Core decisions
(Taken from Revised Lesson Plan)
What?
By the beginning of 3rd grade, many students in my fieldwork classroom were able to perform basic length measurement tasks – using a ruler to measure individual objects with lengths of integer value (either inches or centimeters) – though not all were successful in early assessments. This lesson represents a reinforcement of that work, and an introduction to a more complex understanding of measurement – one which relies on an understanding of length as continuous (an example of this concept could be understanding that multiple objects can be measured together to create a greater length), and which involves continuous units (i.e., all units can be subdivided into smaller units).
This deeper understanding of measurement will involve the integration of multiple skills with which students are practiced – primarily those involving measurement and addition. Students should leave the lesson prepared to determine the lengths of combined objects, using additive (or multiplicative) reasoning. This will promote understanding of length measurement and basic addition; more than this, however, students will also be required to recognize units of measurement as an abstract representation of object length, and be able to add lengths of objects that cannot be directly measured (e.g., measuring the length of two pencils when only one is present). Finally, the lesson will expose students to the concept of continuous units, as it will make use of measurements that are not expressed in whole units; however, because this is not the primary focus of the lesson, it will mostly be limited to multiples of ½ (although there will be opportunities for students to encounter objects whose length is somewhere between one half and one whole). Although it will not be modeled or assessed, this lesson may also encourage the use of estimation – for one component, students will be asked to determine what objects can combine to achieve a given length, a task which will be greatly simplified by beginning with estimation.
To avoid adding extra dimensions/complications to this lesson, I decided to employ only one unit of measurement, so I am using inches throughout this lesson. Although 3rd graders are expected to be able to use both inches and centimeters, I have made the assumption that students will be more comfortable with inches. Because of that, and because measurements in inches will involve smaller numbers, I believe that using inches will help me focus the lesson on the understandings described above (however, once students are becoming comfortable with the above, using centimeters might become a good extension for future lessons).
My overarching inquiry for the Term III assignment involves differentiated instruction, and particularly the use of partner work as a strategy to support students with diverse skill/experience levels. This lesson will include some salient factors related to this overarching inquiry. First, I will not dictate particular strategies for students to use for combining lengths of objects, though I will provide pencils and paper, as well as string; as a result, students asked to determine the length of “three keys” (but only given one key) could use multiple strategies, such as: tracing one key three times in a row, then measuring the total length; measuring the length of one key, then drawing that length three times, then measuring the total length; measuring the length of one key, then adding the numeric value of that length three times; etc. As should be evident, each of these strategies should reveal something different about the students’ thinking about addition and measurement (the first example approximates direct modeling, though it is slightly abstracted; the second one is similar but relies on a more abstract understanding of measurement; the third one involves mental representation as well as a much more abstract understanding of measurement).
Second, the lesson makes significant use of partner work; the activities are designed to support collaboration both by including directions that involve both members, and by including tasks that are extremely difficult without two participants. Third, there are opportunities for differentiation of work within the partner activities. Finally, the lesson will involve many opportunities for students to share their methods and understandings, both within pairs and as a whole group.
Pedagogical Focus:
Although this lesson will make use of all three of the “high-leverage” pedagogical focuses which were provided as options for Term III (assessing student understanding, facilitating mathematical discussion, and selecting and using representations), I will make primary use of the second of these – facilitating mathematical discussion. I want my students to be able to learn from one another, particularly as my grouping will involve a combination of students who are “at grade level” and students who are significantly “below grade level” according to their prior math assessments (both formal and informal). I hope to promote this through a combination of activities designed to encourage student collaboration and discussion in partners, and through whole group discussions where I will rely upon “talk moves” to elicit student ideas, help make them accessible to other students, and use them as opportunities for other students to apply their own inquiry and expand their own understanding.
How?
This lesson will revolve around a series of hands-on activities, each of which is designed to expand upon the previous one in drawing upon increasingly complex understandings of measurement. As teacher, I will provide small amounts of modeling (although I will be prepared to increase my use of modeling if students seem to be struggling), but my primary role will be similar to my role while leading math talks: acting as a facilitator, using “talk moves” to promote student sharing and consideration of alternative methods and solutions.
