Performance Assessment in Mathematics

October 25-26, 2011

Jonathan Katz, Ed. D

Joseph Walter

Day 1

It is the questions that drive mathematics. Solving problems and making up new ones is the essence of mathematical life” (Hersh, 1997).

What is Hersh saying? Do you agree or disagree? Why?

How would you define the idea of a “problem” in mathematics?

How do we see “problems” as connected to our everyday teaching?

How do we see “problems” connected to performance assessment tasks?

Groups of teachers will be given a task/problem to look at. All these tasks/problems have been used with students as either lessons or on-demand problems. Each group will be given fifteen minutes play with the task/problem and have a discussion about the general questions below as well as your specific question.

Each group will then make a five minute presentation to the whole group at large.

General Questions to ponder as you look at these tasks:

How are these tasks/problems helping to develop both conceptual understanding and procedural fluency?

How do these tasks/problems help students to reason quantitatively and construct viable arguments?

Would students find the task/problem interesting? Why or why not? If you found it uninteresting, how would you change the problem?

The Tasks

Playing with Exponents: How do we help students to look for and make use of structure?

Who is Right? How can we use error analysis to help a student critique the reasoning of others?

Multiplying Signed Numbers: How do we help students look for and express regularity in repeated reasoning?

Cops and Robbers: How do we help students to model mathematically?

How Tall is the Building? How do we help students to strategically use a set of tools to understand to solve a problem?

Creating Definition: How do we support student development of precision in mathematics?

Baffling Bacteria: How do we help students to model mathematically?

What is a performance assessment task?

Examples of Tasks…written by ISA coaches and teachers. Take a few minutes to look at the given performance tasks then think about the questions below.

Tina’s Quilt Squares

The Race

Garden Design

The Wonder of Pythagorean Theorem

Will You Ever Reach the Door?

General questions to ponder:

How do we see these performances assessments in relation to the tasks we looked at earlier?

Based on all the work we’ve done so far, how would you describe the qualities that make up a performance task?

Creating a performance task

Each small group will be given the same topic and asked to develop some ideas on how you would create a performance assessment/task on that topic. We will then come together to write a communal performance assessment task.

Some questions you might think about in your work.

Why should students learn this?

What are the big ideas you want your students to think about in relation to this

topic?

What new knowledge do you want them to be able to construct through the

performance assessment experience?

What do you want to assess in this task?

Discussion –The Group Creation of the Task

Presentation – Athena Leonardo, mathematics teacher at Arts and Media Prep

Planning your own assessment – *Explanation of requirements

Brainstorming

Sharing of a template

Writing of a performance task

*The requirements for this performance tasks is that you can choose any domain within the common core standards and you must include at least two of the practice standards.

Laws of Exponents

Generalize a mathematical idea

Look at the following sets of powers. Find the solutions to each, changing all decimals to fractions. (You can use a calculator)

2353103

2252102

2151101

2050100

2-15-110-1

2-25-210-2

2-35-310-3

What patterns do you see?

What questions do you have?

Are there any conjectures you can you make?

How can you prove if your conjectures will hold true for all cases?

Who is right?

Tenisha and Jose were given a problem by their teacher. They were asked to simplify a rational expression. In their work, they were left with:

X + 1

X - 3

Tenisha said, “I think we’re finished. Jose said, “No I think we can still simplify.” He did the following:

X + 11

X - 3= -3

Who was correct? Why do you say that?

(For the teacher: Questions you might ask students at different stages of their process. Is there any way of proving who is right or wrong? Can you find a way of disproving Jose’s method? Can you find a mathematical reason why this simplification is incorrect?)

For further investigation: Can you simplify this problem? Why or why not?

How is it similar and how is it different from the previous problem?

3(x - 1)

4(x - 1)

Multiplication of Signed numbers

Task 1: You already know the answer to +4 times +3. Look at the following; observe the given results and following the pattern write what the results need to be.

+4+4+4+4+4+4+4+4+4

+3+2+1 0-1-2-3-4-5

+12+8+4 0

Observe your results and write down any conjectures you might have.

Using what you learned in the previous computation and predict the following multiplications.

+4+3+2+1 0-1-2-3-4

-3-3-3-3-3-3-3-3-3

Write down your observations about the patterns any new conjectures you might have.

Do you think your conjectures will hold true for multiplication of all signed numbers? why or why not?

Create another method to find out how to multiply signed numbers and see if you get the same results.

How can patterns help us to explain multiplication of signed numbers?

Cops and Robbers Problem

Robin Banks robs a bank and drives off. A short time later he passes a truck stop at which police officer, Willie Katchup is dining. Willie receives a call from his dispatcher and takes off in pursuit of Robin. Two minutes after he passes the truck stop, Robin is 1.5 km away. (When Robin passes the truck stop his time is zero minutes and his distance from the truck stop is zero km.)

Willie takes off and six minutes after Robin has passed, he is 2 km from the truck stop. Seven minutes after Robin passed; Willie is 4 km from the truck stop. Do you think Willie will catch Robin? How do you know? Show all your evidence for your thinking.

How tall is Your SchoolBuilding?

1) What do you estimate the length of the classroom to be?

2) How did you come up with your estimate?

