Transcript PILMtg.020212.AHS - Silicon Valley Mathematics Initiative

Thoughts about
Mathematical Sense-Making:
Where we’re heading with the
Common Core Standards and
Smarter Balanced Assessments
and How to Get Ready for It
Alan H. Schoenfeld
University of California
Berkeley, CA, USA
[email protected]
Today’s Activities
1. Playing with some mathematics
2. What’s coming down the pipe –
Common Core Standards,
Smarter Balanced Assessments
3. Thoughts about what to look for in
productive mathematics classrooms
4. Q&A at any time
5. Lunch
1. Let’s Play!
David says,
If you draw in the two diagonals of
a quadrilateral, you divide the
quadrilateral into four equal areas.
The question: Is David’s claim
(a)Always right?
(b)Sometimes right?
(c)Never right?
Evaluating Statements About Length & Area
Student Materials
Always, Sometimes,
or
Never
True?
Card Set A: Always, Sometimes, o
A
Cutting Shapes
B
Slid
When you cut a piece off a shape
you:
If you slide the
left to right:
(a) Reduce its area.
(b) Reduce its perimeter.
(a) Its area stay
(b) Its perimete
C
D
Rectangles
Medi
h & Area
Student Materials
Beta Version
Always,
Sometimes,
orTrue?
Never
Always,
Sometimes,
or Never
apes
shape
es
B
Sliding a Triangle
If you slide the top corner of a triangle from
left to right:
(a) Its area stays the same.
(b) Its perimeter changes.
D
Medians of a Triangle
True?
you:
left to right:
(a) Reduce its area.
(b) Reduce its perimeter.
(a) Its area s
(b) Its perime
C
D
Always, Sometimes, or Never True?
Rectangles
Me
P
Draw a diagonal of a rectangle and mark any If you join ea
point on it as P. Draw lines through P, parallel midpoint of t
to the sides of the rectangle. The two shaded you get all ha
rectangles have:
(a) Equal areas.
(b) Equal perimeters.
E
Square and Circle
F
Midpo
ape
If you slide the top corner of a triangle from
left to right:
(a) Its area stays the same.
(b) Its perimeter changes.
Always, Sometimes, or Never True?
D
Medians of a Triangle
e and mark any If you join each vertex of a triangle to the
ough P, parallel midpoint of the opposite side, the six triangles
The two shaded you get all have the same area.
rcle
F
Midpoints of a Quadrilateral
Draw a diagonal of a rectangle and mark any If you join e
point on it as P. Draw lines through P, parallel midpoint of
to the sides of the rectangle. The two shaded you get all
rectangles have:
Always,
Sometimes,
or Never True?
(a) Equal
areas.
(b) Equal perimeters.
E
Square and Circle
If a square and a circle have the same
perimeter, the circle has the smallest area.
F
Midp
If you join t
quadrilater
one half the
le and mark any If you join each vertex of a triangle to the
hrough P, parallel midpoint of the opposite side, the six triangles
The two shaded you get all have the same area.
Always, Sometimes, or Never True?
ircle
F
the same
smallest area.
If you join the midpoints of the sides of a
quadrilateral, you get a parallelogram with
one half the area of the original quadrilateral.
Midpoints of a Quadrilateral
David says,
If you draw in the two diagonals of
a quadrilateral, you divide the
quadrilateral into four equal areas.
The question: Is David’s claim
(a)Always right?
(b)Sometimes right?
(c)Never right?
Student Work 1
If you draw in the two diagonals of a
quadrilateral, you divide the
quadrilateral into four equal areas.
12
Student Work 2
If you draw in the two diagonals of a
quadrilateral, you divide the
quadrilateral into four equal areas.
13
Some Questions. Did You Have To
• Make sense of problems and persevere in
solving them?
• Reason abstractly and quantitatively?
• Construct and critique viable arguments?
• Model with mathematics?
• Use appropriate tools strategically?
• Attend to Precision?
• Look for and make use of structure?
• Look for and express regularity in
repeated reasoning?
More Questions.
• Was there honest-to-goodness math in
what we did?
• Did you engage in “productive struggle,”
or did I dumb it down to where you
didn’t?
• Who had the opportunity to engage? A
select few, or everyone?
• Who had a voice? Did people get to say
things, develop ownership?
• Did instruction find out what you know,
build on it?
