2. Goals for Our Session
• Help to calm fears concerning Common Core State Standards (CCSS)
• Common Core Instructional Shifts in Education
• Implementation of CCSS (what, why, how and when)
• Analyzing a Performance Based Assessment
• Mathematical Content vs. Mathematical Practices
• The Structure of a Lesson
• How the Implementation of a Task Impacts Learning
• Assessing and Advancing Questions
• How CCSS Ties to TEAM Evaluation
1
4. Transitional Plan for 2012-2013
Six Key Instructional Shifts
MATH:
1.
Focus strongly where the Standards focus
2.
Coherence: think across grades, and link to major topics within grades
3.
Rigor: require conceptual understanding, procedural skill and fluency, and application
with intensity.
ELA:
1.
Building knowledge through content-rich nonfiction and informational texts
2.
Reading and writing grounded in evidence from text
3.
Regular practice with complex text and its academic vocabulary
3
5. Shift One: Focus
Strongly where the Standards focus
•Significantly narrow the scope of content and deepen how time and
energy is spent in the math classroom
•Focus deeply only on what is emphasized in the standards, so that
students gain strong foundations
4
6. It starts with Focus
• The current U.S. curriculum is “a mile wide and an inch deep.”
• Focus is necessary in order to achieve the rigor set forth in the
standards.
• Hong Kong example: more in-depth mastery of a smaller set of
things pays off
5
7. The shape of math in A+ countries
Mathematics topics intended at
each grade by at least
two-thirds of A+ countries
Mathematics topics intended
at each grade by at least twothirds of 21 U.S. states
1 Schmidt,
Houang, & Cogan, “A Coherent Curriculum: The
Case of Mathematics.” (2002).
6
8. Shift Two: Coherence
Think across grades, and link to major topics
within grades
• Carefully connect the learning within and across grades so that
students can build new understanding onto foundations built in
previous years.
• Begin to count on solid conceptual understanding of core
content and build on it. Each standard is not a new event, but
an extension of previous learning.
7
9. Shift Three: Rigor
In the major work, equal intensity in conceptual
understanding, procedural skill/fluency, and
application
•The CCSSM require a balance of:
– Solid conceptual understanding
– Procedural skill and fluency
– Application of skills in problem solving situations
•This requires equal intensity in time, activities, and resources in
pursuit of all three
8
10. Implementation of Common Core State Standards
Compliments other Work Underway
Student readiness for
postsecondary education
and the workforce
(WHY we teach)
Common Core State
Standards provide a
vision of excellence
for WHAT we teach
TEAM provides a
vision of excellence
for HOW we teach
9
11. Our Transition to Common Core Standards is Central
to Strengthening Tennessee’s Competitiveness
Only 21% of adults in
TN have a college
degree
TN ranks 46th in 4th
grade math and 41st in
4th grade reading
nationally
Tennessee’s
Competitiveness
54% of new jobs will
require postsecondary education
Only 15% of high
school seniors in TN
are college ready
Source: “Projections of Jobs and Education Requirements Through 2018” (The Georgetown University Center on Education and the
Workforce), 2011 NCES NAEP data, ACT
10
12. What are the Common Core State Standards?
The Standards . . .
• are aligned with college and work expectations;
• are clear, understandable and consistent;
• include rigorous content and application of knowledge through high-order skills;
• build upon strengths and lessons of current state standards;
• are informed by other top performing countries, so that all students are
prepared to succeed in our global economy and society; and
• are evidence-based.
http://corestandards.org/about-the-standards
11
13. Common Core State Standards are narrower…
# of TN Standards for 3rd Grade Math:
# of Common Core Standards for 3rd Grade Math:
25
113
There are 1,119 Tennessee ELA standards not covered in the Common Core
CLE
CU
GLE
SPI
Grand Total
45
501
109
464
1119
12
14. …and deeper.
3rd Grade Math
3OA.3 (Operations and Algebraic thinking): Use multiplication and division within 100 to solve word problems in
situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with
a symbol for the unknown number to represent the problems.
