![Page 1: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/1.jpg)
Worcester State University
Concrete Pictorial Abstract
Richard Bisk Ph.D.
Professor of Mathematics [email protected]
https://sites.google.com/site/singmathproject/
Singapore’s Approach to Math Instruction
![Page 2: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/2.jpg)
Worcester State University
To find presentation slides on the web
Search for: singmathproject
Then select presentations
Will also post on NCTM website.
![Page 3: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/3.jpg)
Worcester State University
Background
• 35 years teaching college students.
• 25 years working with K-12 teachers and their students.
• 15 years working with math textbooks from Singapore. Their students are top math performers in international studies.
![Page 4: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/4.jpg)
Worcester State University
“Singapore” Textbooks in the US
Primary Mathematics
Math In Focus (Consulting Author)
Both published by Marshall Cavendish Education.
“The students are provided with the necessary learning experiences beginning with the concrete and pictorial stages, followed by the abstract stage to enable them to learn math meaningfully.”
![Page 5: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/5.jpg)
Worcester State University
CPA Approach
• Based on the work of US Psychologist Jerome Bruner.
• “I shall call the three modes of representation mentioned earlier enactive representation, iconic representation, and symbolic representation. Their appearance in the life of the child is in that order, each depending upon the previous one for its development, yet all of them remaining more or less intact throughout life—…”
• Bruner: The Course of Cognitive Growth (1964)
![Page 6: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/6.jpg)
Worcester State University
Bruner, J. S. (1960). The Process of education.
“ It has also been pointed out … that the
method of discovery would be too time-
consuming for presenting all of what a
student must cover in mathematics.”
(page 21)
![Page 7: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/7.jpg)
Worcester State University
My Take
Concrete Abstract
It’s a continuum
![Page 8: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/8.jpg)
Worcester State University
This Talk
Examples from:
1. The Books.
2. Asian Classrooms.
3. My Classrooms.
4. Me.
![Page 9: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/9.jpg)
Worcester State University
C→P→A
Concrete: ?
Pictorial: ││││││││
Abstract: 8
![Page 10: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/10.jpg)
Worcester State University
Abstraction
• Gives mathematics its power.
• But abstraction without understanding??
• Leads to confusion.
![Page 11: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/11.jpg)
MODEL DRAWING
and
C→P→A
The model drawing approach takes students from
the concrete to the abstract stage via an
intermediary pictorial stage.
Many complex problems are easier to solve with
algebra….
if you understand the algebra!
![Page 12: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/12.jpg)
Worcester State University
Addition
![Page 13: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/13.jpg)
Worcester State University
x + 11= 14
![Page 14: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/14.jpg)
![Page 15: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/15.jpg)
Worcester State University
Division
Suppose Trish scored x goals.
Then Shawn scores 3x goals
x + 3x = 36
4x = 36
x = 9
![Page 16: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/16.jpg)
Example (grade 4): Josh spent 2/5 of his
money on a present for his Dad. The present
cost $12. How much money did he have at first?
5 units = 30 Josh started with $30.
12
Josh’s
Money ?
2 units = 12
1 unit = 6
![Page 17: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/17.jpg)
5 units = 30.
12
Josh’s
Money
?
2 units = 12
1 unit = 6
212
5x
Suppose Josh started with x dollars.
16
5x
30x
![Page 18: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/18.jpg)
Worcester State University
Grade 1 - Number Bonds
![Page 19: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/19.jpg)
Worcester State University
![Page 20: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/20.jpg)
Worcester State University
Number Bonds
![Page 21: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/21.jpg)
Worcester State University
Number Bonds
![Page 22: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/22.jpg)
Worcester State University
![Page 23: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/23.jpg)
Worcester State University
Progression to Abstract
![Page 24: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/24.jpg)
Worcester State University
Grade 2
![Page 25: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/25.jpg)
Worcester State University
![Page 26: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/26.jpg)
Worcester State University
![Page 27: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/27.jpg)
Worcester State University
![Page 28: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/28.jpg)
Worcester State University
![Page 29: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/29.jpg)
Worcester State University
Still Concrete; but More Abstract
![Page 30: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/30.jpg)
Worcester State University
Seoul – Grade 3
![Page 31: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/31.jpg)
Worcester State University
Same Problems with Chips
![Page 32: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/32.jpg)
Worcester State University
Fractions
• Where we lose too many students.
• They don’t understand what a fraction is.
• They learn algorithms without first learning meaning.
![Page 33: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/33.jpg)
Worcester State University
Common Core – Grade 3
CCSS.Math.Content.3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Understand a fraction 1/3 as the quantity formed by 1 part when a whole is partitioned into 3 equal parts; understand a fraction 2/3 as the quantity formed by 2 parts of size 1/3.
![Page 34: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/34.jpg)
Worcester State University
MIF – 2B
![Page 35: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/35.jpg)
Worcester State University
Common Core – Grade 3
CCSS.Math.Content.3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
0 1/2 1
![Page 36: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/36.jpg)
Worcester State University
Adding Fractions
![Page 37: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/37.jpg)
Worcester State University
Multiplying Fractions
![Page 38: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/38.jpg)
Worcester State University
Multiplying Fractions
![Page 39: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/39.jpg)
Worcester State University
Multiplication
![Page 40: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/40.jpg)
Worcester State University
Multiplication
• 23 x 45
40 5
20
3 120
800 100
15
![Page 41: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/41.jpg)
Worcester State University
Multiplication
• 2 1/3 x 4 1/2
4 1/2
2
1/3 4/3
8 1
1/6
![Page 42: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/42.jpg)
Worcester State University
Calculus Source: Swokowski: Calculus – Classic Edition
![Page 43: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/43.jpg)
Worcester State University
![Page 44: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/44.jpg)
Worcester State University
![Page 45: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/45.jpg)
Worcester State University
![Page 46: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/46.jpg)
Worcester State University
![Page 47: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/47.jpg)
Worcester State University
𝐴 𝑥 𝑑𝑥
𝑎
0
![Page 48: Concrete Pictorial Abstract - Confex · •Based on the work of US Psychologist Jerome Bruner. •“I shall call the three modes of representation mentioned earlier enactive representation,](https://reader030.vdocuments.net/reader030/viewer/2022040522/5e7caa0313b71c7a0241fe58/html5/thumbnails/48.jpg)
Worcester State University
Bruner: The Process of Education
The teacher is not only a communicator but a model. Somebody who does not see anything beautiful or powerful about mathematics is not likely to ignite others with a sense of the intrinsic excitement of the subject.