NCTM13 SupportingReasoningProofCollaboration HO€¦ · Is#it#true?Always?Supporting#...
Transcript of NCTM13 SupportingReasoningProofCollaboration HO€¦ · Is#it#true?Always?Supporting#...
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Is it true? Always? Supporting Reasoning-‐and-‐Proof Focused Collaboration among Teachers
2013 NCTM Annual Conference Denver, Colorado
Nicole Miller Rigelman * Portland State University [email protected]
http://goo.gl/ys4Qd
In this session we will: * Unpack what it is meant by reasoning-‐and-‐proving and consider what it looks like to prompt such thinking. * Explore tools to support teacher collaboration with selecting tasks, planning, and observing/examining classroom practice and student thinking with an eye on supporting students with developing convincing arguments.
Session Overview
* Growing consensus in the community that it should be “a natural, ongoing part of classroom discussions, no matter what topic is being studied,” (NCTM, 2000, p. 342).
What gets in the way? * It is difficult for teachers and student. * We have limited conceptions of what counts as “reasoning and proving.”
Why focus on reasoning and proof in your PLC?
Instructional programs [preK-‐12] should enable students to develop and evaluate mathematical arguments and proofs.
-‐ NCTM, 2000, p. 56
[In grades 3-‐5] mathematical reasoning develops in classrooms where students are encouraged to put forth their own ideas for examination. Teachers and students should be open to questions, reactions, and elaborations from others in the classroom. Students need to explain and justify their thinking and learn how to detect fallacies and critique others thinking.
-‐ NCTM, 2000, p. 188
Reasoning and Proof
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…both plausible and flawed arguments that are offered by students create an opportunity for discussion. As students move through the grades, they should compare their ideas with others’ ideas, which may cause them to modify, consolidate, or strengthen their arguments or reasoning. Classrooms in which students are encouraged to present their thinking, and in which everyone contributes by evaluating one another’s thinking, provide rich environments for learning mathematical reasoning.
-‐ NCTM, 2000, p. 58
Reasoning and Proof
In the domain of Number and Operation… * Computation strategy. Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another. * Computation algorithm. A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly.
-‐ K-‐5 Progressions, Number and Operations in Base Ten, 2011, p. 3
Reasoning and Proof
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 mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning
- Common Core State Standards for Mathematics, 2010, pp. 6-7
Standards for Mathematical Practice Mathematics Task Framework
Cognitive Demand Design Set-‐Up Implementation Student Learning
Stein et al, 1998
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Mathematical task as represented in
curricular/ instructional materials
Mathematical task as set up by the teacher in the classroom
*Task features *Cognitive demands
Mathematical task as implemented by students in the
classroom *Enactment of task
features *Cognitive processing
Students’ Learning outcomes
Relationships among various task-‐related variables and students’ learning outcomes. Henningsen and Stein, 1997, p. 528; Stein and Smith, 1998, p. 270
Factors influencing setup *Teachers’ goals
*Teachers’ knowledge of subject matter
*Teachers’ knowledge of students
Factors influencing students’
implementation *Classroom norms
*Teachers’ instructional dispositions
*Students’ learning dispositions
Mathematics Task Framework
* Jamie’s family visited their grandmother who lives 634 miles from their house. On the first day they drove 319 miles. How many miles did they have left to drive the second day?
From Investigations in Number, Data, and Space, Russell & Economopoulos, 2008
Visiting Grandma
Task Analysis Guide
Memorization Procedures with Connections • involve either reproducing previously learned facts, rules,
formulas, or definitions or committing facts, rules, formulas or definitions to memory.
• cannot be solved using procedures because a procedure does not exist or because a time frame in which the task is being completed is too short to use a procedure.
• are not ambiguous. Such tasks involve exact reproduction of previously seen material, and what is to be reproduced is clearly and directly stated.
• focus students’ attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideas.
• suggest explicitly or implicitly pathways to follow that are broad general procedures that have close connections to underlying concepts.
• usually are represented in multiple ways, such as visual diagrams, manipulatives, symbols, and problem situations. Making connections among multiple representations helps develop meaning.
• require some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with conceptual ideas that underlie the procedures to complete the task successfully and that develop understanding.
Procedures without Connections Doing Mathematics
• are algorithmic. Use of the procedure either is specifically called for or is evident from prior instruction, experience, or placement of the task.
• require limited cognitive demand for successful completion. Little ambiguity exists about what needs to be done and how to do it.
• have no connection to the concepts or meaning that underlie the procedure being used.
• are focused on producing correct answers instead of on developing mathematical understanding.
• require no explanations or explanations that focus sole on describing the procedure that was used.
• require complex and non-‐algorithmic thinking -‐-‐ a predictable, well-‐rehearsed approach or pathway is not explicitly suggested by the task, task instructions, or a worked-‐out example.
• require students to explore and understand the nature of mathematical concepts, processes, or relationships.
• demand self-‐monitoring or self-‐regulation of one’s own cognitive processes.
• require students to access relevant knowledge and experiences and make appropriate use them in working through the task.
• require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.
Henningsen and Stein, 1997, p. 528; Stein and Smith, 1998, p. 270
Task Analysis Guide
Procedures with Connections • focus students’ attention on the use of procedures for the purpose
of developing deeper levels of understanding of mathematical concepts and ideas.
• suggest explicitly or implicitly pathways to follow that are broad general procedures that have close connections to underlying concepts.
• usually are represented in multiple ways, such as visual diagrams, manipulatives, symbols, and problem situations. Making connections among multiple representations helps develop meaning.
• require some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with conceptual ideas that underlie the procedures to complete the task successfully and that develop understanding.
