james e newton elementary mathematics for washington edtpa submission...

James E Newton

Elementary Mathematics for Washington

EdTPA submission

May 25, 2015

Table of Contents

Context for Learning 1

Lesson Plans for Learning Segment 3

Instructional Materials 13

Assessments 27

Planning Commentary 29

Instruction Commentary 34

Student Working Samples 36

Assessment Commentary 39

Evaluation Criteria 44

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Elementary Mathematics Context for Learning Information

About the School Where I am Teaching:

1. In what type of school do you teach?

a. Rural

2. What grade levels are at your site?

a. K-5

3. List any special features of your school or classroom setting that will affect your teaching

in this learning segment:

a. Public School District

b. Cooperating Mentor Teacher (CMT)

4. Describe any district, school, or cooperating teacher requirements or expectations that

might affect your planning or delivery of instruction, such as required curricula, pacing

plan, use of specific instructional strategies, or standardized tests:

a. Engaged New York Curriculum

About the Class Featured in this Assessment:

1. How much time is devoted each day to mathematics instruction in your classroom?

a. 2 hours/day

2. Is there any ability grouping or tacking in mathematics? No

3. Identify any textbook or instructional program you primarily use for mathematics:

a. Engaged New York

4. List other resources you use for mathematics instruction in class:

a. PowerPoint

b. Classroom Whiteboard

c. Personal Whiteboards

d. Markers

e. Document Camera

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About the Students in the Class Featured in this Assessment:

1. Grade-level: 3

2. Number of:

a. Students in the class: 24

b. Males: 15, Females: 9

3.

Students with Specific Learning Needs

IEP/504 Plans: Classifications/Needs

Number of Students Supports, Accommodations, Modifications, Pertinent IEP Goals

Reading and Writing 1 Provide oral explanations for directions and simplified text for word problems.

Other Learning Needs Number of Students Supports, Accommodations, Modifications

Underperforming Students 3 Close monitoring w/ CMT

Students Needing Greater Challenges

1 Extensions of in-class tasks via PowerPoint

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Lesson Plans for Learning Segment

Lesson 1:

Launch: 5 Min. How will you start the lesson to engage and motivate students in learning?

I will begin by greeting students, followed by providing them with evidence that positions them as mathematicians. The first slide will provided a written summary of this evidence for students to follow along.

Instruction: 30 min. What will you do to engage students in developing understanding of the lesson objectives? How will you link the new content (skills and concepts) to students’ prior academic learning and their personal/cultural and community assets? What will you say and do? What questions will you ask? How will you engage students to help them understand the concepts? What will students do? How will you determine if students are meeting the intended learning objectives?

I will begin by introducing students to data. Students will be tasked to collect data by tallying the class’s favorite colors on a tally chart. Students will then plot this data onto a picture graph. Students will think-pair-share to brainstorm ways data can be collected in an organized way before the activity begins. Students will then record data by discovering the favorite colors of their classmates. Students will include themselves in this tally. Today, you will be exploring another mathematical idea. This idea involves a certain kind of information called data. Some of you may remember this word used before. Data is information that is recorded in order to answer certain questions. How can we keep track of our data in an organized way? Using pictures or a picture graph, we will graph the data we collected. Find the key, which tells you the value of a unit, on each picture

graph. What is different about the keys on these two picture graphs? Students will engage understanding of concepts by way of exploring their classmates’ favorite colors. Students will ask, “What is your favorite color?” Students will then explore plotting this data onto a picture graph. Students will tally their classmate’s favorite colors. Students will plot their tally charts into a picture graph. Students will fill out corresponding questions located on their tally chart and picture graph worksheets.

Structured Practice and Application: 40 min. How will you give students the opportunity to practice so you can provide feedback?

Students will complete questions corresponding to the tally chart and picture graph activities. Their progress will be monitored during this session. Misconceptions will be addressed as they

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How will students apply what they have learned? How will you determine if students are meeting the intended learning objectives?

occur. Worksheets will be graded promptly and returned to students. Students will complete a second picture graph, replicating the first one, using a key that represents 2 units rather than 1 unit. Students will be monitored for misconceptions and learning opportunities through the application practice. Worksheets will be graded promptly. Exit tickets will provide additional insights into additional questions or misconceptions students might have regarding this lesson’s activities.

Closure: 10 min. How will you end the lesson?

This lesson will conclude with a lesson debrief, reviewing what the students had learned, discovered, or explored. Students will then have an opportunity to write what they had explored onto a 3x5 index card. Students will also write any corresponding questions they might have regarding the lesson onto this index card.

Differentiation/Planned Support How will you provide students access to learning based on individual and group needs? How will you support students with gaps in their prior knowledge that is necessary to be successful in this lesson?

Students with reading and writing IEPs will be provided access to learning by way of having verbal explanations for directions provided to them. Underperforming students will be closely monitored by the CMT and lesson facilitator. Students needing greater academic challenges will be prompted to tackle extensions of the items provided through the day’s lesson. Students will be supported with gaps in prior knowledge by first exploring tally charts, which is review from previous grades. They will also be supported by physically moving to engage with other students in order to complete their tally chart. Task instructions, resources, and review of prior concepts will also be provided via the projector and whiteboard.

