Transcript 3 - GuidedMathLTPSWiki

Core Content State Standards
… for Learning
… as Learning
Math in Grades 3-12
Daniel J. Heisey, Ph.D.
Mathematics Coordinator
NJ Department of Education
[email protected]
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Fear and Angst
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Presentation Roadmap
• NAEP – gatekeeper for NCLB
• Conceptual understanding of fractions
• Core Content State Standards [CCSS]
– Grade 3 understand fractions as numbers and
build fractions from unit fractions
– Grade 4 add and subtract fractions with like
denominators and generate decimal equivalents
– Grade 5 multiply fractions (including mixed
numbers) and divide fractions in special cases
• Diagnostics and learning progressions
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NAEP Data
Reading and Math Gap
(ETS summary, 2010)
African American, Latino, and poor of all races
vs.
their wealthier, mostly white peers
– By 4th grade, they are 2 years behind
– By 8th grade, they are 3 years behind
– By 12th grade, they are 4 years behind
High School Dropouts (75% in jail are school dropouts)
68% of six graders score basic (or worse) in math
Only 17% of H.S. seniors are proficient in math
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NAEP Released Item
What fraction of the figure is shaded?
Answer: _______________
Did you us a calculator on this question? ______
Grade 4 NAEP 2007
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22% incorrect
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NAEP Released Item
Luis is making a game spinner. He wants the chance
of landing on red (R) to be twice the chance of
landing on blue (B). Show how he could label his
spinner.
Number of blues ____
Number of reds ____
Grade 4 NAEP 2007
59% incorrect
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Gene Wilhoit
Gene Wilhoit is the executive director of the
Council of Chief State School Officers [CCSSO].
This video was made in November 2010.
It is a presentation to the U. S. Congress about
the federal role in the state-initiated Common
Core initiative … a historical perspective.
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Gene WilHoit in Congress
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Fraction beginnings: Part vs. Whole
Grade 2: CCSS 2.G.3
Partition circles and rectangles into two or four equal shares and
describe shares using words halves, thirds, half of, third of, etc.,
and describe the whole as two halves, three thirds, four fourths.
Recognize equal shares of identical wholes need not have the
same shape.
Numerals for fractions (1/2, 1/3, 1/4, etc.) are not used at grade 2.
Two fractions are compared only if they refer to the same whole.
Grade 3:
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.
Note: The above is not a complete map (e.g. proportional relationships).
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Critical areas in grades 3-5 -- plus grade 6 ….. HANDOUT #1
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The first mention of multiplying
or dividing fractions in NJCCCS
was “implied” in this single
standard in grade 6
>>>>>>>>>>>
There is no lead up, and no
progression of ideas. The grade
5 NJCCCS is not very helpful.
In Grade 5:
4.1.5 B. Numerical
Operations
2. Construct, use, and explain
procedures for performing
addition
and subtraction with fractions
and decimals with:
•Pencil-and-paper
•Mental math
•Calculator
In Grade 6:
4.1.6 B. Numerical
Operations
2. Construct, use, and explain
procedures for performing
calculations with fractions and
decimals with:
•Pencil-and-paper
•Mental math
•Calculator
NJCCCS vs. CCSS 4.NF.3 & 4.NF.4 …………... HANDOUTs #2 & #3
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Common Core demands we revamp the mile-wide, inch-deep
approach in curriculum and textbooks.
Key moments in the curriculum (like 4.NF.4) demand that we slow
down and devote more time to allow for reasoning / thinking /
discussing as well as the necessary hard work and practice.
CCSS gives three years to the division of fractions thread:
• In grade 4, we multiply a fraction by a whole number.
• In grade 5, we multiply a fraction by a fraction and we divide a unit
fraction by a whole number or a whole number by a unit fraction.
• Finally, in grade 6, we divide a fraction by a fraction.
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(1) Build on prior work of multiplying whole numbers.
(2) Build fractions from unit fractions:
5/4 means 5 x ¼. It is also ¼ + ¼ + ¼ + ¼ +
¼, which builds from unit fractions using additive
reasoning. (see 4.NF.3)
(3) We achieve the more complex case 5 x ¾ by
saying, 5 times 3/4 equals (5 times 3) fourths
equals 15 fourths. This stresses properties of
operations, making arithmetic a rehearsal for algebra.
