Math Vignette Cooking the Turkey Problem

Transcription

1 Math Vignette Cooking the Turkey Problem A third grade class is beginning to work with multiplication. The purpose of this lesson is to use an open-ended investigation to develop students ideas, strategies, and models for multiplication through a problem that is based in a familiar context. Background: On the previous day, the teacher had posed a real-life problem to the class for which she requested their assistance. The teacher told the students that she had invited a large group of people over to her house for Thanksgiving dinner. The problem she posed was to find out how much it would cost to buy a large turkey (24 pounds) if it costs $1.25 per pound. The students discussed and solved this problem in pairs and then reported out their solutions and solutions methods to the whole group. Launching the problem: The video clips we will watch happen on the next day. The teacher again gathers the class together on the rug and poses a second, related turkey problem. The teacher wants to make sure her dinner is cooked properly. According to her favorite cookbook, a turkey of this size should cook for 15 minutes per pound. She again sends the students off to work in pairs or threes to solve this problem. Students may use any strategy or approach they choose and must record their solution and their method on a large sheet of poster paper that will later be shared with the class. Students at work: Four groups of students are seen working on the problem. 1. Hannah M. and Julia: They split 15 into 10 and 5 and add groups of 10 and groups of 5 separately. 2. Kenneth, Marlon, and Sam: They keep track of the number of pounds each fifteen minutes represents and skip count by 15s in a list that resembles a table when they include their labels. 3. May and Rafe: They use a doubling and halving strategy ( = 30 and 24 2 = 12) 4. Nate and Nellie: They put two fifteens together to make thirty minutes and count by that group. They count by 30s (minutes) while simultaneously counting by 2s (pounds). Students discuss solutions: After students have solved the problem and created posters showing their solutions and strategies, the teacher brings the group back together on the rug and asks some groups to share with the class. This portion of the class is called the Math Congress. (See the transcripts for a complete representation of the next clips.) We see Amber and Vicky present their poster. After they share, the teacher promotes discussion of their method by asking other students to explain Amber and Vicky s method. Next the teacher chooses Marlon, Kenneth, and Sam to present because she sees a connection between their method and the method used by Amber and Vicky. Finally, she asks May and Rafe to share because their method used a shortcut that included an important mathematical idea: the inverse relationship between doubling and halving. Since other students had not used this thinking and were somewhat confused by it, the teacher allowed this discussion to go longer than the others until many students indicated that they understood the strategy. 1 Vignette based on clips chosen from: Cameron, A., Hersch, S.B., & Fosnot, C.T. (2005). Turkey investigations: A context for multiplication, grades 3-5. Portsmouth, NH: Heinemann.

2 Characteristics of Effective Instruction Math Vignette: Individual Recording Sheet Evidence of engaging students with the mathematics content Evidence of creating an environment conducive to learning 2 Vignette based on clips chosen from: Cameron, A., Hersch, S.B., & Fosnot, C.T. (2005). Turkey investigations: A context for multiplication, grades 3-5. Portsmouth, NH: Heinemann.

3 Evidence of ensuring access for all students Evidence of use questioning to monitor and promote understanding Evidence of helping students make sense of the mathematics content 3 Vignette based on clips chosen from: Cameron, A., Hersch, S.B., & Fosnot, C.T. (2005). Turkey investigations: A context for multiplication, grades 3-5. Portsmouth, NH: Heinemann.