Lesson 2:


I will begin this lesson by reviewing what was explored previously. Questions and misconceptions posed by students via exit tickets in the previous lesson will also be explored with supplemental examples.

Instruction: 30 min. What will you do to engage students in developing understanding of the lesson

Students will explore tape diagrams. Students will be tasked to write a number sentence that corresponds with the example provided in the PowerPoint slide. Students will then explore an

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objectives? How will you link the new content (skills and concepts) to students’ prior academic learning and their personal/cultural and community assets? What will you say and do? What questions will you ask? How will you engage students to help them understand the concepts? What will students do? How will you determine if students are meeting the intended learning objectives?

application word problem that utilizes tape diagrams. Students will be tasked to model tape diagrams using the data provided on the next slide. Students will think-pair-share to compare similarities and differences between the tape diagram and the second picture graph explored the previous lesson. Students will then flip their personal whiteboards so that the tape diagrams are now vertical.

Write a multiplication sentence that represents the total value of the tape diagram. What is the total value of the tape diagram? Reisha played in three basketball games. She scored 12 points in Game 1, 8 points in Game 2, and 16 points in Game 3. Each basket she made was worth 2 points. She uses tape diagrams with a unit size of 2 to represent the points she scored in each game. How many total units of 2 will it take to represent the points she scored in all three games? How are these vertical tape diagrams similar to the picture graphs you made yesterday? Now, suppose each unit on this graph has a value of 4 points instead of 2 points. How many units will I draw to represent Reisha’s points in Game 1? How do you know? Remember to look at the original data we collected and recorded in order to answer this question. Students will use personal whiteboards to explore tape diagrams. Volunteers will share and present their thinking to their classmates. Students will also utilize think-pair-share in order to provide additional access points. Students will model vertical tape diagrams based off of the data provided through the slide show. Students will be monitored while they practice tape diagrams via personal whiteboards for misconceptions. Students will then continue questions that expand on the concept of tape diagrams via worksheet in the structured practice and application segment.

Structured Practice and Application: 40 min. How will you give students the opportunity to practice so you can provide feedback? How will students apply what they have learned?

Students will complete questions corresponding to the tape diagram activities. Their progress will be monitored during this session. Misconceptions will be addressed as they occur. Worksheets will be graded promptly and returned to students. Students will design their own tape diagrams. They will model alignment and measurement scale.

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How will you determine if students are meeting the intended learning objectives?

Students will be monitored for misconceptions and learning opportunities throughout this application practice. Worksheets will be graded promptly. Exit tickets will provide additional insights into additional questions or misconceptions students might have regarding this lesson’s activities.





Lesson 3:



Instruction: 30 min. What will you do to engage students in developing understanding of the lesson objectives?

Students will explore an application problem involving the number of fish inside of 5 tanks at Sal’s Pet Store. The vertical scaled bar graph will be drawn on the whiteboard and Tank B’s information will be missing. Students will discover the total number of fish in one of the tanks, they will then draw a tape diagram to represent 30 fish for Tank B, and then they will calculate how many more finish are in Tank B than in Tank A and D combined. After this, students will be tasked to replicate

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How will you link the new content (skills and concepts) to students’ prior academic learning and their personal/cultural and community assets? What will you say and do? What questions will you ask? How will you engage students to help them understand the concepts? What will students do? How will you determine if students are meeting the intended learning objectives?

this graph on their worksheets before transforming it into a scaled bar graph. Students will be provided time to accomplish this task on their own before convening as a whole group. This lesson builds off of the conceptual understanding provided in the previous lesson by tasking students to model tape diagrams. Students will erase the units of value inside of the tape diagrams and construct a scale bar graph to indicate the number of fish inside of the fish tanks. What are you noticing about this scale and what we have explored so far about graphs? Why is alignment important? What do the numbers on the scale tell us? What do the labels tell us? What’s a good title for this graph? How is this scaled graph different from the vertical tape diagram on the whiteboard? Students will use personal whiteboards to model the missing information for Tank B via tape diagram. Volunteers will share and present their thinking to their classmates. Students will write number sentences associated with the data provided on the whiteboard and constructed on their personal whiteboards. Students will also utilize think-pair-share in order to provide additional access points. Students will practice vertical tape diagrams and build a conceptual connection between this application and scaled bar graphs. They will then construct a horizontal scaled bar graph using the same information. Students will think-pair-share similarities and differences between scaled bar graphs, tape diagrams and picture graphs. Students will be monitored while they practice tape diagrams via personal whiteboards for misconceptions. Students will then explore vertical scaled bar graphs via worksheet as a whole group and continue to explore scaled bar graphs as an independent task via template 2 in the structured practice and application segment.

Structured Practice and Application: 40 min. How will you give students the opportunity to practice so you can provide feedback? How will students apply what they have learned?