There were 28 cookies on a plate.
Five children each ate 1 cookie.
Two children each ate 3 cookies.
One child ate 5 cookies.
The rest of the children each ate 2 cookies.
Then the plate was empty
How many children ate 2 cookies? Use multiplication equations
and other operations, if needed, to show how you found your
answer.
Jane thinks this question can be solved by dividing 28 by 2. She
is wrong. Explain using equations and operations why this is not
possible.
Source: University of Pittsburgh, Copyrighted
13
16. How will the Common Core State Standards affect
Teaching in Tennessee?
• Creativity and Flexibility
• Deeper Engagement
• Smaller Number of Standards
• Critical Thinking and Problem Solving
15
18. 2012-2013 Assessment Plan, Math 3-8
Student performance on the Constructed Response
Assessments will not affect teacher, school, or district
accountability for the next two years.
October
• CRA 1
(paper and online
option, scored by
teachers in Field
Service Center
region, reported by
school team)
February
• CRA 2
(paper and online
option, scored by
teachers in Field
Service Center
region, reported
by school team)
May
• Official Constructed
Response Assessment
(paper-based
only, scored by
state, results reported
in July)
17
19. We will narrow the focus of the TCAP and
expand use of Constructed Response Assessments
2011-2012
2012-2013
2013-2014
2014-2015
TCAP
Constructed Response
PARCC
We will remove 15-25% of SPIs that are not reflected in
Common Core State Standards from the TCAP NEXT year.
The specific list of SPI’s will be shared on May 1.
We will expand the constructed response assessment for all
grades 3-8, focused on the TNCore focus standards for math.
NAEP
NAEP
18
20. The Common Core State Standards
19
The standards consist of:
The CCSS for Mathematical Content
The CCSS for Mathematical Practice
19
21. Overview of Activity
• Analyze and discuss the CCSS for mathematical content and
mathematical practices.
• Analyze the PBA in order to determine the way the assessment is
assessing the CCSSM.
• Discuss the CCSS related to the tasks and the implications for
instruction and learning.
• Discuss what it means to develop and assess conceptual
understanding.
2
20
23. 2012 – 2013 Tennessee Focus Clusters Grade 5
• Extend understanding of fraction equivalence and ordering.
• Build fractions from unit fractions by applying and extending
previous understanding of operations of whole numbers.
2
22
24. Analyzing Assessment Items (Private Think Time)
One assessment item has been provided:
48 Gumdrops Task
• Solve the assessment item.
• Make connections between the standard(s) and the
assessment item.
2
23
25. 1. 48 Gumdrops Task
Two children are sharing 48 gumdrops.
Jessica says, “I want 2/4 of the set of 48 gumdrops.”
Samuel says, “I want 2/3 of the set of 48 gumdrops.”
a.
Is it possible for Jessica and Samuel to each have a fraction of the
gumdrops that they want?
Answer: ____________________
b.
If you respond yes, use diagrams and equations to explain how you know
they can each receive the share of gumdrops they want. If you respond
no, use diagrams and equations to explain why the children cannot receive
the number of gumdrops they each want.
24
26. Discussing Content Standards (Small-Group Time)
With your small group, discuss the connections
between the content standard(s) and the
assessment item.
2
25
27. The CCSS for Mathematical Content: Grade 5
Number and Operations – Fractions
5.NF
Apply and extend previous understandings of multiplication and division to
multiply and divide fractions.
Apply and extend previous understandings of multiplication to
5.NF.4
multiply a fraction or whole number by a fraction.
Interpret multiplication as scaling (resizing), by:
5.NF.5
Explaining why multiplying a given number by a fraction greater than
5.NF.5b
1 results in a product greater than the given number (recognizing
multiplication by whole numbers greater than 1 as a familiar case);
explaining why multiplying a given number by a fraction less than 1
results in a product smaller than the given number; and relating the
principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of
multiplying a/b by 1.
Solve real-world problems involving multiplication of fractions and
5.NF.6
mixed numbers, e.g., by using visual fraction models or equations to
represent the problem.
