Geometric Thinking in the Middle Grades

Geometric Thinking in the Middle Grades

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 Grades (FGT)

• 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), published by Heinemann, Inc.

Do Math with Colleaguesand Reflect

Have Students Do Math and Collect

Student Work

Analyze Student Work with

Colleagues and Reflect

FGTFGT’s Approach to ’s Approach to Professional DevelopmentProfessional Development

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, WestEdMark 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 RelationshipsReasoning 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?

5

3

5

3

10

6

10

6

2 2

4

4

Generalizing Geometric Ideas 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 InvariantsInvestigating 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 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.

-10 -5 5 10

14

12

10

8

6

4

2

-2

Tommy’s “7 Possibilities”

(4,12)

(0,6) (4,6)

(0,12)

(4,7)

(4,8)

(4,9)

(4,10)

(4,11)

Ana’s Procedure

Ana’s Procedure

(4,12)(0,12)

(4,6)(0,6)

(4,7)

(4,8)

(4,9)

(4,10)

(4,11)

(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.

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.”

Working Definition of Similarity

Static vs dynamic and connecting similarity to linearity

x 2

1

3

2

6

x K

11xK

3 3xKy

x

1/3 1/3

y/x =1/3

y =1/3 x

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 add

on? • 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 Fostering Mathematics Success of English Language LearnersEnglish Language Learners

Mark Driscoll, EDC Mark Driscoll, EDC mdriscoll@edc.orgDaniel Heck, Horizon Research, Inc. Daniel Heck, Horizon Research, Inc. dheck@horizon-research.com

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.