Day 1
Rigorous/High Level Task
Math & Science Collaborative at the Allegheny
Intermediate Unit
1
Goals
Understand the importance of engaging students in
rigorous and relevant tasks and activities.
• Recognize the characteristics of rigorous, high-
level tasks in mathematics
• Understand the importance of engaging
students in high-level tasks in order to more
deeply learn the content and see relevance to
the real world.
Math & Science Collaborative at the Allegheny
Intermediate Unit
2
Goals
Develop a deep understanding of the PA Core
Standards, Keystone Assessment Anchors and Eligible
Content,
 Understand the importance of engaging
students in the Standards for Mathematical
Practice as the means of learning important
content.
• Understand the focus and coherence of the PA
Core standards.
• Become familiar with Learning Progressions as
narrative documents describing the progression
of a topic across grade levels, informed both by
research on children's cognitive development
and by the logical structure of mathematics
Math & Science Collaborative at the Allegheny
Intermediate Unit
3
Goals
Write Curriculum and Plan Lessons/Units
• Utilize the different Curriculum Maps from PA and
other states to organize curriculum writing
• Utilize the Learning Progressions as “touchstone
documents” to assist with curriculum writing
• Select high-level tasks to include in lessons/units from
a variety of vetted resources
Math & Science Collaborative at the Allegheny
Intermediate Unit
4
Analyzing Mathematical Tasks
“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 and Briars, 1995
Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of
Pittsburgh, 2003
What are mathematical tasks?
We define mathematical tasks as a set of problems
or single complex problem the purpose of which is to
focus students’ attention on a particular
mathematical idea.
Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of
Pittsburgh, 2003
Why focus on mathematical tasks?
 Tasks form the basis for students’ opportunities
to learn what mathematics is and how one does
it;
 Tasks influence learners by directing their
attention to particular aspects of content and by
specifying ways to process information; and
 The level and kind of thinking required by
mathematical instructional tasks influences what
students learn.
Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of
Pittsburgh, 2003
Comparing Two Mathematical Tasks
Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of
Pittsburgh, 2003
Comparing Two Mathematical Tasks
Solve Two Tasks:
Hundreds, Tens and Ones
Muffles Truffles
Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of
Pittsburgh, 2003
10
Muffles Truffles
Here are the truffles that Muffles’ assistant Patricio needs
to package:
■ 218 raspberry truffles
■ 132 strawberry truffles
■ 174 dark chocolate truffles
■ 83 vanilla truffles with cinnamon and nutmeg
■ 126 green truffles with pistachios
■ 308 truffles with pecans and caramel
■ 97 butterscotch crunch truffles covered in milk chocolate
■ 22 truffles with white and dark chocolate swirls
■ 44 chocolate-covered cherry truffles
■ 46 almond and raisin truffles
Muffles Truffles
• How many boxes does Patricio need for each flavor? How
many leftovers of each kind will there be?
• Is there a shortcut way to know how many boxes of each
kind he needs to pack and how many leftovers there will
be for the assortment boxes?
• How many assortment boxes can he make?
• Muffles sells his fancy truffles for $1.00 each so his boxes
of truffles cost $10 each. How much money will he collect
if he sells them all?
Comparing Two Mathematical Tasks
How are Hundreds, Tens and Ones
and Muffles Truffles the same and
how are they different?
Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of
Pittsburgh, 2003
Comparing Two Mathematical Tasks
Do the differences between Muffles Truffles and
Hundreds, Tens and Ones matter?
Why or Why not?
Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of
Pittsburgh, 2003
Comparing Two Mathematical Tasks
“Not all tasks are created equal, and different
tasks will provoke different levels and kinds of
student thinking.”
Stein, Smith, Henningsen, & Silver, 2000
Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of
Pittsburgh, 2003
Comparing Two Mathematical Tasks
“The level and kind of thinking in which students
engage determines what they will learn.”
Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997
Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of
Pittsburgh, 2003
Components of a Math Task
 Developing the Context
 Supporting the Investigation
 Preparing for the math congress
 Facilitating the math congress
 Integrating mini-lessons, games and routines
Math & Science Collaborative at the Allegheny Intermediate Unit
Components of a Math Task
 Developing the Context
 Can use stories, situations (realistic or fictional), contexts,
models
 Children work to explore and make sense of the situations
 They try out strategies to solve and make sense of the use of
the strategies
 They explore and generate patterns
 They generalize
 And “mathematize”




Supporting the Investigation
Preparing for the math congress
Facilitating the math congress
Integrating mini-lessons, games and routines
Math & Science Collaborative at the Allegheny Intermediate Unit
Muffle’s Truffles
 Use blank paper to record your own thinking about
solving the task. Actually do the work to solve the
problem yourself. Think as a learner.
