves (Common Core) and I think the teacher did an outstanding job of translating the “new” way of doing things to her students.
Math Observation #2
Summary:
The students were given a fraction problem to work on independently. [You have 24 cakes that you need to frost. Each cake uses ¾ frosting. How much frosting is needed to frost all cakes?] The students were instructed to draw a picture and then use that picture to create an equation. The same process as the previous day’s lesson was implemented: after students had drawn their pictures, they were able to discuss within their table groups. Then the teacher held a whole class discussion and students were asked to come to the board to show and explain their thinking. This problem was just a review on fractions for the class. After the class was done with this problem, the teacher did a whole class review on the powers of ten. She showed different equations that reflected the different powers or ten and had the students tell her what they remembered. The teacher touched on exponents and their use in writing different equations. Once she was satisfied that the students remembered sufficiently, she had them relate what they know to decimals. She showed a series of equations and had the students talk within their groups about what they noticed. Then they shared with the whole class the things they noticed. After that, she had them look for relationships. The students talked within their groups and then came back as a whole class to share their thoughts on the relationships.
Would you characterize the lesson as attending to procedural or conceptual knowledge?
I would characterize this lesson as attending to conceptual knowledge. The first part of the lesson, the students were given a type of problem that they had seen before. The teacher instructed them to create a picture that represented the problem and then create an equation. The students had to think about what they knew about fractions and how to represent that and then they were asked to create an algorithm to represent their drawing. In the second part of the lesson, very little “math” was done. The students were revisiting a mathematical concept they covered earlier in the year (powers of ten) and the teacher showed equations involving decimals and had them talk about the relationships they saw. The students weren’t asked to do math, they had a discussion on what they saw.
Describe how students were actively involved in the mathematics lesson.
The students were actively involved throughout this lesson. First, they constructed pictures and talked within their groups about the problem. They were allowed to come to the board to explain their pictorial representation of the problem and the teacher held a whole class discussion about what the pictures represented. Second, the entire 2nd half of the math block was discourse. The teacher led the students on a discovery where they looked at equations and talked to each other about what they noticed. They first talked about just the things they saw. Then they talked to each other to try and find relationships between the equations. The students participated in small group and whole group discussion.
Describe how communication was integrated in the mathematics lesson.
My cooperating teacher designs her mathematics lessons to promote communication. The majority of her lessons begin with students talking within their table groups. Then they share their thoughts to the rest of the class. Each student is encouraged to share his/her opinion either within their groups or with the whole class. While students are within their table groups, the teacher walks around the classroom and talks with them. She listens to the students that don’t readily volunteer their thoughts during whole class discussion and gets them to communicate with their peers.
Describe how the teacher utilized questioning during the mathematics lesson.
The entire second part of this lesson was just questioning. The teacher used questioning to get an understanding of what the students remembered about powers of ten. She made sure they remembered enough so that when they were shown decimal equations, they would see the powers of ten relationships. The students weren’t asked to solve anything during the last part of the lesson, but they were asked a lot of questions that made them think about what they were seeing.
Describe how the teacher assessed the students’ knowledge during the lesson.
The teacher used only informal assessments during this lesson. The teacher walked around the room throughout the lesson to check for understanding. She asked questions and had the students explain their thinking. She only moved on once she felt they answered sufficiently and could explain their thinking.
Reflection:
What was effective about this lesson?
I think there were 2 aspects of this lesson that were effective. The teacher asked strategic questions about powers of ten. This way the students were able to refresh their memories of powers of ten before having to answer any problems using them. Also, because the teacher didn’t expect them to answer anything, the students were more willing to state their opinions and thoughts to what they were seeing. The teacher helped them remember older information (powers of ten using whole numbers) and they talked about how to apply it to new information (powers of then using decimals).
If you were teaching this lesson, what might you do differently?
The only think I would do differently would’ve been to not start with a problem that didn’t have any relationship to the rest of the lesson. The fraction problem wasn’t related to the previous day’s lesson or the next day’s lesson. I know she had the students do the problem because she wanted them to have more practice with drawing representations before using a standard algorithm. However, the content within the problem had nothing to do with powers of ten or decimals. If it was worded differently 0.75 instead of ¾, I would consider having this problem as a warm up for the lesson.