4 For Facilitator Use Only Math Vignette Cooking the Turkey Problem Notes on the Elements of Effective Instruction Vignette based on clips chosen from: Cameron, A., Hersch, S.B., & Fosnot, C.T. (2005). Turkey investigations: A context for multiplication, grades 3-5. Portsmouth, NH: Heinemann. Sequence of clips to be shown: Section from CD Clip number Time Continuing the Investigation Clip 11 2:01 Children at Work: Cooking the Turkey Hannah M. and Julia Children at Work: Cooking the Turkey Kenneth, Marlon, and Sam Children at Work: Cooking the Turkey May and Rafe Children at Work: Cooking the Turkey Nate and Nellie Math Congress Group 1 Group 2 Group 3 Clip 17 1:51 Clip 20 1:19 Clips 16 and 17 2:20 0:30 Clips 14 and 15 1:30 0:25 Clip 25 5:30 Clip 26 2:16 Clip 27 10:36 TOTAL VIDEO 28:18 Evidence of engaging students with the mathematics content The teacher chose a scenario from a real (or at least believable) context to frame the problem. The need to buy and cook food for a large group is a situation that students have likely encountered and/or can relate to. The use of a real cookbook increases student engagement as does the notion that she is personally seeking students help to solve her real-life problem. Evidence that the students are engaged can be seen in their conversations both during the work time and during the math congress. During the math congress the teacher says, There is something similar between yours and Amber and Victoria s [method] but she does not tell them what the similarity is. Instead she engages them in the process of identifying the connection. Evidence of creating an environment conducive to learning Students work in pairs or small groups assigned by the teacher to solve the problem. Purposeful partner selection minimizes the likelihood that any students will be excluded from the process. By gathering students together on the rug for the posing of the problem as well as the subsequent discussion, the teacher creates a friendly atmosphere that is inclusive and engaging. During the 4 Vignette based on clips chosen from: Cameron, A., Hersch, S.B., & Fosnot, C.T. (2005). Turkey investigations: A context for multiplication, grades 3-5. Portsmouth, NH: Heinemann.

5 student sharing, the teacher sits on the floor as a member of the classroom community. This increases the student-to-student interaction and diminishes the tendency for the conversation to be teacher-dominated. She offers encouragement that is authentic and based on students work and conversations. For example, after listening to and then restating Hannah M. and Julia s approach she said, You guys have a really good idea here. Evidence of ensuring access for all students Because the problem is posed in a context that students can relate to, there are many access points for students. The teacher does not specify a solution strategy nor does she put a time-limit on students. In her conversations with individual groups and during the math congress, she very often orchestrates the discussion in a way that ensures all students stay engaged. For example when working with Nate and Nellie, she asks Nellie about what Nate had said, Do you understand what he s doing? During the math congress she offers multiple opportunities for different students to restate or rephrase the explanations that others have given. She also allows and encourages student-student questions, particularly when students are sharing with the class. The classroom climate appears to be one in which questions are welcomed and students are expected to interact respectfully. Evidence of use questioning to monitor and promote understanding The classroom interactions that the teacher has with students are characterized by her listening and asking questions. She rarely makes a statement, and when she does it is often to paraphrase or restate the idea that a student has shared. Reviewing the transcripts and the clips provides evidence that more of the teacher utterances are questions than statements. Some examples: From students at work: (Clip 17) You said you have the 24 tens, right? So how many fives do you need?...how are you going to add all those fives on? (Clip 14) So every two you are going to be counting by 30s?...Can you write down here how you figured out the hours, so I can see? From math congress: (Clip 27) Ben, do you think you understand? It s a little tricky. Who thinks they can explain? Emma? (Clip 25) And how would you know the next number to circle? Evidence of helping students make sense of the mathematics content 5 Vignette based on clips chosen from: Cameron, A., Hersch, S.B., & Fosnot, C.T. (2005). Turkey investigations: A context for multiplication, grades 3-5. Portsmouth, NH: Heinemann.