Source: http://www.p12.nysed.gov/ciai/mst/math/standards/revisedg4.html
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Fraction progressions …
• Grade 3: Develop an understanding of fractions as numbers.
• Grade 4: Understand fraction equivalence and ordering.
Build fractions from unit fractions and apply and extend
previous understandings of operations on whole numbers.
Use decimal notation for fractions and compare fractions.
• Grade 5: Use equivalent fractions to add and subtract
fractions (including mixed numbers w unlike denominators).
Apply and extend previous understandings of multiplication
and division to multiply and divide fractions.
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Thinking about 3/4 …
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Unit Fractions … 1/b … as building blocks
Grade 3: Students name the “numeral” for each unit fraction as
1/2, 1/3, 1/4, etc. Fractions are built from unit fractions.
3/4 is 3 copies of 1/4 … 3/4 = 1/4 + 1/4 + 1/4
4/5 is 4 copies of 1/5 … 4/5 = 1/5 + 1/5 + 1/5 + 1/5
Note: No need to introduce proper or improper fractions. The
quantity 5/3 is the sum of 5 parts of a whole divided into 3
equal parts. Students can easily find 5/3 on the number line.
Grade 3: Students identify “equivalent” fractions as having the
same size or the same position on the number line.
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Domain: Number – Fractions
Cluster: Develop understanding of fractions as numbers.
3.NF.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.
3.NF.2 Understand a fraction as a number on the number line;
represent fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by
defining the interval from 0 to 1 as the whole and partitioning it
into b equal parts … each part has size 1/b and the endpoint of
that part based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by
marking off a lengths 1/b from 0. Recognize that the resulting
interval has size a/b and that its endpoint locates the number
a/b on the number line.
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Domain: Number – Fractions
Cluster: Develop understanding of fractions as numbers.
3.NF.3 Explain equivalence of fractions in special cases, and
compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are
the same size, or at the same point on a number line.
b. Recognize and generate simple equivalent fractions, e.g.,
1/2 = 2/4, 4/6, = 2/3 … explain using visual fraction models.
c. Express whole numbers as fractions, & recognize fractions
equivalent to whole numbers.
Examples: Express 3 as 3/1 & recognize 6/1 = 6; locate 4/4 and 1 at the same
point on a number line.
d. Compare two fractions with the same numerator or
denominator by reasoning about their size.
Comparisons need to refer to the same whole.
Use symbols >, =, < for comparisons … explain using visual fraction models.
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Conceptual Understanding
¾
What happens to the value of a fraction if:
-- the numerators is increased by 1 ?
-- the denominator is decreased by 1 ?
-- the denominator is increased by 1 ?
-- the value of the numerator and denominator are doubled ?
Which comparison is true? … Explain WHY using a number line.
½<¾
versus
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Grade 4 Fractions: Equivalent Fractions
Grade 4: CCSS 4.NF.1 Explain why a fraction a/b is
equivalent to a fraction (n x a)/(n x b) using fraction
models, with attention to how the number and size
of the parts differ even though the two fractions are
the same size. Use this principle to recognize and
generate equivalent fractions.
This fundamental property of equivalent fractions
impacts much of the computation and procedural
work in grade 4 …
Comparing … adding … subtracting … finite decimals …
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Grade 4 : Fraction Questions
Create an area model for 2/3 = (4 x 2)/(4 x 3).
How can 2/5 + 1/3 be represented as a length?
Part of a whole vs. Part of a part.
Convert mixed numbers to improper fractions.
Multiply fractions by whole numbers.
What visual fraction model shows 0.2 > 0.17 ?
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Area models
multiply whole numbers 14 x 6
Each block shown represents 1/6
multiply unit fractions 1/2 x 1/3
1/3 of 1/2 = 1/6
1/2 of 1/3 = 1/6
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Multiply Fractions
2/3 x 4/5 = 8/15
(Area Model)
(2-dimensional arrays)
2/3 of 15/15 = 10/15
4/5 of 15/15 = 12/15
either
… or
8/15 is 4/5 of 10/15
8/15 is 2/3 of 12/15
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Multiply mixed numbers
(area model)
A typical grade 5 problem 4 ½ x 7 ¼ = ??
