### Geometric Thinking - Teachers Development Group

```Geometric Thinking in the
Geometric Thinking in the
Why is it important?
1. Complements Algebraic Thinking
“Broadly speaking I want to suggest that geometry is that part of
mathematics in which visual thought is dominant whereas
algebra is that part in which sequential thought is dominant.
This dichotomy is perhaps better conveyed by the words ‘insight’
versus ‘rigour’ and both play an essential role in real
mathematical problems.
The educational implications of this are clear. We should aim to
cultivate and develop both modes of thought. It is a mistake to
overemphasize one at the expense of the other and I suspect
that geometry has been suffering in recent years” (p. 29).
From Sir Michael Atiyah’s 1982 essay “What is
geometry?”, reprinted in Pritchard, C. (2003).
The changing shape of geometry: Celebrating a century
of geometry and geometry teaching. Cambridge:
Cambridge University Press.
2. Strengthens Algebraic Thinking
Example:
Linear functions are central to algebra. Graph a
linear functions and you get a straight line.
Why?
To fully understand why, one needs to
understand how similar triangles relate to
each other.
3. Expands Learning Opportunities for
English Learners
Multimodal communication
• “I folded the patty paper this way.” (shows with
dotted lines on a square drawn to represent the
piece of patty paper)
• “It goes like this.” (with a sweeping circular gesture)
Language intensive
• “This parallelogram is the same as that
parallelogram.”
• “Our method will work for any triangle.”
Influence of Three Projects
(DRK12-funded)
• Fostering Geometric Thinking in the Middle
• Learning and Teaching Geometry (LTG)
• Fostering Mathematics Success of ELLs
(FMSELL)
My way of thinking about LTG and
FGT as complements
FGT: What does geometric thinking look like as it
develops?
LTG: What does teaching to develop geometric
thinking look like?
FGT Book and FGT Toolkit (FGTT),
FGT’s Approach to
Professional Development
Do Math with
Colleagues
and Reflect
Have Students Do
Math and Collect
Student Work
Analyze Student
Work with
Colleagues and
Reflect
FGT’s Approach to Thinking about Geometric
Thinking – Provide a Lens and a Language:
The Geometric Habits of Mind
• Reasoning with Relationships
• Generalizing Geometric Ideas
• Investigating Invariants
• Balancing Exploration with Reflection
LTG
Videocases for Learning and Teaching Geometry:
Promoting an Understanding of Similarity
Nanette Seago, WestEd
Mark Driscoll, EDC
Project Overview
• Beginning 2nd year of a 5-year project
• Develop 5 videocase modules:
– 1 foundation module
– 4 extension modules
LTLF vs LTG
• Learning and Teaching Geometry grew out of the
work of Learning and Teaching Linear Functions
(Seago, Mumme & Branca, 2004)
• Similarities
– Videocase based
– Mathematically focused
– Modular: Foundation module & extension modules
• Differences
– Mathematical focus and problems vetted by
mathematicians prior to videotaping
– Multiple video of a variety of teachers teaching the
same lesson
FGT’s Approach to Thinking about Geometric
Thinking – The Geometric Habits of Mind
• Reasoning with Relationships
• Generalizing Geometric Ideas
• Investigating Invariants
• Balancing Exploration with Reflection
Reasoning with Relationships
Actively looking for relationships and using the
relationships to help understanding or problem solving.
Relationships…
 Include: congruence, similarity, parallelism, etc.
 Could be within or between geometric figures
 Could be in 1, 2, or 3 dimensions
Internal questions
(?’s that problem solvers ask themselves)
 "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 one?"
 "What if I think about this relationship in a different dimension?“
 “Can symmetry help me here?”
Reasoning with Relationships Example
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)
Characterized by…
 wondering if “I have them all,”
 wanting to understand and describe the "always"
and the "every" related to geometric phenomena.
Internal Questions
 "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?"
Generalizing Geometric Ideas Example
In squares, do the diagonals always intersect at a
90-degree angle?