 Share your solution strategies with your small group
Math & Science Collaborative at the Allegheny Intermediate Unit
Anticipating
 Actively envision how students might mathematically
approach the instructional task or tasks that they will
work on
 Involves developing considered expectations about
how students might mathematically interpret a
problem, the array of strategies – both correct and
incorrect – that they might use…and how those
strategies relate to the mathematical concepts,
representation, procedures and practices…
 5 Practices for Orchestrating
Productive Mathematics Discussions
by Margaret Smith and Mary Kay Stein
Math & Science Collaborative at the Allegheny Intermediate Unit
Muffle’s Truffles
 Do some Anticipating
 Together, think as teachers. Strategize about other
solution strategies that students in your grade 3 or 4
classroom might use.
 Record these strategies for yourself.
 Now, examine the student work samples.
 Were you able to anticipate these strategies? If not, no
worries. Just by doing the anticipating, you are more
prepared to deal with unanticipated strategies.
Math & Science Collaborative at the Allegheny Intermediate Unit
Components of a Math Task
 Developing the Context
 Supporting the Investigation – AKA: Monitoring
 Facilitator observes strategies
 Listens to discussions
 Confers with pairs or small groups
 Ask questions and make comments (not leading ones)


Help me understand your method
What made you decide to use that strategy?
 Preparing for the math congress
 Facilitating the math congress
 Integrating mini-lessons, games and routines
Math & Science Collaborative at the Allegheny Intermediate Unit
Sample student dialogue
 Examine the snip-it of conversation on the
Conferring with Students at Work handout. This
is from Toni’s monitoring of the students while
they work.
 What is Toni doing to assist his/her students?
 What isn’t Toni doing to assist his/her students?
Math & Science Collaborative at the Allegheny Intermediate Unit
Components of a Math Task
 Developing the Context
 Supporting the Investigation
 Preparing for the math congress
 Have children talk with partners about what they want
to share (can use posters or paper with doc camera)
 Might do a gallery walk
 Facilitator decide what ideas to have shared and what
order to have the ideas shared
 AKA: Selecting and Sequencing
 Facilitating the math congress
 Integrating mini-lessons, games and routines
Math & Science Collaborative at the Allegheny Intermediate Unit
Components of a Math Task
 Developing the Context
 Supporting the Investigation
 Preparing for the math congress
 Facilitating the math congress
 NOT a whole class share
 Designed to push the math development of students
 Many possible structures to use
 Make sure to ask questions
 Help students make connections among ideas, among
strategies, among representations, etc.
 Integrating mini-lessons, games and routines
Math & Science Collaborative at the Allegheny Intermediate Unit
Math Congress
 Examine the sample excerpt from the Math Congress
in Toni’s class, entitled A Portion of the Math Congress
 What is Toni doing to assist his/her students?
 What isn’t Toni doing to assist his/her students?
Math & Science Collaborative at the Allegheny Intermediate Unit
What is the math in this task?
CONTENT
 Place Value patterns, especially with groups of ten
 Unitizing
 Quotative division – Finding how many groups
Math & Science Collaborative at the Allegheny Intermediate Unit
Math Congress
 What would you focus on in the Math Congress for
this task?
 Unitizing
 Place value
 Connection in this task is the fact that the number of
tens (the unit)is the number of full boxes
 Might move from least to most efficient strategies
Math & Science Collaborative at the Allegheny Intermediate Unit
What is the math in this task?
PRACTICES
 Read the elementary elaborations (draft) of the
Standards for Mathematical Practices.
 Which Practices were STRONGLY exhibited in the
Muffles Truffles task and debrief? How?
 Some of them will NOT be strongly exhibited.
Math & Science Collaborative at the Allegheny Intermediate Unit
Some general “questions”
 Who can put what Sarah just said into your own




words?
Who has a question or comment for Daniel?
Who agrees with Ashley, but used a different strategy?
Who still needs convincing that Carmine’s strategy
will work?
Will Bailey’s strategy always work? How do you know
for sure?
