1. Revision of Modes of Representation or Cognitive Levels
Dr. Enrique Ortiz
School of Teaching, Leaning & Leadership
University of Central Florida

2. Cognitive Levels (CPAV)
Concrete
Non-Digital
Dynamic
(Non-static)
Manipulative Models: Using objects or manipulative materials.
Pictures: Using pictures or representations of objects or manipulative materials.
Pictorial, Representational
or Semi-concrete
Digital or Non-Digital
Non-Dynamic
(Static)
Abstract
Digital or Non-Digital
Non-Dynamic
(Static)
Written Symbols/Oral Language: Using written words, symbols, sign language or verbal communication.
Virtual
Digital or Non-Digital
Non-Dynamic
(Static)
Virtual Manipulative Models: Using Apps or Applets that contain virtual objects or virtual manipulative materials. Video not included.

3. CPAV
Concrete (it never digital: This cognitive level involves the use of real concrete objects or manipulative materials to model mathematical ideas: Dynamic (Non-static), and Non-digital.
Pictorial (maybe digital, but non-dynamic and non-interactive): This cognitive level involves the use of pictures, drawings or digital images (which are non-dynamic and static versions used in electronic files, including Webpages, Interactive Whiteboard files, Apps or Applets) of concrete objects or manipulative materials to model mathematical ideas: Non-dynamic (Static), Digital or Non-Digital.
Abstract (may be digital, but non-dynamic and non-interactive): This cognitive level involves the use of written words or symbols, verbal communication, Braille, sign language, or digital text (which may appear in electronic files like Webpages, Interactive Whiteboard files, word-processed files, databases, spreadsheets, Apps or Applets) to model mathematical ideas: Non-dynamic (Static), Digital or Non-Digital.
Virtual (digital): This cognitive level involves the digital-dynamic use of electronically developed manipulatives, objects, pictures, numbers or symbols; and must be digital and dynamic):
Virtual-Concrete (D2C): Digital-Dynamic (Non-Static) versions of objects or manipulative materials (e.g., Apps or Applets)
Virtual-Pictorial (D2P): Digital-Dynamic (Non-Static) versions of pictures,
Virtual-Abstract (D2A): Digital-Dynamic (Non-Static versions involving abstract level. They may be digitally manipulated on a computer screen or other electronic device (e.g., tablets, iPads, iPhones or Interactive Whiteboards) to model mathematical ideas.

4. Interaction Between Levels
Note: The symbol " " used within cognitive levels indicates a combination of levels and that the relationship between levels could go in either directions (for example, Concrete  Abstract is the same as indicating that the relationship is either Concrete to Abstract or Abstract to Concrete).
Concrete
Non-digital
Dynamic
Abstract
Digital or Non-Digital
Non-Dynamic
Pictorial, (Representational, Semi-Concrete)
Virtual:
Virtual-Concrete (D2C)
Virtual-Pictorial (D2P)
Virtual-Abstract (D2A)

5. Learning Levels Characteristics
Level/Type
Digital
Non-Digital
Dynamic
Non-Dynamic
Concrete (C) x
Pictorial (P)
Abstract (A)
Virtual-Concrete (D2C)
Virtual-Pictorial (D2P)
Virtual-Abstract (D2A)

6. Concrete to Pictorial Concrete: Teacher original using real objects.
Pictorial: Student translation using crayons to color (drawing or print).

7. Pictorial to Concrete
Pictorial: Teacher original using crayons to color (drawing or print).
Concrete: Student translation using real objects.

8. Concrete to Abstract C D
Concrete: Teacher original using real objects.
Abstract: Student translation using symbols (written or printed). C D

9. Abstract to Concrete C D
Abstract: Teacher original using symbols (written or printed). C D
Concrete: Student translation using real objects.

10. Pictorial to Abstract C D Pictorial: Teacher original using drawings.
Abstract: Student translation using symbols (written or printed). C D

11. Abstract to Pictorial C D
Abstract: Teacher original using symbols (written or printed). C D
Pictorial: Student translation using drawings.

12. Abstract to Virtual (or Abstract to Virtual)
National Library of Virtual Manipulatives:
NCTM Illuminations: Resources for Teaching Mathematics:

13. Your Turn 1: What level or combination of levels are represented by this graph made of real cubes to represent number of students who prefer each color?
Select the level or levels that is/are involved in this task:
Concrete (Non-digital, Dynamic)
Pictorial (Digital or Non-digital, Non-Dynamic or Static)
Abstract (Digital or Non-digital, Non-Dynamic or Static)
D2-Concrete (Digital-Dynamic, Non Static)
D2-Pictorial (Digital-Dynamic, Non-Static)
D2-Abstract (Digital-Dynamic, Non-Static)

14. Your Turn 1 Answer: Concrete and Abstract – Concrete (real cubes)
– Abstract (color names, and values for columns of cubes)

