Fostering Mathematical Discourse

1. Fostering Mathematical Discourse

2. Defining Mathematical Discourse

3. What is mathematical discourse?What teacher and student behaviors occur in a classroom where the teacher promotes discourse? Brainstorm

4. Defining Mathematical Discourse • Discourse: written or spoken communication or debate • Oxford Dictionary

5. 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

6. What does the Common Core say? Understanding Mathematics These 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

7. What does Common Core say? Common Core Standards for Mathematical Practice • 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?

8. Standards for Math Practice 3. 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.

9. 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.

10. Teaching Practices and the Teacher’s Role

11. 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.”

12. 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!

13. 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?

14. 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

15. 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

16. Practice 5: Orchestrating the Discourse The Five Practices Model The teacher’s role is to: • anticipatestudent responses to challenging mathematical tasks; • monitorstudents’ work on and engagement with the tasks; • selectparticular students to present their mathematical work; • sequencethe student responses that will be displayed in specific order; and • connectdifferent students’ responses and connect the responses to key mathematical ideas.

17. 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?

18. 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.

19. 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

20. Experience Mathematical Discourse from a Student’s Perspective

21. 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.

22. Example Discourse: The Good, the Bad, and the Ugly

23. Example Discourse – the Good, the Bad, and the Ugly Read 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?

24. 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?

25. Keys to Mathematical Discourse

26. The Keys to Mathematical Discourse • Authentic, Rich Tasks • Level of Questioning

27. The only reasons to ask questions are: • 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. • (Black et al., 2004)

28. Question Prompts and Stems

29. Question Analysis • Revised Bloom’s Taxonomy • Question Analysis Activity

30. 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

31. 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

32. 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

33. 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

34. Typical Problem The children in the Wright family are aged 3, 8, 9, 10, and 5. What is their average age?

35. Better Problem There are five people in a family and their average age is 7. What might their ages be?

36. Typical Problem Round 11.8 to the nearest whole number.

37. Better Problem My coach timed me running 100 meters in about 12 seconds. What numbers might have been on the stopwatch?

38. How About This? There are 6 birds and 2 cats. If the answer is. . . a. 20 b. 8 c. 4 What could the question be?

39. Better Questions, Better Results

40. Rich Tasks – Where do I find them? • Core Plus is full of them!!! • List of resources on the training wiki

41. Assessing Discourse

42. Basic Rubric for Assessing Levels of Discourse in a Math Classroom http://www.nctm.org/publications/mt.aspx?id=8594

43. Observation Tools • Scripting of Questions/Question Analysis Tool • Classroom Discourse Data Tool • Student Discourse Observation Tool • Video Modeling • OMLI Classroom Observation

44. Wednesday PLT Time • How can we team within the department to be more intentional about creating discourse?

45. 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)