Utilizing Mathematical Tasks

1. Utilizing Mathematical Tasks Transitioning to the Common Core Train the Trainer Sessions April 30, 2014 Hilary Dito & Pam Tyson

2. The emphasis is on students doing mathematics and demonstrating learning. Standards for Mathematical Practices To achieve equity it is essential to consider: • selection of tasks, • assessment of tasks, and • student learning environment

3. Standards for Mathematical Practices Mathematically proficient Students … • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning.

4. What are Mathematical Tasks? • A mathematical task is a problem or set of problems that focuses students’ attention on a particular idea and/or provides an opportunity to use mathematical practices. • Calls for problem formulation, problem solving, and mathematical reasoning based on evidence • Encourages collaboration and discussion • Enables teacher to provide descriptive qualitative feedback • Stimulates students to make connections and develop a coherent framework for mathematical ideas

5. Mathematical Task: Odd Numbers 7th grade Task

6. Odd numbers • After you complete the problems, compare your work with a partner’s work. What would you change and why?

7. www.InsideMathematics.org

8. Tasks as they appear in curricular materials Student learning The Nature of Tasks Used in the Classroom … Will Impact Student Learning!

9. Tasks as enacted by teachers and students Tasks as they appear in curricular materials Tasks as set up by teachers Tasks as they appear in curricular materials Student learning Student learning But, what teachers do with the tasks matters, too! Stein, Grover & Henningsen (1996) Smith & Stein (1998) Stein, Smith, Henningsen & Silver (2000) The Mathematical Tasks Framework

10. Re-examine the task through student work samples Mathematical Task: Odd Numbers 7th grade Task

11. Odd numbers • What does student work tell you about the understanding of mathematics?

12. Student A

13. Student B

14. Comparing Students A and B • Do both students have the same understanding of patterning? • How do these students differ in the way they see the patterns? • What next steps could you take to deepen their understanding?

15. Student D

16. Student F

17. Student H

18. Comparing Students D, F and H • What mathematical reasoning skills do the students use to answer the questions? • How can the responses be used to guide student learning? • What strategies can be used to assess students’ understanding?

19. In the classroom… • Protocols for setting up your classroom • What is the students’ role? • What is the teachers’ role?

20. In the classroom… • If you were to do this task with your students, what would you need to do ahead of time to get them ready? • How might you encourage the sharing of approaches and strategies? • What evidence would show that students were learning?

21. Writing Mathematical Explanations What Constitutes a Complete & Strong Response

22. Reflecting on Learning to Write • On your own, think about when you were first learning to write (this could be a single lesson/assignment or a process that developed over time). • Discuss with a partner: • Some of the particulars about your experience? • In general, what kind of writing do you do today? What kinds of things do you need to write about?

23. What the Skill of Writing Is and Isn’t • Is a transaction, rather than a fixed product • Is a skill that can be learned and fine tuned • Isn’t in a kit or a program • Isn’t separate from reading • Classrooms are brimming with voices waiting to be heard, it is figuring out how to master the format of communication that needs to be developed.

24. Why Write in Math Class? • The Common Core State Standards (CCSS) emphasize literacy in all subject areas, including mathematics. • The 3rd Mathematical Practice Standard in CCSS in Mathematics calls for students to construct viable arguments and critique the reasoning of others • Smarter Balanced Assessment Consortium (SBAC) Tests will evaluate students based on their ability to communicate their reasoning in mathematics: • Mathematics Claim 3: Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.

25. General Aspects of Clear Written Communication • Organized • Well developed • Audience-aware • Free of mechanical and grammatical errors

26. Choose one of the Student’s Responses and Rewrite It & to Make It clearer • Work individually • Focus on providing clear, written explanations of your thinking • In pairs • Compare and contrast your answer with your partners’ • Together rewrite your answer by combining the strengths of both of your explanations • List/Articulate the features of your written response that make it a strong response • Be ready to share what you noticed

27. Evaluating Written Math Reponses According to Three Categories • Accurate Understanding of Mathematical Concepts (content) • Planning and Using Strategies to Solve a Problem (process) • Explaining Mathematics Actions and Thinking Through Writing (explanation)

28. Examining the Rubric Categories C. Math Explanation Write what I did and why Explain each step Use math words and strategy names Write final answer in a complete sentence • Accuracy of Answer • Figure out correct answer • Solve with no mistakes • Math Process or Strategies • Use all important information • Show all steps • Make a drawing • Provide a key

29. Parker, R. and Breyfogle, M Lynn, Learning to Write About Mathematics. Teaching Children Mathematics (Sept 2011), pp. 90-99.

30. Score your Response • With your partner examine the rubric and discuss the revisions that you would now make to your response.

31. Did your Response ... • Include that the sum of the first 13 consecutive odd numbers totaled 169? • Include finding the square root to help you with the pattern (13 x 13)? • Include an explanation and not just numbers and symbols? • Merely add on from the previous answer without thinking about the attributes of the pattern? • Include the description of any patterns?

32. Pam’s Attempt • In Kate’s pattern the number of smaller squares added to complete the larger square in each step increased by two. She started with 1 square and then in the second step she added 3 squares and in the third she added 5 squares. So in the third step, a total of 1 + 3 + 5 or 9 smaller squares were used. • The number of terms (first odd numbers) in the addition is equal to the step number. • The total number of smaller squares is also equal to the step number squared. In step 3 the total number of squares was 32 = 9. This makes sense because in each step, Kate completed the square with the side equal to the step number. • To have 169 smaller squares, I would construct a square that was (the square root) 13 on each side.

33. Writing is a Process • Have students write and re-write frequently. • More than a half, maybe as much as two-thirds of my life as a writer is rewriting. I wouldn't say I have a talent that's special. It strikes me that I have an unusual kind of stamina - John Irving • There is no great writing, only great rewriting.Justice Brandeis • Have students work together … • To externalize and construct their writing • To rehearse their structure • To help writers to elaborate • Offer students feedback that matters. • Provide students with and train students on the rubric • Provide ongoing feedback to successive drafts • Give students well written texts (i.e., models). Have them read the mentor text, analyze it, and emulate it.

34. Instructional Tips • Be patient. Students need time to become comfortable with writing about their thinking • Tell students that you understand that writing about mathematics is new and different from them • Provide timely, descriptive, on-going feedback • Require that they regularly write for a specific amount of time