(Lesson Design)
(Final Version)
What?
Since I know that the students that I selected to participate in my small group math lesson are fairly advanced, I want to work with the same subject matter that we are learning in their math class but at a slightly higher level. Having observed their math lessons and gone through their math investigations book, it seems like the students are mainly working on separate result unknown problems. (i.e. Rosa had 15 pennies. She spent 5 pennies. How many pennies did she have left?, Rosa had 10 shells. She gave 6 of them to Max. How many shells does Rosa have left?, Kim saw 20 ducks on the pond. Then 9 ducks flew away. How many ducks were still in the pond?)
Chapin and Johnson (2006) talk about how textbooks often include one or two different types of addition and subtraction problems, even though "to become proficient and competent users of mathematics, students need to be able to solve all types of problems" (p. 56). I believe my classroom mentor is teaching the whole class result known subtraction word problems because she just introduced subtraction and some of the students are struggling. According to Chapin and Johnson (2006), "In general, when the result is unknown, students are more likely to be able to make sense of the relationships." (p. 57) After looking at my six students' work and doing individual conferences, I feel like introducing change unknown problems is an appropriate next step.
Since change unknown problems are presumably new to the students, I do not want the students to worry about "finding the answer" to the problems. My goal is to help this group of students see that word problems and number sentences usually have three components (start, change, and end) and introduce them math problems that require you to find the change rather than the end number or what the problem equals.
How?
I will begin this lesson by directly modeling a word problem. Although I believe many students in this group are beyond direct modeling based on my individual conferences and observations, Janine and I talked about how it would be helpful for the students to see a representation of the type of problems we will be working on.
I want to begin the lesson by providing a lot of guidance and support, because I will be introducing a new concept. I will model the questions that I would ask myself using the problem that we acted out. Students will be able to refer to the questions on the poster throughout the lesson to help them understand the problem. The questions are:
During the second example problem, I will ask guiding questions to help the students gather the relevant information, but I will ask them to fill in the start, change, and end chart and then the number sentence by themselves. I thought students might have an easier time filling in the chart because the headings (i.e. "start", "change", and "end") remind the students to think about which number corresponds with which piece according to the word problem. I am trying to have a gradual release model because I am hoping that they will be able to complete more of the task independently as they progress. After they write down the numbers in the table, I will call on a couple of students to share what they have. I will ask the students if they agree or disagree and they will talk about why they agree or disagree as well as their thought process.
Next, the students will work together to create their own word end unknown and change unknown problems in pairs. Lastly, each student will solve a few word problems independently so that I can see where each student is in their thinking about the different types of unknowns in subtraction problems.
Why?
Although I thought that a lesson around the measurement and data or geometry Common Core Standards might be engaging for students, I felt like that would not fit with our current math focus and goals. Our class has spent a considerable amount of time learning about addition and different addition strategies. We recently introduced subtraction. Since I wanted my small group lesson to be aligned with what the students are learning in math class, I decided to focus on subtraction as well.
My classroom mentor specifically mentioned how she is more focused on the students at the lower end of the spectrum. Knowing this, I wanted to work with a group of students who may not receive the individual attention and challenges that they need to further develop their mathematical understanding. I felt like I could challenge these students to solve different math problems than what they were used to working with. I thought introducing the students to a new type of word problem and emphasizing the process of solving the problem rather than finding the answer to the problem might be new and interesting for them.
Instead of changing the numbers so that the math was more challenging, I chose to change the type of problem to help the students develop their flexibility with different kinds of subtraction and addition problems. Being able to identifying the different components of number sentences based on word problems is essential for solving the problem, but it also may help students see the relationship between the components, specifically, and help them see the relationship between addition and subtraction more generally. For example, if you know the end number is 37 and the start number was 62, you can solve for the unknown by doing 62 - 37 = the change. If you know the change number is 37 and the end number is 25, you can make the subtraction problem into an addition problem of 37 + 25 = the start number. I want these students to develop as mathematicians by engaging in a lesson that emphasizes the mathematical practices laid out in the Common Core State Standards.
My classroom mentor liked the idea of giving the students the number sentence and asking them to create a word problem, so she did this as a mini-lesson with the whole class. The students that she called on were able to create word problems that went with the number sentence. I thought this was a valuable activity because one student initially began saying an addition word problem instead of a subtraction word problem, but then she realized that it did not make sense. She thought about it and changed it into a subtraction problem. Although I recognize that students represent word problems through number sentences, I think having the students create the word problems can help me see how students understand what the operations mean in real-life and how they might use math in a real world situation. Since this is a relatively new concept, I do not want students to feel pressured to come up with the problems on their own, which is why it is a partner activity.
I also felt like it was important to show the students that mathematics can be a collaborative subject that involves both solving and identifying problems that might occur in the real world. I want students to be excited about learning and doing math. I want students to have opportunities to collaborate and communicate with each other about math and their mathematical thinking. Although the students only had a brief introduction to the change unknown separate problem type, I thought that talking about it with a classmate might help.
Pedagogical Focus
I chose to focus on using a representation to make mathematics meaningful in hopes that my students would be able to use this representation to draw connections between mathematical concepts later on. The representation that I chose was the start, change, and end table. I introduced this table as a way for students to represent subtraction word problems with different types of unknowns. I thought that using a representation would help students see the difference between end unknown and change unknown problems. It was my hope that the students would eventually use this table to examine the relationship between addition and subtraction.
