1. Concrete Representational Abstract
The Continuum
Nick Freathy
Math Instructional Coach
Francesca Reinhard
Nick
Welcome: And Set-up the ABC trio by having folks identify themselves
Room Set-up:
Table Boxes
Pull out sticky-notes and spead around table
Blue papers, one per table
CRA Continuum on tables
Other materials to have ready:
1 minute for welcome and introductions
Also, ask participants to quickly introduce themselves to their table group

2. Francesca
1 minute

3. EGUSD Strategic Goals
All students will have access to standards aligned curriculum and receive high quality instruction to promote college, career, and life readiness and eliminate the achievement gap.
High-Quality
Classroom
Instruction &
Curriculum
All students will benefit from instruction guided by assessment results (formative, interim and summative) and continuous programmatic evaluation.
Assessment,
Data Analysis,
& Action
All students will have an equitable opportunity to learn in a culturally responsive, physically, and emotionally healthy and safe environment.
Wellness
Francesca
1 minute
Family
& Community
Engagement
All students will benefit from programs and services designed to inform and involve family and community partners.

4. What You Will Gain
Learn the Concrete - Representational - Abstract continuum and understand why it is important for student learning
Analyze the Go Math text through the lense of turning tasks into experiences that address the Concrete - Representational - Abstract continuum for student learning
Learn Google Classroom basics
Francesca
1 minute

5. Questions and Wonderings
Throughout the process, please write all of your questions on post-it notes and place on the blue paper. We will answer them and send an with the information.
Nick
1 minute
Prep: Colored piece of paper on each table, and folks put their post-it questions on that paper. Then we will collect it later.

6. C→ R → A Concrete The Doing Stage Representational The Seeing Stage
Abstract
The Symbolic Stage
Concrete and pictorial representations should be used at all grade levels
Creates sense-making strategies for unfamiliar problems.
Helps students begin to see patterns that emerge in the Representational and Abstract stages.
Provides a visual and organized representation of the concrete
Helps students visualize the mathematics
The patterns that students notice through representation launch the generalization needed for understanding the abstract
Provides a shorter, efficient way to represent numerical operations
Helps students to interpret the problem using the relationship between the concrete, representational, and abstract
Nick:
3 minutes
We will discuss the advantages of each one, but overall the Abstract is what we have seen the most in our books, yet what we know about human learning is that the abstract becomes a mere procedure that can only be used with familiar problems without having any sensemaking from the concrete, representational, or both.
ABC Trio:
Which of these do you feel most comfortable using as a teacher?
Which of these do you feel you use as a learner?

7. Why Use C→ R → A?
Research-based studies show that students who use concrete materials
Develop more precise and more comprehensive mental representations
Often show more motivation and on-task behavior
Understand mathematical ideas
Better apply these ideas to life situations
Harrison & Harrison, 1986; Suydam & Higgins, 1977
Nick
1 minute
Note the more on task behavior

8. How do we turn this task into a C-R-A experience for kids?
An Everyday Math Task
Nick has 45 followers on his YouTube channel and increases his followers by ten per day. Dave has 15 YouTube channel followers, but plans to increase that by fifteen followers per day. Write two equations to find who has more followers after a week.
Francesca
1 minute
How do we turn this task into a C-R-A experience for kids?

9. Concrete→ Representational → Abstract
Acting
Nick has 45 followers on his YouTube channel and increases his followers by ten per day. Dave has 15 YouTube channel followers, but plans to increase that by fifteen followers per day. After a week, who will have more followers?
Manipulatives
Drawing
Francesca
3 minutes
The question changed to allow for entry into the concrete. Often the text leads students to the abstract.
ABC Trio:
Take a minute to discuss which of these you already do and are there any other concrete activities that you facilitate with your students? Call on people to share- be clear- they could pertain to this activity or a different standards aligned activity

10. Concrete→ Representational → Abstract
Tables
Nick has 45 followers on his YouTube channel and increases his followers by ten per day. Dave has 15 YouTube channel followers, but plans to increase that by fifteen followers per day. After a week, who will have more followers?
Day Number
Nick’s number of followers
Dave’s number of followers 45 15 1 55 30 2 65 3 75 60 4 85 5 95 90 6
105 7
115
120
Francesca
3 minutes
In this case the concrete is being organized into a representation of the concrete
From the patterns we see on the table and graph, students are more prepared to enter the abstract phase of the continuum because they have made meaning of it
ABC Trio:
Take a minute to discuss which of these you already do and are there any other representational activities that you facilitate with your students? Call on people to share-be clear- they could pertain to this activity or a different standards aligned activity
Graphs

11. Concrete→ Representational → Abstract
Algebraic approaches (expressions and equations)
Algebraic Approach
Nick has 45 followers on his YouTube channel and increases his followers by ten per day. Dave has 15 YouTube channel followers, but plans to increase that by fifteen followers per day. After a week, who will have more followers?
Francesca
1 minute
In this case the patterns from the representational are being organized into an abstract representation
From the patterns we see on the table and graph, students are more prepared to enter the abstract phase of the continuum because they have made meaning of it

12. Another Everyday Math Task
Which is greater?
Nick
1 minute (actually 10 seconds)
Ask the question...and then lead it to the next slide.
How do we turn this task into a C-R-A experience for you?

