Rigor Breakdown:
Conceptual Understanding
Grades 3–5
Sequence of Sessions
Overarching Objectives of this February 2013 Network Team Institute
· Participants will explore GK—M5 and G3—M5 and be prepared to train others to teach these modules.
· Participants will examine the K–5 progressions documents and the sequence of standards foundational to developing an understanding of fractions and be prepared to train others to enact cross-grade coherence of fraction development in the classroom.
· Participants will deepen their study of rigor by examining its relationship to coherence using examples from GK—M5 and G3—M5 and be prepared to train others to promote coherent and balanced instruction.
· Participants will explore a natural development of RDW and tape diagraming skills and be prepared to build proficiency in students and teachers new to the process.
High-level Purpose of this Session
· Examine the conceptual understanding component of rigor in G3—M5.
· Explore the conceptual understanding component of rigor through selected content of grades 4 and 5.
· Explore how cross-grade coherence is accessible through the conceptual understanding component of rigor.
· Recognize opportunities to emphasize the Standards of Mathematical Practice during activities that promote conceptual understanding.
Related Learning Experiences
· This session is the beginning of a three-part series on the components of rigor, which will provide examples of all three components of rigor in G3—M5 and give guidance to participants for enacting rigor in grades 4 and 5.
· This session, preceded by a session on coherent development of fractions across the grades, also explores the expression of coherence through the conceptual understanding component of rigor.
· Participants will build upon their work in this session during the next two sessions focusing on the other components: procedural skill and fluency, and application.
Key Points
· Conceptual understanding can be promoted in a variety of ways including use of concrete and pictorial models, real-world contexts, conceptual questioning, and writing/speaking about understanding.
· Each of these ways for promoting conceptual understanding can be used to coherently bridge gaps in prerequisite knowledge.
· Content knowledge directed by the standards and the progressions is required to provide coherent and balanced instruction.
· Many activities that promote conceptual understanding, also address the Standards of Mathematical Practice.
Session Outcomes
What do we want participants to be able to do as a result of this session?
· Recognize conceptual understanding activities in G3—M5.
· Select concrete and pictorial materials for use in lessons related to the development of fractions across the grades.
· Compose conceptual questions that promote coherence and conceptual understanding in the classroom.
· Understand and convey to others the importance of content knowledge directed by the standards and progressions in delivering coherent and balanced instruction.
How will we know that they are able to do this?
· Participants will report preparedness for implementing activities that promote conceptual understanding, across the grades, as described in A Story of Units.
· Participants will share their grasp of conceptual understanding implementation and its importance with colleagues, providing examples of how it is implemented in A Story of Units.
Session Overview
Section / Time / Overview / Prepared Resources / Facilitator PreparationOpening / 0:00-0:05
(05 min) / · Link to previous sessions; frame the session, referencing the agenda
· Provide an overview of conceptual understanding as defined in the Shifts
· Review the flow and objectives of this session / · PowerPoint Presentation / · Review session notes and PowerPoint presentation.
Conceptual Understanding – Concrete and Pictorial Models & Real-World Contexts / 0:05-0:45
(40 min) / · Identify concrete and pictorial model use that promotes conceptual understanding in G3—M5.
· Select concrete and pictorial model use to promote understanding in selected topics of grades 4 and 5.
· Consider adaptions to the models that bridge gaps in prerequisite knowledge
· Reflect on ensuring coherence across grades PK– 5. / · Grade 3—Module 5
· Fraction Strips
Conceptual Understanding – Conceptual Questioning & Writing and Speaking About Understanding / 0:45-1:10
(25 min) / · Identify use of conceptual questions (oral and written) that promote conceptual understanding in the modules.
· Select conceptual questions to promote understanding in selected topics of grades 4 and 5.
· Consider additional questions that assess prerequisite knowledge, consolidate prerequisite knowledge, and bridge gaps in prerequisite knowledge. / · Video Clip: Inches and Centimeters
· Grade 3—Module 5 / · Download and review clip.
Closing / 1:10-1:15
(5 min) / · Summarize key points
· Reflect on the role of conceptual understanding in coherent and balanced instruction
TOTAL TIME / 75 min
Session Roadmap
Opening
Time: 0:00-0:05
· Link to previous sessions; frame the session, referencing the agenda.
· Provide an overview of conceptual understanding as defined in the Shifts.
· Review the flow and objectives of this session.
Materials used include:
· Session PowerPoint
(SLIDE 1) We have just completed a thorough walk through of the standards and related progressions documents that lay the foundation and then continue the work of Grade 3—Module 5: Fractions as Numbers on the Number Line. Now we will begin a series on the three components of rigor.
(SLIDE 2) We will start with conceptual understanding, and then continue with fluency and applications.
(SLIDE 3) Those of you attending our last NTI may be wondering how these sessions will differ from the sessions presented in November. Let’s begin by looking at our new objectives.