A composite model showing 4 areas
7
¼
4
½
4 ½ x 7 ¼ = (4 x 7) + (½ x 7) + (4 x ¼) + (½ x ¼)
32 ⅝ = 28 + 3 ½ + 1 + ⅛
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Length model :: Unit Fractions
How can 2/5 + 1/3 be represented as a length?
Draw a model of 2/5 plus 1/3 using the number line.
0
1
Reference: Lamon, Susan. Teaching Fractions and Ratios for
Understanding, 2nd Edition, New York, NY: Routledge, 2008.
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Length model :: Unit Fractions
What fraction is located at the point “X”?
0
X
2/3
Reference: Lamon, Susan. Teaching Fractions and Ratios for
Understanding, 2nd Edition, New York, NY: Routledge, 2008.
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Part of a whole vs. Part of a part
Part of a part: Bill had 2/3 of a cup of juice. He drank
1/2 of his juice. How much juice did Bill have left?
This problem cannot be solved by subtracting
2/3 - 1/2 because the 2/3 refers to a cup …
but the 1/2 refers to the amount of juice Bill
had and not to a cup of juice.
A similar problem: If ¼ of a garden is planted with daffodils, 1/3
with tulips and the rest with vegetables, what fraction of the
garden is planted with flowers?
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Mixed and Improper Fractions
4 2/5 = 4 + 2/5 = 1 + 1 + 1 + 1 + 2/5
Tricks are not helpful to students with memory problems,
think about the concept and purpose to the calculations
“Four and two – fifths”
… “and” says to add …
4 + 2/5 = 5/5 + 5/5 + 5/5 + 5/5 + 2/5 = 20/5 + 2/5 = 22/5
Trick: The whole number times the fraction’s denominator
plus the numerator equals the new numerator. The number in
the denominator does not change.
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Multiplication of fractions by a whole number
4 2/5 is not the same as 4 x 2/5
Tricks are not always helpful …
“Four times two fifths”
… “times” means add 2/5 four times
4 x 2/5 … 2/5 + 2/5 + 2/5 + 2/5 = 8/5
TRICK: Multiply the whole number by the numerator and
that product becomes the new numerator in the answer.
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Visual models for decimal fractions
The “tenth” scale vs. the “hundredths” scale …
0.2 < 0.17 is not true
2 is “less than” 17 but …
0.2 = 0.20 which is 0.03 greater than 0.17
Compare all fractions by referencing the same part ...
0
0.1
0.2
1/4
0.3
0
0.10
0.20
0.25
0.30
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How do Fraction Models Differ Across Grades
Concrete Manipulatives: Grades K-2
Representational Drawings: Grades 2-3
Abstract Number Procedures: Grades 3-4
1. There were 16 apples. Rhonda ate 1/4 of them. How many
are left?
2. Mom used one-third of 12 eggs. How many are left?
3. Tom ate 4 hazelnuts, which was 1/8 of the nuts. How many
were there in the first place?
4. Lisa used $5 to buy a gift which took 1/3 of her savings. How
much did she have in the beginning? How much does she
have now?
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The Khan Academy – a resource
The web-access to this video library is free and individual
videos can be purchased for 99¢. The videos show only a
blackboard and chalk writing. The invisible “teacher” explains
the math by audio.
Not the lessons are not
aligned to the CCSS skills.
For example, the video is
multiplying fractions (a
grade 5 CCSS skill) with
integer math included
(a grade 6 CCSS skill).
