Richard Bisk Professor Mathematics Department · PDF fileSingapore Math Overview with ... Richard Bisk Professor Mathematics Department Worcester State University ... CCSS.Math.Content.3.NF.A.1

Singapore Math Overview with

Common Core Connections

Richard Bisk Professor

Mathematics Department Worcester State University

[email protected]

Dr. Richard Bisk - [email protected]

Warnings about the presenter

He talks ahead of his slides.

He may have too many slide; so he might skip over a few. They will be posted on the website.

He will talk about the Common Core tomorrow too.


Singapore Books in the U.S.

Three versions, all published by Marshall Cavendish Education:

a. Primary Math: US Edition California Standards Edition Common Core Edition

c. Math In Focus – HMH Adaptation of “My

Pals are Here”


My Career

• Trained as a mathematician.

• Worked entirely at teaching colleges and universities.

• Many students (25-50%) unprepared for college level math courses.

• Few (<10%) prepared for calculus.


College Readiness Why do many students come to higher education

with significant mathematical weaknesses that limit their career options?

Weak foundation that goes back to elementary and

middle school. Led to an interest in working with K-8 teachers and

their students. Content based PD is my passion. Use books from

Singapore because the math is so clear and coherent.

Why the interest in Singapore?

a. TIMSS Studies - 1995, 1999, 2003, and 2007, 2011.

b. National Math Panel Report - 2008

c. Common Core State Standards Initiative (CCSSI) - 2010


TIMSS – 2011

South Korea

Singapore

Taiwan

Hong Kong

Japan

Russia

Israel

Finland

United States

England

International

613

611

609

586

570

539

516

514

509

507

500

Grade 8

Singapore

South Korea

Hong Kong

Taiwan

Japan

Northern Ireland

Belgium

Finland

England

Russia

International

606

605

602

591

585

562

549

545

542

542

500

Grade 4


Instruction in Singapore is in English

National Math Panel

• Even in elementary school, the U.S. is not among the world leaders; only 7% of U.S. fourth-graders scored at the advanced level in TIMSS, compared to 38% of fourth-graders in Singapore, a world leader in mathematics achievement. (page 4)

• In elementary school textbooks in the United

States, easier arithmetic problems are presented far more frequently than harder problems. The opposite is the case in countries with higher mathematics achievement, such as Singapore. (page 26)


Common Core Standards

The composite standards [of Hong Kong, Korea and Singapore] have a number of features that can inform an international benchmarking process for the development of K–6 mathematics standards in the US.

(Second paragraph of introduction- quoted from: Ginsburg, Leinwand and Decker, 2009)


Common Core Standards

In general, the US textbooks do a much worse job than the Singapore textbooks in clarifying the mathematical concepts that students must learn. Because the mathematics concepts in [U.S.] textbooks are often weak, the presentation becomes more mechanical than is ideal. We looked at both traditional and non-traditional textbooks used in the US and found this conceptual weakness in both.

(first page of introduction – Red portion from March, 2010 draft – quoted from Ginsburg et al., 2005)


Mathematics Curriculum Framework Ministry of Education 2007

Mathematical Problem Solving

Concepts

Numerical Algebraic

Geometrical Statistical

Probabilistic Analytical

Reasoning, communication & connections Thinking skills & heuristics Application & modelling

Numerical calculation Algebraic manipulation

Spatial visualization Data analysis

Measurement Use of mathematical tools

Estimation

Monitoring of one’s own thinking Self-regulation of learning

Beliefs Interest

Appreciation Confidence

Perseverance


Mathematical Practices - Common Core

1. Make sense of problems and persevere in

solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique

the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated

reasoning. Dr. Richard Bisk - [email protected]

Mathematical Problem Solving

Concepts

NumericalAlgebraic

GeometricalStatistical

ProbabilisticAnalytical

Reasoning, communication & connectionsThinking skills & heuristicsApplication & modelling

Numerical calculationAlgebraic manipulation

Spatial visualizationData analysis

MeasurementUse of mathematical tools

Estimation

Monitoring of one’s own thinkingSelf-regulation of learning

BeliefsInterest

AppreciationConfidence

Perseverance

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

This Presentation

Connect the Common Core Standards to examples from the Singapore Books.


Reason abstractly and quantitatively.

C→P→A

Concrete: ?

Pictorial: ││││││││

Abstract: 8

Abstraction

• Gives mathematics its power.

• But abstraction without understanding??

• Leads to confusion.


Look for and make use of structure

MP7: “Mathematically proficient students

look closely to discern a pattern or structure … students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property.”


• CCSS.Math.Content.3.OA.B.5 …Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find

8 × 7 as

8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56.


http://www.corestandards.org/Math/Content/3/OA/B/5




Grade 1 – Common Core

• CCSS.Math.Content.1.OA.B.4 Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.

• CCSS.Math.Content.1.OA.C.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.

Use strategies such as making ten ( 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14);



http://www.corestandards.org/Math/Content/1/OA/C/6


• CCSS.Math.Content.2.OA.B.2 Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers.



Number Bonds

part 5

8 whole

part 3




Number Bonds


Number Bonds



Progression to Abstract


Making Ten

part 7

10 whole

part 3


Ten Frame




Compensation

7 + 8

3 + 5


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.


http://www.corestandards.org/Math/Content/3/NF/A/1


This reinforces: “Understand subtraction as an unknown-addend problem. “




3.OA.2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.

3.OA.6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.


Number of Objects


Number of Shares



Use place value understanding and properties of operations to perform multi-digit arithmetic.

• 4.NBT.4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.







Singapore – Grade 2




5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.


If time


• CCSS.Math.Content.5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1


http://www.corestandards.org/Math/Content/5/NF/B/7

If time




6.RP.3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.


Tape Diagrams

Also called:

• bar diagrams

• model drawing

• bar models


Apply and extend previous understandings …

• CCSS.Math.Content.7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


http://www.corestandards.org/Math/Content/7/NS/A/1/

-8 + 10


Use properties of operations to generate equivalent expressions.

• CCSS.Math.Content.7.EE.A.1

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.


http://www.corestandards.org/Math/Content/7/EE/A/1/

7A


3(2x+4) = 6x+12

Common Core - First sentence

For over a decade, research studies of mathematics education in high performing countries have pointed to the conclusion that the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country.


“A mathematically logical, uniform national framework that develops topics in-depth at

each grade guides Singapore’s mathematics system.

“…. The framework covers a relatively small number of topics in-depth and carefully sequenced grade-by-grade, ….”


AMERICAN INSTITUTES FOR RESEARCH Report prepared for the U.S. Department of Education (2005): “What the United States Can Learn From Singapore’s World-Class Mathematics System.”

Summary of Key Connections

• Focused, coherent, rigorous standards.

• Emphasis on conceptual understanding and procedural fluency.

• Early learning of mathematics emphasizes number and operations in base 10.

• Model drawing as a problem solving technique and precursor to algebra.


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Richard Bisk Professor Mathematics Department · PDF fileSingapore Math Overview with ... Richard Bisk Professor Mathematics Department Worcester State University ... CCSS.Math.Content.3.NF.A.1