Henningsen and Stein, 1997, p. 528; Stein and Smith, 1998, p. 270
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0. Setting Goals and Selecting the task
1. Anticipating (e.g., Fernandez & Yoshida, 2004; Schoenfeld, 1998)
2. Monitoring (e.g., Hodge & Cobb, 2003; Nelson, 2001; Shifter, 2001)
3. Selecting (Lampert, 2001; Stigler & Hiebert, 1999)
4. Sequencing (Schoenfeld, 1998)
5. Connecting (e.g., Ball, 2001; Brendehur & Frykholm, 2000)
The Five+ Practices
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Orchestrating Productive Mathematical Discourse Chart for Monitoring, Selecting, Sequencing, and Connecting Student Thinking
Strategy Work of Specific Students Sequence Compare
A.
B.
C.
D.
Adapted from Smith and Stein (2011).
Orchestrating Productive Mathematical Discourse Chart for Monitoring, Selecting, Sequencing, and Connecting Student Thinking
Strategy Work of Specific Students Sequence Compare
A.
3 A and B, A and C
B. Number line connect to add up
1
C. Add up to nearest hundred
2 C and B
D.
5 D and F
E. Base Ten Pieces connect to take away showing regrouping**
6 delay if
not enough
time
F. Base Ten Pieces or number line connect to take away too many (320) and add 1 back
4 Ask students for connections
they see
Adapted from Smith and Stein (2011).
** By the end of the unit, the target is: 4.NBT4 Fluently add and subtract multi-digit numbers using the standard algorithm. Since this is the first lesson on multi-digit subtraction, this is not the main emphasis.
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-‐ Boaler & Humphreys, p. 37, 2005
! !
!
Examining Student Thinking
& Teacher Moves
* Type 1 – Answering, Stating, or Sharing
* Type 2 – Explaining
* Type 3 – Questioning or Challenging
* Type 4 – Relating, Predicting, or Conjecturing
* Type 5 – Justifying or Generalizing -‐ Weaver, Dick, & Rigelman, 2005
Discourse Types
* Type 1 – Answering, Stating, or Sharing A student gives a short right or wrong answer to a direct question or makes a simple statement or shares work that does not involve an explanation of how or why. * Type 2 – Explaining * Type 3 – Questioning or Challenging * Type 4 – Relating, Predicting, or Conjecturing * Type 5 – Justifying or Generalizing
-‐ Weaver, Dick, & Rigelman, 2005
Discourse Types
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* Type 1 – Answering, Stating, or Sharing * Type 2 – Explaining A student explains a mathematical idea or procedure by describing how or what he or she did but does not explain why. * Type 3 – Questioning or Challenging * Type 4 – Relating, Predicting, or Conjecturing * Type 5 – Justifying or Generalizing
-‐ Weaver, Dick, & Rigelman, 2005
Discourse Types
* Type 1 – Answering, Stating, or Sharing * Type 2 – Explaining * Type 3 – Questioning or Challenging A student asks a question to clarify his or her understanding of a mathematical idea or procedure or makes a statement or asks a question in a way that challenges the validity of an idea or procedure. * Type 4 – Relating, Predicting, or Conjecturing * Type 5 – Justifying or Generalizing
-‐ Weaver, Dick, & Rigelman, 2005
Discourse Types
* Type 1 – Answering, Stating, or Sharing * Type 2 – Explaining * Type 3 – Questioning or Challenging * Type 4 – Relating, Predicting, or Conjecturing A student makes a statement indicating that he or she has made a connection or sees a relationship to some prior knowledge or experience or makes a prediction or a conjecture based on an understanding of the mathematics behind the problem. * Type 5 – Justifying or Generalizing
-‐ Weaver, Dick, & Rigelman, 2005
Discourse Types
* Type 1 – Answering, Stating, or Sharing * Type 2 – Explaining * Type 3 – Questioning or Challenging * Type 4 – Relating, Predicting, or Conjecturing * Type 5 – Justifying or Generalizing A student provides a justification for the validity of a mathematical idea or procedure or makes a statement that is evidence of a shift from a specific example to the general case.
-‐ Weaver, Dick, & Rigelman, 2005
Discourse Types
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For more information about the Student Discourse Observation Protocol see, Weaver, D. & Dick, T. (September 2006). Assessing the Quantity and Quality of Student Discourse in Mathematics Classrooms, Year 1 Results. Paper presented at Math Science Partnership Evaluation Summit II, Minneapolis, MN. Available at: http://ormath.mspnet.org/index.cfm/14122.
Discourse Analysis Tool Discourse Types Predicted Actual Type 1 – Answering, Stating, or Sharing A student gives a short right or wrong answer to a direct question or makes a simple statement or shares work that does not involve an explanation of how or why.
Type 2 – Explaining A student explains a mathematical idea or procedure by describing how or what he or she did but does not explain why.
Type 3 – Questioning or Challenging A student asks a question to clarify his or her understanding of a mathematical idea or procedure or makes a statement or asks a question in a way that challenges the validity of an idea or procedure.
Type 4 – Relating, Predicting, or Conjecturing A student makes a statement indicating that he or she has made a connection or sees a relationship to some prior knowledge or experience or makes a prediction or a conjecture based on an understanding of the mathematics behind the problem.
Type 5 – Justifying or Generalizing A student provides a justification for the validity of a mathematical idea or procedure or makes a statement that is evidence of a shift from a specific example to the general case.
Implications
As you reflect on what you heard in this session, what are take aways… * For your classroom?
* For your professional learning community?
* For your work supporting teachers in either setting?
Questions?
Nicole Miller Rigelman * Portland State University [email protected]
http://goo.gl/ys4Qd