13. Another Everyday Math Task
Which is greater?
On your own, or with your Trio:
Use sticky-notes to generate as many ways to show this problem.
Consider: Some C’s, R’s, and A’s
Feel free to share/compare with your group.
Nick
20 minutes
C R A

14. Another Everyday Math Task
Which is greater?
As a table group:
Share your post-it notes and then discuss where you can place each one along the continuum.
Nick
20 minutes
Give 3-4 minutes for strategy generation
Prompt people to begin placing on the coninuum with partner
Give 3-4 minutes for partners to share for group comparison and discussion. Combine continuums with group to bring all ideas together.
If we have time: Have a group share their continuum- listen to their conversations and choose a group with a good one to put theirs on the board. Have them put it on the board before the group discussion begins. Have them share their reasoning and where they struggled as a team.
As people share note that it is a continuum, not 3 separate buckets or constructs.
Ask- how much time should you spend on eache area?
C R A

15. Grab a Document with Classroom
Visit classroom.google.com
We are using Google Classroom today because it is an easy way for me to quickly distribute google docs/slides/sheets/forms to anyone with an EGUSD account.
Francesca
7 minutes

16. Francesca
1 minute

17. Francesca
1 minute

18. 5lubmeb Once you’re in the class, minimize your Chrome Browser.
Francesca
3 minutes

19. Access a Document from the Announcement
Francesca and Nick
4 minutes
Graphic organizer- multiple representations lesson

20. Bring it Back Choose tasks from an upcoming lesson:
Where do they fall on the continuum?
What could you do to create a CRA for them?
How will you phrase this for students?
Feel free to use post-it notes as reminders for yourself in your book
Be ready to share what you find and create!
Nick
30 minutes
Potentially logon to your my.hrw.com to view all teacher resources digitally.
C R A

21. Quiet Reflection Time
Write what are you already doing that targets the CRA Continuum in your classroom.
Write how you will plan to utilize the CRA Continuum in the future.
Francesca
5 minutes

22. Give One - Get One - Ask One
With your Trio:
Give one: Share the answers to your reflection.
Get one: Listen to your partners’ reflection response.
Ask one: Pose a question to your partners (clarifying or probing) about their reflection.
Francesca
5 minutes

23. Capitalizing on Fraction Baggage
Turning the baggage into rolly-luggage
Deb Stetson
Sacramento State Math Project, Director
Nick Freathy
Math Instructional Coach
Francesca Reinhard
During the break, put out two envelopes, A and B, for each trio
1 minute for welcome and introductions
Also, ask participants to quickly introduce themselves to their table group

24. Our Final Destination
Learn the progression of fractions and key math models and strategies students are exposed to in prior grades.
Discover how the fraction progressions support middle school content.
Nick
1 minute
Learn progression of fractions and models kids use in prior grades

25. Questions and Wonderings
Throughout the process, please write all of your questions on post-it notes and place on the blue paper. We will answer them and send an with the information.
Nick
1 minute
Prep: Colored piece of paper on each table, and folks put their post-it questions on that paper. Then we will collect it later.

26. Continuing with the Continuum
With your trio...
Open envelope A and place the cards along the continuum
C R A
Francesca
15 minutes for this activity, (5 for each envelope and 5 to debrief)
Have 14 sets of envelopes made
Francesca to create sorting activity off of what Nick does with the grades 3-6 documents

27. Continuing with the Continuum
With your trio...
Keep the cards in order along the continuum
Open envelope B and match the slips with the cards
Make your best guess!
Spying is encouraged
C R A
15 minutes for this activity,
Francesca to create sorting activity off of what Nick does with the grades 3-6 documents
Were any of them obvious?
Were there any that you found to be tough and argued over?

28. Understanding the Baggage Grade by Grade
Nick - pass out the packets to each person
3 minutes
Key point to say to make transition to this activity:
Teaching new material and not always sure how to connect it, and we have kids that have trouble and we aren’t sure what they have had, not how they had it. So today we are digging into that….
Pass out doc
Walk through structure of doc
Explain that they are now in a group of 4 people. Make note of your group of 4 as noted by the COLOR on the page.
Then make note of the NUMBER. The number refers to which grade you will be reading.
Send into grade-level groups and explain that they will get instructions as soon as they have moved and are in groups

29. Understanding the Baggage Grade by Grade
Individually…
Read the grade-level (key concepts and models)
Be prepared to share what stands out
Which key concepts stand out to you?
What do you notice about the models?
Nick
5 minutes for reading

30. Understanding the Baggage Grade by Grade
Prepare for the next conversation
Group Protocol…
Choose a timekeeper
Each person gets 1.5 minutes to share
Which key concepts stood out to you?
What did you notice about the models?
Group discuss: What connections do you see to middle school mathematics?
Nick
10 minutes
When you are having your discussion about how it connects to your grade level, feel free to comment on what someone else said
When you prepare, you can jot a couple thoughts down

