Transcript Fostering Geometric Thinking in the Middle Grades

Fostering Geometric
Thinking in the Middle
Grades
Education Development
Center
Newton, MA
Horizon Research,
Inc.
Chapel Hill, NC
Project supported by the National Science Foundation under Grant ESI-0353409
Knowledge for Teaching Geometry
• Geometry content central to the middle
grades
• Development of student geometric thinking
related to this content
• Related student conceptual hurdles and
what research says about them
• Middle-grades geometry as groundwork for
high school geometry
Fostering Geometric Thinking (FGT)
• Identify productive ways of thinking in geometry (GHOMs)
• Create professional development materials based on GHOMs
– 40 hours  20 two-hour sessions
– Group-study materials
– Structured Exploration Process (Kelemanik et al. 1997)
• Stages: Doing mathematics; Reflecting on the mathematics; Collecting
student work; Analyzing student work; Reflecting on students’ thinking
– G-HOMs framework is a lens for analysis
• Create a multimedia book
Fostering Geometric Thinking (FGT)
• Field Test
Research Questions: In what ways does the use of
professional development materials that target
students’ geometric ways of thinking…
• Q1: …increase teachers’ content knowledge and
understanding of student thinking in geometry and
measurement?
• Q2: …affect instructional practice in geometry?
FGT Field Test
• Groups randomly assigned to 2 conditions:
Treatment (N=137), Wait-Listed Control
(N=140)
• Measures
– Geometry Survey—pre/post
• Multiple choice geometry problems
• Open-ended questions about approaches to a problem
• Open-ended questions about analyzing student work
– Observations
• Classrooms and teacher groups
– Embedded task
Geometric Habits of Mind (G-HOMs)
• Reasoning with Relationships
• Investigating Invariants
• Generalizing Geometric Ideas
• Balancing Exploration with Reflection
Reasoning with Relationships
Actively looking for relationships (e.g.,
congruence, similarity, parallelism,
etc.), within and between geometric
figures, in 1, 2, 3 dimensions, and
using the relationships to help
understanding or problem solving.
Internal questions (that is, questions
problem solvers ask themselves)
Include:
 "How are these figures alike?"
 "In how many ways are they alike?"
 "How are these figures different?"
 "What else here fits this description?"
 "What would I have to do to this object to make it
like that object?"
 "What if I think about this relationship in a
different dimension?“
 “Can symmetry help me here?”
Which two make the best pair?
Generalizing Geometric Ideas
Generalizing in mathematics is “passing from the
consideration of a given set of objects to that of
a larger set, containing the given one." (Polya)
This GHOM is characterized by wondering if “I
have them all,” and by wanting to understand
and describe the "always" and the "every"
related to geometric phenomena.
Internal questions include:
 "Does this happen in every case?"
 "Why would this happen in every case?“
 "Have I found all the ones that fit this
description?"
 "Can I think of examples when this is not true,
and, if so, should I then revise my
generalization?”
 "Would this apply in other dimensions?"
In squares, the diagonals always intersect in 90degree angles:
Investigating Invariants
An invariant is something about a situation
that stays the same, even as parts of the
situation vary. This habit of mind shows up, for
example, in analyzing which attributes of a
figure remain the same and which change
when the figure is transformed in some way
(e.g., through translations, reflections,
rotations, dilations, dissections, combinations,
or controlled distortions).
Internal questions include:
 "How did that get from here to there?"
 "Is it possible to transform this figure so it
becomes that one?"
 "What changes? Why?"
 "What stays the same? Why?"
 "What happens if I keep changing this figure?"
 "What happens if I apply multiple
transformations to the figure?"
“No matter how much I collapse the rhombus, the
diagonals still meet at a right angle!”
Balancing Exploration with Reflection
Trying various ways to approach a problem
and regularly stepping back to take stock. This
balance of "what if.." with "what did I learn from
trying that?" is representative of this habit of
mind. Often the “what iffing” is playful
exploration tempered by taking stock.
Sometimes it is looking at the problem from
different angles—e.g., imagining a final state
and reasoning backwards.