(SLIDE 4) To address the first objective, we’ll first take a look at examples of conceptual understanding from G3—M5.
Next, we’re really going to hone in on implementing conceptual understanding ourselves. We’ll do this by going through the related content from grades 4 and 5 and strategizing on how to best provide conceptual understanding of these topics.
The third objective is the exploration of how cross-grade coherence is accessible through the conceptual understanding component of rigor.
Specifically we will want to consider how to:
· Follow the learning path outlined in the progressions and delineated by the standards
· And how to use that awareness to make choices that bridge gaps in prerequisite knowledge.
During all this, we will continue to look for meaningful opportunities to incorporate the Standards of Mathematical Practice.
(SLIDE 5) Let’s begin by revisiting the definition of conceptual understanding. Here we have an abbreviated excerpt from the Instructional Shifts. Take 30 seconds to read it and consider how conceptual understanding is evidenced. What does it look like and sound like when this is successfully occurring?
“Teachers teach more than ‘how to get the answer’ and instead support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations as well as writing and speaking about their understanding.”
(SLIDE 6) Now, let’s spend two minutes working with your partner to describe what it looks like / sounds like when conceptual understanding is occurring in the classroom. Share specific examples with your partner. (Select volunteers to share their reflections.)
(SLIDE 7) In the last NTI we explored conceptual understanding in a few ways: using concrete and pictorial models combined with real-world contexts, and using conceptual questioning to get students speaking and writing about understanding. Today we will re-examine these in groups of two.
(SLIDE 8) We’ll start by examining concrete and pictorial models and contextual situations. And then dig in to conceptual questioning to get students writing and speaking about understanding.
(Click to advance animation, revealing the THREE bullets.)
For each of these two we will go through the same three activities: examining examples from the Grade 3 module, selecting strategies for standards in grades 4 and 5, and then considering ways to bridge gaps in prerequisite knowledge.
So let’s get started with a lesson enactment of conceptual understanding using models and real-world contexts from Grade 3—Module 5.
Conceptual Understanding – Concrete and Pictorial Models and Real-World Contexts
Time: 0:05-0:45
· Engage in a lesson enactment of activities from lessons in G3—M5 that promote conceptual understanding using models and/or contexts.
· Select models and/or contexts to promote understanding for topics in fraction development from grade 4 or 5.
· Consider adaptions to models and/or contexts that bridge gaps in prerequisite knowledge.
· Reflect on ensuring coherence across grades PK–5.
Materials used include:
· Grade 3—Module 5
· Fraction Strips
(SLIDE 9) As an example of models and contexts from G3—M5 we will enact the conceptual understanding portion of Lesson 14. As we go through the example, reflect on what mathematical practice(s) you see occurring.
Introducing the number line model.
Measure a Line of Length 1 Whole:
· Draw a horizontal line with your ruler that is a bit longer than 1 of your fraction strips.
· Place a whole fraction strip just above the line you drew.
· Make a small mark on the left end of your strip.
· Label that mark 0 above the line. This is where we start measuring the length of the strip.
· Make a small mark on the right end of your strip.
· Label that mark 1 above the line. If we start at 0, the 1 tells us when we’ve travelled 1 whole length of the strip.
Measure the Unit Fractions:
· Place your fraction strip with halves above the line.
· Make a mark on the number line at the right end of 1 half. This is the length of 1 half of the fraction strip.
· Label that mark 12. Label 0 halves and 2 halves.
· Repeat the process to make measure and make other unit fractions on a number line.
Draw Number Bonds to Correspond to the Number Lines:
· What do both the number bond and number line show?
· Which model shows you how big the unit fraction is in relation to the whole? Explain how.
· How do your number lines help you to make number bonds?
(SLIDE 10) Take 1 minute now to consolidate your thoughts on the reflection question.
“What mathematical practice(s) do you see occurring in this exercise?”
Then you will have time to share your thoughts with a partner at your table.
(Allow 1 minute for silent, independent reflection.)
Now, turn and talk with a partner at your table about your response to this.
(Allow 2 minutes for participants to turn and talk, and then facilitate a discussion. Allow three people to share. You may build on their comments, supplying that,
· The very use of models is preparing students to become proficient at MP.4.
· Reflecting on, ‘which model lends itself to seeing the relative size of the parts’ is preparing students for MP.5.
· The action of making their number line model can be an opportunity for MP.7 and MP.8 as students notice relationships between fractions and their placement in relation to each other and to the whole, they may begin to generalize shortcuts of building a number line with fractional units.
· The act of making their number line model can also be an opportunity to practice MP.6 as students use care in accurately aligning their fraction strips with their number line and in placing the marks for each fraction on the number line.)
(Allow 1 minute for independent reflection.)
You can see from this example how important the role of models are in both creating conceptual understanding of the symbols we use to denote fractional quantities and how models also play a big role in implementing mathematical practices, not only in MP.4, but in setting students up for success with other mathematical practices.
