Welcome to …
TODAY’S AGENDA
• SMP Qualities
• 5 Practices for Mathematical Discourse
• Shapes and Relationships
• Mosaics of Shapes and How Children Develop an
Understanding of Geometry
• Lessons and Reflections
Bring your ideas…
• As a group of professionals we have made a
commitment to helping children attain success
in life and a voice in the world.
– Many times the best part of these kinds of
professional development is simply the chance to
share ideas, raise questions, and work with other
practitioners to improve our own understandings
and practice.
– Please bring your stories of children’s learning
with you.
Our Socio-mathematical Norms
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Listen intently when someone else is talking avoiding distractions
Persevere in problem solving; mathematical and pedagogical
Solve the problem in more than one way
Make your connections explicit - Presentation Ready
Contribute by being active and offering ideas and making sense
Limit cell phone and technology use to the breaks and lunch unless
its part of the task.
 Be mindful not to steal someone else’s “ice cream”
 Respect others ideas and perspectives while offering nurturing
challenges to ideas that do not make sense to you or create
dissonance.
 Limit non-mathematical and non-pedagogical discussions
Presentation Norms
• Presenters should find a way to show mathematical
thinking, not just say it
• Presenters should indicate the end of their
explanation by stating something like “Are there any
questions, discussion, or comments?”
• Others should listen and make sense of presenters’
ideas.
• Give feedback to presenters, extend their ideas,
connect with other ideas, and ask questions to clarify
understandings
The Standards for Mathematical Practice
Student Reasoning and Sense Making about Mathematics
Let’s list as many qualities as we can of the kinds of
mathematically proficient student behaviors that
exemplify the SMP’s.
1.
2.
3.
4.
5.
6.
7.
8.
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.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
5 Practices
• Session One: Introducing the 5 Practices
Volume Task
• Consider the following question individually
then share with your group.
• Why does length times width times height (l x
w x h) make sense as a way of finding the
volume of a rectangular prism?
• When would l x w x h NOT work for volume?
Solve Volume Task
• Solve the ”Volume Task” individually then
share your ideas with a partner.
• Whole group discussion of task
Reading & Reflection
• Read pages 7 – 12.
• Reflect individually then share with a partner.
• Use the questions on the handout to guide
your discussion.
Reading Discussion
• Discuss the question(s) assigned to your group
and be prepared to present your responses to
the group.
• Presentation of questions & whole group
discussion.
Break Time
Representing Relationships
Using Venn Diagrams
Constructing a Venn Diagram
P = set of parallelograms
Rh = set of rhombuses
Re = set of rectangles
S = set of squares
T = set of trapezoids
Challenge: Where would you put the
set of kites in your Venn diagram?
All, Some, or No
1. _______ rectangles are squares.
2. _______ squares are rectangles.
3. _______ rhombuses are squares.
4. _______ squares are rhombuses.
5. _______ squares are parallelograms.
6. _______ parallelograms are squares.
7._______ rhombuses are rectangles.
8._______ trapezoids are parallelograms.
Lunch
Mosaics of Shapes and How Children
Develop an Understanding of
Geometry
Geometry Mosaic
• Cut out all of the pieces
• Determine the most descriptive name for
each of the pieces
• Let’s try putting pieces together to form
new shapes
A Little Psychology …
So, how do children come to know and
understand “shapes” and their attributes?
The Van Hiele
Model
Teaching and Learning
Geometry
Pierre Van
Hiele (1959)
• Why do students
struggle so much
in geometry?
Stages of Development
• Level 0 – Visualization (whole
shapes)
Stages of Development
• Level 1 – Analysis
(characteristics/properties)
Stages of Development
• Level 2 – Informal Deduction
(interrelationships)
Stages of Development
• Level 3 –
Deduction
(postulates,
theorems,
proof)
Stages of Development
• Level 4 –
Rigor (other
geometric
systems)
Properties of the Levels
• Development is sequential
• Advancement is dependent upon
instruction
• Linguistics typical of each level
• Mismatch between instruction/textbooks
and the level at which students function
How many
triangles
do you
see?
From Focus on Reasoning and Sense
Making (NCTM)…
“Thinking, questioning, and justifying should occur whenever
students encounter a situation that is new to them, both within
and outside of the school setting, and not only when a proof is
required in geometry class. We rarely need to line up statements
and reasons in our daily life, but we frequently need to provide a
rationale … as to why we do or say particular things. Reasoning and
sense making should be regular parts of the geometry curriculum,
with or without formal proof writing. All students … must be able
to reason and make appropriate decisions. Geometry provides an
environment that can allow and encourage students to ‘practice’
the process of reasoning and sense making, for the benefit of all.”
% of 12th Graders?
• International
49%
• United States
19%
Discussion Question
What are the implications of the Van
Hiele’s research on how we plan our
lessons?
Practice/Assessment
Let’s visit a tangrams puzzle online app to
practice manipulating shapes and using them to
create other shapes:
http://www.abcya.com/tangrams.htm
Ohio’s New Mathematics Standards Geometry
Kindergarten
• Identify and describe shapes
• Describe, compare, create, and compose shapes
Grades 1, 2, and 3
• Reason with shapes and their attributes
Grade 4
• Draw and identify lines and angles, and classify
shapes by properties of their lines and angles
Ohio’s New Mathematics Standards Geometry
Grade 5
• Graph points on the coordinate plane to solve realworld and mathematical problems
• Classify two-dimensional figures into categories
based on their properties
Break Time
Math Content for our Classrooms
• Each day we will spend time with grade level
teams making lesson plans.
• Each of us will make one plan that is part of a unit
of plans the grade level team is working on.
• Each plan must have the following:
– Connected mathematics content focus from Ohio’s
Mathematics Learning Standards
– A focus SMP
– Designed to Orchestrate Productive Mathematics
Discussions (The 5 Practices)
Math Content for our Classrooms
Three checks must be made for the completion of lesson plans:
Check 1) Consult with Sandy and/or Mary about the mathematics
content of the lesson and explain to her its mathematical
appropriateness. When the lesson is complete Sandy, our resident
mathematician, will sign off on its content (SMC’s).
Check 2) Consult with Sherry about the design of the lesson to
promote mathematical discourse (5 Practices). Sherry must sign off
on the lessons discourse elements.
Check 3) Consult with Dr. Matney about the design of the lesson
having a focus Standard for Mathematical Practice. Dr. Matney must
sign off on the lessons mathematics proficiency elements (SMP’s)
?Questions about COMP Lesson Plans?
Air of Appreciation
We want to pass on to each generation a sense
of learning how to appreciate life, others, and
learning.
Let’s spend some time sharing one thing or
experience that we appreciate:
Examples: I appreciated when Ray didn’t give up on solving that
hard problem. It encouraged me to keep thinking for myself to
make sense of it.
Time of Reflection
1) On a sticky note tell us one thing
you learned today.
2) Tell us one think you liked or one
thing you are still struggling with.
Stay Safe
• Please help us put the room in proper order.
• Please leave your name tents for next time.