Investigating Invariants
An invariant is something about a situation that stays the
same, even as parts of the situation vary.
For example, this habit of mind shows up when 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
 "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?"
Investigating Invariants Example
“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.
A balance of "what if.." with "what did I learn from trying
that?" is representative of this habit of mind.
Often the “what if-ing” 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
 "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?"
Reflect on Geometric Thinking
Through Problem Solving
• Part of an FGT problem
• First, you work on it and we discuss your
thinking (Look for examples of the geometric
habits of mind)
• Next, we watch two students talk about their
thinking (Look for examples of the geometric
habits of mind)
Two vertices of a triangle are located at (0,6) and (0,12).
The area of the triangle is 12 units2.
14
12
10
8
6
4
2
-10
-5
5
-2
10
Tommy’s “7 Possibilities”
(0,12)
(4,12)
(4,11)
(4,10)
(4,9)
(4,8)
(4,7)
(0,6)
(4,6)
Ana’s Procedure
Ana’s Procedure
(0,12)
(4,12)
(4,11)
(4,10)
(4,9)
(4,8)
(4,7)
(0,6)
(4,6)
(4,5)
LTG Project Goals
•
•
The focus of the LTG materials is to support teachers in recognizing and appreciating a
dynamic, transformational view of similarity, and geometry in general.
The materials will be designed to help teachers develop a diagnostic view of student thinking
about similarity, in the sense that they are
–
–
better prepared to recognize their students emerging understanding of similarity concepts
better equipped with a repertoire of instructional strategies to foster their students’ understanding of
these concepts.
Working Definition of Similarity
Two figures are similar if one is the same as an
enlargement or reduction of the other.
– “the same” refers to “congruent” or “can be matched by
rotating, translating, and/or reflecting.”
– “an enlargement or reduction” refers to “stretching or
shrinking” or “dilating a figure where the distance from the
center of dilation can be multiplied by a fixed scale factor.”
Static vs dynamic and connecting similarity to
linearity
1/3
1/3
2
1
3
6
x2
y =1/3 x
y/x
1xK
1
=1/3
y
3
3xK
x
xK
Potential Foundation Module Mathematical Storyline
Exploring Congruence: Laying Similarity Groundwork
Goal: Unpack “sameness”
Introducing Similarity through Transformations
Goal: Unpack what is the same and different in similar figures
Implications of Similarity
Goals: (1) Explore ration/proportions within and across figures
(2) Explore preservation of angles
Further Implications of Similarity
Goals: (1) Explore the relationship of the area of similar figures, leading into 3-D (2) Explore
linearity and slope
Closure
Goal: Pull together the big ideas of static and dynamic understanding of similarity
Hannah’s questions
• What do you think about those dimensions?
• What do you mean the line goes up by the same
rate?
• Where are you looking when you say that?
• How does that help you look at that (Brian’s picture)?
• Is that not possible with rectangles?
• What’s the difference, then?
• Do you think those are similar rectangles if you just
• What is the origin acting like, then?
• What about their figure is different from yours that
would make you think it is a higher rate?
• What do you think it would look like if we had drawn
the rectangles with the longer dimensions here?
Fostering Mathematics Success of
English Language Learners
Mark Driscoll, EDC [email protected]
Daniel Heck, Horizon Research, Inc.
[email protected]
An efficacy study of FGTT among teachers of
ELLS
FMSELL Research Questions
• Does participation in FGTT increase teachers’
geometric content knowledge?
• What effects does teachers’ participation in FGTT
have on their attention to students’ thinking and
mathematical communication when teachers
analyze student work?
• What effects does teachers’ participation in FGTT
have on instructional practices, especially those
known to benefit ELLs?
• What impact on ELLs’ problem-solving strategies
is evident when teachers participate in FGTT?
We need research sites!!!
Please see me afterwards if you think you
can convene a group of 6 or more
teachers (grades 5-10) interested in
helping us research connections between
teachers improving their understanding
of geometric thinking and their impact
on English learners.
```