1. Critical Learning Phases In Mathematics: Foundations for 3rd Grade
As you settle in . . .
Put a sticker in the column that best describes your familiarity with Kathy Richardson’s Critical Learning Phases.
Critical Learning Phases In Mathematics: Foundations for 3rd Grade
Ashley Zatt & Stefanie Bordeaux
Orange County Schools

3. Session Goals Understand the definition of a critical learning phase
Understand the critical learning phases for grades 2-3
Identify examples of the phases in student work
Learn ways to assess students’ progress in the phases
Create activities that link assessment to instruction

4. Read the quote below and jot down any reactions you have on a sticky note.
“For each major mathematical idea, there are certain understandings that must be in place to ensure that children are not just imitating procedures or saying words they don’t really understand. These understandings must be in place, or at least developing, in order for a child to benefit from particular instructional experiences. These mathematical insights are predictable and universal. I have labeled these understandings, Critical Learning Phases.”
- Kathy Richardson

5. What are the critical learning phases?
Are NOT
ARE
skills directly taught to children
procedures children memorize
children looking to the teacher for the “right” answer
Crucial mathematical ideas that students must understand if we want them to find meaning in math and learning.
Understandings that are developed through purposeful instruction that builds on what students already know.
Children looking to their own ability to think through math and make sense of problems.
The gateway to students being successful in math!

6. What do students need to know to be successful with place value and addition and subtraction?
What would you expect students to do to show they understand?

7. “If place value concepts are to be meaningful, children need to know more than what digit is in the tens place and what digit is in the ones place.”
- Kathy Richardson

8. Critical Learning Phases: Place Value (2-DIGIT)
Understanding . . .
Ten as a unit
The structure of 1 ten and some ones
The structure of tens and ones
The structure of hundreds, tens, and ones (+ thousands)

9. Ten as a unit
Students unitize a ten instead of counting by ones. They organize into tens and represent 10 with ten objects, instead of one. Students must understand that a ten is both one ten and ten ones.
The structure of 1 ten and some ones
Students combine 1 ten and a number of ones up to 9 without needing to count.
Students decompose numbers from into 1 ten and leftover ones.
The structure of tens and ones
Students count groups of tens. They are able to identify that 38 is 3 groups of tens instead of 4 groups of ten. They also understand that it is 3 groups of ten (not 30 groups) and can distinguish between number of groups of ten and number altogether.
Students know the total amount instantly when the number of tens and ones is known.
Students know the number of tens and amount of ones leftover from any number of ones.
Students know 10 more and 10 less for any 2-digit number.
The structure of hundreds, tens, and ones (+ thousands)
Same as above, plus students know the number of hundreds that can be made from any group of tens, and determines total value by reorganizing into all possible hundreds, then all possible tens with leftover ones.

10. Using a Learning Trajectory
Richardson’s Critical Learning Phases is one learning trajectory related to number sense and other number concepts
Use formative assessments to pinpoint a child’s place on this trajectory.
Tailor instruction based on assessments.
Use formative assessments regularly to chart students’ progress along this trajectory.
Keep in mind that your students are individuals.

11. Ten as a Unit
18 - Show with the counters what the 1 in this number means.
Model the number 14. What part of your model shows the meaning of the 1?

12. Assessing to Determine the Critical Learning Phase
Use student interviews to determine the students’ phase along the trajectory.
The assessments are conducted in short one-on-one student/teacher interviews. This format is critical since “we learn most about how our students think and what they can do when we sit beside them and observe their mathematical work”. Because of the ages and capabilities of young students, mathematics assessment should rely more on observation and conversations and less on writing. “What students write on paper offers only a glimpse of what they know and think” (Richardson, Assessing Math Concepts).

13. Structure of 1 ten and some ones
Count this group of cubes (amount from ). How many groups of ten could you make with these cubes? How many cubes would you have leftover?
I have one group of ten cubes and these 6 leftover cubes. How many cubes do I have? (Observe if students count all, count on, or just know.)