6 The teacher uses the context of the problem to make sure students understand the results their mathematical methods produce. For example, after Vicky and Amber share their solution (Clip 25), the teacher asks, And what is the 360? The teacher returns to the context of the problem to make sure the students understand the relationship of the numerical answer they have found and the situation of the problem. During the whole class sharing, the teacher purposefully chooses the students who will share and the order in which they will share to encourage connections between methods. For example, she has Marlon, Kenneth, and Sam present after Amber and Vicky because she wants to reinforce the connection between jumps on the number line and a skip counting strategy. Since one pair of students used a strategy that involves a deeper level of mathematical content knowledge (doubling and halving) the teacher makes a point to have that group share. She introduces their approach by saying, And I would like May and Rafe to come up. And May and Rafe came up with an interesting shortcut, and I d like you to explain. The discussion that follows their presentation causes the rest of the class to interact with and explain this strategy and the mathematics that underlie it. 6 Vignette based on clips chosen from: Cameron, A., Hersch, S.B., & Fosnot, C.T. (2005). Turkey investigations: A context for multiplication, grades 3-5. Portsmouth, NH: Heinemann.

7 Math Vignette Best-Buy Problem A sixth grade class is beginning to work with ratio and proportion. The purpose of this lesson is to use an open-ended investigation to develop models for thinking about ratios and equivalent fractions. The teacher gathers the whole class together and tells the students that he needs their help to solve a problem he encountered in his life. The teacher tells them that he has recently gotten a kitten from an animal shelter and the kitten needs to eat a special kind of food. This food is sold at two neighborhood stores and the teacher would like to know which store offers the best buy on cat food. Bob s store sells 12 cans of kitten food for $15.00 and Maria s store sells 20 cans of the same food for $ After the teachers poses the problem he invites students to brainstorm their initial ideas about how they might determine which store offers the best deal. The teacher acknowledges each comment and rephrases them without making any value judgments. Next the teacher sends the students off to work in pairs to solve this problem. Students may use any strategy or approach they choose and must record their solution and their method on a large sheet of poster paper that will later be shared with the class. Three pairs of students are seen working on the problem: 1. Helaina and Lucy: They think of $1 per can as the overall price. At each store this leaves $3 extra to be divided among the total number of cans. They divide the $3 among 12 cans and $3 among 20 cans to get the price per can. 2. Andres and Zach (sitting across from Helaina and Lucy): They figure the price for 60 cans at each store and see which store had the lower total for 60 cans. 3. Dylan and Tristan: They make a list showing the cost for various numbers of cans using the original values given in the problem as the starting point. Their list says Bob s Maria s 12 cans = $ cans = $ cans = $ cans = cans = $ cans = $ can = $ can = $1.15 After students have solved the problem and created posters showing their solutions and strategies, the teacher brings the group back together and asks some groups to share with the class. This portion of the class is called the Math Congress. (See the transcript for a complete representation of the next clip.) We see Dylan and Tristan present their poster. After they share, the teacher asks who understands their strategy. When most students indicate that they understand, the teacher asks them to turn to a partner and explain 1 Vignette based on clips chosen from: Cameron, A., Jacob B., Fosnot, C.T., & Hersch, S.B., (2006). Working with the ratio table: mathematical models. Portsmouth, NH: Heinemann.

8 Dylan and Tristan s method. Next the teacher asks one student to ask Dylan and Tristan a question that came up in the partner conversation. Following this question, another student asks a question to the presenters and continues to ask questions to clarify her understanding of their process. Finally, the teacher records the strategy that Dylan and Tristan shared using a T-chart as a way to generalize about the process and see the mathematics and mathematical relationships in the numbers that extend beyond the context of this problem. 2 Vignette based on clips chosen from: Cameron, A., Jacob B., Fosnot, C.T., & Hersch, S.B., (2006). Working with the ratio table: mathematical models. Portsmouth, NH: Heinemann.

9 Characteristics of Effective Instruction Math Vignette: Individual Recording Sheet Evidence of engaging students with the mathematics content Evidence of creating an environment conducive to learning 3 Vignette based on clips chosen from: Cameron, A., Jacob B., Fosnot, C.T., & Hersch, S.B., (2006). Working with the ratio table: mathematical models. Portsmouth, NH: Heinemann.

10 Evidence of ensuring access for all students Evidence of using questioning to monitor and promote understanding Evidence of helping students make sense of the mathematics content 4 Vignette based on clips chosen from: Cameron, A., Jacob B., Fosnot, C.T., & Hersch, S.B., (2006). Working with the ratio table: mathematical models. Portsmouth, NH: Heinemann.

