Johannah Nikula Project Director

Education Development Center (EDC) jnikula@edc.org

Finding the Standards for Mathematical Practice in Geometry

The Implementing the Mathematical Practice Standards project is supported by the National Science Foundation under Grant No. DRL 1119163. Any opinions, findings, and conclusions or recommendations expressed are those of the author and do not necessarily reflect the views of the National Science Foundation.

As you arrive…

• Review the green handout that describes the Standards for Mathematical Practice (SMP).

• Talk with a neighbor about how you could complete one of these sentences:

– An example of [**insert one of the SMP**] in geometry is…

– Something I need to learn more about the SMP in geometry is…

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Goals

• Build awareness and understanding of the SMP connected to geometry

• Cultivate capacity to identify these SMP in students’ geometric thinking

• Engage in instructional planning to support geometric thinking aligned with the SMP

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Plan for Session

• Math Task: Engage in a geometry task then discuss how our habits of mind aligned with the SMP.

• Student Dialogue: Analyze an excerpt of student dialogue to understand SMP in students’ thinking.

• Planning: Begin using a protocol focused on the SMP to consider instructional questions & strategies.

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Illustrations of the SMP

• 30+ Illustrations developed and reviewed to date (mathpractices.edc.org; see bookmark with QR code)

• Illustrations include math tasks, student dialogues, teacher reflection questions, math overviews, and student questions

• Development included input and review from: – Grades 5-10 mathematics teachers – Mathematicians – Other mathematics educator advisors

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Exploring an Illustration: Parallelogram Vertices

1) Work on geometry task (individual then pairs) 2) Sharing/debriefing mathematics of task (full group)

3) Read and discuss student dialogue and mathematical

overview.

4) Plan instruction based on mathematics task.

Parallelogram Vertices: Math Task

Given three vertices of a parallelogram at A (1, 2), B (4, 1), and C (5, 3), where can the fourth vertex be located?

• Work individually for 4 min, then in pairs. • Keep track of thinking, questions, wrong turns, etc.

Parallelogram Vertices: Discussing Mathematics

What strategies did you use to explore the problem? How did you start the problem? What conjectures or questions do you have at this point?

Parallelogram Vertices: Reflecting on the SMP

What evidence of the SMP did you see in your own or

your colleagues’ work on the task?

Exploring an Illustration: Parallelogram Vertices

1) Work on geometry task (individual then pairs) 2) Sharing/debriefing mathematics of task (full group)

3) Read and discuss student dialogue and mathematical

overview.

4) Plan instruction based on mathematics task.

Student Dialogues

• Dialogue between three fictitious middle grades or high school characters (either Sam, Dana, and Anita, or Chris, Lee, and Matei) working on a mathematics task

• The dialogues are intended to: – Clarify the meaning of particular SMP by showing what student

discourse and thinking could be

– Illustrate key ideas about the Standards for Mathematical Practice (SMP) in context using specific mathematical content

– Serve as an artifact to promote discussion among educators about the SMP, about mathematics, and about issues of teaching practice.

Student Dialogues

Given the intention to illustrate the meaning and key ideas of the SMP: The students…

– Plausible student thinking, but the discourse may not always sound realistic.

– The student characters are “caricatures,” intended to illustrate particular types of thinking and discussion.

The teacher…

– A teacher voice is intentionally not included.

– Discussion of whether/how a teacher might intervene, or of how to promote similar thinking with your students, are productive avenues for discussion.

Reading the Student Dialogue

• 3 volunteers read the student dialogue out loud

• Re-read the dialogue individually:

– Try to understand how the students are thinking about the task. – Then, note places where the students seem to be engaging in any of

the Standards for Mathematical Practice.

Discuss Student Dialogue in Small Groups

1. What evidence do you see of “MP 5: Use appropriate tools strategically” in the Student Dialogue?

2. What did using graph paper as opposed to blank paper allow the students to do? How/why can graph paper be considered a tool? Find evidence in the dialogue of how using graph paper contributed to the students’ thinking.

