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Extreme Math Makeover: State
Assessment and Reporting in the
Era of the Common Core.
Mathematics
Department
Heads of the
West Chicago
Suburbs
November 14, 2011
mathematical modeling
Max McGee
[email protected]
1
Extreme Math Makeover: State
Assessment and Reporting in the
Era of the Common Core.
Mathematics
Department
Heads of the
West Chicago
Suburbs
November 14, 2011
mathematical modeling
Max McGee
[email protected]
2
Extreme Math Makeover: State
Assessment and Reporting in the
Era of the Common Core..
Mathematics
Department
Heads of the
West Chicago
Suburbs
November 14, 2011
mathematical modeling
Max McGee
[email protected]
3
Example from ACT

A car averages 27 miles per gallon. If gas costs
$4.04 per gallon, which of the following is
closest to how much the gas would cost for this
car to travel 2,727 typical miles?
A. $ 44.44
B. $109.08
C. $118.80
D. $408.04
E. $444.40
Examples from OECD (PISA test)
http://www.oecd.org/dataoecd/63/35/37464175.pdf
Example from OECD (PISA test)

For this question you need
to draw together
information from Figure 1
and Figure 2. The
disappearance of the
rhinoceros, hippopotamus,
and auruchs from Saharan
rock art happened:
A.
B.
C.
D.
At the beginning of the
most recent Ice Age.
In the middle of the
periods when Lake Chad
was at its highest level
After the level of Lake
Chad had been falling for
over a thousand years
At the beginning of an
uninterrupted dry period
One-third of the PISA mathematics
items are open-ended response
Advance Organizers

Common Core
Standards and
instructional practices

Assessment

Assessment Reporting
What are your impressions of
the Common Core Standards?
Common Core Standards –
Mathematical Practices
 Make
sense of problems and
persevere in solving them
 Reason abstractly and quantitatively
 Construct viable arguments and
critique the reasoning of others
 Model with mathematics
Common Core Standards –
Mathematical Practices (cont.)
 Use
appropriate tools strategically
 Attend to precision
 Look for and make use of structure
 Look for and express regularity in
repeated reasoning
Mathematical Practices
Describe ways in which students ought to engage with
the subject matter as they grow in mathematical
maturity and expertise throughout their school years.
Designers of curricula, assessments, and professional
development should all attend to the need to connect
the mathematical practices to mathematical content in
mathematics instruction.
Mathematical Content
… a balanced combination of procedure and
understanding. Expectations that begin with the word
“understand” are often especially good opportunities to
connect the practices to the content. Students who lack
understanding of a topic may rely on procedures too
heavily.
Grade 6 Mathematics
Multiple-Choice Sample
Assessment Objective 10.6.05:
Solve problems involving the
probability of a simple event,
including representing the
probability as a fraction, decimal,
or percent.
14
1. x = y
2. 2x = 2y
3. 2x + y = x + 2y
4. 2x + y - 2y = x + 2y - 2y
5. 2x + y - 2y = x
6. 2x + y - 2y - y = x - y
7. 2x - 2y = x - y
8. 2(x - y) = x - y
9. 2(x - y)
(x - y)
10. 2(x - y)
(x - y)
11.
2=1
(x - y)
=
=
(x - y)
(x - y)
(x - y)
Huh?
Here is an algebraic equation
that proves 2=1. Each step
contains a new operation.
Which step, if any, is
incorrect?
Advance Organizers

Common Core
Standards and
instructional practices

Assessment

Assessment Reporting
How about those assessments?
“… in examining the secondary standards,
curriculum, and assessments in Massachusetts,
California, and Texas, we discovered that it was
not the standards or even the curricula that
drove teachers and schools nuts, it was the
tests. The tests did NOT measure the standards,
particularly the higher order ones, well or even
at all. And they certainly did not do justice to the
curriculum.”
- David Pearson, 2010
Assessment “of” learning, “for”
learning, and “to” learn!