2. Thoughts about
Mathematical Sense-Making:
Where we’re heading with the
Common Core Standards and
Smarter Balanced Assessments
and How to Get Ready for It
Not Sense-Making:
How many two-foot boards can be
cut from two five-foot boards?
National Assessment of
Educational Progress, 1983:
An army bus holds 36 soldiers. If 1128
soldiers are being bussed to their training
site, how many buses are needed?
29%
31R12
18%
31
23%
32
30%
other
Kurt Reusser asks 97 1st and
2nd graders:
There are 26 sheep and
10 goats on a ship.
How old is the captain?
76 students "solve" it,
using the numbers.
H. Radatz gives non-problems
such as:
Alan drove the 12 miles from his
house in Berkeley to the Tilden
Early Childhood Center at 3 PM.
On the way he picked up 2 friends.
Sense-Making
What happens when you
add two odd numbers?
7
+
9
7
+
9
7
+
9
The Challenge:
To make sense of:
- The (Common Core) Standards
- High Stakes Assessment and what it’s
likely to mean in California
- Formative Assessment as a
mechanism for making good stuff
happen in our classrooms.
Let’s start with context.
The Common Core State Standards
in Mathematics (CCSSM) now exist.
But what do they mean?
Huh?
What do you mean, what do they mean?
The words are there on the page…
Remember Alice and Humpty
Dumpty?
Here’s WC Fields as Humpty Dumpty in
the 1933 film “Alice in Wonderland”
“When I use a word,” Humpty
Dumpty said in rather a
scornful tone, “it means just
what I choose it to mean –
neither more nor less.”
And so it is with Standards
(Common Core or otherwise)
What defines the Standards?
In today’s high stakes context,
it’s the assessments.
And in California, that’s meant
the CST.
Why is this such a problem?
WYTIWYG
But, the CST is going away…
So things will change.
How, and what might we do?
That’s the rest of the
conversation.
First, the Standards:
Content and Practices
(Alan’s Biased Predictions)
Content: Getting Richer
Practices: A BIG Opportunity
The Practices in CCSS-M:
• Make sense of problems and persevere
in solving them.
• Reason abstractly and quantitatively.
• Construct and critique viable arguments
• Model with mathematics
• Use appropriate tools strategically
• Attend to Precision
• Look for and make use of structure
• Look for and express regularity in
repeated reasoning.
Remember the “processes” in
the ‘89 NCTM Standards:
Mathematics as Problem Solving
Mathematics as Communicating
Mathematics as Reasoning
Mathematics as Connections
Remember the goals of the 1992
CA Mathematics Framework:
Mathematical Power
Mathematical Performance
Large Assignments
Complete Work
Remember NCTM’s (2000)
Principles and Standards:
Five Content Standards:
Number & Operations
Algebra
Geometry
Measurement
Data Analysis and Probability
Remember NCTM’s (2000)
Principles and Standards:
And Five Process Standards:
Problem Solving
Reasoning and Proof
Communication
Connections
Representation
It’s no exaggeration to say that
all of these things “count” in
the Common Core Standards.
But will they count in California?
It’s looking like the answer is
YES
And the reason is …
Assessment
Specifically, the Smarter
Balanced Assessment
Consortium (SBAC)
http://www.k12.wa.us/smarter/
(Just google SBAC)
Here are some of the
headlines.
Four Major Claims [Dimensions for Assessment]
for the SMARTER Balanced Assessment
Consortium’s assessments of the
Common Core State Standards for Mathematics
Claim #1 - Students can explain and apply mathematical concepts and
interpret and carry out mathematical procedures with precision and
fluency.
Claim #2 - Students can solve a range of complex well-posed problems
in pure and applied mathematics, making productive use of knowledge
and problem solving strategies.
Claim #3 - Students can clearly and precisely construct viable
arguments to support their own reasoning and to critique the
reasoning of others.
Claim #4 - Students can analyze complex, real-world scenarios and can
construct and use mathematical models to interpret and solve
problems.