15. Your Turn 2: What level or levels are represented by a number line (printed on paper)?1 2 4 7 6 5 8 9 3
Select the level or levels that is/are involved in this task:
Concrete (Non-digital, Dynamic)
Pictorial (Digital or Non-digital, Non-Dynamic or Static)
Abstract (Digital or Non-digital, Non-Dynamic or Static)
D2-Concrete (Digital-Dynamic, Non Static)
D2-Pictorial (Digital-Dynamic, Non-Static)
D2-Abstract (Digital-Dynamic, Non-Static)

16. Your Turn 2 Answer: Abstract and Pictorial.
Pictorial (printed line with tick marks)
Abstract (written or printed number names or numerals)

17. Your Turn 3: What level or levels are represented by this graph (printed on paper)?
Select the level or levels that is/are involved in this task:
Concrete (Non-digital, Dynamic)
Pictorial (Digital or Non-digital, Non-Dynamic or Static)
Abstract (Digital or Non-digital, Non-Dynamic or Static)
D2-Concrete (Digital-Dynamic, Non Static)
D2-Pictorial (Digital-Dynamic, Non-Static)
D2-Abstract (Digital-Dynamic, Non-Static)

18. Your Turn 3 Answer: Pictorial and Abstract
– Pictorial (squares and bars)
– Abstract (number symbols or numerals, and color names)

19. Your Turn 4: What level or levels are represented by a frequency table (printed on paper)?
Select the level or levels that is/are involved in this task:
Concrete (Non-digital, Dynamic)
Pictorial (Digital or Non-digital, Non-Dynamic or Static)
Abstract (Digital or Non-digital, Non-Dynamic or Static)
D2-Concrete (Digital-Dynamic, Non Static)
D2-Pictorial (Digital-Dynamic, Non-Static)
D2-Abstract (Digital-Dynamic, Non-Static)

20. Your Turn 4 Answer:
Pictorial-Abstract (number symbols or numerals, and color names).
The table is an organizing tool in this case, and could be considered semi-concrete or pictorial representation.

21. Your Turn 5: What level or levels are represented by the Base-ten blocks below (real objects) and numeral written on the right side? Flats = hundreds, Longs = tens, Units = ones)
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Select the level or levels that is/are involved in this task:
Concrete (Non-digital, Dynamic)
Pictorial (Digital or Non-digital, Non-Dynamic or Static)
Abstract (Digital or Non-digital, Non-Dynamic or Static)
D2-Concrete (Digital-Dynamic, Non Static)
D2-Pictorial (Digital-Dynamic, Non-Static)
D2-Abstract (Digital-Dynamic, Non-Static)

22. Your Turn 5 Answer: Concrete and Abstract
– Abstract (number symbols or numerals)
– Concrete
Note: Base-ten blocks are proportional materials.
This is because the representative piece for 10 is actually 10 times larger than the piece that represents 1.
Other examples of proportional materials: Pre-bundled sticks in singles, groups of 10, and groups of 100; and Meter, decimeter and centimeter sticks.
They should be used before non-proportional materials like coins.
Also, notice that counters such as beans and buttons are the most concrete proportional materials or the less abstract. Ten counters can be put into a cup to represent the number10. With Base-ten blocks the grouping is permanent.

23. Your Turn 6: What level or levels are represented by the Base-ten blocks below (real objects)? The students uses the Base-ten blocks, draws them on paper and writes the numerals. Flats = hundreds, Longs = tens, Units = ones)
Select the level or levels that is/are involved in this task:
Concrete (Non-digital, Dynamic)
Pictorial (Digital or Non-digital, Non-Dynamic or Static)
Abstract (Digital or Non-digital, Non-Dynamic or Static)
D2-Concrete (Digital-Dynamic, Non Static)
D2-Pictorial (Digital-Dynamic, Non-Static)
D2-Abstract (Digital-Dynamic, Non-Static)

24. Your Turn 6 Answer: Concrete - Pictorial - Abstract
– Abstract (number symbols or numerals)
– Pictorial (drawing of Base-ten blocks)
– Concrete (real Base-ten blocks).
Note: Base-ten blocks used here are a variation in the abstraction level of the proportional model.
• In the previous example, grouping is permanent, but verification is still possible through matching or counting because one-to-one correspondence still exists.
• Un-scored Base-ten blocks like the ones used in this last example require more abstraction on the part of the child and could serve as a transition to non-proportional materials, but they are still concrete and proportional.

