Transcript How do you define geometric thinking?

How do you define geometric
thinking?
FGT Staff
• Mark Driscoll, Principal Investigator
• Rachel Wing DiMatteo, Research Associate
• Johannah Nikula, Research Associate
• Mike Egan, Curriculum Specialist
Session Agenda
I. Why Geometry?
II. FGT’s Approach to Professional
Development
III. Reflect on Geometric Thinking Through
Problem Solving
IV. Analyze Student Work
V. Preliminary Findings of the Project
VI. Conclusion and Questions
Why Geometry?
“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.
Merging Algebraic and Geometric
Thinking: Excerpts from Wu’s
Curricular Proposals*
On Algebra Curriculum: “Students need to be totally at ease in moving
between the geometric data of a straight line and the algebraic data of a
linear equation. This cannot happen if they are never taught similar
triangles before embarking on the study of linear equations and their
graphs, and have never been exposed to the explanation of why the
equation of a line is linear and why the graph of a linear equation is a line”
(pp. 3-4).
On Geometry Curriculum: “A dilation with center at the origin O and scale
factor r (r ≠ 0) is a transformation of the plane that sends a point (a, b) to
the point (ra, rb)...Once [the students] buy into this concept of dilation,
they are ready for the definition two figures to be similar if one figure is
congruent to a dilated version of the other” (p. 6).
* From Wu, H. (2005). Key mathematical ideas in grades 5-8. Paper presented at the Annual Meeting of the National Council of
Teachers of Mathematics, Anaheim. Available: http://math.berkeley.edu/~wu.
Are We Neglecting Geometry in
Schools?
• U.S. 8th graders’ weakest performance areas in the Trends in
International Mathematics and Science Study (TIMSS) were geometry
and measurement
• U.S. 12th graders posted the lowest geometry scores of any
participating country on the TIMSS assessment
• Data from the Programme of International Student Assessment
(PISA) study also reveals that U.S. students are weakest in geometry
• TIMMS data shows that U.S. 8th graders receive proportionally less
geometry instructional time than students in most comparison
countries
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1Mullis,
I.V.S., et al. (2001). Mathematics benchmarking report – TIMSS 1999 8th grade. Chestnut Hill, MA: International Study Center, Boston College.
2Mullis,
I.V.S., et al. (1998). Mathematics and science achievement in the final year of secondary school: IEA’s Third International Mathematics and Science Study.
Chestnut Hill, MA: International Study Center, Boston College.
3Ginsburg,
A., et al. (2005). Reassessing U.S. international mathematics performance: New findings from the 2003 TIMSS and Pisa. Washington DC: American
Institutes for Research.
Should We Increase Middle School
Students’ Exposure to Geometry?
“Since geometry correlated highly with [important higher-order thinking
skills], geometry may be something of a gateway skill to the teaching of
higher order mathematics thinking skills...These findings suggest that the
curriculum in the United States should put more emphasis on teaching
geometry, because geometry may enable teaching of important
mathematical thinking skills....Surprisingly, algebra did not correlate with
these mathematical thinking skills but was related to computational skills.
Thus, educators concerned with designing effective mathematics curricula
might ask: Is the emphasis on algebra in current U.S. mathematics curricula
sufficient to effectively teach logical reasoning and higher level judgmental
skills to this age group [of 8th graders]?” (pp. 922-923)*
*Tatsuoka,
K.K., et al. (2004). Patterns of diagnosed mathematical content and process skills in TIMMS-R across a sample of 20 countries. American
Educational Research Journal, 41(4), 901-926.
How Can We Prepare Middle Grade Teachers
to Effectively Teach Geometry?
FGT’s Approach to Professional
Development – Overview
Written professional development materials
* Developed with funding from the National Science Foundation
(NSF Grant ESI-0353409)
* For use by a facilitator with a group of teachers
* Intended for grades 5-10 teachers
* 40 hours of PD  broken into 20 two-hour sessions.
* Currently in second year of field testing
FGT’s Approach to Professional
Development – Guiding Structures
Structured Exploration Process (experience today)
Geometric Habits of Mind (G-HOMs) Framework (examples today)
Additional Supports (Questioning, Language, Cognitive Demand)
FGT’s Approach to Professional
Development – SE Process
Structured Exploration Process (Kelemanik et al. 1997)
Stage 1: Doing mathematics
Stage 2: Reflecting on the mathematics
Stage 3: Collecting student work
Stage 4: Analyzing student work
Stage 5: Reflecting on students’ thinking
FGT’s Approach to Professional
Development – The G-HOMs
Generalizing Geometric Ideas
Reasoning with Relationships
Investigating Invariants
Balancing Exploration and Reflection
Generalizing Geometric Ideas
Wanting to understand and describe the "always" and the "every"
related to geometric phenomena.
