Strategies for Infusing Instruction with Mathematical Practices

Samuel OttenUniversity of Missouriottensa@missouri.edu

National Council of Teachers of MathematicsRegional Conference in Louisville, KYNovember 8th, 2013

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IntroductionMath education is more about

what we have students doing than it is about what content they are learning.◦For example, if we’re teaching

content standard N-RN.1, are we going to have students sit quietly and receive information or are we going to activate students as thinkers and problem solvers?

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IntroductionResearch over several decades

has shown that the way students engage in math class impacts their attitudes and their learning of content. (Boaler & Staples, 2008; Hiebert & Grouws, 2007; Stein, Grover, & Henningsen, 1996)

The mathematical practices are the official encapsulation of what students should be doing.

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IntroductionKEY QUESTION: What can we be

doing as teachers to infuse these practices into our teaching?

Or… What more can we be doing to better infuse these practices into our teaching?

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Session OverviewQuick look at the practicesLevels of Implementation

◦Classroom Culture◦Discourse Patterns◦Teacher Questions & Discourse

Moves◦Task Design and Selection

Conclusion

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Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in

repeated reasoningCommon Core State Standards for Mathematics (2010)

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MP6. Attend to PrecisionTwo types of attention to precision

◦Numerical or measurement Clever estimation (and awareness that it’s

an estimate) Awareness of exact answers Significant digits and measurement error

◦Language Precise definitions Say what you mean and mean what you

say (both in words and in symbols) Communication Process Standard

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MP7. Look for and Use StructureStudents must first realize that there is

structure to be found or else they won’t know to look for it. Math makes sense.

Looking for structure is a habit of mind that can be very helpful for learning.◦The structures themselves are often the

key mathematical ideas that we want students to see.

◦Structures also often unlock problems or can be the basis of reasoning.

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MP7. Look for and Use StructureExamples of structures

◦Components of algebraic expressions◦Factors of polynomial coefficients◦Symmetries in graphs or geometric objects

Process-Object distinction (Sfard, 1991)

◦What is initially learned as a process (e.g., taking a square root, using a function rule) eventually becomes a mathematical object in its own right (e.g., a radical term, a function that can be added, multiplied, or composed with other functions).

This practice also involves students shifting perspective and seeing the bigger picture.

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M8. Look for and Express Regularity in Repeated ReasoningNoticing repetitions or regularity

◦ Most patterns/repetitions in mathematics are not coincidental

◦ Expressing a repetition or pattern (in multiple ways?) can be a vehicle for moving mathematical ideas forward.

Most common form of this practice is having students generalize and make conjectures◦ Leads nicely to Practice 3: Constructing

viable arguments.

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M8. Look for and Express Regularity in Repeated ReasoningReasoning itself can have regularity

(e.g., problem types, inverse operations, proof approaches)◦ Great topic for Review or Going Over

HomeworkMetacognition

◦ stepping out and looking at the outcomes and process of reasoning

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Are MP7 and MP8 distinct?

I say “yes,” others say “no.” The answer may not matter because the important thing is for practices to be happening, not identifying which specific practice it might be.

But here’s my take…◦MP7 (Structure) focuses on mathematical

objects, whereas MP8 (Regularity) focuses on repetitions in process or thinking.

◦Although distinct, they do often co-occur.

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CLASSROOM CULTURE

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General Characteristics of Classroom Culture

• Safe environment to share ideas

• Errors or confusions are met with excitement as learning/thinking opportunities

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Math-Specific Characteristics of Classroom Culture

• Careful thinking is pervasive

• Students have openings and time to communicate their mathematical ideas (and to consider or respond to other’s ideas)

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Culture Should Not Be…

• Answer-focused

• Correctness-focused

• Rushed

Grouws et al. (2013) regarding “coverage”

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Learning from One Another

• What strategies have you found successful in promoting a practices-oriented classroom culture?

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DISCOURSE PATTERNS

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Initiate-Respond-Evaluate

• (I) Teacher asks a question• (R) Students gives an answer• (E) Teacher evaluates the answer

• The predominance of this interaction pattern tends to emphasize answers (R) and correctness (E). Can be efficient but also makes things feel “on the clock.”

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Funneling• Interaction wherein a person

(teacher) asks a series of questions but the questions themselves contain the important mathematical ideas and the student’s answers are low-level or unrelated to the important ideas. Example: Solving 6x + 18 = 36 – 12x

• The asker is coopting the practices and lowering the cognitive demand on the other(s).

Herbel-Eisenmann & Breyfogle (2005)

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Focusing

• Rather than funneling, a person (teacher) asks questions designed to focus the student’s attention on the important mathematical ideas or on something that is likely to help the student move forward.

• The asker is offering help but still leaving the mathematical practices for the student.

Herbel-Eisenmann & Breyfogle (2005)

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Going Over HomeworkThe typical discourse of homework review in

middle school and high school math classrooms involves attention on one problem at a time.