Students will complete template 2, which will be a horizontal version of the scaled bar graph modeled during earlier instruction. Their progress will be monitored during this session. Misconceptions will be addressed as they occur. Worksheets will be graded promptly and returned to students. Students will complete a horizontal scaled bar graph. They will model alignment and measurement scale.

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How will you determine if students are meeting the intended learning objectives?

Students will be monitored for misconceptions and learning opportunities throughout this application practice. Worksheets will be graded promptly. Exit tickets will provide additional insights into additional questions or misconceptions students might have regarding this lesson’s activities.





Lesson 4:



Instruction: 30 min. What will you do to engage students in developing understanding of the lesson objectives? How will you link the new content (skills and concepts)

Students will preview a scaled bar graph that displays a total number of students who prefer different assortments of ice cream. Students will engage in a whole group discussion and answer questions that correspond with the task. Students will recognize the scaled bar graph from the previous lesson. They will also recognize the types of ice cream flavors

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to students’ prior academic learning and their personal/cultural and community assets? What will you say and do? What questions will you ask? How will you engage students to help them understand the concepts? What will students do? How will you determine if students are meeting the intended learning objectives?

listed in the application problem. Take a moment to read the information on the graph. What do you see? Let’s read the first question. Where should I begin? What is my first step? Where do I plot my points? What is an equation that shows the total number of students who voted for butter pecan, vanilla, and chocolate? Students will be engaged in think-pair-share to answer the questions corresponding with the bar graph. Students will also refer to the whiteboard for an example and replica of the scaled bar graphs explored in the previous lesson regarding Sal’s Pet Store. Students will think-pair-share and respond to questions associated with the application problem. Student responses will be monitored during this task. Misconceptions will be addressed as learning opportunities as they occur.

Structured Practice and Application: 40 min. How will you give students the opportunity to practice so you can provide feedback? How will students apply what they have learned? How will you determine if students are meeting the intended learning objectives?

Students will complete questions #1-3, modeling vertical and horizontal scaled bar graphs. Their progress will be monitored during this session. Misconceptions will be addressed as they occur. Worksheets will be graded promptly and returned to students. Students will construct a vertical and horizontal scaled bar graph. They will model alignment and measurement scale. Students will be monitored for misconceptions and learning opportunities throughout this application practice. Worksheets will be graded promptly. Exit tickets will provide additional insights into additional questions or misconceptions students might have regarding this lesson’s activities.



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Lesson 5:



Instruction: 30 min. What will you do to engage students in developing understanding of the lesson objectives? How will you link the new content (skills and concepts) to students’ prior academic learning and their personal/cultural and community assets? What will you say and do? What questions will you ask?

Students will review previously explored concepts. They will begin by recognizing line plots from their previous school year. Next, they will review a bar graph showing the number of minutes four students spent practicing piano. Students will then delve into constructing bar graphs using a scale of their own choosing. Students begin by reviewing line plots, which is a concept learned from previous school years. Then they will review scaled bar graphs, similar to what they have explored in the previous two lessons. Today, we are going to continue to work with scaled bar graphs, but before we do that, we are going to review another type of data that you all might be familiar with from last year, which is a line plot. Look to the screen. Does this line plot look familiar? This bar graph shows how many minutes 4 children spent practicing piano. Did Ryan practice for more or less than 30 minutes? More or less than 40 minutes? What is the fraction of

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How will you engage students to help them understand the concepts? What will students do? How will you determine if students are meeting the intended learning objectives?

the time between 30 and 40 did Ryan practice piano? How many minutes is that then? How long did Kari spend practicing piano? Who practiced the longest? Who practiced the least about of time? Write a number sentence that shows how much longer Brian practiced than Kari. Write a number sentence that shows how many total minutes Kari and Liz spent practicing piano. Students will be asked to think-pair-share similarities and differences between the line plot, scaled bar graph and the scaled bar graph scaled to the 100s. Students will write number sentences associated with the line plot, scaled bar graph, and missing units for the final scaled bar graph. Students will plot data on scaled bar graphs. Students will be monitored while they design number sentences for line plots and scaled bar graphs via personal whiteboards for misconceptions and learning opportunities. Students will then explore scaled bar graphs via worksheet using a scale that has a numerical value (100s) than explored previously.

Structured Practice and Application: 40 min. How will you give students the opportunity to practice so you can provide feedback? How will students apply what they have learned? How will you determine if students are meeting the intended learning objectives?

Students will complete question #1 on their worksheets, involving a vertical scaled bar graph. Their progress will be monitored during this session. Misconceptions will be addressed as they occur. Worksheets will be graded promptly and returned to students. Students will complete a horizontal scaled bar graph. They will model alignment and measurement scale. They will also choose the numerical value of their associated scales. Students will be monitored for misconceptions and learning opportunities throughout this application practice. Worksheets will be graded promptly. Exit tickets will provide additional insights into additional questions or misconceptions students might have regarding this lesson’s activities.


This lesson will conclude with a lesson debrief, reviewing what the students had learned, discovered, or explored. Students will then complete a post-assessment worksheet.