3. What evidence do you see of “MP 7: Look for and make use of structure” in the Student Dialogue? What evidence do you see of other SMP?

4. Given points (2,1), (6,1), and (2,-2), how many parallelograms can you make? How many rectangles? Explain your reasoning.

(see handout for additional questions)

Mathematical Overview

What surprises you or resonates with you in the Mathematical Overview?

Exploring an Illustration: Parallelogram Vertices

1) Work on geometry task (individual then pairs) 2) Sharing/debriefing mathematics of task (full group)

3) Read and discuss student dialogue and mathematical

overview.

4) Plan instruction based on mathematics task.

Parallelogram Vertices: Planning Protocol

• Identify goals related to SMP and mathematical content

• Anticipate student thinking – productive avenues and potential challenges

• Plan questions to ask students and strategies to support students to engage in the SMP

Planning Instruction to Support SMP Focus on Question #4

In pairs or small groups, create two lists:

1) Anticipate student thinking on this geometry task.

2) List questions & supports that will create and develop opportunities for students to engage in the SMP.

(Think about particular groups of students – e.g., at your grade level, students who are ELs, etc.)

Choose one or more as teacher ?s MP 1: Make sense of problems and persevere in solving them. • What am I trying to find out? • Which pieces of information in this problem might be useful? • What do I know that seems useful for this problem? • How might I get started? • What do I expect will happen if I try this? • What have I tried so far? What have the results taught me? • What I tried didn’t work. How can I make use of what I did find out? • Are there other possible answers or other possible questions? • What can I do with this information (even if I’m not sure that will answer the question)?

MP 3: Construct viable arguments and critique the reasoning of others. • Is my claim true for all conditions or only some? How do I know? • What are my assumptions? Do I need them all? What do I already know? What conjectures do I have? • Now that I have a conclusion, how can I justify it? • Could I interpret the same steps or facts differently? If so, how? If not, why not? • Can I think of any counter-examples that would show why this conclusion is not always true? • Are the steps I took logical? • Can I follow the reasoning in someone else’s argument?

Choose one or more as teacher ?s MP 7: Look for and make use of structure. • What do I know or notice about the relationship between parts of this geometric drawing? • What is a different way of looking at this situation? • What part of this mathematical situation or problem is complicated, and how can I simplify that part to

understand the rest of the situation or problem? • How are these the same and different?

MP 5: Use appropriate tools strategically. • What tools would help me experiment (e.g., try more cases) more efficiently? • What tools would help me make more accurate measurements or calculations? • What tools would help me to represent and reason about the mathematical ideas in this problem?

MP 6: Attend to precision. • Could others follow the directions or process I’m describing without any other information? • What other words or drawings could I use to explain my thinking? • Do we mean the same thing by this term or phrase or symbol? • What definitions am I relying on for this explanation? Do they need to be stated? • Have I provided a final answer that makes sense for the question? • Have I been careful enough in carrying out the steps of this procedure?

Planning Instruction to Support SMP Focus on Question #4

In pairs or small groups, create two lists:

1) Anticipate student thinking on this geometry task.

2) List questions & supports that will create and develop opportunities for students to engage in the SMP.

(Think about particular groups of students – e.g., at your grade level, students who are ELs, etc.)

Planning Instruction to Support SMP

Full Group: 1) What do you notice in the planned questions and supports?

2) What is a key idea about planning you are taking away?

Implementing the Mathematical Practice Standards

Questions?

mathpractices.edc.org

(QR code on bookmark)

Takeaway handouts = Reading about the SMP in Geometry

Lesson planning protocol (we worked on #4)

Thank you!

(If you are interested in learning more talk to me after the session or email me at jnikula@edc.org or the project at mathpractices@edc.org)

Disclaimer The National Council of Teachers of Mathematics is a public voice

of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

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