An alternative discourse pattern is to look for patterns across problems, compare/contrast problems, or attend to the mathematical ideas of the assignment as a whole.

This alternative leads to learning gains and can promote practices such as MP1, MP7, and MP8. Otten, Herbel-Eisenmann, &

Cirillo (in press)Jitendra et al. (2009)

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Learning from One Another

• What experiences have you had with focusing interactions or other discourse patterns that promote the practices?

• In what ways do you structure your homework review to promote the practices?

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TEACHER QUESTIONS & DISCOURSE MOVES

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Types of Teacher Questions

Inauthentic Questions• The asker already knows the answer• Example: “What is the y-intercept of that graph?”• Function: Mini-quiz of responder’s knowledge

Authentic Questions• The asker does not already know the answer• Example: “How did you think about that graph?”• Function: Invite the responder into dialogue• More aligned with the infusion of the math

practices

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A Simple Fact

“Why” questions from teachers…

…lead to “Because” responses from students.

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Teacher Discourse Moves (TDMs)

Inviting student participation WaitingRevoicingAsking students to revoiceProbing a student’s thinkingCreating opportunities to engage

with another’s reasoning www.mdisc.org

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Teacher Discourse Moves (TDMs)

This set of discourse moves can be used to increase the quantity of talk in the classroom and also channel that talk in mathematically productive directions.

The original “talk moves” have been tied to significant learning gains in urban districts in math and in English!

www.mdisc.orgChapin, O’Connor, & Anderson

(2009)

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Learning from One Another

What questions or discourse moves have you used to promote the mathematical practices?

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TASK DESIGN & SELECTION

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High Cognitive Demand Tasks

Can provide opportunities to engage in the mathematical practices.

Smith & Stein (1998)

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Doing Mathematics Tasks… Require complex and nonalgorithmic thinking—not

predictable or well-rehearsed approaches. Require students to explore and understand

mathematical concepts, processes, or relationships. Demand self-monitoring or self-regulation of one’s

own thinking. Require students to access relevant knowledge and

experiences and make appropriate use of them in working through the task.

Require students to analyze the task and actively examine task constraints that may limit possible solutions.

Require considerable cognitive effort and may involve some level of anxiety because of the unpredictable nature of the solution process.

Smith & Stein (1998)

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Low Cognitive Demand Tasks…can also be great opportunities to

engage in the mathematical practices, especiallyMP8 – Look for and express regularity in repeated reasoning◦As students complete exercises or execute

procedures, they can be thinking about… Short-cuts Patterns Generalizations

Attending to these things can also raise the cognitive demand as things play out.

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Reversing ProblemsMost problem-types have a

canonical “direction.” For example…◦Start with an equation and find x.◦Start with a Given & To Prove and write a

proof.◦Start with some information and find the

missing information◦Start with a series and express the pattern

Reversing that direction can be a great way to infuse the mathematical practices into a lesson

Background on the TaskGrades 6–7Standards for Mathematical Practice

◦ MP1: Problem Solving◦ MP6: Attend to Precision◦ MP7: Look For and Make Use of Structure

Content◦ 6.SP.3 – Recognize that a measure of center for a numerical

data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

◦ 6.SP.5c – Summarize numeral data sets, such as by giving quantitative measures of center and variability, as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context of the data.

◦ Note: “Mode” is not explicitly in the Common Core Standards.

Reversed Data Set TaskMake up a set of eight numbers

that simultaneously satisfy these constraints:◦Mean: 10◦Median: 9◦Mode: 7◦Range: 15

http://mathpractices.edc.org

Reflecting on our workWhat are differences between

this problem and one that gives a data set and asks for the statistical measures?

How do the differences impact students’ engagement in the practices?

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A Few Other Ideas About TasksEngage students in the process

of “well-defining” a problemBuild in time to look back at

students’ work on a task to make explicit to them that they were engaging in mathematical practices

Look across problems to promote practices and deepen learning

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CONCLUSION

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ConclusionThe CCSSM practices are in danger of falling

into the background, with the content standards dominating the foreground

But the practices are arguably the most important aspect of CCSSM in terms of promoting student learning and attitudes toward mathematics

Although we are already implementing the practices in certain ways, we can all continue to improve in this area by focusing on◦ Classroom Culture;◦ Discourse Patterns;◦ Teacher Questions and Discourse Moves; or◦ Task Design and Selection

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Thank you! Boaler, J., & Staples, M. (2008). Creating mathematical futures through an

equitable teaching approach: The case of Railside School. Teachers College Record, 110, 608-645.

Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students' learning. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371-404). Charlotte, NC: Information Age Publishing.

Koestler, C., Felton, M. D., Bieda, K. N., & Otten, S. (in press). Connecting the NCTM Process Standards and the Common Core State Standards for Mathematical Practice to Improve Instruction. Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Governors Association & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC.

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36.

Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33, 455-488.

ottensa@missouri.edu