Apply and extend previous understandings of division to divide unit
5.NF.7
fractions by whole numbers and whole numbers by unit fractions.
Common Core State Standards, NGA Center/CCSSO, 2010
26
28. The CCSS for Mathematical Content: Grade 5
Number and Operations – Fractions
5.NF
Apply and extend previous understandings of multiplication and division to multiply
and divide fractions.
Interpret division of a unit fraction by a non-zero whole number, and
5.NF.7a
compute such quotients. For example, create a story context for (1/3) ÷ 4,
and use a visual fraction model to show the quotient. Use the relationship
between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because
(1/12) × 4 = 1/3.
Interpret division of a whole number by a unit fraction, and compute such
5.NF.7b
quotients. For example, create a story context for 4 ÷ (1/5), and use a visual
fraction model to show the quotient. Use the relationship between
multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5)
= 4.
Solve real-world problems involving division of unit fractions by non-zero
5.NF.7c
whole numbers and division of whole numbers by unit fractions, e.g., by
using visual fraction models and equations to represent the problem. For
example, how much chocolate will each person get if 3 people share 1/2 lb
of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
Common Core State Standards, NGA Center/CCSSO, 2010
27
29. 1. 48 Gumdrops Task
Two children are sharing 48 gumdrops.
Jessica says, “I want 2/4 of the set of 48 gumdrops.”
Samuel says, “I want 2/3 of the set of 48 gumdrops.”
a.
Is it possible for Jessica and Samuel to each have a fraction of the
gumdrops that they want?
Answer: ____________________
b.
If you respond yes, use diagrams and equations to explain how you know
they can each receive the share of gumdrops they want. If you respond
no, use diagrams and equations to explain why the children cannot receive
the number of gumdrops they each want.
28
31. The CCSS for 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.
Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO
30
32. Discussing Practice Standards (Small-Group Time)
With your small group, discuss the connections between
the practice standards and the assessment item.
3
31
33. 1. 48 Gumdrops Task
Two children are sharing 48 gumdrops.
Jessica says, “I want 2/4 of the set of 48 gumdrops.”
Samuel says, “I want 2/3 of the set of 48 gumdrops.”
a.
Is it possible for Jessica and Samuel to each have a fraction of the
gumdrops that they want?
Answer: ____________________
b.
If you respond yes, use diagrams and equations to explain how you know
they can each receive the share of gumdrops they want. If you respond
no, use diagrams and equations to explain why the children cannot receive
the number of gumdrops they each want.
3
32
35. The Structures and Routines of a Lesson
Set Up the Task
Set-Up of the Task
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/
Small-Group Problem Solving
1. Generate and Compare Solutions
2. Assess and Advance Student Learning
MONITOR: Teacher selects
examples for the Share Discuss
based on:
• Different solution paths to
the same task
• Different representations
• Errors
• Misconceptions
SHARE: Students explain
their methods, repeat others’
ideas, put ideas into their own
words, add on to ideas and
ask for clarification.
REPEAT THE CYCLE FOR EACH
SOLUTION PATH
COMPARE: Students discuss
Share Discuss and Analyze Phase
1. Share and Model
2. Compare Solutions
3. Focus the Discussion on Key
Mathematical Ideas
4. Engage in a Quick Write
similarities and differences
between solution paths.
FOCUS: Discuss the
meaning of mathematical
ideas in each representation.
REFLECT: Engage students
in a Quick Write or a
discussion of the process.
34
36. Mathematical Tasks:
A Critical Starting Point for Instruction
There is no decision that teachers make that has a greater
impact on students’ opportunities to learn and on their
perceptions about what mathematics is than the selection
or creation of the tasks with which the teacher engages
students in studying mathematics.
Lappan & Briars, 1995
35
37. Linking to Research/Literature:
The QUASAR Project
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructiona
l materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics
instruction: A casebook for professional development, p. 16. New York: Teachers College Press.
36
38. The Enactment of the Task (Private Think Time)
• Read the Vignettes
• Consider the following question:
What are students learning in each classroom?