Since I know that the students that I selected to participate in my small group math lesson are fairly advanced, I want to work with the same subject matter that we are learning in their math class but at a slightly higher level. Having observed their math lessons and gone through their math investigations book, it seems like the students are mainly working on separate result unknown problems. (i.e. Rosa had 15 pennies. She spent 5 pennies. How many pennies did she have left?, Rosa had 10 shells. She gave 6 of them to Max. How many shells does Rosa have left?, Kim saw 20 ducks on the pond. Then 9 ducks flew away. How many ducks were still in the pond?)
Chapin and Johnson (2006) talk about how textbooks often include one or two different types of addition and subtraction problems, even though "to become proficient and competent users of mathematics, students need to be able to solve all types of problems" (p. 56). I believe my classroom mentor is teaching the whole class result known subtraction word problems because she just introduced subtraction and some of the students are struggling. According to Chapin and Johnson (2006), "In general, when the result is unknown, students are more likely to be able to make sense of the relationships." (p. 57) After looking at my six students' work and doing individual conferences, I feel like introducing change unknown problems is an appropriate next step.
Since change unknown problems are presumably new to the students, I do not want the students to worry about "finding the answer" to the problems. My goal is to help this group of students see that word problems and number sentences usually have three components (start, change, and end) and introduce them math problems that require you to find the change rather than the end number or what the problem equals.
How?
I will begin this lesson by directly modeling a word problem. Although I believe many students in this group are beyond direct modeling based on my individual conferences and observations, Janine and I talked about how it would be helpful for the students to see a representation of the type of problems we will be working on.
I want to begin the lesson by providing a lot of guidance and support, because I will be introducing a new concept. I will model the questions that I would ask myself using the problem that we acted out. Students will be able to refer to the questions on the poster throughout the lesson to help them understand the problem. The questions are:
- What do we know about the situation/problem?
- What was the question asking?
- Is the end result bigger or smaller than the first number?
During the second example problem, I will ask guiding questions to help the students gather the relevant information, but I will ask them to fill in the start, change, and end chart and then the number sentence by themselves. I thought students might have an easier time filling in the chart because the headings (i.e. "start", "change", and "end") remind the students to think about which number corresponds with which piece according to the word problem. I am trying to have a gradual release model because I am hoping that they will be able to complete more of the task independently as they progress. After they write down the numbers in the table, I will call on a couple of students to share what they have. I will ask the students if they agree or disagree and they will talk about why they agree or disagree as well as their thought process.
Next, the students will work together to create their own word end unknown and change unknown problems in pairs. Lastly, each student will solve a few word problems independently so that I can see where each student is in their thinking about the different types of unknowns in subtraction problems.
Why?
Although I thought that a lesson around the measurement and data or geometry Common Core Standards might be engaging for students, I felt like that would not fit with our current math focus and goals. Our class has spent a considerable amount of time learning about addition and different addition strategies. We recently introduced subtraction. Since I wanted my small group lesson to be aligned with what the students are learning in math class, I decided to focus on subtraction as well.
My classroom mentor specifically mentioned how she is more focused on the students at the lower end of the spectrum. Knowing this, I wanted to work with a group of students who may not receive the individual attention and challenges that they need to further develop their mathematical understanding. I felt like I could challenge these students to solve different math problems than what they were used to working with. I thought introducing the students to a new type of word problem and emphasizing the process of solving the problem rather than finding the answer to the problem might be new and interesting for them.
Instead of changing the numbers so that the math was more challenging, I chose to change the type of problem to help the students develop their flexibility with different kinds of subtraction and addition problems. Being able to identifying the different components of number sentences based on word problems is essential for solving the problem, but it also may help students see the relationship between the components, specifically, and help them see the relationship between addition and subtraction more generally. For example, if you know the end number is 37 and the start number was 62, you can solve for the unknown by doing 62 - 37 = the change. If you know the change number is 37 and the end number is 25, you can make the subtraction problem into an addition problem of 37 + 25 = the start number. I want these students to develop as mathematicians by engaging in a lesson that emphasizes the mathematical practices laid out in the Common Core State Standards.
My classroom mentor liked the idea of giving the students the number sentence and asking them to create a word problem, so she did this as a mini-lesson with the whole class. The students that she called on were able to create word problems that went with the number sentence. I thought this was a valuable activity because one student initially began saying an addition word problem instead of a subtraction word problem, but then she realized that it did not make sense. She thought about it and changed it into a subtraction problem. Although I recognize that students represent word problems through number sentences, I think having the students create the word problems can help me see how students understand what the operations mean in real-life and how they might use math in a real world situation. Since this is a relatively new concept, I do not want students to feel pressured to come up with the problems on their own, which is why it is a partner activity.
I also felt like it was important to show the students that mathematics can be a collaborative subject that involves both solving and identifying problems that might occur in the real world. I want students to be excited about learning and doing math. I want students to have opportunities to collaborate and communicate with each other about math and their mathematical thinking. Although the students only had a brief introduction to the change unknown separate problem type, I thought that talking about it with a classmate might help.
Pedagogical Focus
I chose to focus on using a representation to make mathematics meaningful in hopes that my students would be able to use this representation to draw connections between mathematical concepts later on. The representation that I chose was the start, change, and end table. I introduced this table as a way for students to represent subtraction word problems with different types of unknowns. I thought that using a representation would help students see the difference between end unknown and change unknown problems. It was my hope that the students would eventually use this table to examine the relationship between addition and subtraction.
Comments on First Draft of Lesson Design from Janine:
**Note: I did not see the comments on the side or the comments in the text prior to my meeting with Janine or revising my lesson. I was unable to see them in the attachment that was visible in my email inbox. It was not until I finally downloaded the attachment weeks after enacting my revised lesson that I saw the other comments.**
Revised Draft of Lesson Design: