fostering mathematical discourse
DESCRIPTION
Fostering Mathematical Discourse. Defining Mathematical Discourse. What is mathematical discourse? What teacher and student behaviors occur in a classroom where the teacher promotes discourse? . Brainstorm. Defining Mathematical Discourse. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/1.jpg)
S
Fostering Mathematical
Discourse
![Page 2: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/2.jpg)
S
Defining Mathematical
Discourse
![Page 3: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/3.jpg)
What is mathematical discourse?
What teacher and student behaviors occur in a classroom where the teacher promotes discourse?
Brainstorm
![Page 4: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/4.jpg)
Defining Mathematical Discourse
Discourse: written or spoken communication or debate
- Oxford Dictionary
![Page 5: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/5.jpg)
What does NCTM say?
Communication Instructional programs from prekindergarten through grade 12 should enable all students to— Organize and consolidate their mathematical thinking
through communication Communicate their mathematical thinking coherently
and clearly to peers, teachers, and others Analyze and evaluate the mathematical thinking and
strategies of others; Use the language of mathematics to express
mathematical ideas precisely. See more at: http://www.nctm.org/standards/content.aspx?id=322#sthash.rEE2w8Ms.dpuf
![Page 6: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/6.jpg)
What does the Common Core
say?Understanding MathematicsThese Standards define what students should understand and be able to do in their study of mathematics. Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y). Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness. – CCSSM, p. 4
![Page 7: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/7.jpg)
What does Common Core say?
Skim through the Standards. Underline or highlight everything that is related to discourse.
Talk with a shoulder buddy: What stands out to you? What kinds of discourse are already taking place in your classroom? What are areas of need?
Common Core Standards for Mathematical Practice
![Page 8: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/8.jpg)
Standards for Math Practice3. Construct viable arguments and critique the
reasoning of others.Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
![Page 9: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/9.jpg)
Definition of Mathematical Discourse
A process by which students use discourse, both verbal and written, to reflect on the mathematics they have engaged with in order to discover important mathematical concepts and to develop mathematical thinking.
![Page 10: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/10.jpg)
S
Teaching Practices and the Teacher’s
Role
![Page 11: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/11.jpg)
So now that I know what it is, how do I do it?
How to Get Students Talking!: Generating Math Talk That Supports Math Learning by Lisa Ann de Garcia. “Common Core . . . make[s] it clear that
conceptual understanding must be connected to the procedures, and that one way to deepen conceptual understanding is through the communication students have around concepts, strategies, and representations.”
“Children do not naturally engage in this level of talk.”
![Page 12: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/12.jpg)
Practice 1: Talk Moves That Engage Students in
Discourse Revoicing – So you are saying that . . .
Restate someone else’s reasoning – Can you repeat what she just said in your own words?
Apply their own reasoning to someone else’s – What do you think about that? Do you agree or disagree? Why?
Prompt for further participation – Would someone like to add on?
Use wait time!
![Page 13: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/13.jpg)
Practice 2: The Art of Questioning
Help students work together to make sense of mathematics (Practice 1 questions)
Help students rely more on themselves to determine whether something is mathematically correct – How did you reach that conclusion? Does that make sense? Can you make a model and show that?
Help students learn to reason mathematically - Does that always work? Is that true for all cases? Can you think of a counterexample? How could you prove that?
Help students learn to conjecture, invent, and solve problems – What would happen if? Do you see a pattern? Can you predict the next one? What about the last one?
Help students connect mathematics, its ideas and applications – How does this relate to . . .? What ideas that we have learned were useful in solving this problem?
![Page 14: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/14.jpg)
Practice 3: Using Student Thinking to Propel
Discussions Be an active listener Respond neutrally to errors – What do you think about
that? (to whole class) Be strategic about who shares during the discussion Choose ideas, strategies, and representations in a
purposeful way
![Page 15: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/15.jpg)
Practice 4: Set Up a Supportive Environment
Have students facing each other – e.g. desks in groups for partner or small group discussions; students sitting in a circle for whole group
Place visual aids and vocabulary where they can be easily accessed
Create a safe emotional environment where the value is on learning, challenging each other, and working together to solve problems as opposed to just getting the right answer
![Page 16: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/16.jpg)
Practice 5: Orchestrating the Discourse
The Five Practices Model The teacher’s role is to:1. anticipate student responses to challenging
mathematical tasks; 2. monitor students’ work on and engagement with
the tasks; 3. select particular students to present their
mathematical work; 4. sequence the student responses that will be
displayed in specific order; and 5. connect different students’ responses and connect
the responses to key mathematical ideas.
![Page 17: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/17.jpg)
Hold Students Accountable
Explicitly teach students how to engage in each level of discussion: whole group, small group, partnerships Model the behavior – e.g. do a fishbowl of a small
group or partnership discussion, show video clips of discussions and debrief
Address not only content but also behavior when summarizing – I liked how Sarah asked Tom to explain what he meant, That group did a great job with listening to each other, etc.
Do a plus/delta on the discussion – What went well? Where do we need to improve?
![Page 18: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/18.jpg)
Hold Students Accountable
Let them know exactly what they should be saying when they are talking in their partnerships or small groups – Today, when you are talking to your partners and describing ______, I expect to hear you using the words ______.
Let students know what to focus on when someone is sharing a strategy – When Maria is sharing her thinking, I want you to be thinking of how her way is similar to or different from your way.
![Page 19: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/19.jpg)
Hold Students Accountable
Heighten students awareness of themselves as learners through self-evaluation and goal setting have students set and track personal goals related
to participation in mathematical discussions – e.g. exit ticket of a plus/delta on their participation
support students in being open with each other regarding their strengths and weaknesses so they can improve their communication skills and behaviors – e.g. hold a class meeting that focuses on this
![Page 20: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/20.jpg)
S
Experience Mathematical
Discourse from a Student’s
Perspective
![Page 21: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/21.jpg)
What does it feel like?The Tower Problem
Use the blocks to build the fourth tower in the sequence. How many cubes did you use? How many cubes would you need to build the fifth tower? The 12th tower? The 20th tower? The 100th tower? Write a rule to help you find the number of cubes for the nth tower.
Take a break as needed while your group works on this problem.
![Page 22: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/22.jpg)
S
Example Discourse: The Good, the Bad,
and the Ugly
![Page 23: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/23.jpg)
Example Discourse – the Good, the Bad, and the
UglyRead the “Facilitating Discourse” section p. 286-288 of Let’s Talk: Promoting Mathematical Discussions in the Classroom by Catherine C. Stein. Discuss with your shoulder buddy: What is the difference between cognitive and
motivational discourse? Why are both important? What is the difference between low-press and
high-press classrooms? How does the level of “press” affect student learning?
![Page 24: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/24.jpg)
Example Discourse – the Good, the Bad, and the
Ugly
Read “An example of univocal discourse” on p. 322 of Unpacking the Nature of Discourse in Mathematics Classrooms by Eric Knuth and Dominic Peressini.In your group: Identify any missed opportunities (give specific
line number and explain). How could the discourse be improved?
![Page 25: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/25.jpg)
S
Keys to Mathematical
Discourse
![Page 26: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/26.jpg)
The Keys to Mathematical Discourse
Authentic, Rich Tasks Level of Questioning
![Page 27: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/27.jpg)
To PROBE or uncover students’ thinking. • understand how students are thinking about the
problem. • discover misconceptions. • use students’ understanding to guide instruction.
To PUSH or advance students’ thinking. • make connections • notice something significant. • justify or prove their thinking.
The only reasons to ask questions are:
(Black et al., 2004)
![Page 28: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/28.jpg)
Question Prompts and Stems
![Page 29: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/29.jpg)
Question Analysis
Revised Bloom’s Taxonomy Question Analysis Activity
![Page 30: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/30.jpg)
Authentic, Rich Tasks
Current research evidence indicates that students who are given opportunities to work on their problem solving skills enjoy the subject more, are more confident and are more likely to continue studying mathematics, or mathematically related subjects, beyond the age of 16. Most importantly to some, there is also evidence that they do at least as well in standard tests such as GCSEs and A-levels.
http://nrich.maths.org/6299
![Page 31: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/31.jpg)
Authentic, Rich Tasks
Rich tasks (or good problems): are accessible to a wide range of learners, might be set in contexts which draw the learner into the
mathematics either because the starting point is intriguing or the mathematics that emerges is intriguing,
are accessible and offer opportunities for initial success, challenging the learners to think for themselves,
offer different levels of challenge, but at whatever the learner's level there is a real challenge involved and thus there is also the potential to extend those who need and demand more (low threshold - high ceiling tasks),
allow for learners to pose their own problems, allow for different methods and different responses (different
starting points, different middles and different ends),http://nrich.maths.org/5662
![Page 32: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/32.jpg)
Authentic, Rich Tasks
offer opportunities to identify elegant or efficient solutions, have the potential to broaden students' skills and/or deepen
and broaden mathematical content knowledge, encourage creativity and imaginative application of
knowledge. have the potential for revealing patterns or lead to
generalizations or unexpected results, have the potential to reveal underlying principles or make
connections between areas of mathematics, encourage collaboration and discussion, encourage learners to develop confidence and independence
as well as to become critical thinkers. http://nrich.maths.org/6299
![Page 33: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/33.jpg)
How do I incorporate this?
Start Simple (KISS!) Take current problems and make them
better Set a goal: I will incorporate the use of
a rich task once a week, once every two weeks, etc.
Stick with it – it won’t be easy for you or your students; lean on each other in your PLT
Don’t reinvent the wheel – there are plenty of resources out there
![Page 34: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/34.jpg)
The children in the Wright family are aged 3, 8, 9, 10, and 5. What is their average age?
Typical Problem
![Page 35: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/35.jpg)
Better Problem
There are five people in a family and theiraverage age is 7. What might their ages be?
![Page 36: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/36.jpg)
Round 11.8 to the nearest whole number.
Typical Problem
![Page 37: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/37.jpg)
My coach timed me running 100 meters in about 12 seconds. What numbers might have been on the stopwatch?
Better Problem
![Page 38: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/38.jpg)
There are 6 birds and 2 cats. If the answer is . . .
a. 20b. 8c. 4
What could the question be?
How About This?
![Page 39: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/39.jpg)
Better Questions, Better Results
![Page 40: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/40.jpg)
Rich Tasks – Where do I find them?
Core Plus is full of them!!! List of resources on the training wiki
![Page 41: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/41.jpg)
S
Assessing Discourse
![Page 42: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/42.jpg)
Basic Rubric for Assessing Levels of Discourse in a Math Classroom
http://www.nctm.org/publications/mt.aspx?id=8594
![Page 43: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/43.jpg)
Observation Tools
Scripting of Questions/Question Analysis Tool Classroom Discourse Data Tool Student Discourse Observation Tool Video Modeling OMLI Classroom Observation
![Page 44: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/44.jpg)
Wednesday PLT Time
How can we team within the department to be more intentional about creating discourse?
![Page 45: Fostering Mathematical Discourse](https://reader036.vdocuments.us/reader036/viewer/2022081514/5681672c550346895ddbcef2/html5/thumbnails/45.jpg)
Shifting Our Perspective
When students don’t seem to understand something, my instinct is to consider how I can explain more clearly. A better way is to think “They can figure this out. I just need the right question.” - D. Kennedy (2002)
Never say anything a kid can say. - Reinhart (2000)