Valerie Mills NCSM President Oakland Schools, Waterford MI

“Learning is driven by what teachers do in classrooms. Teachers have to manage complicated and demanding situations, channeling personal, emotional, and social pressures of a group of 30 or more youngsters in order to help them learn immediately and become better learners in the future. Standards can be raised only if teachers can tackle this task more effectively.” --Black and Wiliam, 1998

“Learning is driven by what teachers do in classrooms. Teachers have to manage complicated and demanding situations, channeling personal, emotional, and social pressures of a group of 30 or more youngsters in order to help them learn immediately and become better learners in the future. Standards can be raised only if teachers can tackle this task more effectively.” --Black and Wiliam, 1998

Activities in today’s session are designed to support an exploration of  what it means to teach and assess standards that include “understand”;  the features of short answer task "genres" that can be used to teach and assess student understanding of mathematical concepts.  possible implications for teaching and assessing in our classrooms and districts. 4

4.NBT  Generalize place value understanding for multi-digit whole numbers.  Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NF  Extend understanding of fraction equivalence and ordering.  Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.  Understand decimal notation for fractions and compare decimal fractions. 4.MD  Geometric measurement: understand concepts of angle and measure angles.

6 Phil Daro “Understand” is intended to mean that students can explain the concept with mathematical reasoning including concrete illustrations, mathematical representations, and example applications.

Students who understand a concept can: a. Use it to make sense of and explain quantitative situations (“Model with Mathematics” in Practices) b. Incorporate it into their own arguments and use it to evaluate the arguments of others (see “Construct viable arguments and critique the reasoning of others” in Practices) c. Bring it to bear on the solutions to problems (see “Make sense of problems and persevere in solving them”) d. Make connections between it and related concepts - Phil Daro, CC writing team ppt. NCSM  7

 How well do our current short answer tasks assess understanding?  What can we learn from the development of the consortia assessments to improve our items?

10  Compare and contrast the design features of a typical assessment task and a SBAC task.  Working with a partner/small group list the features of the task design that help to assess understanding.  Be prepared to share your thinking.

11  Review the tasks in Set A (yellow) paying attention to the design features of these tasks.  Based on this set of tasks, add to or clarify your description of the design features for this genre of tasks.  How might tasks of this design better assess understanding?

12 Review the new sets of tasks (pink, lavender, and grey).  Describe the design features in each set of tasks.  How might tasks modeled after each of these task “genres” better assess understanding?

13  How do your teaching and assessment tasks support or hinder building and assessing understanding?  What implications do you see for your classroom and/or district?

14 Phil Daro “Understand” is intended to mean that students can explain the concept with mathematical reasoning including concrete illustrations, mathematical representations, and example applications.

Students who understand a concept can: a. Use it to make sense of and explain quantitative situations (“Model with Mathematics” in Practices) b. Incorporate it into their own arguments and use it to evaluate the arguments of others (“Construct viable arguments and critique the reasoning of others” in Practices) c. Bring it to bear on the solutions to problems (“Make sense of problems and persevere in solving them”) d. Make connections between it and related concepts - Phil Daro, CC writing team ppt. NCSM  15

Goals and lessons need to  focus on the mathematics concepts and practices (not on doing particular math problems)  be specific enough that you can effectively gather and use information about student thinking  be understood to sit within a trajectory of goals and lessons that span days, weeks, and/or years “Thinking of understandings as outcomes of solving problems rather than as concepts that we teach directly requires a fundamental change in our perceptions of teaching.” -Hiebert et al. p.22, Making Sense

Describe the mathematical goal(s) that might be reasonably assessed using the each of the tasks we initially compared. Traditional TaskSBAC Task

Standards for Mathematical Content 4.OA Operations and Algebraic Thinking Gain familiarity with factors and multiples. 4. Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Standards for Mathematical Practices 1. Make sense of problems and persevere in solving them  start by explaining to themselves the meaning of a problem and looking for entry points to its solution  analyze givens, constraints, relationships, and goals 3. Construct viable arguments and critique the reasoning of others.  understand and use stated assumptions, definitions, and previously established results in constructing arguments  justify their conclusions, communicate them to others, and respond to the arguments of others. 4. Model with Mathematics  identify important quantities in a practical situation.  analyze those relationships mathematically to draw conclusions.  routinely interpret their mathematical results in the context of the situation.

19  Review tasks in the unit you are currently teaching.  Select an appropriate task you could modify to enhance learning and assessment opportunities around understanding.  Pick a task genre to use as a frame.  Modify the existing task AND identify mathematical goals for your revised task.

How could you modify this problem to incorporate what we learned about teaching and assessing understanding? Task as it appears:

How could you modify this problem to incorporate what we learned about teaching and assessing understanding? Task as modified, connections among representations: How do the goals you are teaching/assessing change with the modified task? a)Represent 493 using the flats, longs, and little cubes in your base ten blocks. b)Write an equation that matches what you have built to represent 493. c)What is the greatest number of longs (i.e., tens) you would need to use if you didn’t have any flats (i.e., hundreds)? d)Explain how you figured out the greatest number of longs (i.e., tens) you would need.

How could you use this set of problems to forward elements of mathematical understanding? Tasks as they appear: How do the goals you are teaching/assessing change with the modified task?

 Sort the equations and explain how they chose to sort.  Solve the problems using tables, or graphs, not just manipulating symbols.

24 Review mathematical goals and the tasks in the unit you are currently teaching.  Select an appropriate task you could modify to enhance learning and assessment opportunities around understanding.  Pick a task category to use as a frame.  Modify the existing task AND identify mathematical goals for your revised task.

How did these activities help you think about what it means to teach and assess mathematical understanding? How might activities such as these be used with teachers to help them reconsider what it means to teach and assess mathematical understanding? 25

Geraldine Devine Dana Gosen Marie Smerigan Valerie Mills