3) What do you estimate the height of the room to be? Describe the method you used to find your answer.

Now you will go outside to the front of the school; otherwise let the class do this activity in the classroom.

THE CHALLENGE:

Estimate the height of your school without any measuring tools. Explain how you determined your estimate.

Now you’re going to be given a set of tools. (a tape measure, a calculator and a clinometer) Your job is to use these tools to figure out the height of the building as accurately as you can.

Show all the work and thinking your group did.

C. Reflect on the results of A and B. What questions arise that you would like the class to talk about?

Creating Definition (Geometry Groups)

You will be given ten geometric figures. In your group classify them in any way that you want. Why did you put them together in these groups? You may put a figure into more than one group.

Create other figures to place into the different groups

Chart your findings. Be sure to make clear your explanations as to your groupings.

Each group will present their findings and other groups will ask questions.

Through these classifications you will create a definition for each of the different groups?

Creating Definition (Algebraic Ideas)

You will be given ten geometric figures. In your group classify them in any way that you want. Why did you put them together in these groups? You may put a figure into more than one group.

Create other figures, table or graphs to place into the different groups

Chart your findings. Be sure to make clear your explanations as to your groupings.

Each group will present their findings and other groups will ask questions.

Through these classifications you will create a definition for each of the different groups?

Note to teachers: This is the non-scaffolded version

Baffling Bacteria

We have studied bacteria and know that they reproduce by fission or simple cell division. Streptococcus bacteria cause the well-known illness, strep throat. The bacteria can reproduce every twenty minutes.

If you inhale a strep bacterium at noon today, how many will have been produced by the time you go home at two o’clock this afternoon? Explain your thinking.

How many will there be by six o’clock p.m.? midnight? Explain your thinking.

Your calculator can’t calculate the number of bacteria that will have been produced by noon tomorrow, so can you find some way to express this number without the aid of the calculator? Explain your thinking.

If a streptococcus bacterium weighs 5.0 x 10-16 pounds, at what time will it weigh as much as you do? Explain your mathematical reasoning.

Note to teachers: This is the scaffolded version

Baffling Bacteria

1. If you inhale a strep bacterium at noon today, how many will have been produced by the time you go home at two o’clock this afternoon?

Fill in the table below to help you organize your data and answer question 1.

Number of Divisions / Number of Bacteria
(Noon) 0 / 1
1
2
3

How many bacteria will there be by six o’clock p.m.? Explain your thinking.

How many at midnight? Explain your thinking.

If a streptococcus bacterium weighs 5.0 x 10-16 pounds, at what time will it weigh as much as you do? Explain your mathematical reasoning.

Preparation for Writing a Performance Assessment Task

Institute for Student Achievement

What is your unit topic(s)?

What are the big ideas in this unit?

What are the mathematical/conceptual understandings you want your students to come away with from this unit?

What are the skills/procedures that students will develop through this unit?

What prior skills/understandings do you expect students to have coming into this unit?

What ideas and skills do you want to assess in the performance task?

List of Questions to Think About WhenWriting a Mathematical Performance Task

What is the context for the task? Why would that context engage students?

What are you going to ask students to do? Why do you want them to do it?

What do you want to learn about your students’ mathematical thinking and understanding from this task?

Does the task ask students to think and wonder about mathematical ideas?

Is the task sufficiently open-ended so that there will be diversity of product?

Will students with different levels of math ability be able to enter into this task and simultaneously find it interesting and challenging?

Do you expect students to communicate their mathematical thinking and their process which might also include rethinking of strategy?

Performance Assessment in Mathematics

October 25-26, 2011

Jonathan Katz, Ed. D

Joseph Walter

Day 2

Welcome back

Reflection on yesterday’s work

Presentation from Teachers on Performance Assessments

Eric Marintsch-mathematics teacher at UrbanActionAcademy
Grace O’Keeffe- mathematics teacher at Hudson H.S.

Continued work writing a performance task

Group discussions on strengthening each member’s performance task

Use of protocol

Discussion: Scoring and norming of student work using the ISA math rubric

How should we approach this work? Qualitative vs. quantitative evaluation?

Next Steps

Implementation of task
Collection of student work
Writing the required task from the DOE

An Inquiry Approach to Looking at a Performance Task

In this session teachers will have the opportunity to begin to look at a task or a performance assessment in relation to developing student mathematical thinking and understanding. This task will be given to students in the next few weeks and we want it to be meaningful to the students and valuable to the teacher.

Step 1

Each presenter will discuss the task in relation to its purpose and what he/she hopes to learn about his/her student’s mathematical thinking. The presenter should then share any concerns/questions he/she has about the task that his/her colleagues might help answer.

Step 2

Each member of the groups looks closely at the task and writes down his/her observations in relation to the teacher’s presentation. If there are any clarifying questions, members of the group should share them at this time.

Step 3

Observations including responses to the presenter’s question(s) are made by each member of the group. This is not the time to make recommendations but rather just to share what you see. Presenting teacher listens without commenting. All take notes of what is said.

Step 4

Presenter responds to observations

Step 5

The whole group will make recommendations to strengthen the task in terms of learning about students’ development as mathematical thinkers.