Total Score for Mathematics
Content and Procedures Score
Grade 3 C&P Sub-scores
Operations & Algebraic Thinking
Number/Ops – Fractions
Measurement & Data
Grade 4 C&P Sub-scores
Operations & Algebraic Thinking
Number/Ops – Base 10
Number/Ops – Fractions
Measurement & Data
Grade 5 C&P Sub-scores
Number/Ops – Base 10
Number/Ops – Fractions
Measurement & Data
Problem
Communicating
Grade 6 C&P Sub-scores
Solving
Reasoning
Number System
Score
Ratio & Proportion
Expressions & Equations
Grade 7 C&P Sub-scores
Number System
Ratio & Proportion
Expressions & Equations
Grade 8 C&P Sub-scores
Expressions & Equations
Functions
Geometry
High School C&P Sub-scores
Number & Quantity
Algebra
Functions
Using
Models
Total Score for Mathematics
Content and
Procedures
Score
Problem
Solving
Score
40%
20%
Communicating Mathematical
Reasoning
Modeling
Score
Score
20%
20%
So, OK… but, what do the
tasks look like?
“Hurdles Race.”
Think of the Content involved:
• Interpreting distance-time graphs in a
real-world context
• Realizing “to the left” is faster
• Understanding points of intersection in
that context (they’re tied at the moment)
• Interpreting the horizontal line segment
• Putting all this together in an explanation
Think of the Practices involved:
• Make sense of problems and
persevere in solving them.
• Reason abstractly and quantitatively.
• Construct viable arguments…
• Model with mathematics…
25% Sale, Part 1
In a sale, all the prices are reduced by 25%.
Julie sees a jacket that cost $32 before the
sale. How much does it cost in the sale?
25% Sale, Part 2
In the second week of the sale, the prices are
reduced by 25% of the previous week’s price.
In the third week of the sale, the prices are again
reduced by 25% of the previous week’s price.
In the fourth week of the sale, the prices are
again reduced by 25% of the previous week’s
price.
Alan says that after 4 weeks of these 25%
discounts, everything will be free. Is he right?
Explain your answer.
Again:
Core content, central practices.
Want to see more?
Check out the SBAC specs
The Mathematics Assessment
Project (google the name or go to
http://map.mathshell.org/materials/)
OK, you say,
But does a difference in
tests really matter?
From an SVMI study of 16,420
kids taking the MARS and SAT-9:
So, how do we prepare kids to
do well on assessments like the
Smarter Balanced Assessments?
(I thought you’d never ask!)
There are resources on the web:
-Mathematics Assessment Project
-Silicon Valley Math Initiative
-Inside Mathematics
- Math Forum
web sites
And, we can do more…
By way of formative assessment.
The purpose of formative
assessments is not simply to show
what students “know and can do”
after instruction,
but to reveal their current
understandings so you can help
them improve.
Important Background Issues
1. Formative assessment is not summative
assessment given frequently!
2. Scoring formative assessments rather
than or in addition to giving feedback
destroys their utility (Black & Wiliam,
1998: “inside the black box”)
3. This is HARD to do. Tools help!
A Tool:
The formative assessment lesson, or FAL:
A rich “diagnostic” situation
and
Things to do when you see the results of
the diagnosis.
We zipped through one. Here’s another.
A Challenge:
We know that students have
many graphing misconceptions,
e.g., confusing a picture of a
story with a graph of the story in
a distance-time graph.
Here’s one way to address the
challenge.
Before the lesson devoted to this topic, we
give a diagnostic problem as homework:
Describe what may have happened. Is the graph realistic? Explain.
We point to typical student misconceptions and
offer suggestions about how to address them…
The lesson itself begins with a
diagnostic task…
Students are given the chance to
annotate and explain…
Follow-up Task: Card Sort
The students make posters.
Card Set B: Interpretations
Card Set A: Distance-Time Graphs
A.
B.
1.
Tom ran from his home to the bus
stop and waited. He realized that
he had missed the bus so he
walked hom e .
2.
Opposite Tom's home is a hill.
Tom climbed slowly up the hill,
walked across the top and then ran
quickly down the other side.
C.
D.
3.
Tom skateboarded from his house,
gradually building up speed. He
slowed down to avoid some rough
ground, but then speeded up again.
4.
Tom walked slowly along the
road, stopped to look at his watch,
realized he was late, then started
running.
E.
F.
5
Tom left his home for a run, but he
was unfit and gradually came to a
stop!
6.
Tom walked to the store at the end
of his street, bought a newspaper,
then ran all the way back.
G.
H.
8.
This graph is just plain wrong.
How can Tom be in two places at
once?
I.