Math & Science Collaborative at the Allegheny Intermediate Unit
Components of a Math Task




Developing the Context
Supporting the Investigation
Preparing for the math congress
Facilitating the math congress
 Integrating mini-lessons, games and routines
 Can be used at the start of a lesson for 10-15 minutes
 Designed to highlight a particular computational
strategy
 Designed to help build fluency
 Might help with mental math
 Make sure to structure the games so that they actually
support strategies and discussions, not just fact practice
Math & Science Collaborative at the Allegheny
Intermediate Unit
Multiplication and Division
Word Problem Types
Rigor – What it is
 Rigor refers to academic rigor

learning in which students demonstrate a thorough,
in-depth mastery of challenging tasks to develop
cognitive skills through reflective thought, analysis,
problem-solving, evaluation, or creativity.
 Rigorous learning can occur at any school
grade and in any subject.
(Rigor/Relevance Framework- International Center for Leadership in
Education, Dr. Bill Daggett)
 3 aspects of Rigor, defined by CCSSM, are:
 Conceptual understanding
 Procedural skill and fluency
 Applications
Math & Science Collaborative at the Allegheny Intermediate Unit
33
Math & Science
Collaborative
Characterizing Tasks
Characterizing Tasks
• Develop a list of criteria that describe the
tasks in each category
Math & Science
Collaborative
• Sort the Tasks into two categories
[high level and low level]
“If we want students to develop the
capacity to think, reason, and problem
solve then we need to start with high-level,
cognitively complex tasks.”
Stein & Lane, 1996
Math & Science
Collaborative
Categorizing Tasks
Categorizing Tasks
• Are all high-level tasks the same?
[Is there an important difference between Tasks
H and I?]
• Are all low-level tasks the same?
[Is there an important difference between Tasks E
and O?]
Math & Science
Collaborative
Levels of Cognitive Demand &
The Mathematical Tasks
Framework
Linking to Literature/Research:
The QUASAR Project
• High-Level Tasks
Math & Science
Collaborative
• Low-Level Tasks
Linking to Literature/ Research:
The QUASAR Project
• memorization
• procedures without connections
• High-Level Tasks
• procedures with connections
• doing mathematics
Math & Science
Collaborative
• Low-Level Tasks
Lower-Level Tasks
Memorization
What are the decimal equivalents for the
fractions ½ and ¼?
Procedures without connections
Convert the fraction 3/8 to a decimal.
Higher-Level Tasks
Procedures with connections
 Using a 10 x 10 grid, identify the decimal
and percent equivalents of 3/5.
Doing mathematics
 Shade 6 small squares in a 4 x 10 rectangle.
Using the rectangle, explain how to
determine:
1. The decimal part of area that is shaded;
2. The fractional part of area that is shaded.
Math Practices
• How does using High Level Tasks allow you to
engage students in the Math Practices?
Linking to Literature/ Research:
The QUASAR Project
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Math & Science
Collaborative
The Mathematical Tasks Framework
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000, p. 4
TIMSS Video Study
FIGURE 1 Types of Math Problems Presented
100
FIGURE 2 How Teachers Implemented
Making Connections Math Problems
100
90
90
84
80
77
80
69
70
70
61
57
59
60
Math & Science
Collaborative
60
54
52
50
50
48
46
41
40
40
30
30
37
31
24
20
17
16
15
20
18
20
19
13
10
10
0
0
Australia
16
Czech Republic
Using Procedures
Making Connections
Hong Kong
Japan
Netherlands
United States
8
0
Australia
Czech Republic
Using Procedures
Making Connections
Hong Kong
Japan
Netherlands
United States
Does Maintaining
Cognitive Demand
Matter?
YES
A.
High
High
High
B.
Low
Low
Low
C.
High
Low
Moderate
Stein & Lane, 1996
Developed under the auspices of the NSF-funded ESP
Project (ESI - 0301962) -- Directed by Margaret Smith,
University of Pittsburgh, 2003
Patterns of Set up, Implementation,
and
Student
Learning
Task Set Up
Task Implementation
Student Learning
• Not all tasks are created equal -- they provided different
opportunities for students to learn mathematics.
• High level tasks are the most difficult to carry out in a
consistent manner.
• Engagement in cognitively challenging mathematical tasks
leads to the greatest learning gains for students.
• Professional development is needed to help teachers build
the capacity to enact high level tasks in ways that maintain
the rigor of the task.
