Fostering Geometric Thinking in the Middle Grades

Fostering Geometric Thinking in the Middle Grades

Horizon Research, Inc.

Chapel Hill, NC

Education Development Center

Newton, MA

Project supported by the National Science Foundation under Grant ESI-0353409

Knowledge for Teaching Knowledge for Teaching GeometryGeometry

• 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 Fostering Geometric Thinking (FGT)Thinking (FGT)

• Identify productive ways of thinking in geometry (G-HOMs)

• Create professional development materials based on G-HOMs– 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 Fostering Geometric Thinking (FGT)Thinking (FGT)

• Field TestResearch 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 TestFGT 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 Geometric Habits of Mind (G-HOMs)(G-HOMs)

• Reasoning with Relationships

• Investigating Invariants

• Generalizing Geometric Ideas

• Balancing Exploration with Reflection

Reasoning with Reasoning with RelationshipsRelationships

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?

5

3

5

3

10

6

10

6

2 2

4

4

Generalizing Geometric Generalizing Geometric Ideas 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 90-degree angles:

Investigating InvariantsInvestigating 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 Balancing Exploration with ReflectionReflection

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 Knowledge for Teaching GeometryGeometry

• 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 Knowledge for Teaching GeometryGeometry

• 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 Knowledge for Teaching GeometryGeometry

• 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 Knowledge for Teaching GeometryGeometry

• 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 Knowledge for Teaching GeometryGeometry

• Middle-grades geometry as groundwork for high school geometry– Proof: explanation, verification,

discovery (De Villiers)– Reasoned conjectures (Herbst)– Multimodal communication (Chval,

Khisty)

Contact InformationContact InformationMark DriscollMark Driscoll

mdriscoll@edc.org

Rachel WingRachel Wingrwing@edc.org

Fostering Geometric Thinking websiteFostering Geometric Thinking websitewww.geometric-thinking.comwww.geometric-thinking.com

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