The students were given a fraction problem to work on independently. [You have 24 cakes that you need to frost. Each cake uses ¾ frosting. How much frosting is needed to frost all cakes?] The students were instructed to draw a picture and then use that picture to create an equation. The same process as the previous day’s lesson was implemented: after students had drawn their pictures, they were able to discuss within their table groups. Then the teacher held a whole class discussion and students were asked to come to the board to show and explain their thinking. This problem was just a review on fractions for the class. After the class was done with this problem, the teacher did a whole class review on the powers of ten. She showed different equations that reflected the different powers or ten and had the students tell her what they remembered. The teacher touched on exponents and their use in writing different equations. Once she was satisfied that the students remembered sufficiently, she had them relate what they know to decimals. She showed a series of equations and had the students talk within their groups about what they noticed. Then they shared with the whole class the things they noticed. After that, she had them look for relationships. The students talked within their groups and then came back as a whole class to share their thoughts on the relationships.
Would you characterize the lesson as attending to procedural or conceptual knowledge?
I would characterize this lesson as attending to conceptual knowledge. The first part of the lesson, the students were given a type of problem that they had seen before. The teacher instructed them to create a picture that represented the problem and then create an equation. The students had to think about what they knew about fractions and how to represent that and then they were asked to create an algorithm to represent their drawing. In the second part of the lesson, very little “math” was done. The students were revisiting a mathematical concept they covered earlier in the year (powers of ten) and the teacher showed equations involving decimals and had them talk about the relationships they saw. The students weren’t asked to do math, they had a discussion on what they saw.
Describe how students were actively involved in the mathematics lesson.
The students were actively involved throughout this lesson. First, they constructed pictures and talked within their groups about the problem. They were allowed to come to the board to explain their pictorial representation of the problem and the teacher held a whole class discussion about what the pictures represented. Second, the entire 2nd half of the math block was discourse. The teacher led the students on a discovery where they looked at equations and talked to each other about what they noticed. They first talked about just the things they saw. Then they talked to each other to try and find relationships between the equations. The students participated in small group and whole group discussion.
Describe how communication was integrated in the mathematics lesson.
My cooperating teacher designs her mathematics lessons to promote communication. The majority of her lessons begin with students talking within their table groups. Then they share their thoughts to the rest of the class. Each student is encouraged to share his/her opinion either within their groups or with the whole class. While students are within their table groups, the teacher walks around the classroom and talks with them. She listens to the students that don’t readily volunteer their thoughts during whole class discussion and gets them to communicate with their peers.
Describe how the teacher utilized questioning during the mathematics lesson.
The entire second part of this lesson was just questioning. The teacher used questioning to get an understanding of what the students remembered about powers of ten. She made sure they remembered enough so that when they were shown decimal equations, they would see the powers of ten relationships. The students weren’t asked to solve anything during the last part of the lesson, but they were asked a lot of questions that made them think about what they were seeing.
Describe how the teacher assessed the students’ knowledge during the lesson.
The teacher used only informal assessments during this lesson. The teacher walked around the room throughout the lesson to check for understanding. She asked questions and had the students explain their thinking. She only moved on once she felt they answered sufficiently and could explain their thinking.
Reflection:
What was effective about this lesson?
I think there were 2 aspects of this lesson that were effective. The teacher asked strategic questions about powers of ten. This way the students were able to refresh their memories of powers of ten before having to answer any problems using them. Also, because the teacher didn’t expect them to answer anything, the students were more willing to state their opinions and thoughts to what they were seeing. The teacher helped them remember older information (powers of ten using whole numbers) and they talked about how to apply it to new information (powers of then using decimals).
If you were teaching this lesson, what might you do differently?
The only think I would do differently would’ve been to not start with a problem that didn’t have any relationship to the rest of the lesson. The fraction problem wasn’t related to the previous day’s lesson or the next day’s lesson. I know she had the students do the problem because she wanted them to have more practice with drawing representations before using a standard algorithm. However, the content within the problem had nothing to do with powers of ten or decimals. If it was worded differently 0.75 instead of ¾, I would consider having this problem as a warm up for the lesson.