Differentiation/Planned Support How will you provide students access to learning

Students with reading and writing IEPs will be provided access to learning by way of having verbal explanations for directions

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based on individual and group needs? How will you support students with gaps in their prior knowledge that is necessary to be successful in this lesson?

provided to them. Underperforming students will be closely monitored by the CMT and lesson facilitator. Students needing greater academic challenges will be prompted to tackle extensions of the items provided through the day’s lesson. Students will be supported with gaps in prior knowledge by first exploring tally charts, which is review from previous grades. They will also be supported by physically moving to engage with other students in order to complete their tally chart. Task instructions, resources, and review of prior concepts will also be provided via the projector and whiteboard.

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Instructional Materials

Lesson 1:

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Lesson 2:

19

Lesson 3:

21

Lesson 4:

24

Lesson 5:

27

Assessments:

Pre-assessment:

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Post-assessment:

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Planning Commentary:

1. Central Focus

a. Describe the central focus and purpose for the content you will teach in the

learning segment: Students will draw on prior knowledge from grade 2 to

generate categorical data from community-building activities. In lesson 1, they

organize data and represent it in a variety of ways, including tally marks, graphics

with one-to-one correspondence, and tables. In lesson 2, students rotate tape

diagrams vertically to help transition students toward creating scaled bar graphs

in lesson 3. This lesson adds complexity to grade 2 materials of picture graphs

by recognizing that a picture or unit on a bar can represent more than single unit.

In lesson 3, students construct the scale on the vertical axis of a bar graph.

Finally, lesson 4 provides students with the opportunity to analyze graphs and to

solve more sophisticated one- and two-step problems.

b. Given the central focus, describe how the standards and learning targets within

your learning segment address conceptual understanding; procedural fluency;

and mathematical reasoning or problem-solving skills: The grade level standard

this central focus covers is 3.MD.3: Draw a scaled picture graph and a scaled bar

graph to represent a data set with several categories. Solve one- and two-step

“how many more” and “how many less” problems using information presented in

scaled bar graphs.

i. This focus makes use of conceptual understanding providing students

with opportunities to represent data sets and the alignment of scaled

measurements to compare data efficiently.

ii. This focus makes use of procedural fluency by orienting students to keys

and scales that represent more than one unit.

iii. Students will be able to use the lessons learned from this lesson to

answer multi-step word problem questions.

c. Explain how your plans build on each other to help students make connections

between facts; concepts; computation/procedures; and mathematical reasoning

or problem-solving strategies to deepen their learning of mathematics:

i. Facts: Students will engage in activities that represent real-world and

culturally relevant data. Students begin by tallying their favorite colors.

They then record this into picture graphs. Students then use tape

diagrams to represent the points made during a basketball game. Then

they will use tape diagrams to show the number of fish in fish tanks at

Sal’s Pet Store before transforming them into scaled bar graphs. Finally,

students tackle multi-step problems to calculate and plot the number of

wing vibrations different insects produce.

ii. Concepts: Students will begin by drawing on prior knowledge regarding

single unit representations. They will then expand single unit

representations into multiple value units, including values of 2s, 5s, 10s

and 100s.

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iii. Computations/Procedures: Students will draw on prior knowledge

regarding multiplication to write number sentences that represent the

information provided via different graphs.

iv. Mathematical reasoning/Problem-solving strategies: Students will use

problem-solving strategies to answer the number of wing vibrations for

various insects.

d. How and when will you give students opportunities to express their

understanding of the learning targets and why they are important to learn? I will

give students opportunities to express their understanding of learning targets and

why they are important to learn most notably through their exit tickets, but also

through classroom discussion periods.

2. Knowledge of Students to Inform Teaching

a. Prior academic learning and prerequisite skills related to the central focus:

Students have used strategies to solve for addition, subtraction, multiplication,

division and for identifying fractions. All students have shown proficiency to

mastery level understanding of multiplication and division concepts as indicated

through sprint sessions.

b. Personal/cultural/community assets related to central focus: Some of the material

these lesson cover are review from grade 2 academia. The lessons provide

personal context by connection them with tangible subjects like tallying favorite

color, the number of pets in a pet store, and displaying the number of points

scored during basketball. All students are able to identify a favorite color. Many

students are interested in sports, while others are more interested in pets.

c. Mathematical dispositions related to the central focus:

i. Most students perceive mathematics as “sensible, useful, and

worthwhile,” but there are three students who have indicated a strong

contempt for the subject.

ii. Most students have shown persistence in applying mathematics to solve

problems. This is evident by their insistence to complete their

assessments and to correct mistakes and tackle misconceptions.

iii. There are two students who frequently indicate a deficient mindset

concerning their ability to learn mathematics. Most students believe they

have the ability to learn mathematics.