"Does 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?”
Reasoning with Relationships
Actively looking for and trying to use relationships (e.g., congruence,
similarity, parallelism, etc.), within and between geometric figures.
"How are these figures alike?"
"In how many ways are they alike?"
"How are these figures different?"
Investigating Invariants
An invariant is something about a situation that stays the same, even
as parts of the situation vary.
"What changes? Why?"
"What stays the same? Why?"
Balancing Exploration and
Reflection
Trying various ways to approach a problem and regularly stepping back
to take stock.
"What happens if I (draw a picture, add to/take apart this picture, work
backwards from the ending place, etc.….)?"
"What did I learn from trying that?"
FGT’s Approach to Professional
Development – The G-HOMs
Generalizing Geometric Ideas
Reasoning with Relationships
Investigating Invariants
Balancing Exploration and Reflection
Reflect on Geometric Thinking
Through Problem Solving
1) Take a few minutes to play with the problem yourself.
2) Continue working on the problem with others at your table.
Goals:
 Explore the problem
 Prepare to analyze students’ thinking about the problem
 Pay attention to how you are working on the problem
Reflect on Geometric Thinking
Through Problem Solving
Two vertices of a triangle are located at (0,6) and (0,12).
The area of the triangle is 12 units2.
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12
10
8
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2
-10
-5
5
-2
10
Reflect on Geometric Thinking
Through Problem Solving
a) What are all possible positions for the third vertex?
b) Explain how you know these vertices create triangles with an area of 12
units2.
c) How do you know there aren’t any more?
Reflect on Geometric Thinking
Through Problem Solving
1) Take a few minutes to play with the problem yourself.
2) Continue working on the problem with others at your table.
Goals:
 Explore the problem
 Prepare to analyze students’ thinking about the problem
 Pay attention to how you are working on the problem
Geometric Habits of Mind elicited
by Finding Area in Different Ways
Generalizing Geometric Ideas
"Does 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 - should I then revise my generalization?”
Reasoning with Relationships
"How are these figures alike?"
"In how many ways are they alike?"
"How are these figures different?"
Investigating Invariants
"What changes? Why?"
"What stays the same? Why?"
Balancing Exploration and Reflection
"What happens if I (draw a picture, take apart this picture, work backwards, etc.….)?"
"What did I learn from trying that?"
Analyze Student Work
1)
What relationships are the students paying
attention to as they try to find triangles with
an area of 12 units2?.
2)
What steps are the students taking toward
generalizing their ideas?
Preliminary Findings
•
Increase in geometric content knowledge,
especially in the area of measurement
•
Increase in attention to students’
mathematical thinking
Transcript Question: What stood out for you
in terms of the work the students and
teachers were doing in this activity?
Before FGT:
“…the teacher was leading the students by the questions she was
asking. I also liked the way the kids had a chance to think about
the problem and come back to answer questions.”
After FGT:
“It was clear that they had worked on the volume of a rectangular
prism and they had filled a rectangular prism with layers of cubes
to find the volume of the prism. They actually took this thought
and applied it to the cylinder…”
Transcript Question: What stood out for you
in terms of the work the students and
teachers were doing in this activity?
Before FGT:
“(The) teacher was not giving any answers, formulas or even
strategies (and) was simply working through a discussion with the
class to see how they solved it and what direction it went. Each
class discussion could be totally different.”
After FGT:
“ --Analyzing cylinders and boxes
--volume of prisms versus cylinders
--understanding of height and radius
--dimensions of shapes”
Importance of Geometry
•
U.S. 8th graders’ weakest performance areas in TIMSS were
geometry and measurement
•
“… geometry correlated highly with [important higher-order
thinking skills], geometry may be something of a gateway skill to
the teaching of higher order mathematics thinking skills
•
TIMMS data shows that U.S. 8th graders receive proportionally
less geometry instructional time than students in most comparison
countries
Will be published by Heinemann in 2008
To find out more about the project:
www.geometric-thinking.org
Mark Driscoll ([email protected])
Mike Egan ([email protected])
Johannah Nikula ([email protected])
Rachel Wing DiMatteo ([email protected])