The first stage of the lesson will involve measurement of individual objects; after some brief modeling of measuring in whole- and half-inch increments, students will be responsible for sorting through a box of objects to identify which objects match given length measurements. The next stage involves combined measurements. First, partners will to identify combinations of objects that together match given length requirements. Second, partners will make measurements of objects that are longer than the length of the rulers, forcing them to find a way to combine the lengths of multiple rulers (which will advance two goals: providing experience with measuring combining objects; and supporting student ability to measure objects larger than 12 inches, which will be useful for later parts of the lesson). Student strategies for this will be reviewed as a full group.
The third and final stage will require students to determine lengths of object combinations that cannot be measured directly – for example, two pencils and three quarters, when only one of each object is available for measurement. It will resemble a number talk in terms of the way that whole group discussion of answers will be facilitated; it will differ from a traditional number talk, however, because students will have the opportunity to work in partners, with materials, to determine answers to the various questions. The final question will require that students determine what objects would need to be added to their heights in order to reach a larger height (specifically, mine). My height will be represented both numerically (number of inches) and visually (tape markers on the floor). The lesson will close with a whole-group discussion of strategies used for this task.
Task: Solve a variety of problems involving measurement of objects, and measurement of combinations of objects (including repeated use of objects of which students only have one copy).
Discourse: Students will have many opportunities for discussion, both in partners and as a whole group. I will act primarily as a facilitator during whole group discussion, although I may become more actively engaged in supporting partner collaboration and/or providing modeling, as needed.
Tools: Worksheets, rulers, pencils, paper, string, minds.
Norms: During partner work, students will be expected to participate and collaborate actively and respectfully – this means both students are engaged in making contributions, and students discuss ideas thoughtfully without discussing them personally. These same conversational norms will apply in full-group discussions, where each student will also be responsible for sharing their understandings in a way that is comprehensible to others, and for listening and being prepared to respond to one another’s contributions.
Why?
Context played a role in my choice of lesson: as with my Science and Literacy lessons, I wanted to choose a topic connected approximately to the curriculum the students will be engaging with elsewhere in the classroom; in this case, I will be timing my lesson to coincide roughly with the beginning of students’ work with measurement in our classroom, which will focus on measurements of length (including both full and half inches and centimeters). Because my lesson will be quite short (~45 minutes), I wanted to ensure that the lesson I teach will be quickly reinforced and connected to other activities they will be doing.
I also chose a lesson that draws upon some of the Common Core Standards for 3rd grade. In this case, I found two relevant 3rd grade standards: CC.2.4.3.A.1, which asks students to “[s]olve problems involving measurement and estimation of…length;” and CC.2.1.3.C.1, which has students “[e]xplore and develop an understanding of fractions as numbers” (not a primary point of this lesson, but nonetheless one that is embedded). However, many if not most of the students in my class began the year substantially below grade level for mathematics (according to both formal and informal assessments); therefore, I thought it valuable to also draw upon 2nd grade standards, to ensure that students have achieved the level of scaffolding with which they were supposed to enter 3rd grade. Two 2nd grade lessons have particular relevance. The first, CC.2.4.2.A.1, has students “[m]easure and estimate lengths in standard units using appropriate tools.” The second, CC.2.4.2.A.6 is perhaps the most central standard embedded in this lesson: “Extend the concepts of addition and subtraction to problems involving length.”
While considerations of curriculum and standards played a role in determining the mathematical content of my lesson, the structure of it drew upon a variety of other influences. As I have already described above, some elements of the structure (particularly the use of partnerships) were guided directly by my personal inquiry and by the pedagogical focus; of course, these elements are also ultimately rooted in my belief in the power of student discussion and collaboration to engage students as active learners and expose them to additional and alternative modes of thinking which they can potentially learn from and make use of. My construction of the lesson was also guided by a desire to give the students an opportunity to make hands-on use of a variety of objects not normally used in math lessons, and to have some degree of choice in which objects are used for various parts of the lesson – in part to enhance their engagement and make the lesson more “practically” grounded, and in part simply because the students have had limited opportunities for hands-on work in our classroom, which I am eager to remedy.
My selection of students is connected to my desire to use this exercise as an opportunity to practice differentiation. I offered all the students in my class the opportunity to select which of my Term III lessons they would like to participate in; I will be teaching this lesson to a 4-student subgroup of those who chose math as their first or second choice (I’ll actually teach it twice, as I had roughly 8 students sign up for math: I will teach it once formally, to be recorded, observed, and analyzed; then again, informally). Pairs within these 4-student groups will be determined based on 1) students who are assessed at different levels in math (with a special attention to pairing students who did poorly on measurement problems in math assessments with those who performed strongly), and 2) students who will hopefully work together with minimal need for behavioral intervention.