3. Supporting Students’ Mathematics Learning

a. Justify how your understanding of your students; prior academic learning and

personal/cultural/community assets (from prompts 2a-b above) guided your

choice or adaptation of learning tasks and materials. Be explicit about the

connections between the learning tasks and students’ prior academic learning,

assets, mathematical dispositions, and research/theory. Richard Milner, Lisa

Delpit, and Maxine Greene talk about how it is essential to provide students with

culturally relevant material in order to invoke learning. Basketball, pets, and

colors are interests that all students share in one capacity or another, making

them ideal for bringing personal relevance into the classroom. In regards to prior

academic learning, Van de Walle delves into the importance of making

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connections to mathematics visible. Addition, subtraction, multiplication, division

and fractions are a few of the mathematical concepts students have explored

throughout the school year and each of these concepts are presented throughout

the five lessons. Students are expected to count by twos and fives, to subtract

fish from a fish tank and indicate the fraction that is represented halfway between

a scaled unit between the values of 30 and 40.

b. Describe and justify why your instructional strategies and planned supports are

appropriate for the whole class, individuals, and/or groups of students with

specific learning needs. Van de Walle and CGI illustrate how learning occurs in

a social context, such that any knowledge that is shared in classroom discussion

belongs to everyone in attendance. Throughout the lessons, students are also

expected to engage in individual tasks after exploring conceptual ideas like

scaled bar graphs and the wing vibrations of insects. The classroom-wide

engagement acts as scaffolding for students to delve into the material on their

own. Additionally, students are brought back together for debriefing sessions, to

review what was explored during the day’s lesson, which is another component

recommended by CGI. Students needing greater academic challenges have

learning extensions that explore different layers of questions provided to them via

the worksheet, by asking them to design word problems or number sentences

based off the data provided in graphs. This extension of learning is

recommended by Van de Walle to ensure that students continue to discovering

new learning opportunities and to expand their thinking. Additionally, these

extensions are provided for all students who wish to pursue and investigate them

further.

c. How will students identify resources to support their progress toward the learning

targets? Students will identify resources to support their progress through think-

pair-share sessions, facilitator questions and guidance, self-exploration, and the

utilization of previous lessons replicated on the whiteboard and/or PowerPoint

slides. Milner, Greene, CGI, and Van de Walle recommend having many access

points for students to delve into the exploration of learning, which these different

items make accommodations for.

d. Describe common mathematical preconceptions, errors, or misunderstandings

within your central focus and how you will address: Common misconceptions

may include the rationale behind having so many different types of graphs or

data allotments for the same thing. I would choose to address this

misconceptions by pushing student reasoning to investigate what types of graphs

might be more appropriate and where. This conceptual understanding of graphs

is a learning standard that students will be expected to comprehend in grade 4.

4. Supporting Mathematics Development Through Language:

a. Language Function – Categorize: One of the language functions that students

are expected to understand as they explore the concepts of graphs is the idea of

categorization. Every graph features lists of categories and students are

expected to group similar ideas, like different Tanks versus that of the number of

Fish and label them correctly on their respective graphs. Moreover, this example

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illustrates how one category refers to numerical scale while the other refers to

something non-numerical.

b. Identify a key learning task from your plans that provides students with

opportunities to practice using the language function identified above. Identify

the lesson in which the learning task occurs. This learning task of categorization

occurs in lesson 3, where students explore and plot data concerning the number

of fish within five fish tanks at Sal’s Pet Store. The number of fish is one

category while Tanks A through E are another. Together, these two categories

represent a holistic category that can be labeled as the Number of Fish in Sal’s

Pet Store.

c. Additional Language Demands:

i. Vocabulary/Symbols: Students must know the definition of the following

words, which include key; enrolled; vertical; horizontal; scale; alignment;

unit; value

ii. Discourse: Students must know how to communicate effectively to peers

for think-pair-share and to communicate understanding of material for the

facilitator to follow. In some events, it may be necessary for the facilitator

or peers to restate what was communicated in order for other students to

build understanding.

d. Language supports: Describe the instructional supports that help students

understand and successfully use the language function and additional language

demands identified in prompts 4a-c.

i. Categorizing will be explored via whole group discussions for deciding

where to label the scale and measured item on bar graphs. Students will

have opportunities to explore this individually. Students with reading and

writing needs will also have instructions read to them and additional

explanations of the expected task provided if necessary.

ii. Vocabulary will be reviewed during whole group discussions, including

the launch and debrief sessions. Vocabulary words will also be included

on the whiteboard or PowerPoint to provide students with additional entry

points.

iii. Discourse will be interlaced throughout the lessons, such that students

are provided with additional points of reference and to break the

monotony of long sessions.