37
37
39. The Enactment of the Task (Small-Group Discussion)
Discuss the following question and cite evidence from
the cases:
What are students learning in each classroom?
38
38
40. The Enactment of the Task (Whole-Group Discussion)
What opportunities did students have to
think and reason in each of the classes?
39
39
41. Linking to Research/Literature:
The QUASAR Project
How High-Level Tasks Can Evolve During a Lesson
•
Maintenance of high-level demands
•
Decline into procedures without connection to meaning
•
Decline into unsystematic and nonproductive exploration
•
Decline into no mathematical activity
40
42. Factors Associated with the Decline and Maintenance
of High-Level Cognitive Demands
Decline
Maintenance
• Problematic aspects of the task
become routinized
• Scaffolds of student thinking and
reasoning provided
• Understanding shifts to
correctness, completeness
• A means by which students can
monitor their own progress is
provided
• Insufficient time to wrestle with the
demanding aspects of the task
• Classroom management problems
• Inappropriate task for a given group
of students
• Accountability for high-level
products or processes not expected
• High-level performance is modeled
• A press for
justifications, explanations through
questioning and feedback
• Tasks build on students’ prior
knowledge
• Frequent conceptual connections
are made
• Sufficient time to explore
41
43. Linking to Research/Literature:
The QUASAR Project
Task Set-Up
Task Implementation
A.
High
High
B.
Low
Low
C.
High
Low
Student Learning
High
Low
Moderate
Stein & Lane, 1996
42
45. Using Questioning During the Exploration Phase
Imagine that you are walking around the room as your groups of students
work on 48 Gumdrops Task – if it were a lesson, not an assessment.
Consider what you would say to the groups who produced responses in
order to assess and advance their thinking about key mathematical ideas,
problem-solving strategies, or use of and connection between
representations.
Specifically, for each response, indicate what questions you would ask:
– to determine what the student knows and understands
(ASSESSING QUESTIONS)
– to move the student towards the target mathematical goals
(ADVANCING QUESTIONS)
44
46. Characteristics of Questions that Support
Students’ Exploration
Assessing Questions
• Based closely on the work
the student has produced.
• Clarify what the student has
done and what the student
understands about what
s/he has done.
• Provide information to the
teacher about what the
student understands.
Advancing Questions
• Use what students have
produced as a basis for
making progress toward the
target goal.
• Move students beyond their
current thinking by pressing
students to extend what they
know to a new situation.
• Press students to think about
something they are not
currently thinking about.
45
48. Gumdrops Task
•Assessing Questions:
•Advancing Questions:
• Tell me what you did.
• Can you write an explanation for how
you used the values to draw a
diagram?
• What does ___ represent?
• Why did you … ?
• How does you diagram represent the
equation (or vice versa)?
• How did you know how to draw your
diagram?
• If Amanda had 1/5 of the gumdrops
and Samuel had 2/3 of the gumdrops,
how many would be leftover?
• Your neighbor says that they have do
enough to share. How could you help
them understand that this conclusion
is incorrect?
47
49. Supporting Student Thinking and Learning
In planning a lesson, what do you think can be gained by
considering how students are likely to respond to a task and
by developing questions in advance that can assess and
advance their learning, depending on the solution path they
choose?
48
50. Reflection
What have you learned about assessing and advancing
questions that you can use in your classroom?
Turn and Talk
49
51. Tennessee Education Rubric
Which of the indicators on the Teacher Education
Rubric are related to the use of
Assessing and Advancing questions?
50
52. Using the Assessment to Think About Instruction
In order for students to perform well on the PBA, what are
the implications for instruction?
• What kinds of instructional tasks will need to be used in
the classroom?
• What will teaching and learning look like and sound like
in the classroom?
51
53. Common Core and the TN Core 4 –
Linking to the TEAM Evaluation Model
This year the TEAM Evaluation will focus on the following four
areas:
1.
Questioning
2.
Academic Feedback
3.
Thinking
4.