(SLIDE 11) Now we have examined one example of models used in G3—M5.
Let’s now have our hand at selecting models and contexts for use in related standards from grades 4 and 5.
(SLIDE 12) Regardless of what grade you teach, find a partner at your table and agree upon whether you want to work with Grade 4 or Grade 5 standards. Get your resources available and ready.
(SLIDE 13) Take 2 minutes to carefully review the standard you shown here for the grade level you chose. (Allow 2 minutes for participants to study the standard.)
(SLIDE 14) We are going to begin to think about lessons to meet the standard you chose, and specifically how to meet the conceptual understanding component for the lessons.
(CLICK TO ADVANCE 1st BULLET.) The first thing we’ll do is decide on what model or models are most suitable – consider both concrete and/or pictorial. Recall the models suggested by the progressions document for 4th and 5th grade, and the models that students used in 3rd grade’s Module 5. These considerations can be supplemented with concrete materials as well. One commonly helpful concrete context is the Hershey bar which, when unwrapped, very naturally depicts the whole separated into four equal parts. Take 2 minutes to discuss your selection of models with your partner. Refer back to the progressions document as needed. (Allow 2 minutes for partners to discuss.)
(CLICK TO ADVANCE 2nd BULLET.) Next discuss with your partner some appropriate contextual situations to be used either as the concept is introduced or later as an application of the concept. Take 2 minutes to discuss. (Allow for 2 minutes of discussion, and then facilitate a discussion where participants are selected to share their ideas for each of the standards. If participants are struggling, suggest the following.)
4.NF.4 examples:
· Recipe’s that need to be doubled, tripled, quadrupled, etc. to feed large groups. (Nice in that this is real world, students this age might make brownies or cookies for bake sale fundraising.)
· Running a track that is a fraction of a km or mile, if the track is run multiple times, how many total miles did the student run?)
5.NF.3 examples:
· Any of a number of situations where you are sharing or otherwise splitting up x number of something among y people or containers, in cases where x > 1 and x is not evenly divisible by y.
o 3 Pizza’s for 4 people.
o 7 gallons of water into 4 vats.
(CLICK TO ADVANCE 3rd BULLET.) Next, work with your partner to create a short vignette, or script, of how you would first introduce the concept in the standard. Use any remaining time to outline a sequence of how that introduction will need to progress to meet the standard fully.
You’ll have 8 minutes to work with your partner. (Allow for about 8 minutes of work.)
Find another pair who worked on the same standard, either at your table or at a nearby table, and exchange vignettes. Provide a least one comment on how the vignette could be improved. (Allow 3 minutes.)
Who feels that the vignette they reviewed for another team was really well done? (Ask the partners identified to read or present their ideas, clarify which standard/grade level they are presenting.)
(SLIDE 15) Next on our agenda is using models and contexts to address gaps in prerequisite knowledge.
(SLIDE 16) Look again at your lessons and consider the prerequisite knowledge required for success with this standard.
As you reflect keep in mind we are focusing now on understanding required, we will revisit this idea again with a focus on skills required in the next session.
Refer back to the Standards and progression documents and the notes you took to guide your discussion. Also consider the understanding that is promoted in Grade 3 Module 5 as a base level of understanding that students should be coming in with, but might not be.
Take 1 minute to reflect and discuss.
(Allow for 1-2 minutes of discussion, and then facilitate a discussion documenting on flip charts the understanding that is required for success in each of the standards that were examined.) If the following are not brought up do so, listing them on the flip-chart:
Foundations supporting 4.NF.4:
· Understanding how fractional parts combine to form whole number and/or fractions greater than 1
· Understand that the fraction 1/b is the quantity formed by 1 part when a whole is partitioned into b equal parts.
· Understanding a fraction a/b as the sum of fractions 1/b
Foundations supporting 5.NF.3:
· Understand that the fraction 1/b is the quantity formed by 1 part when a whole is partitioned into b equal parts.
· Understanding a fraction a/b as the sum of fractions 1/b
· Understand the area model of multiplication and division problems
(SLIDE 17) Now, referring as needed to the progressions and to the models and contexts we’ve seen in G3-M5, how might the models and contexts you chose be adapted or even supplemented with other models and contexts in order to bridge gaps in students’ prerequisite knowledge?
Take 3-4 minutes to discuss with your partner.
(Allow 3-4 minutes for work. Then facilitate a discussion and post ideas on flip chart notes.)
(SLIDE 18) Take one minute now to reflect on what you can share with your colleagues (elevator speech) about how the progressions study of Session 2 has informed your understanding of promoting coherence through models and contexts.
Recall that A Story of Units recommends a finite number of concrete and pictorial models used coherently across the grades.
Conceptual Understanding – Conceptual Questioning / Writing & Speaking About Understanding
Time: 0:45-1:10