J
7.
Tom went out for a walk with
some friends when he suddenly
realised he had left his wallet
behind. He ran home to get it and
then had to run to catch up with
the others.
9.
After the party, Tom walked
slowly all the way home.
.
10.
Make up your own story!
Students work on converting
graphs to tables:
Tables are added to the card sort…
Card Set C: Tables of data
A.
B.
Time
0
1
2
3
4
5
Distance
0
40
40
40
20
0
Time
0
1
2
3
4
5
Distance
0
40
80
60
40
80
Time
0
1
2
3
4
5
Distance
0
20
40
40
80
120
D.
C.
Time
0
1
2
3
4
5
Distance
0
10
20
40
60
120
Time
0
1
2
3
4
5
Distance
0
20
40
40
40
0
Time
0
1
2
3
4
5
Distance
0
45
80
105
120
125
E.
G.
Distance
0
18
35
50
85
120
F.
H.
J. Make this one up!
Time
Distance
0
1
2
3
4
5
6
7
8
9
10
Time
0
1
2
3
3
5
Time
0
1
2
3
4
5
Distance
0
30
60
0
60
120
I
K.
Time
0
1
2
3
4
5
6
7
8
9
10
Time
0
1
2
3
4
5
Distance
120
96
72
48
24
0
Distance
And the class compares solutions together.
The Mathematics Assessment
Project’s goals are to:
• Help students grapple with core content
and practices in CCSSM, and prepare
them for the rich assessments they
should (and it looks like, will) experience;
• Support formative assessment; and
• Do so in “curriculum-embeddable” ways.
We’re building 20 FALs at
each grade from 6 through 10.
They’re FREE, at
http://map.mathshell.org/materials
and I hope they help!
To sum up this part of our
conversation:
The Common Core Standards
and their instantiation in the
Smarter Balanced
Assessments offer a welcome
challenge.
Let’s roll up our sleeves and
work together toward meeting it.
3. Thoughts about what to
look for in productive
mathematics classrooms
What do you want to look for
in a math classroom? What
counts?
Key Questions for Math Classes:
• Was there honest-to-goodness math in
what students and teacher did?
• Did students engage in “productive
struggle,” or was the math dumbed
down to the point where they didn’t?
• Who had the opportunity to engage? A
select few, or everyone?
• Who had a voice? Did students get to
say things, develop ownership?
• Did instruction find out what students
know, and build on it?
Algebra Teaching Study, UC Berkeley/MSU, Alan Schoenfeld and Bob Floden, PIs; http://ats.berkeley.edu
“TRU Math”
(Teaching for Robust Understanding of Mathematics)
Essential Dimensions of
Productive Mathematics Classrooms
Level
1
Important Mathematics
Cognitive Demand
Skills-oriented focus; little or
Content is proceduralized to
no attention to concepts,
where it becomes rote.
connections, practices
Access
Agency: Authority and
Accountability
Uses of Assessment
No evidence of collecting or
No apparent effort to improve Teacher presents information and using student reasoning. (e.g.
access; uneven pattern of
judges student work (I.e., IRE
in IRE sequences or returning
participation.
sequences)
scored papers and not
discussing student thinking)
2
Some attention to
concepts/connectioons,
minimal CCSSM practices
Sudents are supported in
making connections between
Some efforts to invite student
procedures and concepts;
participation
some engagement in
practices
Students have some time to
engage/explain, but their role is
often reactive; the bottom line is
teacher authority.
Student reasoning is elicited
(in class) or referred to (as in
HW or tests), and corrected
when in error.
3
Significant attention to
concepts & connections;
opportunty to develop
practices
Students are supported in
"productive struggle" in
working complex problems
and building understandings
Students are expected and
encouraged to explain and
respond to mathematical ideas.
Student reasoning is referred
to and discussed, sometimes
affecting directions of
classroom discussion.
Important Mathematics
Cognitive Demand
Clear efforts to invite and
support broad student
participation
Access
Agency: Authority and
Accountability
Uses of Assessment
Algebra Teaching Study, UC Berkeley/MSU, Alan Schoenfeld and Bob Floden, PIs; http://ats.berkeley.edu
Two honest questions:
Do these matter?
Can it help to think about them?
You’ve made it to parts 4
and 5:
4.Q & A, for as long as you
can stand it;
5. Lunch!