Summative

Formative

Performance


ISBE, IBHE, and the P-20 Council should lead the
nation in developing pioneering performance
assessment items that are aligned with the revised
Illinois Learning Standards and the 21stCentury Skills.
With the assistance of educators, ISBE should provide
exemplars of performance based assessment items,
student work samples, rubrics, and resource materials
(including videos and web based products) that
educators can use to model and assess the revised
standards and the 21stcentury knowledge and skills at
the district, school, and classroom levels.
The 24 PARCC states have committed to
building a K-12 assessment system that:





Builds a pathway to college and career readiness for
all students,
Creates high-quality assessments that measure the
full range of the Common Core State Standards,
Supports educators in the classroom,
Makes better use of technology in assessments, and
Advances accountability at all levels.
The PARCC design will incorporate four features
designed to significantly improve the quality and
usefulness of large-scale assessments:




Be anchored in college and career readiness,
Enable deeper and richer assessments,
Measure learning and provide information on student
progress throughout the school year, and
Provide timely results.
PARCC will also leverage technology throughout
the design and delivery of the assessment
system. The overall assessment system design
will include a mix of constructed response items,
performance-based tasks, and computerenhanced, computer-scored items. The PARCC
assessments will be administered via computer,
and a combination of automated scoring and
human scoring will be employed.
Grade 6 Mathematics
Multiple-Choice Sample
Assessment Objective 10.6.05:
Solve problems involving the
probability of a simple event,
including representing the
probability as a fraction, decimal,
or percent.
23
To prepare for the ISAT, one might
make a typical test item like this
one for fraction computation

2/3 of 1/4 is:
A.
B.
C.
D.
8/3
3/7
2/12
3/8
24
A Common Core Assessment
item for fractions may look like:

Use a blank sheet of paper to
demonstrate how you figure out what 2/3
of 1/4 is, OR in the space below draw a
model that shows how you figure out what
2/3 of 1/4 is.
25
What is Performance
Assessment?
An assessment that
affords students the
opportunity to
demonstrate
proficiency by
creating, producing,
presenting and/or
performing a
meaningful task.
Common Core Standards –
Third Grade

Measurement and Data 1.
Tell and write time to the nearest minute and measure
time intervals in minutes. Solve word problems involving
addition and subtraction of time intervals in minutes, e.g.,
by representing the problem on a number line diagram.

Operations and Algebraic Thinking 8.
Solve two-step word problems using the four operations.
Represent these problems using equations with a letter
standing for the unknown quantity. Assess the
reasonableness of answers using mental computation and
estimation strategies including rounding.3
Third Grade Performance
Based Assessment
The Flat Stanley Elementary School day starts at 9:00
a.m. and ends at 3:15 p.m. Students have 30 minutes
for lunch and 20 minutes for recess. Make a timeline
to show how long (in hours and minutes) are they in
class each day.
At Jerome Horwitz Elementary, the day starts at 8:30
and ends at 3:30. Students have 40 minutes for lunch
and 20 minutes for recess. Draw a picture to show
long they are in class each day.
(If the school year is 180 days long, write a number
sentence and make a chart to show how much longer
students at Jerome Horwitz spend in school?
Common Core Standards –
Eighth Grade
Statistics and Probability
Construct and interpret scatter plots for bivariate
measurement data to investigate patterns of association
between two quantities. Describe patterns such as
clustering, outliers, positive or negative association, linear
association, and nonlinear association.
Know that straight lines are widely used to model
relationships between two quantitative variables. For
scatter plots that suggest a linear association, informally
fit a straight line, and informally assess the model fit by
judging the closeness of the data points to the line.
Common Core Standards –
Eighth Grade
Use the equation of a linear model to solve problems in
the context of bivariate measurement data, interpreting
the slope and intercept..
Expressions and Equations 5.
Graph proportional relationships, interpreting the unit
rate as the slope of the graph. Compare two different
proportional relationships represented in different ways.
For example, compare a distance-time graph to a
distance-time equation to determine which of two
moving objects has greater speed.
Eighth Grade Performance Based Assessment: Draw
conclusions from this graph and explain your reasoning.
Eighth Grade Performance Based Assessment: Draw a
correlation line for the light blue dots and a correlation line for
the dark blue dots (and triangle), calculate the slopes and
explain why the two lines have similar or different slopes.
Common Core Standards –
Eleventh Grade

Mathematic Modeling
Modeling links classroom mathematics and statistics to
everyday life, work, and decision-making. Modeling is
the process of choosing and using appropriate
mathematics and statistics to analyze empirical
situations, to understand them better, and to improve
decisions. Quantities and their relationships in physical,
economic, public policy, social, and everyday situations
can be modeled using mathematical and statistical
methods. When making mathematical models,
technology is valuable for varying assumptions,
exploring consequences, and comparing predictions with
data.
Common Core Standards –
Eleventh Grade