Problem Solving
52
New Common Core standards are due to roll out in 45 states and 3 territories by 2014. They're designed to get students ready for college and careers by requiring them to think, write and explain their reasoning. They also de-emphasize multiple-choice test questions in favor of written responses.—Toppo, 2012
English and Language Arts (ELA) pilot programs will take place this school year with full implementation of all standards in the 2013-2014 school year.
Common core is what we are going to teach. Student readiness is why we teach, and TEAM is how we teach.
78% of students in Tennessee taking the ACT indicate that they want to go to college, but only 15% are ready. Of all Tennessee students that enter college, 60% have to be remediated.
The Common Core State Standards are a set of standards for math and English Language Arts that were developed by state leaders to ensure that every student graduates high school prepared for college or the workforce. The standards are designed to set clear expectations of what students should know in each grade and subject. They reflect rigorous learning benchmarks when compared to countries whose students currently outperform American students on international assessments.
The common core state standards offer a narrower and deeper approach to teaching and learning.
Have participants read slide to themselves.SAY: Discuss why this is different than what is currently practiced.
Say: CCSS has simplified the way teachers and administrators will read Grade Level Standards. The Domains are larger groups of related standards and Clusters are smaller groups of related standards.
Common Core standards will allow teachers more creativity and flexibility in their teaching while requiring a significant shift in instructional practice. Although there is some alignment between Tennessee’s standards and the Common Core, the Common Core requires a deeper engagement with a smaller number of standards than what the state currently requires. These standards will require new approaches to teaching, and students should expect enhanced rigor in their courses. Specifically, students will be required to master more critical thinking and problem-solving skills.
The October PBA is an assessment as learning. The February PBA is an assessment for learning, and the May PBA is an assessment of learning.
The transition to the Common Core will also include the adoption of new assessments that will measure what students have learned under the new standards. Starting in the 2014-15 school year, these assessments will replace the current Tennessee Comprehensive Assessment Program (TCAP) tests in math and English Language Arts for grades 3-11. These assessments will be administered online and include a summative assessment comprised of a Performance Based Assessment (PBA) and an End-of-Year Assessment (EOY). Tennessee is working with a number of other states in the Partnership for Assessment of Readiness for College and Careers (PARCC) Consortium to design these new assessments
Read the slide
Directions:(SAY) The Tennessee Department of Education has focused the work on two clusters per grade level. Each cluster will include specific standards that
A NOTE TO THE FACILITATOR: Hand out the Participant Packets.Directions:(SAY) Solve each task. Then compare and contrast the task.
In what way does the prompt for the identified item elicit a student response that will demonstrate what s/he knows about specific content within the standard?5.NF.4, 5.NF.7, 5.NF.7bWhat is it about the prompt for the identified item that will require students to use mathematical practices? Make sense of problems and persevere in solving them.Reason abstractly and quantitatively. (When students write equations and then re-contextualize the numbers telling us about the meaning of the amounts in the context of the problem, then we know they have demonstrated this standard.)Construct viable arguments and critique the reasoning of others. (Students must determine if Jessica and Samuel can both have their fraction of the gumdrops and then if they can or can’t, they must explain why.)Model with mathematics. (Students must use a diagram and write an equation to explain how they know the students can or cannot have their share of the gumdrops.)Use appropriate tools strategically. Attend to precision. Look for and make use of structure. (We will know if students understand what it means to take a fraction of a set of 48 gumdrops. We will know if students understand the meaning of the denominator and the numerator.)
Directions:(SAY) Determine the content standards related to the task.Refer to the notes pages for each task for the standards that relate to each task.