25. Select the level or levels that is/are involved in this task:
Your Turn 7: What level or levels are represented by Sieve of Eratosthenes? This virtual manipulative displays a grid containing numbers from 2 to 200. You can use it to explore patterns and relationships involving multiples. Using this virtual manipulative you may: - Remove multiples of a number - Show multiples of numbers - Reset the workspace - Choose how many rows to display
Select the level or levels that is/are involved in this task:
Concrete (Non-digital, Dynamic)
Pictorial (Digital or Non-digital, Non-Dynamic or Static)
Abstract (Digital or Non-digital, Non-Dynamic or Static)
D2-Concrete (Digital-Dynamic, Non Static)
D2-Pictorial (Digital-Dynamic, Non-Static)
D2-Abstract (Digital-Dynamic, Non-Static)

26. Your Turn 7 Answer: Abstract and D2-Abstract
– Abstract (written or printed number names or numerals) (Digital or Non-digital, Non-Dynamic or Static)
– D2-Abstract (digital, and dynamic and non-static manipulation of numbers to get factors and multiples)
Pictorial (Digital or Non-digital, Non-Dynamic or Static): Table format.

27. Select the level or levels that is/are involved in this task:
Your Turn 8: What level or levels are represented by Rectangle Multiplication of Fractions? Use this virtual manipulate to graphically demonstrate, explore, and practice multiplying fractions. The grid shows two fractions multiplied together by showing one fraction in red on the left and another in blue on the bottom of a grid. The area of the overlapping region shown in purple is the product (result of multiplying) the fractions.
Select the level or levels that is/are involved in this task:
Concrete (Non-digital, Dynamic)
Pictorial (Digital or Non-digital, Non-Dynamic or Static)
Abstract (Digital or Non-digital, Non-Dynamic or Static)
D2-Concrete (Digital-Dynamic, Non Static)
D2-Pictorial (Digital-Dynamic, Non-Static)
D2-Abstract (Digital-Dynamic, Non-Static)

28. Your Turn 8 Answer: Abstract, Pictorial, D2-Pictorial, D2-Abstract.
Pictorial (number line, and printed line with tick marks)
D2-Pictorial (ability to manipulate columns and rows with colors, but without movement of pieces)
Abstract (written or printed number names or numerals)
D2-Abstract (arrows that can change fractional parts)

29. Select the level or levels that is/are involved in this task:
Your Turn 9: What level or levels are represented by Fraction Pieces? This virtual manipulative can be used to work with fractional parts of a circle or a square. Using this virtual manipulative you may: - Add blocks to the workspace - See a fraction name for a block - Remove blocks from the workspace
Select the level or levels that is/are involved in this task:
Concrete (Non-digital, Dynamic)
Pictorial (Digital or Non-digital, Non-Dynamic or Static)
Abstract (Digital or Non-digital, Non-Dynamic or Static)
D2-Concrete (Digital-Dynamic, Non Static)
D2-Pictorial (Digital-Dynamic, Non-Static)
D2-Abstract (Digital-Dynamic, Non-Static)

30. Your Turn 9 Answer:
Abstract, Virtual-Pictorial, Virtual-Concrete, Pictorial.
D2-Concrete (movable Dynamic-Digital pieces)
D2-Pictorial (movable colors, can change colors): Dynamic-Digital
Pictorial (circle for pieces) Non-dynamic-Digital
Abstract (written or printed number names or numerals): Non-Dynamic-Digital

31. Your Turn 10: What level or levels are represented by the chip trading material (real pieces using chips) below? Red chips = ones, Blue chips = tens, Yellow Chips = hundreds
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Select the level or levels that is/are involved in this task:
Concrete (Non-digital, Dynamic)
Pictorial (Digital or Non-digital, Non-Dynamic or Static)
Abstract (Digital or Non-digital, Non-Dynamic or Static)
D2-Concrete (Digital-Dynamic, Non Static)
D2-Pictorial (Digital-Dynamic, Non-Static)
D2-Abstract (Digital-Dynamic, Non-Static)

32. Your Turn 10 Answer: Abstract  Concrete
– Abstract (number names or numerals)
– Concrete (chips)
Note: Chip trading materials are considered non-proportional materials.
• In non-proportional materials, the same object is sometimes used to represent place values.
• They are challenging because they necessitate a focus on position, color, size, or value of objects.
• The abacus and coins are also considered non-proportional materials.

33. Possible Areas for Cognitive Levels or Combinations
Every learning activity can be represented as a point somewhere within this diagram.
C = Concrete, P = Pictorial, V = Virtual and A = Abstract C A V C P A P
C  A
P  A V A C V P
A  V
C  V
P  V
C  P

34. Revised Modes of Representation or Cognitive Levels
Pictures
Manipulative Models
Written Symbols and/or Oral Language
Real-World Situations
Virtual Manipulative Materials (Apps &Applets)

35. Your Turn
Describe an effective mathematics activity that you have experienced as a student or used as a teacher.
Match the activity with of one of the cognitive levels or combination of cognitive levels.
Be ready to share with the class. Don’t tell what level(s) is represented to others until we will try to figure the level represented in your activity.

36. The End