31. Adding the Wheels In your color-group of 4…
Share your grade level findings (2 minutes per person)
Group discuss: In our own classrooms...
How can we build on these models & strategies to teach 7th and 8th grade content?
Keep in mind the shift to applying fractional fluency for 8th grade content
How can step back into these models & strategies to address students’ gaps?
Nick
10 minutes
How can you use these models and strategies to:
build from there with your 7th or 8th grade content?
address gaps some student have, or to

32. Ponder...
Private Think Time (PTT): What are the 2 biggest challenges with fractions for your students?
Share with a partner.
Deb
5-1 minutes total
Some desired responses to eavesdrop for and to address:
Students perceive a fraction as 2 numbers → Can you ideas for sequence of problems could be used as number talks: Building Fraction Sense
Students don’t know why they need to get common denominators when adding or subtracting. → same size pieces, same unit
Students lose the meaning of fractions when get to algebra, like x/4 or 5/p or x/p. → same size pieces, same unit
Each of those can launch discussions.

33. Digging into the Baggage
PTT...
Make sense of
Deb
2 minutes
Setting up the video:
Before we watch, let’s try the problem. Take 30 seconds of private think time, then with a parnter show how you would make sense of that problem on a whiteboard.
Listen to your elbow partner’s method, and compare to your own.
Video Info:
7th graders in the Bay Area several years ago.
A teacher is going to give students this problem to solve. It’s something that “should” be prior knowledge for students. But the teacher wants students to make meaning of division and what they know before jumping into new content.
Give participants 1 minute to do the problem and then share their representation.

34. TSA Scan Teacher moves:
What questions or prompts are used to spread thinking?
Student thinking:
What concrete, representational, and abstract strategies are present in the room?
Cathy Humphreys Video
Deb
8 minutes + 5 minutes gained from shortening Ponder and Looking Ahead
Explain the two lenses, and give partners time to decide which lens they are taking.
Show video for total of 8 minutes 16 seconds in 3 chunks, skipping between.
1st chunk: (2 minutes) Leslie, Claire, Cathy frames question.
Start time:0:48
End Time: 2:45
PTT: what is Cathy probably thinking here? What options might she have?
What about the kid’s thinking, where is each strategy on the continuum? Leslie: C, R, A?
Claire: C, R, A ?
2nd chunk: 3 minutes) (Sam, Michael, Cathy, Ben)
Start time: 3:20
End Time: 6:20
3rd chunk: (3 minutes 16 seconds) (Cheryl, Cathy, Christine)
Which kids had concrete ways? Which Representational? Which were abstract?
What do you do for the Christines of class?
What did Cathy mean a “numbers way”?
Scribe student work on the board, or chart paper, as it happens on the video.
Options- you can split it into sections and share teacher and student moves, or show the entire clip before sharing. It’s up to you, Deb.

35. TSA Scan At your table, share your observations. Reflect:
Based on your observations, what are some options for NEXT teacher moves?
What could be used as an opening task for tomorrow?
Deb
5 + 2 minutes if can have it from extra time given.
NEXT teacher moves could be for the whole class, small group, or a specific student.

36. Using the Baggage: Grade 7
Deb
-->What if we give them the option of doing one of these spots that we identified, or to sort through their books and find one in next couple weeks as the next slide suggests.
2 minutes
Column 1 = Desired strategies by on-grade level 7th grader
Column 2 = If kids are struggling, then identify what can you connect to from previous grades
There a few different ways to implement each strategy. Try and create multiple applications of each strategy

37. Using the Baggage: Grade 8
Deb (8th grade questions have a different approach than 7th, so talk about why there is a shift in focus here)
2 minutes
Make sure instructions are clear in terms of what participants should do in each column. Consider paraphrasing questions for each column.
8th grade has shift away from explicitly teaching/learning operations on fractions...and toward using their fluency with fractions to approach other concepts and problems applying their fractional understanding.
For the kids that have not concept of this, what is interfering with them being able to do this? They are not owning the definition of a fraction in their heads (2/4=½)
If there is time, or with groups that are ready, nudge them to consider this...What if you considered a conceptual way to go after fraction busting, built on a conceptual notion of fractions. How might you build a way based on concept of fractions, to solve that equation by keeping the fractions or by not keeping the fractions (busting), but work to provide opportunities to build the notion of fraction busting onto the concept of fraction, instead of busting as a way to not have to deal with fractions. So a chance to suggest reframing that fraction busting is only possible when one understands all those denominators are different sized pieces. That leads to 2 choices: 1) gather enough of all of them (multiply by LCD), or change all the pieces to the same size (get all denominators common, and then work with equation of numerators)

38. Pause and Look Ahead Sort through the next module and consider:
What’s coming up?
Representations they should use?
How to bridge gaps or to build from understandings?
Deb
15-3 minutes
Sort through next model
What’s coming up that builds upon fractions?
Which representations should students be using?
What can I apply from today to bridge the gaps or to build from understandings students do have?

39. Final Destination Stand up, Hand up, Pair up
Consider one or two things you learned, thought about, and you want to implement. Be ready to share.
Francesca
5 minutes
For second partner: Before you share what you got out of today, share what you heard your first partner say