14. Structure of tens and ones
How many cubes do you think I have here? (Show an amount from cubes.) Count these cubes into groups of ten. How many cubes are there? (Observe if students know the amount of cubes immediately.)
I have 42 cubes here. How many groups of ten could I make with these cubes?
How many cubes would I have if you gave me 10 more? What if I had 10 less?

15. Structure of hundreds, tens, and ones
What number is represented in the model below? (Show a model of 3 hundreds, 4 tens, and 6 ones.)
What is ?
If I have 15 tens, how many hundreds could I make? How many tens would be leftover?
What is the value of the model below? It has 3 hundreds, 24 tens, and 17 ones. (Show a model of the amount above with base ten blocks.)

16. Video
What do you notice in these videos? Where would you place this student in understanding place value?
Video Link Video Link #2
Proficient
Needs additional practice
Needs teacher support
Observational Checklist

17. Next Steps: Linking Assessment to Instruction
Potential activities may include:
Small group activities
Independent Work Stations
1 on 1 Math Conferences with students

18. Sample Work Stations Lots of Lines Measuring Things in the room
Building Stacks
Race to 100
Race to Zero
Task Cards

19. Lots of Lines
Using all types of lines and regular/irregular shapes, students will measure the length of the lines and the area of the paper shapes.
After measuring with unifix cubes, students will group the cubes into tens and ones to determine the length and area in relationship to how many cubes.

20. Measuring Myself and Measuring Things
Using unifix cubes, students will measure objects in the classroom or they will measure each other.
After measuring, students will group the unifix cubes into groups of tens and ones to determine the measurement of the object.

21. Build A City
Students take turns rolling a die and building a stack on their side of the game board.
They start a new stack with each roll.
When the board is filled, the students take apart their stacks and organize into tens and ones to count the total.
Students spin the more/less spinner to determine if the student with more cubes or less cubes wins the round!

22. Dash to 100 and First to Zero
Using unifix cubes, students will roll a dice to either add the designated amount to 0 or subtract the designated amount from 100.
Students will work to be the first student to reach 100 or 0.
Students will need to group and ungroup ones to create groups of ten.

23. WorkStation Question Stems
Is there a way for you to organize your counters so it is easy for you to keep track of what you have?
How many tens do you think you can make?
Now that you have this many tens, do you have another idea of what it could be?
I see you have organized your cubes into tens and ones. How many do you have so far?
Now that you know how many tens and leftovers there are, does that help you know how many altogether?
What did you notice?

24. Two-Digit Addition and Subtraction

25. Think about the strategies you use to solve these mentally.

26. Critical Learning Phases: Addition and Subtraction
Adding Numbers to 20
Subtracting Numbers to 20
Adding Numbers to 100
Subtracting Numbers to 100
Adding and Subtracting 3-Digit Numbers
Adding and Subtracting 4-Digit Numbers

27. Adding & Subtracting Numbers to 100
Adds multiples of ten to two-digit numbers without counting
Adds 2 numbers up to 100 by reorganizing them into tens and leftover ones
Subtracts multiples of ten from two-digit numbers without counting
Subtracts from quantities to 100 by breaking apart tens when necessary, and reorganizing into remaining tens and leftovers

28. Adds multiples of ten without counting
If I had 42 cubes and added 30 more, how many cubes would I have?
What is ?

29. Adds 2 numbers using tens and ones
With models
I have 38 cubes here (show 3 tens and 8 ones). If I added 17 more cubes, how many would I have?
I have 36 cubes (3 tens and 6 ones). If I added 38 more cubes, how many would I have?
Without models
How many cubes are here? (Show 3 tens and 8 ones and then cover.) If I add 24 cubes, how many cubes would I have?
Show equation Solve this equation.

30. Think about the strategies you use to solve these mentally.

31. Subtracting Numbers to 100
What is ?
Show 45 (4 tens and 5 ones). If I take away 27 cubes, how many would I have?
Show the equation What is the solution to this equation?

32. Gallery Walk In what areas are the students proficient?
What strengths can be used to grow the student?
What areas need improvement?