Internal questions include:
 "What happens if I (draw a picture, add to/take
apart this picture, work backwards from the
ending place, etc.….)?"
 "What did that action tell me?"
 “How can my earlier attempts to solve the
problem inform my approach now?”
 "What intermediate steps might help?"
 "What if I already had the solution….What
would it look like?"
Sketch if it’s possible (or say why it’s impossible):
A quadrilateral that has exactly 2 right angles and
no parallel lines
"I'll work backwards and imagine the figure has
been drawn. What can I say about it? One
thing: the two right angles can't be right next to
each other. Otherwise, you’d have two parallel
sides. So, what if I draw two right angles and
stick them together…."
Knowledge for Teaching Geometry
• Geometry content central to the middle
grades
• Development of student geometric thinking
related to this content
• Related student conceptual hurdles and
what research says about them
• Middle-grades geometry as groundwork for
high school geometry
Knowledge for Teaching Geometry
• Geometry content central to the middle
grades
• Three FGT content strands: Properties of
geometric objects; Geometric transformations; and
Measurement of geometric objects
• Example: a problem from the last of these
strands
Finding Area in Different Ways
• Work on the problem (both parts, if you have
time)
• Compare with 1 or 2 people around you. What
content is featured here? Can you identify some
of the G-HOMs you were using, or hear in the
account of others’ thinking?
One group’s thinking
"We figured the area of the pentagon was 25
square units, by cutting it up into little triangles.
Then we thought 'Well, that is the area of a 5x5
square. Let's try to make that square out of the
pentagon.' Then we started cutting and rotating
and matching pieces."
An indicator of Balancing exploration and
reflection
“Describes what the final state would look like."
Knowledge for Teaching Geometry
• Development of student geometric thinking
related to this content
– Looking for potential, not just deficit
– G-HOM framework as a lens
– Internal questions as bridges to practice
Looking at student work
• In pairs or trios, look at the samples and discuss,
citing evidence:
– Where might there be potential in the thinking?
– Are there any signs of conceptual
misunderstandings?
– What G-HOMs seem to be at play in the thinking?
(Use your G-HOM descriptors and lists of internal
questions.)
Knowledge for Teaching Geometry
• Related student conceptual hurdles and
what research says about them
– Research summaries for the 3 content strands
– A questioning framework, with particular
attention to assessing questions
Example: Understanding area
• Mental structuring of space is essential to
understanding area. Students go through several
stages of structuring space.
• Even as they move through the stages, they may
not connect their structuring of space to the
area formula.
They may just see computing the area of a 4- unit
by 7-unit rectangle as 4 times 7
rather than an array of 4 rows of 7 units in each
row.
Type
Purpose
Examples
Orienting
To focus students’ attention
on what is important in the
problem and/or on
particular ways to approach
the problem
What is the problem asking?
Would tangrams be useful here?
Do you think comparing those two sides
might help?
Which triangles are they asking you to
compare?
Assessing
To gauge students’
understanding of their
statements and actions while
problem solving
What do you think "congruent" means in
the problem statement?
Why did you fold the patty paper like that?
How did you arrive at this answer?
How do you know this is a rectangle?
Advancing
To help students extend
their thinking toward a
deeper understanding of the
problem
How could you convince a skeptic that the
figure you've made is a parallelogram?
What if you didn’t know the measure of
this angle?
How would you solve the problem without
graph paper?
What types of triangles will this work for,
and why?
Transcript
• First consider the problem the students are engaged in.
How would you think about it?
• Read the transcript once to understand the flow of the
interaction.
• Using the Questioning Framework, find a teacher
question you think is an Assessing question.
• Compare notes with neighbor. Discuss “What is the
payoff for the teacher in asking this and other
questions here?” “What are the costs?”
Knowledge for Teaching Geometry
• Middle-grades geometry as groundwork for
high school geometry
– Proof: explanation, verification, discovery (De
Villiers)
– Reasoned conjectures (Herbst)
– Multimodal communication (Chval, Khisty)
Contact Information
Mark Driscoll
[email protected]
Rachel Wing
[email protected]
Fostering Geometric Thinking website
www.geometric-thinking.com
.