What?
By the beginning of 3rd grade, many students in my fieldwork classroom were able to perform basic length measurement tasks – using a ruler to measure individual objects with lengths of integer value (either inches or centimeters) – though not all were successful in early assessments. This lesson represents a reinforcement of that work, and an introduction to a more complex understanding of measurement – one which relies on an understanding of length as continuous (an example of this concept could be understanding that multiple objects can be measured together to create a greater length), and which involves continuous units (i.e., all units can be subdivided into smaller units).
This deeper understanding of measurement will involve the integration of multiple skills with which students are practiced – primarily those involving measurement and addition. Students should leave the lesson prepared to determine the lengths of combined objects, using additive (or multiplicative) reasoning. This will promote understanding of length measurement and basic addition; more than this, however, students will also be required to recognize units of measurement as an abstract representation of object length, and be able to add lengths of objects that cannot be directly measured (e.g., measuring the length of two pencils when only one is present). Finally, the lesson will expose students to the concept of continuous units, as it will make use of measurements that are not expressed in whole units; however, because this is not the primary focus of the lesson, it will mostly be limited to multiples of ½ (although there will be opportunities for students to encounter objects whose length is somewhere between one half and one whole). Although it will not be modeled or assessed, this lesson may also encourage the use of estimation – for one component, students will be asked to determine what objects can combine to achieve a given length, a task which will be greatly simplified by beginning with estimation.
To avoid adding extra dimensions/complications to this lesson, I decided to employ only one unit of measurement, so I am using inches throughout this lesson. Although 3rd graders are expected to be able to use both inches and centimeters, I have made the assumption that students will be more comfortable with inches. Because of that, and because measurements in inches will involve smaller numbers, I believe that using inches will help me focus the lesson on the understandings described above (however, once students are becoming comfortable with the above, using centimeters might become a good extension for future lessons).
My overarching inquiry for the Term III assignment involves differentiated instruction, and particularly the use of partner work as a strategy to support students with diverse skill/experience levels. This lesson will include some salient factors related to this overarching inquiry. First, I will not dictate particular strategies for students to use for combining lengths of objects, though I will provide pencils and paper, as well as string; as a result, students asked to determine the length of “three keys” (but only given one key) could use multiple strategies, such as: tracing one key three times in a row, then measuring the total length; measuring the length of one key, then drawing that length three times, then measuring the total length; measuring the length of one key, then adding the numeric value of that length three times; etc. As should be evident, each of these strategies should reveal something different about the students’ thinking about addition and measurement (the first example approximates direct modeling, though it is slightly abstracted; the second one is similar but relies on a more abstract understanding of measurement; the third one involves mental representation as well as a much more abstract understanding of measurement).
Second, the lesson makes significant use of partner work; the activities are designed to support collaboration both by including directions that involve both members, and by including tasks that are extremely difficult without two participants. Third, there are opportunities for differentiation of work within the partner activities. Finally, the lesson will involve many opportunities for students to share their methods and understandings, both within pairs and as a whole group.
Pedagogical Focus:
Although this lesson will make use of all three of the “high-leverage” pedagogical focuses which were provided as options for Term III (assessing student understanding, facilitating mathematical discussion, and selecting and using representations), I will make primary use of the second of these – facilitating mathematical discussion. I want my students to be able to learn from one another, particularly as my grouping will involve a combination of students who are “at grade level” and students who are significantly “below grade level” according to their prior math assessments (both formal and informal). I hope to promote this through a combination of activities designed to encourage student collaboration and discussion in partners, and through whole group discussions where I will rely upon “talk moves” to elicit student ideas, help make them accessible to other students, and use them as opportunities for other students to apply their own inquiry and expand their own understanding.
How?
This lesson will revolve around a series of hands-on activities, each of which is designed to expand upon the previous one in drawing upon increasingly complex understandings of measurement. As teacher, I will provide small amounts of modeling (although I will be prepared to increase my use of modeling if students seem to be struggling), but my primary role will be similar to my role while leading math talks: acting as a facilitator, using “talk moves” to promote student sharing and consideration of alternative methods and solutions.
The first stage of the lesson will involve measurement of individual objects; after some brief modeling of measuring in whole- and half-inch increments, students will be responsible for sorting through a box of objects to identify which objects match given length measurements. The next stage involves combined measurements. First, partners will to identify combinations of objects that together match given length requirements. Second, partners will make measurements of objects that are longer than the length of the rulers, forcing them to find a way to combine the lengths of multiple rulers (which will advance two goals: providing experience with measuring combining objects; and supporting student ability to measure objects larger than 12 inches, which will be useful for later parts of the lesson). Student strategies for this will be reviewed as a full group.