5. Monitoring Student Learning:

a. Describe how your planned formal and informal assessments will provide direct

evidence for you and your students to monitor their conceptual understanding,

computational/procedural fluency, and mathematical reasoning or problem-

solving skills throughout the learning segment.

i. Informal: During whole group activities, which include the launch, debrief,

and instructional time, students will have opportunities to volunteer their

understanding of concepts. Additionally, there will be small tasks asking

students to record their thinking regarding the writing of number

sentences, the drawing of tape diagrams and scaled bar graphs, their

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justifications and/or strategies for solving multi-step word problems

involving the wing vibrations of insects. Independent tasks and exit

tickets will also act as informal assessments to gauge student

understanding concluding every lesson. These informal assessments will

help determine the direction of instruction the following day should include

in addition to the expected planning.

ii. Formal: Concluding this one-week of lessons, students will complete a

post-assessment worksheet that closely aligns with material first

introduced to them via the pre-assessment. It will include additional

information that is relevant to what was learned throughout the week,

such as that of defining data, the importance of graphs, and constructing

a scaled bar graph.

b. Explain how the design or adaptation of your planned assessments allows

students with specific needs to demonstrate their learning.

i. Any questions that require reading or writing will be provided in words that

are developmentally appropriate for the student(s). Additionally, the

formal assessment will allow for the option of allowing students to answer

the question using pictures if necessary. Furthermore, the final question

on the formal assessment requires the student to have conceptual

understanding of scaled bar graphs, such that reading and writing are

less demanding, allowing for the student to display their conceptual

understanding of scale, alignment and graphs using the allocation of data

appropriately.

c. Describe when and where you will elicit student voice (oral and written) during

instruction to raise awareness in both you and the students of where students are

relative to the learning targets.

i. I will be utilizing student voice during launch, instruction and debrief

sessions, such that students are provided opportunities to ask questions,

share their understanding, and to allow for students to learn from more

capable peers. Written feedback will also be provided in the context of

their in-class assessments upon their completion. Feedback will also be

provided via their exit tickets during the launch session of every lesson.

d. What tools and strategies will students use to monitor their own learning process

during the learning segment?

i. Students will monitor their own learning process by the completion of

independent and whole-class tasks, in addition to orienting themselves to

more capable peers during think-pair-share and the lessons’ debrief.

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Instruction Commentary:

1. Which lesson or lessons are shown in the video clip?

a. Lesson 1

2. Promoting a Positive Learning Environment:

a. How did you demonstrate mutual respect for, rapport with, and responsiveness to

students with varied needs and backgrounds, and challenge students to engage

in learning? I began the lesson by orienting students to previous lessons.

Because it had been some time since I had facilitated a lesson with them, I felt it

was necessary to remind students about what we worked on last time and why I

am facilitating a lesson for them. I then launched into examples of how I have

seen every student in the classroom utilize various strategies involving

mathematics, such that I position students as being mathematicians in this

classroom setting. This recognition is essential because there are many

students who do not feel confident in their abilities for learning and understanding

mathematical concepts. During this instructional period, I allowed for students to

provide commentary and build dialogue to express what they currently

understand about the concept of data. I also took the opportunity of orienting

students to examples of data that are not mathematical in nature, such as

observations made by students through the physics of sound unit. Doing these

acts demonstrated mutual respect for, rapport with, and responsiveness to

students.

3. Engaging Students in Learning:

a. Describe your strategies to elicit student expression of their understanding of the

learning target and why they are important: The strategies that I utilized to elicit

student expression included orienting them to aspects of prior knowledge, think-

pair-share, and a whole group activity involving the physical collecting of data.

b. Explain how this instruction engaged students in developing understanding of

mathematical concepts: In this instruction, students were tasked to explore

different features of collecting data and organizing that data efficiently. Students

relied on prior knowledge about simple graphs or graphic organizers for

distributing data effectively. The whole group activity of collecting and recording

data is the first step in orienting students to recording data symbolically, such that

the symbols or keys represent more than a single unit of numerical value.

c. Describe how your instruction linked students’ prior academic learning and

personal, cultural, and community assets with new learning: Throughout the year,

students have used various types of graphic organizers to represent and

communicate data. Students’ prior knowledge of grouping and multiplication

provided them with conceptual understanding of grouping tally marks into

bundles of fives. Students’ prior knowledge of tally charts, along with their

personal interest in sharing their favorite color, is utilized to allow students entry

into learning about symbolic representation with a numerical value greater than

one unit, preparing students for learning about scaled bar graphs in the following

lessons.

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4. Deepening Student Learning during Instruction:

a. Explain how you elicited and build on student responses to promote thinking and

develop understandings of mathematical concepts: During the instruction, we

reviewed the concept of tally marks. The students engaged in a think-pair-share

to describe why it might be useful to use the fifth tally to cross out the previous 4

tally marks in a sequence. I asked for one volunteer to share their reasoning,

which she described that it allows the individual to count easily by fives. I

responded by confirming her response, acknowledging that counting by fives is

easier than counting individual tally marks.

b. Explain how you and the students used representations to support students’

understanding and use of mathematical concepts: Students engaged in an

activity to tally their classmates’ favorite colors using the tally marks described

previously. This activity followed immediately after the think-pair-share session

described in 4a. This activity supports students’ understanding of and use of

mathematical concepts by providing context for students to follow that correlates

with the think-pair-share session.