11 For Facilitator Use Only Math Vignette Best Buy Problem Notes on the Elements of Effective Instruction Vignette based on clips chosen from: Cameron, A., Jacob B., Fosnot, C.T., & Hersch, S.B., (2006). Working with the ratio table: mathematical models. Portsmouth, NH: Heinemann. Sequence of clips to be shown: Section of CD Clip number Time Developing the Context Clip 2 2:11 Children at Work Helaina and Lucy Clip 11 (Helaina and Lucy) Clip 21 (Helaina and Lucy talking with two boys) Children at Work Dylan and Tristan Clip 12 1:14 Math Congress Group 2 Clip 60 Clip 67 Clip 69 Clip 74 Math Congress Group 2 Clip 81 Clip 82 1:33 2:34 4:51 1:18 0:24 3:13 1:01 2:22 TOTAL VIDEO 20:00 Evidence of engaging students with the mathematics content The teacher uses a scenario from a real (or at least believable) context to frame the problem. The desire to determine a good deal and the need to do comparison shopping is familiar to students at this age. The teacher also uses the notion that he is personally seeking the students help to solve his problem to increase the level of engagement. Because the teacher did not model any particular method or even suggest a starting point for this problem, this lesson engages students in doing the intellectual work of making sense of and representing this mathematical situation. By interacting with a real-world problem situation, students maintain a connection to the numbers and what they represented in this situation. The numbers in this problem were carefully chosen to engage students in many possible strategies and ways of thinking about the problem: estimation to make the cost-per-can at each store easy to think about division and the distributive property using the remainders from the division as the point of comparison ($3 in each case) use of factors and multiples (e.g., 4 is a common factor of 12 and 20) connections between landmark fractions and decimal (money) equivalents (Note to facilitator: See pg 49 of Cameron, A., Jacob, B., Fosnot, C.T., & Hersch, S (2006), Working with the Ratio Table: Mathematical Models for a more detailed treatment of potential student strategies and how the problem numbers support them.) 5 Vignette based on clips chosen from: Cameron, A., Jacob B., Fosnot, C.T., & Hersch, S.B., (2006). Working with the ratio table: mathematical models. Portsmouth, NH: Heinemann.

12 Evidence of creating an environment conducive to learning Students work in pairs or small groups assigned by the teacher to solve the problem. Purposeful partner selection minimizes the likelihood that any students will be excluded from the process. By gathering students together for the posing of the problem as well as the subsequent discussion, the teacher created a friendly atmosphere that is inclusive and engaging. During the student sharing, the teacher acts as a facilitator, not as the authority or information-giver. This increases the student-to-student interaction and diminishes the tendency for the conversation to be teacher-dominated. Having students share multiple approaches to the problem allows the students to notice connections between strategies and to make generalizations about solution paths. The teacher has students explain Dylan and Tristan s method to a partner as a way to solidify their thinking about this method and to generate questions that might further clarify this method. The teacher uses the ideas that came from their solutions to develop a mathematical model: the ratio table. The teacher represents one solution shared by the students (Dylan and Tristan s method) in a format that supports the development of the ratio table model (using a T-chart). In this way the teacher is connecting what students did in the lesson with the mathematics he intended they learn. Evidence of ensuring access for all students After the teacher poses the problem he has a few students share their initial thinking before sending them off to solve the problem on their own. Children use imprecise language and have only partially formed ideas at this point. You could think about how many more cans you are getting at Maria s store and if that makes a difference in the price, but that kind of also reflects on how much each can costs. You could try and see if you can make like an equivalent fraction maybe like 3 goes into 12 and it goes into 15 and you can see if like a common fraction or something. You could use like the 15 and 5 goes into it so you might say 5 out of the 12 cans might be how much money. Having this brief brainstorming time provides students who might have been struggling with how to get started a few ideas they might use to find an initial starting point. During the math congress the teacher often checks to make sure that students understand the logic and process that other students are explaining. He frequently has students turn to their partners and explain their understanding of something that has just been shared. In another instance the teacher adds to a whole class discussion by saying, Is there anybody else who feels 6 Vignette based on clips chosen from: Cameron, A., Jacob B., Fosnot, C.T., & Hersch, S.B., (2006). Working with the ratio table: mathematical models. Portsmouth, NH: Heinemann.