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COMMON CORE MATH
STANDARDS
Clearer, Fewer, Higher
No more …
… content a mile wide and an inch deep
… standards that require “un-packing”
… skills re-taught (spiraled) in the next grades
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Addition of Whole Numbers
7
6
Grade Level
5
4
3
2
1
0
American Institute for Research
Steve Leinwand
States
Culminating Learning Expectation
Intermediate Expectations
Initial Learning Expectation
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NJCCCS
CCSS
% of Standards/ CPI’s
100%
90%
80%
70%
Data, Prob, & Stat
60%
Algebra
50%
Geometry
40%
Number
30%
20%
10%
0%
2
3
4
5
6
7
8
12
2
Grade Levels
3
4
5
6
7
8
12
Grade Levels
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The CCSS Format of Standards
Domain
(Topic)
Grade
Level
Standard
Algebra
Symbol
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CCSS Developer Commentary
http://www.americaschoice.org/uploads/Comm
on_Core_Standards_Resources/PhilDaro_Mat
hStandards/PhilDaro_MathStandards.html
In this video Phil Daro applauds Common Core
developers as being mindful of common sense
skill levels selection “less is more” idea.
http://successatthecore.com/teacher_development_featured_video.aspx?v=44
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CCSS Assessments coming 2014-15
In 2014-15, assessments for the Common Core will be
administered via the internet.
The Partnership for Assessment of Readiness for College and
Careers [PARCC] is an alliance of states working together to
develop common assessments serving nearly 25 million
students. PARCC’s work is funded through a four-year 185
million dollar grant from the U.S. DOE.
PARCC’s partners include 200 higher education institutions
and is led by its member states and managed by Achieve.
PARCC’s goal is to ensure that all students graduate from high
school college and career ready. (See www.parcconline.org )
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Kinds of Assessments
•
Readiness
•
Benchmarks
•
Diagnostics
• Common Core is a new paradigm. Teachers will teach to
mastery so that all students can demonstrate mastery.
• Diagnostic assessments will be needed to manage the
intervention events for struggling math students.
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ASSESSMENT NOTE RE: Student Progress
“It is impossible for a norm-referenced test to
align with standards. Norm-reference tests
tell you how well students are doing
compared to each other …
Standards mean that student progress must
be compared to the standard, not to how well
or poorly others do.”
• Quoted from … Workshop: Teaching to Academic Standards
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Readiness Test Items
1. A batch of muffins requires 2/3 cups of sugar. A loaf of
zucchini bread requires 3/4 cups of sugar. How many cups of
sugar are needed to make one batch of each recipe?
a. 5/7
b. 1 5/12
c. 1 1/2
d. 1 7/12
2. What is 4 divided by one-half?
a. 1/2
b. 1/8
c. 2
d. 8
3. If AX = B and A, B, and X are not equal to zero, then ______ ?
a. X = A divided by B
b. X = A minus B
c. X = B divided by A
d. X = B times A
4. Which number is largest ?
a. 0.72
b. 0.080
c.
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d. 6/7
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Diagnostic Test Items
Find the equivalent common fraction.
1. 1/4
a) 11/14
b) 3/5
c)
2/8
d) 4/1
a) 6/5
b)
c)
15/16
d) 5/11
a) 7/4
b) 13/4
c)
7/14
d) 3/14
a) 32/5
b)
c)
6/5
d) 2/32
2. 5/6
50/60
3. 1 3/4
4. 3 2/5
17/5
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Diagnostic Testing
Diagnostic tests assess fewer skills – “less is more”
Skills are ordered as a learning progression
-- Four test items assess each skill
-- Each Item has similar difficulty
-- 3 or 4 items correct demonstrates mastery
Guessing is NOT a factor: guess 1 out of 4
guess 2 out of 4
guess 3 out of 4
guess 4 out of 4
25%
6.25%
1.6%
0.4%
… very few false positives !
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Part 1 Equivalent common fractions [4]
Part 2 Place value [4]
Part 3 Equivalent decimals [4]
Part 4 Add & subtract [4]
Part 5 Multiply & divide [4]
• Student scores and number correct per skill (left tally matrix)
• Student mastery indicators (right OK matrix)
Name
Score
1 2 3 4 5 %
Jim E
11
2
4
0
3
2
55
Mark G
9
0
2
0
4
3
45
Sue T
14
3
4
0
4
3
70
Mike W
10
0
3
0
4
3
Jill D
12
1
4
1
3
Paul R
12
0
4
1
Dan H
11
2
4
0
1
2
OK
3
4
5
OK
O-E
2
OK
OK
1
OK
OK
OK
2
50
OK
OK
OK
3
3
60
Ok
OK
OK
3
3
3
60
OK
OK
OK
3
3
2
45
OK
OK
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0
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Classroom Learning Progressions
A learning progression is a carefully
sequenced set of building blocks that students
must master en route to mastering a more
distant curricular aim.