5. Analyzing Teaching:

a. What changes would you make to your instruction- for the whole class and/or for

students who need greater support or challenge – to better support student

learning of the central focus? During this lesson, I tasked students to think-pair-

share effectively strategies for collecting and organizing data. Upon review of the

video, I discovered that some students where simply repeating the word graphs

for the duration of the 30 seconds provided to them. One change that I would

utilize here after the think-pair-share is to not only ask students to communicate

what they and their partners discussed, but to illustrate physical samples of their

ideas onto the whiteboard beside me. This was a missed opportunity that could

have provided students with additional entry points into the learning target

concerning graphs and measurement.

b. Why do you think these changes would improve student learning? Support your

explanation with evidence of student learning and principle from theory and/or

research: Having many entry points into the learning target of any lesson is an

important factor to consider because knowledge gaps concerning the use of

graphic organizers or graphs may exist amongst several students. Additionally,

some students may not recognize that their ideas of organization are legitimate

strategies for engaging in mathematical concepts. Van de Walle, CGI, and

Richard Milner make notable references about how additional entry points for any

lesson are essential for learning growth.

36

Student Work Samples:

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Assessment Commentary:

1. Analyzing Student Learning:

a. Identify the specific learning targets and standards measured by the assessment

you chose for analysis:

i. Learning Targets: Students will be able to draw a scaled bar graph to

represent a data set with several categories (3.MD.3). Students will be

able to attend to precision by labeling axes on graphs to clarify the

relationship between quantities and units and attend to the scale on the

graph to precisely interpret the quantities involved (MP.6).

b. Provide a graphic or narrative that summarizes student learning for your whole

class:

Name Pre-assessment Post-assessment Growth

Student 1 Partial Met Yes

Student 2 Partial Partial No

Student 3 Partial Not Met No





Student 8 Not Met Met Yes

Student 9 N/A Met N/A

Student 10 Not Met Partial Yes

Student 11 Not Met Met Yes






Student 17 Not Met Not Met No




Student 21 Met Met No

Student 22 Not Met N/A N/A


Student 24 Met Met No

c. Provide a graph or narrative that summarizes student understanding of their own

learning progress:

i. After every lesson, students engaged in whole group debrief to review

what was learned or explored throughout the day. Following this debrief,

students summarized and highlighted some of the key components of

what they learned or explored throughout the day’s instruction through an

exit ticket on a 3x5 index card, allowing for students who were unable to

share their knowledge to indicate what they had learned. Students also

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took the opportunity to submit questions or reveal misconceptions during

this opportunity.

d. Use evidence found in the 3 student work samples, student self-reflections, and

the whole class summary to analyze the patterns of learning for the whole class

and differences for groups of individual learners relative to:

i. Conceptual Understanding: The student work samples provided

evidence of a varied amount of conceptual understanding. Most students

illustrated a strong understanding of scale, similar to student 1’s sample,

but were often missing the labeling of categories. A handful of students

displayed partial understanding of creating a scaled bar graph, similar to

student 2’s sample, which the student chose to use the table to draw a

scale with a one-to-one unit correspondence. Two students had

incomplete assessments, such that conceptual understanding cannot be

assessed. One of the two who had incomplete post-assessments

students had indicated partial conceptual understanding regarding the

usefulness of data in their pre-assessment, but was unable to complete

the post-assessment.

ii. Procedural Fluency: Neither pre- or post-assessment gauged procedural

fluency, however, all students were tasked to write corresponding number

sentences involving data sets provided during instruction via their

personal whiteboards. All students indicated procedural fluency by

answering question #1b from lesson 4’s independent task/materials,

which asked for students to write a number sentence tallying the total

number of students who are enrolled in four different classes.

iii. Mathematical Reasoning or Problem-Solving Skills: The post-assessment

had an opportunity for students to elicit understanding of mathematical

reasoning by way of indicating the usefulness of data and graphs, which

could be explained by using words or pictures. Students indicated

mathematical reasoning and problem-solving skills by calculating the total

number of wing vibrations of various insects during lesson 4. As a whole

group, students were able to solve for the missing variable of the first part

of the multi-step problem in order to solve for the missing variable in the

second part.

2. Feedback to Guide Further Learning:

a. In what form did you submit your evidence of feedback for the 3 focus students?

i. Written directly on work samples

b. Describe what you did to help each student understand his/her performance on

the assessment.

i. Student 1 met the criteria of conceptual understanding for the post-

assessment. I acknowledged their use of scale and labeling of

categorizations. I also highlighted an opportunity for learning regarding

the inclusion of a title for the graph itself.

ii. Student 2 partially met the criteria of conceptual understanding for the

post-assessment. In their work, they tied data to graphs when defining

41

data, but they did not indicate why data itself is useful. I responded to this

by asking, “How? What does the data tell you?” as a means to orient the

student to think deeper about the significance of data. In his second

response regarding graphs and their usefulness, he indicated that graphs

are used for tracking. In my response, I pushed for deeper understanding

by asking what they had meant regarding tracking and to expand on that

idea further. Finally, the student drew a bar graph that showed a one-to-

one unit scale correspondence to the data provided. I referenced that his

use of expanding beyond the scale of the graph was creative and that I

am wondering how they might choose to draw a graph that would stay

within the boundaries provided.

