West Virginia’s Next Generation
Mathematics Content Standards
and Objectives
Spring Conference for Federal Program Directors
and
Chief Instructional Leaders
Waterfront Place Hotel, Morgantown, WV
March 10, 2011
Lynn Baker, Math Science Partnership Coordinator
John 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 miseducation” 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-asusual 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 Mathematics
Nuts 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 Standards
Standards
Counting &
Cardinality
Operations &
Algebraic
Thinking
Number and
Operations in
Base Ten
Measurement &
Data
Geometry
Number &
Operations
Fractions
K
1
2
3
4
5
• 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.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?
•
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 reallife 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.
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 Mathematics
Nuts and Bolts
Please Compute These Differences
(-3) – (4)
-7
(17) – (4)
13
(-2) – (6)
-8
(-7) – (-12)
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)
-4
-3
-2
-1
1
0
1
1 space
2
3
4
What is the difference between 3 and 4?
How far (how many spaces) between
them?
Relate Subtraction (finding differences)
to the Number Line
(4) – (3)
7
7
spaces
-4
-3
-2
-1
0
1
2
3
4
What is the difference between -3 and 4?
How far (how many spaces) between
them?
Compare WV CSOs to CCSS
NxG WV CSOs 7th Grade –
Number Systems
21C WV CSO
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.
M.O.6.1.9 develop and test
hypotheses to derive the rules for
addition, subtraction,
multiplication and division of
integers, justify by using realworld examples and use them to
solve problems
High School and the Common Core
NxG 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*
Geometry
Algebra II Conceptual
Mathematics
Transition Math for Seniors
Electives:
Algebra III
Trigonometry
Probability and Statistics
Pre-Calculus
Calculus
Other college level math courses
*Available
NxG CSOsPathways in
West Virginia
Math I*
Math II
Math III(STEM)
Math III (LA)
Options for the required fourth math
credit:
Math IV
Transition Math for Seniors
Advanced Mathematical Modeling
STEM Readiness Mathematics
Technical Readiness Mathematics
AP Calculus
AP Statistics
Other college level math courses
in 8th grade
The Key to Drive Successful
Implementation
Teacher
Professional
Development and
On-Going Support
Mathematical Practices
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.