-- James Popham, ASCD, 2007
Classroom learning progressions are:
– often “customized”;
– not etched in stone;
– often referred to as learning trajectories; and,
– needed to find where students are at.
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Fractions and Word Problems
A scoop holds ¾ of a cup. How many scoops of bird
seed are needed to fill a bird feeder that holds 3 cups?
Show how to use pictures to solve this problem.
… or explain your solution in words.
Number line
l__ı__ı__ı__l__ı__ı__ı__l__ı__ı__ı__l
1
1
1
2
2
2
3
3
3
4
4
4
Arrays
1 2 3 1
2 3 1 2
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Length model … explained …
Draw a model of 2/5 plus 1/3 using the number line.
0
← 2/5 →
↑ ← 1/3 → ↑
11/15
1
count unit fractions … 1/5 = 3/15 … 1/3 = 5/15
therefore ... 2/5 + 1/3 = 6/15 + 5/15 = 11/15
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Length model … explained …
What fraction is located at the point “X”?
0
X
2/3
1
2/3 equals 10 units
1/3 equals 5 units
1 equals 15 units = 15/15
count 6 unit fractions of 1/15 to the “X” = 6/15 = 2/5
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ONE FINAL NOTE:
BEWARE OF MISCONCEPTIONS
+
=
Does the above sketch represent 4/6 + 4/6 = 8/12 ?
… what is wrong with this picture?
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MISCONCEPTION FIX
+
=
A major goal for K-8 math needs to be a
proficiency with fractions … students
lost in math can recover if we can find a
way to teach them where they are at.
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Bob Moses Re: The Algebra Project
Since 1985 Robert Moses has been building a
consortium of Southern schools with the aim
of graduating more students in the poor rural
areas of Alabama and Mississippi.
http://flash.unctv.org/ncnow/ncn_rmoses_081408.html
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References
Anderson, Charles W. and Mohan, Lindsey. “Learning Progressions to Inform the Development of
Standards,” Michigan State University, in Learning Progressions and Standards, Vol 2: AERA, 2009
Bahr, Damon L. and Kathleen, and DeGarcia, Lisa Ann. Elementary Mathematics is Anything but
Elementary: Content and Methods from a Developmental Perspective, Paperback ISBN 0618928170:
Cengage Brain, 2008. (Amazon $100.00)
Clements, Douglas H. and Sarama, Julie Learning and Teaching Early Math New York, NY: Routledge, 2008
English, Lyn D. and Graeme S. Halford. Mathematics Education: Models and Processes, Mahwah, NJ:
Lawrence Erlbaum Associates, 1995.
Lamon, Susan. Teaching Fractions and Ratios for Understandin” 2nd Edition, New York, NY: Routledge,
2008.
Lamon, Susan. MORE In-depth Discussion of the Reasoning Activities in Teaching Fractions and Ratios for
Understanding 2nd Edition, New York, NY: Routledge, 2009
Lamon, Susan. “Rational Numbers and Proportional Reasoning: Toward a Theoretical Framework for
Research.” In Handbook of Research on Mathematics Teaching and Learning, edited by Douglas
Grouws, pp. 629-67. Reston, VA: National Council of Teachers of Mathematics, 2007.
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References
Loveless, Tom, “The Misplaced Math Student Lost in Eighth-Grade Algebra.” Brown Center on Educational
Policy, 2008
National Mathematics Advisory Panel. The Final Report of the National Mathematics Advisory Panel.
Washington, DC: US Department of Education, 2008.
PARCC Math Frameworks for Grades 3 – 11 (76 pages) http://www.parcconline.org/parcc-content-frameworks
Popham, James W., (2007, April). “The Lowdown on Learning Progressions.” Educational Leadership, 64(7), pp.
83-84
Sanders, S., Riccomini, P. J., & Witzel, B. S. (2005). “The algebra readiness of high school students in South
Carolina: Implications for middle school math teachers.” South Carolina Middle School Journal.
Van de Walle, John A. Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA:
Pearson Education, 2007.
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