33. Sample Work Stations Addition Partner Add It Add ‘Em Up Activities
Subtraction
Partner Take Away
Roll and Subtract
Task Cards

34. Partner Add It
Using tens and ones boards, students place unifix cubes on the board and below the board to create two 2-digit numbers.
Students work together to add the two numbers represented by the unifix cubes on the board.
Then, students can discuss and model with the cubes.

35. Add ‘Em Up Activities
Students use unifix cubes to measure several objects in the room.
Then, students use various strategies to add the different amounts together to find the total amount.

36. Partner Take Away
Using tens and ones boards, a student places unifix cubes on the board and both students write the number.
The second partner decides how many cubes to take away and both partners write that amount.
The second partner removes that number of cubes, putting them below the board and covering the amount left.
Together, the partners figure out how many cubes are under the paper.

37. Roll and Subtract
Students roll a tens cube and ones cube and write the amount created.
They roll again and create a subtraction problem.
They solve using unifix cubes and place value boards.

38. Work Station Question Stems
Can you show me?
How did you figure that out?
What if we had 10 more?
What if we had 10 less?
Do you know a different way to do that?
Do you have enough to make another ten?
Do you need to break up a ten?
How many tens do you have now? How many leftovers?
How many altogether?
Can you find out without counting all of them?

39. Incorporating CLPs Into Classroom Instruction
Pop Tab Unit
Created with a service-learning lens
Focus on place value, addition, and subtraction
Adapted on work by Catherine Fosnot

40. Frequently Asked Questions
How will I ever have time to do all these diagnostic interviews?
How do I help students when they are all at different critical learning phases?
When do I provide instruction in the needed area?
How do I know that my students are doing the activities correctly?
What if a student is stagnant in their growth?
How do I organize student stations if they are all working on different things?

41. How do I support this type of instruction in my classroom?
Jot down the current structure of your math block.
Consider:
What are you already doing that supports this type of individualized assessment and instruction?
What aspects could you tweak to better support this type of individualized assessment and instruction?
Here are three proposed math block structures to help you implement instruction to support Richardson’s Critical Learning Phases:

42. Structure 1
Focus Mini-Lesson: Direct Instruction and Whole Group Practice
minutes
Include number talk, concept, or skill development
Independent Stations/Guided Math Groups
minutes
Independent Stations - AMC or related to mini-lesson
Guide students to focus on learning while working
Teacher observes students, does 1 on 1 conferring with students, or works in small groups with students using math questioning to push students to deeper thinking and explaining their math reasoning
Group Share
Expose math thinking across room
Add to/develop anchor charts about our thinking
Discussion of “big ideas”
Synthesis of new understandings

43. First Half of Math Block
Structure 2
First Half of Math Block
Concepts or skills within district’s curriculum and Common Core, math talk, and problem solving
Second Half of Math Block
AMC Workstations
Conference with students working independently or pull one small group
* This model can also be split into thirds - ⅓ CGI problem with math talk, ⅓ concept or curriculum, ⅓ workstations.

44. Structure 3
For initial introduction of AMC stations in a classroom, a teacher may choose to start with a “Workstation Day” once a week. For example, Monday through Thursday is the curriculum work that relates to the Common Core and Friday is a Workstation Day with AMC independent stations at student working levels.
From Phelps, Humphries, & Lowman (2014) under the direction of Barbara Bissell for Kenan Fellows

45. Questions? Make and Take!

46. References
Phelps, K. A. G., Humphries, A., & Lowman, C. (2014, October). Making math count: A project from the Kenan Fellows Program in conjunction with the Department of Public Instruction. B. Bissell (Ed.). Presented at NCCTM 2014 State Mathematics Conference, Greensboro, NC.
Richardson, K. (2012). How children learn number concepts: a guide to the critical learning phases. Bellingham, WA: Math Perspectives.
Richardson, K. (1999). Developing number concepts: book three: place value, multiplication, and division. Parsippany, NJ: Dale Seymour Publications.