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. Dr. Richard Bisk - [email protected] 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”
Dr. Richard Bisk - [email protected] 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.
Dr. Richard Bisk - [email protected] 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
Dr. Richard Bisk - [email protected] TIMSS – 2011 Grade 4 Grade 8 Singapore 606 South Korea 613
South Korea 605 Singapore 611
Hong Kong 602 Taiwan 609
Taiwan 591 Hong Kong 586
Japan 585 Japan 570 InstructionNorthern Ireland in Singapore562 is Russiain English 539 Belgium 549 Israel 516
Finland 545 Finland 514
England 542 United States 509
Russia 542 England 507
International 500 International 500 Dr. Richard Bisk - [email protected] 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)
Dr. Richard Bisk - [email protected] 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)
Dr. Richard Bisk - [email protected] 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)
Dr. Richard Bisk - [email protected] Mathematics Curriculum Framework Ministry of Education 2007 Beliefs Interest Monitoring of one’s own thinking Appreciation Self-regulation of learning Confidence Perseverance
Numerical calculation Mathematical Algebraic manipulation Reasoning, Spatial visualization Problem communication & Data analysis Solving connections Measurement Thinking skills & Use of mathematical tools heuristics Estimation Concepts Application & modelling Numerical Algebraic Geometrical Statistical Probabilistic Analytical Dr. Richard Bisk - [email protected]
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] Beliefs Interest Monitoring of one’s own thinking Appreciation Self-regulation of learning Confidence Perseverance
Numerical calculation Mathematical Algebraic manipulation Reasoning, Spatial visualization Problem communication & Data analysis Solving connections Measurement Thinking skills & Use of mathematical tools heuristics Estimation Concepts Application & modelling Numerical Algebraic Geometrical Statistical 1. Make sense of problems and persevere in Probabilistic solving them. Analytical 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.
Dr. Richard Bisk - [email protected]
Reason abstractly and quantitatively.
C→P→A
Concrete: ?
Pictorial: ││││││││
Abstract: 8
Abstraction
• Gives mathematics its power.
• But abstraction without understanding??
• Leads to confusion.
Dr. Richard Bisk - [email protected]
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.”
Dr. Richard Bisk - [email protected] • 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.
Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] 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);
Dr. Richard Bisk - [email protected] Grade 2 – Common Core
• 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.
Dr. Richard Bisk - [email protected] Number Bonds
part 5 8 whole part 3
Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Number Bonds
Dr. Richard Bisk - [email protected]
Number Bonds
Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Progression to Abstract
Dr. Richard Bisk - [email protected] Making Ten
part 7 10 whole part 3
Dr. Richard Bisk - [email protected] Ten Frame
Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Compensation
7 + 8
3 + 5
Dr. Richard Bisk - [email protected] 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.
Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] This reinforces: “Understand subtraction as an unknown-addend problem. “
Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Grade 3 – Common Core 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.
Dr. Richard Bisk - [email protected] Number of Objects
Dr. Richard Bisk - [email protected] Number of Shares
Dr. Richard Bisk - [email protected] Common Core – Grade 4
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.
Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Singapore – Grade 2
Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Common Core – Grade 5
5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
Dr. Richard Bisk - [email protected] If time
Dr. Richard Bisk - [email protected] • 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
Dr. Richard Bisk - [email protected] If time
Dr. Richard Bisk - [email protected] Dr. Richard Bisk - [email protected] Common Core – Grade 6
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.
Dr. Richard Bisk - [email protected] Tape Diagrams
Also called:
• bar diagrams • model drawing • bar models
Dr. Richard Bisk - [email protected] 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.
Dr. Richard Bisk - [email protected] -8 + 10
Dr. Richard Bisk - [email protected] 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.
Dr. Richard Bisk - [email protected] 7A 3(2x+4) = 6x+12
Dr. Richard Bisk - [email protected] 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.
Dr. Richard Bisk - [email protected] 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.”
“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, ….”
Dr. Richard Bisk - [email protected]
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.
Dr. Richard Bisk - [email protected]