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Concrete Pictorial AbstractSingapore’s Approach to Math InstructionRichard Bisk Ph.D.Professor of .com/site/singmathproject/Worcester State University

To find presentation slides on the webSearch for: singmathprojectThen select presentationsWill also post on NCTM website.Worcester State University

Background 35 years teaching collegestudents. 25 years working with K-12teachers and their students. 15 years working with mathtextbooks from Singapore. Theirstudents are top math performersin international studies.Worcester State University

“Singapore” Textbooks in the USPrimary MathematicsMath In Focus (Consulting Author)Both published by Marshall Cavendish Education.“The students are provided with the necessary learningexperiences beginning with the concrete and pictorialstages, followed by the abstract stage to enable them tolearn math meaningfully.”Worcester State University

CPA Approach Based on the work of US Psychologist Jerome Bruner. “I shall call the three modes of representationmentioned earlier enactive representation, iconicrepresentation, and symbolic representation. Theirappearance in the life of the child is in that order, eachdepending upon the previous one for its development,yet all of them remaining more or less intactthroughout life— ” Bruner: The Course of Cognitive Growth (1964)Worcester State University

Bruner, J. S. (1960). The Process of education.“ It has also been pointed out that themethod of discovery would be too timeconsuming for presenting all of what astudent must cover in mathematics.”(page 21)Worcester State University

My TakeConcreteAbstractIt’s a continuumWorcester State University

This TalkExamples from:1. The Books.2. Asian Classrooms.3. My Classrooms.4. Me.Worcester State University

C P AConcrete:Pictorial:Abstract:? 8Worcester State University

Abstraction Gives mathematics its power. But abstraction withoutunderstanding? Leads to confusion.Worcester State University

MODEL DRAWINGandC P AThe model drawing approach takes students fromthe concrete to the abstract stage via anintermediary pictorial stage.Many complex problems are easier to solve withalgebra .if you understand the algebra!

AdditionWorcester State University

x 11 14Worcester State University

DivisionSuppose Trish scored x goals.Then Shawn scores 3x goalsx 3x 364x 36x 9Worcester State University

Example (grade 4): Josh spent 2/5 of hismoney on a present for his Dad. The presentcost 12. How much money did he have at first?Josh’sMoney?122 units 121 unit 65 units 30Josh started with 30.

Josh’sMoney?122 units 122x 1251 unit 61x 655 units 30.x 30Suppose Josh started with x dollars.

Grade 1 - Number BondsWorcester State University

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Number BondsWorcester State University

Number BondsWorcester State University

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Progression toAbstractWorcester State University

Grade 2Worcester State University

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Still Concrete; but More AbstractWorcester State University

Seoul – Grade 3Worcester State University

Same Problems with ChipsWorcester State University

Fractions Where we lose too many students. They don’t understand what a fraction is. They learn algorithms without first learning meaning.Worcester State University

Common Core – Grade 3CCSS.Math.Content.3.NF.A.1 Understand a fraction 1/b as thequantity formed by 1 part when a whole is partitionedinto b equal parts; understand a fraction a/b as the quantityformed by a parts of size 1/b.Understand a fraction 1/3 as the quantity formed by 1 part when awhole is partitioned into 3 equal parts; understand afraction 2/3 as the quantity formed by 2 parts of size 1/3.Worcester State University

MIF – 2BWorcester State University

Common Core – Grade 3CCSS.Math.Content.3.NF.A.2 Understand a fraction as anumber on the number line; represent fractions on anumber line diagram.01/21Worcester State University

Adding FractionsWorcester State University

Multiplying FractionsWorcester State University

Multiplying FractionsWorcester State University

MultiplicationWorcester State University

Multiplication 23 x 4540520380010012015Worcester State University

Multiplication 2 1/3 x 4 1/241/221/3814/31/6Worcester State University

CalculusSource: Swokowski: Calculus – Classic EditionWorcester State University

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𝑎𝐴 𝑥 𝑑𝑥0Worcester State University

Bruner: The Process of EducationThe teacher is not only acommunicator but a model.Somebody who does not seeanything beautiful or powerfulabout mathematics is not likely toignite others with a sense of theintrinsic excitement of the subject.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,