• Being cognizant of the factors that lead to maintenance of
the cognitive demands of the task leads to higher students
achievement
Math & Science
Collaborative
Conclusion
Math Standards
Math & Science
Collaborative
• Reexamine some of the high level tasks from the
card sort and discuss the SMPs and content
standards that are there.
Additional Articles and Books about the
Mathematical Tasks Framework
Boston, M.D., & Smith, M.S., (in press). Transforming secondary
mathematics teaching: Increasing the cognitive demands of instructional
tasks used in teachers’ classrooms. Journal for Research in Mathematics
Education.
Stein, M.K., Grover, B.W., & Henningsen, M. (1996). Building student
capacity for mathematical thinking and reasoning: An analysis of
mathematical tasks used in reform classrooms. American Educational
Research Journal, 33(2), 455-488.
Stein, M. K., & Lane, S. (1996). Instructional tasks and the
development of student capacity to think and reason: An analysis of the
relationship between teaching and learning in a reform mathematics
project. Educational Research and Evaluation, 2(1), 50 - 80.
Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and
student cognition: Classroom-based factors that support and inhibit highlevel mathematical thinking and reasoning. Journal for Research in
Mathematics Education, 28(5), 524-549.
Math & Science
Collaborative
Research Articles
Additional Articles and Books about the
Mathematical Tasks Framework
Practitioner Articles
Smith, M.S., & Stein, M.K. (1998). Selecting and creating mathematical
tasks: From research to practice. Mathematics Teaching in the Middle
School, 3(5), 344-350.
Henningsen, M., & Stein, M.K. (2002). Supporting students’ high-level
thinking, reasoning, and communication in mathematics. In J. Sowder & B.
Schappelle (Eds.), Lessons learned from research (pp. 27 – 36). Reston VA:
National Council of Teachers of Mathematics.
Smith, M.S., Stein, M.K., Arbaugh, F., Brown, C.A., & Mossgrove, J. (2004).
Characterizing the cognitive demands of mathematical tasks: A sorting task.
In G.W. Bright and R.N. Rubenstein (Eds.), Professional development
guidebook for perspectives on the teaching of mathematics (pp. 45-72).
Reston, VA: NCTM.
Math & Science
Collaborative
Stein, M. K., & Smith, M.S. (1998). Mathematical tasks as a framework for
reflection. Mathematics Teaching in the Middle School, 3(4), 268-275.
Additional Books about the
Mathematical Tasks Framework
Books
Smith, M.S., Silver, E.A., Stein, M.K., Boston, M., Henningsen, M., & Hillen,
A. (2005). Cases of mathematics instruction to enhance teaching
(Volume I: Rational Numbers and Proportionality). New York: Teachers
College Press.
Smith, M.S., Silver, E.A., Stein, M.K., Henningsen, M., Boston, M., &
Hughes,E. (2005). Cases of mathematics instruction to enhance teaching
(Volume 2: Algebra as the Study of Patterns and Functions). New York:
Teachers College Press.
Smith, M.S., Silver, E.A., Stein, M.K., Boston, M., & Henningsen, M.
(2005). Cases of mathematics instruction to enhance teaching (Volume 3:
Geometry and Measurement). New York: Teachers College Press.
Math & Science
Collaborative
Stein, M.K., Smith, M.S., Henningsen, M., & Silver, E.A. (2000).
Implementing standards-based mathematics instruction: A casebook for
professional development. New York: Teachers College Press.
Additional References Cited in This
Slide Show
Hiebert, J., Carpenter, T.P., Fennema, D., Fuson, K.C., Wearne, D.,
Murray, H., Olivier, A., Human, P. (1997). Making sense: Teaching
and learning mathematics with understanding. Portsmouth, NH:
Heinemann.
Lappan, G., & Briars, D.J. (1995). How should mathematics be
taught? In I. Carl (Ed.), 75 years of progress: Prospects for school
mathematics (pp. 131-156). Reston, VA: National Council of Teachers
of Mathematics.
Stigler, J.W., & Hiebert, J. (2004). Improving mathematics teaching.
Educational Leadership, 61(5), 12-16.
TIMSS Video Mathematics Research Group. (2003). Teaching
mathematics in seven countries: Results from the TIMSS 1999 Video
Study. Washington, DC: NCES.
Math & Science
Collaborative
Boaler, J., & Staples, M. (2008). Creating mathematical futures
through an equitable teaching approach: The case of Railside
School. Teachers College Record, 110(3), 608-645.