13 like they understand the system that Dylan and Tristan used that maybe could put into their words? Maybe another voice in here would help clarify this. Evidence of using questioning to monitor and promote understanding The classroom interactions that the teacher has with students are characterized by his listening and asking questions. He rarely makes a statement, and when he does it is often to paraphrase or restate the idea that a student has shared. Reviewing the transcripts and the clips provides evidence that more of the teacher s utterances are questions rather than statements. For example, the teacher says, So are there any other ways we might think about this [problem]? Does that make sense? How many understand what Dylan just said here? Questions about that? Are you saying you re wondering about this jump from 3 cans to 1 can? I just want to clarify what you are asking. Evidence of helping students make sense of the mathematics content In this lesson, students are not only to solve the problem and decide which store offers the best buy but also to justify their choice and to explain their reasoning. We can see this in both the group-to-group sharing and the group-to-class sharing of solutions. Students give their solution along with an explanation of their reasoning. They are also free to challenge or question other students in order to understand a strategy or way of thinking that is different from their own. For example, in the whole group sharing Jeremy asks Dylan and Tristan, How did you know that you would stop at, like $1.25? Lucy asks a series of questions which is resolved when the teacher points out that the boys are talking about 3 cans and Lucy is talking about $3. There is an expectation in the classroom that students take responsibility for their learning by both asking questions when they don t understand something and by supporting and justifying their own work both in writing and verbally. 7 Vignette based on clips chosen from: Cameron, A., Jacob B., Fosnot, C.T., & Hersch, S.B., (2006). Working with the ratio table: mathematical models. Portsmouth, NH: Heinemann.

14 Math Vignette Second Year Algebra Polynomials Lab An 11 th and 12 th grade class is beginning to work with polynomial functions through a lab exploration. The purpose of this lab is to use an open-ended investigation to help students understand the general shape of polynomial functions. Questions they will be exploring include: What are some predictable patterns in the graphs of polynomial functions? Where will the graphs rise and fall? Where will they cross the x- and y-axes? Through the course of the lab, students will also apply what they already know about linear and quadratic functions, as well as some procedures and skills they have learned (substitution, factoring, finding intercepts, relating tables of values to equations and graphs). The teacher begins the class by reviewing some vocabulary students have already encountered that relate to polynomial functions. Next she hands out the lab assignment and allows students to work alone, in pairs, or small groups of their own choosing. The teacher s intention is to have students uncover patterns and make generalizations on their own. The lab sheet includes several questions that point out what students should notice, but the questions do not take them directly to a conclusion. While the students work, the teacher captures some of the significant questions and observations that she overhears in student interactions. Rather than stopping the interactions to point these out or have a whole class discussion, she simply reminds students that they can look at the board at any time to see if there is an idea or question there that prompts their thinking. Finally, we see a clip that takes place three weeks later at the end of the unit. On this day students are asked to solve and discuss problems that involve polynomial functions. They are then assigned a partner with whom they are to make a poster of one problem that will be posted for other students to use as a study aide as they prepare for a unit assessment. 1 Vignette based on clips chosen from: Coalition of Essential Schools. CESEssential Visions, Disc 1, Classroom Practice, 2005.

15 Characteristics of Effective Instruction Math Vignette: Individual Recording Sheet Evidence of engaging students with the mathematics content Evidence of creating an environment conducive to learning 2 Vignette based on clips chosen from: Coalition of Essential Schools. CESEssential Visions, Disc 1, Classroom Practice, 2005.