Statistics and Probability
Making Inferences and Justifying Conclusions 6.
Evaluate reports based on data.
Use probability to make decisions 7.
(+) Analyze decisions and strategies using
probability concepts (e.g., product testing, medical
testing, pulling a hockey goalie at the end of a game).
High School Performance
Based Assessment
A regional city has had lots of problems with
gangs and violence over the years. The
mayor, chief of police, and city council need
your help. Review the available data,
analyze and model these data to give the
city a plan to reduce violence. After you
complete your analysis and model, prepare
a news release for the mayor briefly outlining
your proposals that recommend a campaign
strategy to curb the violence.
Performance
Based
Assessment
In the American League playoffs, Joe Girardi, the
Yankees manager made a practice of
intentionally walking a man when the first Texas
hitter doubled. Was this smart to do? Would it
be smart with one out? With two outs? Prepare
a presentation explaining your conclusion.
What model can you develop to assure that
all schools spend an equitable amount on
instruction per pupil?
Enrolled
% low
EAV pp
Tax Rate
% Pr Tx
Inst pp
Evanston
2895
39%
867 K
$2.10
84.3%
$11,542
Thornton
Fractional
1879
36%
285 K
$3.00
60.8%
$6,340
Quincy
1362
36%
106 K
$4.20
50.6%
$5,789
Example from ACT

What is the value of x when 2x + 3 = 3x – 4 ?
A. – 7
B. - 1/5
C. 1
D. 1/5
E. 7
Example from the U.K.
Example from ACT

What is the greatest common factor of 42, 126,
and 210 ?
F.
G.
H.
J.
K.
2
6
14
21
42
Example from Finland: Basic
A solution of salt and water contains 25 per cent salt.
Diluted solutions are obtained by adding water. How
much water must be added to one kilogram of the
original solution in order to obtain a 10 per cent
solution? Work out a graphic representation which
gives the amount of water to be added in order to get
a solution with 2–25 % of salt. The amount of water
(in kilograms) to be added to one kilogram of the
original solution must be on the horizontal axis; the
salt content of the new solution as a percentage
must be on the vertical axis.
Example from Finland: Advanced
In a society the growth of the standard of living
is inversely proportional to the standard of living
already gained, i.e. the higher the standard of
living is, the less willingness there is to raise it
further. Form a differential-equation-based
model describing the standard of living and
solve it. Does the standard of living rise forever?
Is the rate of change increasing or decreasing?
Does the standard of living approach some
constant level?
Example from ACT

The length, in inches, of a box is 3 inches less
than twice its width, in inches. Which of the
following gives the length, l inches, in terms of
the width, w inches, of the box?
F. l = w + 3
G. l = w + 3
H. l = w – 3
J. l = 2w + 3
K. l = 2w – 3
Example from Australia


Your task is to design a space to store
enough stackable chairs to seat all the staff
and students in your school.
You will:
1)
2)
3)
Follow a series of steps to help you design a
suitable space
Use a research journal to record your ideas and
rough working
Write a report on the process and solutions
Example from Australia (continued)


Develop mathematical models for each
dimension of a stack of chairs
To help you think about the practicalities of
storing chairs, use your model to find:
1)
2)
The greatest number of chairs in one stack that
can fit into a storage area with a 4m high ceiling
The number of stacks that fit across a 3.2 m
wide area if there are 10 chairs in each stack
Grade 6 Mathematics
Multiple-Choice Sample
Assessment Objective 10.6.05:
Solve problems involving the
probability of a simple event,
including representing the
probability as a fraction, decimal,
or percent.
47
48
49
La Jolla Spa
Vacation or a
Cool Convertible
Mustang
50
Autographed Copy of NCLB
51
Another State Mandate
52
Create a model that illustrates or
demonstrates why one of the following four
answers is the correct one.
What should the contestant do to increase his or
her chance of winning the car and spa vacation?
A.
B.
C.
D.
Stick with his or her first choice
Switch, i.e. select the other door
Either stick or switch--it doesn’t
matter because the odds are 50-50
Go with his or her “gut” feeling
53
Still don’t believe it? Try the
“Monty Hall Problem” yourself
http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html
http://math.ucsd.edu/~crypto/Monty/monty.html
Or just Google Let’s Make a Deal or Monty Hall Problem
54
Let's Make a Deal
Behind one of these doors is a car.
Behind the other two is a goat.
Click on the door that you think the car is behind.
55
Advance Organizers

Common Core
Standards and
instructional practices

Assessment

Assessment Reporting
Intel’s White Paper on Assessment
The growing move to promote assessment of, for and
as learning, rather than seeing testing as a separate
disjointed element of the education enterprise, may
provide opportunities for strengthening the teaching
and learning of 21st century skills, as well as their
assessment.
The growing emphasis on school-based performance
assessments in many countries appears to strengthen
teaching where teachers learn more deeply about how
to enact standards by participating in scoring and/or
reviewing student work.
Questions and comments welcome
Max McGee
224.234.6129 (cell)
[email protected]