The third and final stage will require students to determine lengths of object combinations that cannot be measured directly – for example, two pencils and three quarters, when only one of each object is available for measurement. It will resemble a number talk in terms of the way that whole group discussion of answers will be facilitated; it will differ from a traditional number talk, however, because students will have the opportunity to work in partners, with materials, to determine answers to the various questions. The final question will require that students determine what objects would need to be added to their heights in order to reach a larger height (specifically, mine). My height will be represented both numerically (number of inches) and visually (tape markers on the floor). The lesson will close with a whole-group discussion of strategies used for this task.
Task: Solve a variety of problems involving measurement of objects, and measurement of combinations of objects (including repeated use of objects of which students only have one copy).
Discourse: Students will have many opportunities for discussion, both in partners and as a whole group. I will act primarily as a facilitator during whole group discussion, although I may become more actively engaged in supporting partner collaboration and/or providing modeling, as needed.
Tools: Worksheets, rulers, pencils, paper, string, minds.
Norms: During partner work, students will be expected to participate and collaborate actively and respectfully – this means both students are engaged in making contributions, and students discuss ideas thoughtfully without discussing them personally. These same conversational norms will apply in full-group discussions, where each student will also be responsible for sharing their understandings in a way that is comprehensible to others, and for listening and being prepared to respond to one another’s contributions.
Why?
Context played a role in my choice of lesson: as with my Science and Literacy lessons, I wanted to choose a topic connected approximately to the curriculum the students will be engaging with elsewhere in the classroom; in this case, I will be timing my lesson to coincide roughly with the beginning of students’ work with measurement in our classroom, which will focus on measurements of length (including both full and half inches and centimeters). Because my lesson will be quite short (~45 minutes), I wanted to ensure that the lesson I teach will be quickly reinforced and connected to other activities they will be doing.
I also chose a lesson that draws upon some of the Common Core Standards for 3rd grade. In this case, I found two relevant 3rd grade standards: CC.2.4.3.A.1, which asks students to “[s]olve problems involving measurement and estimation of…length;” and CC.2.1.3.C.1, which has students “[e]xplore and develop an understanding of fractions as numbers” (not a primary point of this lesson, but nonetheless one that is embedded). However, many if not most of the students in my class began the year substantially below grade level for mathematics (according to both formal and informal assessments); therefore, I thought it valuable to also draw upon 2nd grade standards, to ensure that students have achieved the level of scaffolding with which they were supposed to enter 3rd grade. Two 2nd grade lessons have particular relevance. The first, CC.2.4.2.A.1, has students “[m]easure and estimate lengths in standard units using appropriate tools.” The second, CC.2.4.2.A.6 is perhaps the most central standard embedded in this lesson: “Extend the concepts of addition and subtraction to problems involving length.”
While considerations of curriculum and standards played a role in determining the mathematical content of my lesson, the structure of it drew upon a variety of other influences. As I have already described above, some elements of the structure (particularly the use of partnerships) were guided directly by my personal inquiry and by the pedagogical focus; of course, these elements are also ultimately rooted in my belief in the power of student discussion and collaboration to engage students as active learners and expose them to additional and alternative modes of thinking which they can potentially learn from and make use of. My construction of the lesson was also guided by a desire to give the students an opportunity to make hands-on use of a variety of objects not normally used in math lessons, and to have some degree of choice in which objects are used for various parts of the lesson – in part to enhance their engagement and make the lesson more “practically” grounded, and in part simply because the students have had limited opportunities for hands-on work in our classroom, which I am eager to remedy.
My selection of students is connected to my desire to use this exercise as an opportunity to practice differentiation. I offered all the students in my class the opportunity to select which of my Term III lessons they would like to participate in; I will be teaching this lesson to a 4-student subgroup of those who chose math as their first or second choice (I’ll actually teach it twice, as I had roughly 8 students sign up for math: I will teach it once formally, to be recorded, observed, and analyzed; then again, informally). Pairs within these 4-student groups will be determined based on 1) students who are assessed at different levels in math (with a special attention to pairing students who did poorly on measurement problems in math assessments with those who performed strongly), and 2) students who will hopefully work together with minimal need for behavioral intervention.