iii. Student 3 was one of two students who did not complete the task. He

partially responded to the first question by indicating that data is

information, but they did not indicate its usefulness. The remaining

questions were left unanswered. In my response, I recognized that this

student might require additional time to complete the task asked of him

and that he had managed a good start so far. I supplemented this

response by asking him to give an example of what data might be used

for and how might he illustrate data using the table provided on the post-

assessment.

c. Explain how feedback provided to the 3 focus students addresses their individual

strengths and needs relative to the learning targets measured:

i. Student 1 is a high performer in academic tasks. This student likely

recognizes their status as a student. Although the inclusion of a title for

the graph was not part of the grading criteria, the recommendation of

including a relevant title for their graph is an indication that there is more

to learn.

ii. Student 2 is a mid-level performer in academic tasks. This student likely

recognizes that more scaffolding is necessary in order to learning to occur

depending on the context. My feedback for this student provides them

with probing questions that push their thinking beyond that of knowledge-

level thinking and that there is an expectation of justifying why they chose

to write what they had.

iii. Student 3 is a struggling student in academic tasks. This student has a

tendency to require additional time in order to comprehend what is asked

of them in order to complete any given task. The partial response to the

first question indicates that the student has some understanding of the

material being presented to them, but it also indicates that they may be

unsure about their response. My feedback positions the student

competently in this regard by providing him with recognition that what

they have completed so far is a good start. The additional feedback is

also useful because it will orient the student to complete the scaled bar

graph using the data table provided, which is the primary variable to

gauge students’ understanding.

42

d. I will continue to utilize extension questions to further students’ conceptual

understanding and procedural fluency of mathematical concepts. In the following

lessons, students will have more opportunities to construct scaled graphs and

working with scale by constructing rulers.

3. Evidence of Language Understanding and Use:

a. Explain and provide evidence for the extent to which your students were able to

use or struggled to use language to develop content understanding:

i. Students 2 and 3, as indicated by their sample works, struggled to define

data and graphs, along with their respective usefulness.

4. Using Assessment to Inform Instruction

a. Based on your analysis of student learning presented in prompts 1b-d, describe

next steps for instruction to impact student learning:

i. For the whole class: My next step for instruction to impact student

learning is to provide additional replicas of previous lessons for students

to draw from during the launch phase of every lesson. Additional

misconceptions will be addressed, such as the attention to scale beyond

that of a one-to-one correspondence by providing replicas of past lessons

involving picture graphs and scaled bar graphs. Vocabulary words will

also be provided on the whiteboard, along with their descriptions. Lesson

extensions will also ask students to provide examples of how a particular

vocabulary word might be utilized given certain parameters like the data

from a table.

ii. For the 3 focus students and other individuals/groups with specific needs:

Additional extensions will be provided to enhance the learning of all three

students and will appropriately complement the lesson tasks that are

asked of them. Procedural fluency and mathematical reasoning and

problem-solving will be featured alongside the conceptual understanding

of drawing graphs and measurements to scale. Focusing questions will

be provided to students that are able to provide surface deep information

regarding their material, like student 2, such that conceptual

understanding is elicited. Additional assistance in the way of articulating

the parameters of the provided questions or task will be made available

for students like Student 3 who otherwise require additional time in order

to reach comprehension.

b. Explain how these next steps follow from your analysis of student learning and

student self-reflections. Support your explanation with principles from research

and/or theory.

i. Van de Walle and Richard Milner describe that the best learning

environment for students who struggle is to provide them with high

expectations and access to equitable learning objectives, such that all

students are learning the same grade-level material. The reason for the

use of learning extensions and focusing questions is to provide all

students, include gifted students and mid-level students, like students 1

and 2, respectively, access to measures that will enhance their

43

conceptual understanding of the material. Students like that of student 3

require differed articulation of expectations in order for the task’s

objectives to be completed, ensuring that the individual is able to

comprehend the parameters of the task.

44

Evaluation Criteria:

1) What is data? Why is it useful?

Data is information. It is useful because it provides information that can be measured or

observed. (Student answers may vary.)

2) What are graphs? Why are they useful?

Graphs are a type of graphic organizer that is used to organize and articulate data. It is

useful because it provides information that be measured or observed in a medium that is easy

to read. (Student answers may vary.)

3) Draw a scaled bar graph using the table below:

Students’ Favorite Colors

Green Yellow Red Blue

8 4 12 14

16 14 12 10

8 6 4 2 0

Students’ Favorite Colors

Green Yellow Red Blue Colors

Nu

mb

er o

f St

ud

ents

Works Cited

Carpenter, Thomas P., ed. Children’s Mathematics: Cognitively Guided Instruction. Portsmouth, NH:

Heinemann, 1999. Greene, Maxine. Teacher as Stranger; Educational Philosophy for the Modern Age. Belmont, Calif:

Wadsworth Pub. Co, 1973. Milner IV, H. Richard. Start Where You Are, But Don’t Stay There. Cambridge, Massachusetts:

Harvard Education Press, 2010. Van de Walle, John A., Karen Karp, and Jennifer M. Bay-Williams. Elementary and Middle School

Mathematics: Teaching Developmentally. 8th ed. Boston: Pearson, 2013.

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