West Virginia’s Next Generation Mathematics Content Standards

and ObjectivesSpring Conference for Federal Program Directors

and Chief Instructional Leaders

Waterfront Place Hotel, Morgantown, WV March 10, 2011

Lynn Baker, Math Science Partnership CoordinatorJohn Ford, Title I Mathematics Coordinator

Lou Maynus, Mathematics Coordinator

WV’s Next Generation Mathematics Content Standards and Objectives

• Why do we need yet another set?

• Sets of state and national math standards have come and gone in the past twenty years.

• So, how are these different?

• These standards are truly the next generation of standards - their demand on teachers' content knowledge is substantial.

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

Previous sets of standards focused on:

(1) whether to include a certain mathematical topic (e.g., the long division algorithm, logarithm),

(2) whether certain activities receive the correct emphasis (e.g., use of manipulative or use of estimation),(3) whether to do topic x in grade n (e.g., x = data and n = 3, or x = algebra and n = 8).

The underlying assumption has been that the mathematics of the school curriculum is well understood and it is only a matter of putting all the pieces together in the right way.

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

K-12 Mathematics is NOT well understood and it is MUCH MORE THAN a matter of lining up the pieces in

the right way.

• WV’s next generation set of standards are written to ensure depth of understanding of the required topics in mathematics.

• Getting the math right is a serious issue. If we don't get it right, our students cannot learn. Garbage in, garbage out. We as a nation have been suffering from this “mathematics mis-education” for decades.

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

You have all heard…

• “I’m just not a math person”• “I taught it, they just don’t get it”• “They don’t know their facts” • “If they only knew fractions, they could do

algebra”

• These are all manifestations of garbage in, garbage out.

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

We cannot teach what we do not know. We must KNOW the content.

Knowing a concept means knowing its precise statement, when it is appropriate to apply it, how to prove that it is correct, the motivation for its creation, and, of course, the ability to use it correctly in diverse situations. We cannot claim to know the mathematics of a particular grade without also knowing a substantial amount of the mathematics of three or four grades before and after the grade in question.

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

• Let us look at a simple topic: adding fractions. A set of state standards, long regarded as one of the best, has this to say:Grade 5. Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals.Grade 6. Students calculate and solve problems involving addition, subtraction, multiplication, and division.

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

• That is all. No need to go into details because we all know what to do, right?

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

• Another set of standards from a state that takes great pride in its work has this to say about adding fractions:Grade 4. Use fraction models to add and subtract fractions with like denominators in real-world and mathematical situations. Develop a rule for addition and subtraction of fractions with like denominators.Grade 5. 1. Add and subtract decimals and fractions, using efficient and generalizable procedures, including standard algorithms. 2. Model addition and subtraction of fractions and decimals using a variety of representations.

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

• Again, no need to go into details because we all know what to do, right?

• Wrong!

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

• Adding is supposed to “combine things". The concept of “combining" is so simple to children that it is always taught at the beginning of arithmetic.

• But did you see any “combining" in the preceding description of how to add ⅞ to ⅚ ?

• If children have made the effort to master the addition of whole numbers as “combining things”, they should rightfully expect the addition of fractions to the same. So how can they learn this hard to figure out procedure?

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

• NxG WVCSOs realizes that the business-as-usual kind of standards will not improve math education. So it approaches the addition of fractions as a progression from the simple to the complex, and spreads it across grades 3-5 to allow things to sink in.

• Its aim is to make students see that “adding” is “combining things"

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

• Altogether, these standards guide students through three grades to get them to know the meaning of adding fractions: Addition is putting things together, even for fractions, and the logical development ends with the formula

a/b + c/d = (ad + bc)/bd.• There is no mention of Least Common

Denominator. This is as it should be.

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

• Teachers have to be aware how a child learns about “combining things", and more importantly, have to know the mathematics so that they can teach in a way that respects the child's intuition about “combining things".

• The same can be said for the teaching of fractions and whole number in general. This will requires extensive professional development.

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

• Professional development (PD) unfortunately means different things to different people at the moment.

• We must provide PD that teaches deeply the basic topics of the mathematics we teach with precision, reasoning, and coherence.