16 Evidence of ensuring access for all students Evidence of using questioning to monitor and promote understanding Evidence of helping students make sense of the mathematics content 3 Vignette based on clips chosen from: Coalition of Essential Schools. CESEssential Visions, Disc 1, Classroom Practice, 2005.

17 For Facilitator Use Only Math Vignette Second Year Algebra Polynomials Lab Notes on the Elements of Effective Instruction Vignette based on clips chosen from: Coalition of Essential Schools. CESEssential Visions, Disc 1, Classroom Practice, Sequence of clips to be shown: Main Menu Less Is More, Depth Over Coverage Second Year Algebra: Polynomials Coaching a Lab Investigation Description of Clip Counter Time Launching the lab, teacher s rationale 00:00 3:20 3:20 Students working in groups, teacher 3:20 11:27 8:20 describes other typical days in her class, teacher talks about the value of going deeper on fewer topics. Day 2: Students continue to work in small 11:40-16:40 5:00 groups, teacher talks about her big picture goals for her students, Three weeks later: Students make posters 16:40-23:47 to demonstrate/share their solutions and their thinking in preparation for a unit assessment. The posters are displayed in a poster gallery in the hall. TOTAL VIDEO 20:00 Evidence of engaging students with the mathematics content The teacher talks about her goal that students become autonomous in their thinking. She wants them to think like mathematicians rather than like high school students taking a math class. The students view the lab directions as prompts to get them started thinking and observing. They make a connection between this lab in math class and a lab you might do in science class in which you have to gather data to lead to a conclusion. Students are seen posing their own questions such as, What if we try smaller numbers? Students are seen making conjectures and testing them with each other. One student says, We go in-depth on fewer topics and try to determine why. Evidence of creating an environment conducive to learning Students work individually, in pairs, or small groups at their choosing to complete the lab. During the work time, the teacher makes a deliberate effort not to interrupt student thinking with teacher announcements. Rather, she keeps a running record on the board of important student observations and questions. The teacher acts as a facilitator, not as the authority or informationgiver. This increases the student-to-student interaction and diminishes the tendency for the conversation to be teacher-dominated. The students talk about the value in having less teacher- 4 Vignette based on clips chosen from: Coalition of Essential Schools. CESEssential Visions, Disc 1, Classroom Practice, 2005.

18 talk and more student-student interaction. Students ask and answer each other s questions without waiting for a teacher-led interaction. Evidence of ensuring access for all students The teacher s expectation is that students will find the patterns and make generalizations on their own. She maintains high expectations for all students and provides classroom supports (group interactions, access to teacher assistance) when students run into difficulty, but she does not rescue them. The teacher points out to the students that the depth of their thinking will be greater if they talk amongst themselves about the content. The teacher assigns partners in the final phase when students demonstrate their learning by making and presenting posters for problems that are similar to the ones they will see on their assessment. By doing this in partners, she provides an opportunity for pairs to continue learning through the discussion that will formalize their thinking. Evidence of using questioning to monitor and promote understanding This video does not capture many instances of the teacher asking questions. One piece of evidence we have is that the teacher talks about her practice of answering their questions with questions. Evidence of helping students make sense of the mathematics content Students make connections between this lab and a previous lab on exponential functions. The teacher focuses on big ideas rather than the small procedural details. At the end of class the teacher polls the students to find out their level of understanding. In this way she is encouraging them to reflect on their learning. Three weeks later the teacher wraps up the unit of study by having students work on additional problems and create a poster gallery showing what they know. It is her intention to hold them accountable for all the learning that has taken place over the course of the unit. After making the poster, one student made the connection between many ideas of graphs and equations including intercepts and substitution. She pointed out how it all came together for her as a result of the teacher s expectation that each group present to the entire class. She said, The pressure was kind of on, which indicates that in this classroom environment she felt responsible for her own learning. 5 Vignette based on clips chosen from: Coalition of Essential Schools. CESEssential Visions, Disc 1, Classroom Practice, 2005.

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