In what way does the prompt for the identified item elicit a student response that will demonstrate what s/he knows about specific content within the standard?5.NF.4, 5.NF.7, 5.NF.7b
Directions: Show the list of mathematical practices. Ask participants which of these mathematical practices students will have an opportunity to use when solving the task. (See reference to the practices on the notes pages of each of the tasks.)Probing questions and possible student responses (in italics): What does it mean to look for and make use of structure? Refer to one of the tasks to help us make sense of what this practice means. (Consider the 48 Gumdrops Task. If students must determine if the two students can each have their share of the 48 gumdrops, this means that the student must demonstrate that s/he understands that the denominator tells the number of equal groups that 48 will be partitioned into and the top number tells how many of the equal groups that s/he will get.)What does it mean to reason abstractly and quantitatively? Refer to one of the tasks to help us make sense of what this practice means. Which problems require students to use this practice? Which do not require students to use this practice and why? (The 48 Gumdrops Task permits us to assess this practice because the students have to pull out the quantities, manipulate them and then re-contextualize them explaining what the amounts mean in the context of this problem.)A fraction of a set in the 48 Gumdrops Task. We also get an opportunity to determine if students can write equations. Attend to precision means more than just getting the correct answer. What else does it mean?Why might we want students to have opportunities to use repeated reasoning? How does it help to support learners?
Directions:(SAY) Referring to your mathematical practice handout, discuss with your group which practice standards are being addressed by the assessment item.
What is it about the prompt for the identified item that will require students to use mathematical practices? Make sense of problems and persevere in solving them.Reason abstractly and quantitatively. (When students write equations and then re-contextualize the numbers telling us about the meaning of the amounts in the context of the problem, then we know they have demonstrated this standard.)Construct viable arguments and critique the reasoning of others. (Students must determine if Jessica and Samuel can both have their fraction of the gumdrops and then if they can or can’t, they must explain why.)Model with mathematics. (Students must use a diagram and write an equation to explain how they know the students can or cannot have their share of the gumdrops.)Use appropriate tools strategically. Attend to precision. Look for and make use of structure. (We will know if students understand what it means to take a fraction of a set of 48 gumdrops. We will know if students understand the meaning of the denominator and the numerator.)
Directions:(Say) When a high-level task is selected, it does not always remain high level as it is implemented in the classroom. Research from the QUASAR Project indicates that one of four things can occur during implementation:Maintenance of high-level demands: The cognitive demands of the task are maintained and students have a chance to think and reason about mathematics. Decline into procedures without connection to meaning: What began as a high-level task becomes a procedural task. Rather than students thinking and reasoning about mathematics, they end up applying a rule or procedure without understanding why or how it works. This can happen when students become frustrated with the challenges, and pressure the teacher to tell them how to solve the problem or to give them the answer. It can also happen when the teacher breaks down the high-level task into procedural steps.- Decline into unsystematic exploration: Although the high-level task is not proceduralized, students do not make progress toward the mathematical goal of the lesson. They may be working on mathematics, but it does not move them further toward the goal. For example, if the goal of a lesson was to explore the dimensions of a rectangle that would maximize the area for a fixed perimeter of a rectangle, students might spend the entire period building rectangles and measuring but never focusing on maximizing the area.- Decline into NO mathematical activity: The focus of the lesson is no longer on mathematics. For example, consider a lesson in which students present graphs they have made based on a collection of data on favorite ice cream. Instead of discussing the scale, what the graph tells us in terms of student preference, etc., the focus becomes the color that was used, why they liked chocolate ice cream best, etc.
Directions:Give participants a moment to read the slide before you ask the probing questions on this slide. If you have the time, have participants Turn and Talk with a partner before beginning the questions.Possible responses to first question about tasks (in italics): The tasks would have to give students opportunities to use the mathematical practices.The tasks should start with different representations, like the assessment tasks do, and ask for different representations to be used.The tasks would need to access multiple content standards, like the assessment tasks do.Students would need to write about the concept explored by the task in some way.In fact, the tasks probably would need to look a lot like the assessment tasks do.Possible responses to second question about teaching and learning (in italics): Teachers will need to engage students in a way that permits them to regularly use the mathematical practices when solving problems. Well-designed tasks will help that. Students will need the opportunity to learn to justify themselves orally so they can write about their justifications on the tasks and assessments.Since the tasks need to begin from different representations so that students are comfortable using tables, graphs, equations, etc., students need to be encouraged to make connections among these representations in class.So, teachers will need to consider how to arrange the sequence of instructional tasks so all these things happen!