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

• There have been criticisms that such detailed specifications in the standards on how to teach many topics are an imposition of pedagogical ideology on the teaching of mathematics. You now know, of course, that such criticisms can only come from people who don't recognize mathematics when they see it.

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

The detailed prescriptions in WVNGMS define the learning progressions and complexity of mathematical content required for career and college readiness in the 21st century.

Maynus - Adapted with permission from Hung-Hsi Wu, co author of CCSS

Mathematics

K-5 MathematicsNuts and Bolts

• Provide greater focus and coherence.• Are based on what is known today about how

students’ mathematical knowledge, skill, and understanding develop over time.

• Focuses on the development of mathematical understanding and procedural skills using rich mathematical tasks.

K-5 Mathematical StandardsStandards K 1 2 3 4 5

Counting & Cardinality

Operations & Algebraic Thinking

Number and Operations in

Base TenMeasurement &

Data

Geometry

Number & Operations Fractions

• M.2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

2 + 2 + 2 = 6

Mathematical Content Moved to a Different Grade

• M.O.K.4.6 identify the name and value of coins and explain the relationships between:

– penny – nickel– Dime

• M.O.1.4.6 identify, count, trade and organize the following coins and bill to display a variety of price values from real-life examples with a total value of 100 cents or less.

– penny – nickel – dime – quarter – dollar bill

• M.O.2.4.7 identify, count and organize coins and bills to display a variety of price values from real-life examples with a total value of one dollar or less and model making change using manipulatives.

• M.O.3.4.5 identify, count and organize coins and bills to display a variety of price values from real-life examples with a total value of $100 or less and model making change using manipulatives.

• M.2.MD.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ￠symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

WV Content No Longer Included in Standards

• M.O.K.4.4 use calendar to identify date and the sequence of days of the week.

6-12 MathematicsNuts and Bolts

Please Compute These Differences

(-3) – (4)

(17) – (4)

(-2) – (6)

(-7) – (-12)

-7

13

-8

5

Remember a Rule

Subtraction means “add the opposite (additive inverse)”; so 4 – 3 means 4 + (-3) = 1

4 – (-3) means 4 + (3) = 7 (Since the rule for adding is: signs same, find

the sum, signs different, find the difference)

Relate Subtraction (finding differences) to the Number Line

-4 -3 -2 -1 0 1 2 3 4

Relate Subtraction (finding differences) to the Number Line

-4 -3 -2 -1 0 1 2 3 4

(4) – (3) 1

What is the difference between 3 and 4? How far (how many spaces) between them?

1 space

Relate Subtraction (finding differences) to the Number Line

-4 -3 -2 -1 0 1 2 3 4

(4) – (3)

What is the difference between -3 and 4? How far (how many spaces) between them?

7 spaces

7

Compare WV CSOs to CCSS

NxG WV CSOs 7th Grade – Number Systems

M.7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a number line.

(c) Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (-q). Show the distance between two numbers on the number line is the absolute value of their difference, and apply this in real world contexts.

21C WV CSO

M.O.6.1.9 develop and test hypotheses to derive the rules for addition, subtraction, multiplication and division of integers, justify by using real-world examples and use them to solve problems

High School and the Common CoreNxG CSOs are organized by conceptual category, not by courses. Our work has been to group the

standards into courses and the courses into pathways.

Categories:

Number and Quantity

Algebra

Functions

Geometry

Modeling

Probability and Statistics

West Virginia will be using this

pathway

What’s Different in High School?Current High School Pathways

Algebra I*

GeometryAlgebra II Conceptual

MathematicsTransition Math for SeniorsElectives:

Algebra III Trigonometry

Probability and StatisticsPre-Calculus

CalculusOther college level math courses

NxG CSOsPathways in West Virginia

Math I*

Math IIMath III(STEM)Math III (LA)Options for the required fourth math

credit:Math IV

Transition Math for SeniorsAdvanced Mathematical Modeling

STEM Readiness MathematicsTechnical Readiness Mathematics

AP CalculusAP Statistics

Other college level math courses

*Available in 8th grade

The Key to Drive Successful Implementation

Teacher Professional

Development and On-Going Support

Mathematical Practices1. 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.