Transcript Learning from Children’s Geometric Thinking Math Alliance October 19, 2010 Chris Guthrie, Beth Schefelker, & Melissa Hedges.

Learning from Children’s
Geometric Thinking
Math Alliance
October 19, 2010
Chris Guthrie, Beth Schefelker,
& Melissa Hedges
Learning Intentions and
Success Criteria
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Learning Intentions
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We are learning to
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Examine the geometric thinking and reasoning children
exhibit and classify it according to van Hiele’s level of
geometric thought.
Success Criteria
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We will be successful when
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we recognize that “geometry is more than definitions; it is
about describing relationships and reasoning.” –PSSM.
we can begin to identify “next steps” to move children to
more developed levels of geometric thought.
Tricky Triangle Task
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Select two pieces of student work each and
place them in the middle of the table.
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One paper of a regular education student. (Identify
with yellow dot.)
One paper of a student with special education
need. (Identify with blue dot.)
Take turns silently viewing each piece.
Tricky Triangle Task
As you review student work, note the
following on a post-it:
1.
2.
Ideas or student thinking that pique your
interest – ideas that please you or concern
you.
Estimated van Hiele level
Place post-it on back of paper and pass to a
neighbor.
Sharing perspectives
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As a group decide on 4 papers you would like to
discuss as a group.
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Discuss each paper one at a time.
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Review the post-its placed on the back for each
paper and respond to the following:
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How is student reasoning similar?
How is student thinking different?
How do students with and without disabilities compare?
Decide on the van Hiele level for each student and
support your decision
Post your results!
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Review the spread sheet you brought.
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Note how many students you have at each level.
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Differentiate between students with disabilities and
students without.
Post that number on the appropriate poster. Include
the grade level on your post-it.
Reflecting on class results…
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What patterns surface as we review this
data?
What “red flags” surface around student
understanding as you reflect on this data?
How do students with and without disabilities
compare?
Strategic Instruction
Characteristics of the van Hiele levels
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The levels are sequential.
The levels are not age dependent.
Geometric experience is the greatest single factor
influencing advancement through the levels.
Students required to wrestle with objects of thought
that have not been constructed at the earlier level may
be forced into rote learning and achieve only
temporary and superficial success.
Strategic Instruction
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What next steps might you take with each
student to move them to next level?
Use your van Hiele articles to help you.
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Pre-level 0 → Level 0
Level 0 → Level 1
Level 1 → Level 2
PreLevel 0 to Level 0
Level 0 to Level 1
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Teachers should involve students in lots of sorting and classifying
of shapes.
Students need to identify and express similarities and differences
between shapes.
As children surface properties such as symmetry, numbers of
sides and corners, etc. appropriate geometric terms are
introduced by the teacher.
As sorting tasks continue teachers should challenge students to
use features (sides, angles, etc.) to classify shapes.
Students should be provided numerous, and focused,
opportunities to draw, build, make, put together, and take apart
shapes in both 2-D and 3-D.
Level 0 to Level 1
Accurate use of Level 0 and Level 1
vocabulary and Level 1 and Level 2
understanding of geometric properties
on behalf of the teacher is critical!
Level 1 to Level 2
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Focus more on properties of shapes rather than on simple
identification of shapes. As new geometric concepts are
learned, the number of properties that figures have can be
expanded. (e.g., sort quadrilaterals first with “2 pairs of
opposite sides are parallel” then with “2 pairs of opposite
sides are congruent.” What do we notice?)
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Apply ideas to all classes of figures (e.g., all rectangles, all
prisms) rather than individual models. Example: Find ways to
sort all possible triangles into groups. From these groups
define types of triangles.
Level 1 to Level 2
Teachers must be comfortable guiding
students through classifying shapes,
identifying and defining properties, and
using geometric vocabulary with meaning.
Review the Triangle Task Project
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Now that you know where your students’
thinking lies…
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Summarize our discussion from today in a 1.5-2
page paper due November 16. Follow guidelines
in syllabus (p. 6).
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Between now and the beginning of March
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Keep a log identifying the instruction you provide relative
to students’ understanding of polygons, specifically
triangles, and your reflections on this instruction.
Resource Binder Big Idea #1
Description, Classification, and Analysis of Polygons
We will all use the Triangle Task Project for this first
Big Idea.
Page 5 of the syllabus:
 As a learner of mathematics
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As a teacher of mathematics
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How is what you are learning supporting your teaching?
Identifying student levels of understanding.
What have we learned from the assessments?
Research/Trace how the Big Idea is presented in the
textbook program you use. (This is the instructional
implications portion of your Triangle Task.)
Quick Images – Video Viewing
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What did you notice about the children’s
responses?
How did students “form the mental image” of
the shape?
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Spatial abilities
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Spatial orientation
Spatial visualization
What did they notice that helped them
remember and draw each image?
Quick Images
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Show the image for 3 seconds
Hide the image
Draw what you saw
Show the image again and revise your
drawing
Sharing your image
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Share what you drew with your neighbor
Explain how you approached “remembering”
your image.
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What aspect of shape did you note first?
Were you noticing the way parts of the shape fit
together, or were you looking at the shape as a
whole?
Discussing the cases
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Turn each question over, one at time.
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Facilitator reads question to the group and
ensures that each participant offers their
thinking.
Whole group debrief:
 What mathematical questions does this
chapter raise for you?
Focus Questions: Chapter 1
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The first paragraph of this chapter (p. 5) lists a variety of
ways that children might describe a 3-D object. Find
examples in the cases that illustrate each of the
approaches.
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In lines 397-406 of Alexandra’s case 5, the children
describe a triangular prism. What is each child focusing
on? What might that indicate about that child’s thinking?
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Earlier in Alexandra’s case (lines 380-388) the students
begin to notice relationships between the geometric
blocks, rather than simply describe them. Talk about the
thinking of each child. How do they express the
relationships they see?
Focus questions: Chapter 1
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In case 2, lines 163-178, Molly’s students use
a wooden block (3-D object) to explain their
thinking about a triangle drawn on paper ( a
2-D object). What aspects of these figures
and objects are significant to them?
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In Paul’s case 6, we see children considering
a cone (lines 482-488) and a sphere (lines
503-505). Examine and discuss the various
ways children comment on these objects.
Homework
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Due November 2
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Due Nov 18
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Collaboration Goal
Finish reading Section 10.1
Try Quick Images twice with your students.
Triangle Task Project 1.5-2 pages write up.
Planning ahead for Resource Binder – Big
Idea #1
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Find, and become familiar with, the units/lessons
in your textbook that support students’ reasoning
around polygons.
Bring your textbook materials to class on
November 2