Golden Circle

Math is all around us.  Helping students understand math is cool.  Students gain confidence when they see the "how" of solving math problems.

Getting students to feel the "why" of learning math is the key to keeping students engaged in learning.

Learn the way you want to learn, learn the way you learn best.

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Use the link below to find the full research on Performance Assessment Strategies.  I created a one-page summary which is below the link.


Link:

Assessment Strategies to Inform Science and Mathematics Instruction

Performance Assessment Research (Excerpted) [1]

Performance assessments typically focus on the process of solving problems or completing complex tasks.  The emphasis is on what students can do, not just what they know. 

Performance assessments allow students to show how they reach their solutions, explain their answers, and demonstrate their ability to synthesize.  This assessment strategy provides teachers with information about students’ thinking and reasoning (Parke & Lane, 1996).

Performance assessments can provide information that is more instructionally relevant than paper-and-pencil tests.  Teachers gain a more complete picture of students’ understandings and abilities.  This in turn leads to better decisions about student misconceptions and errors, and provides better assessment of the instructional process (Stenmark, 1991).

In order to understand how to help students improve their work, teachers need information on the processes that students use.  It is impossible to determine what elements of a teacher’s instruction lead to which student results.  However, teachers can use performance assessments to develop hypotheses about how their students go about creating an adequate or superior product or performance (Herman, Aschbacher, & Winters, 1992).  Using this information, they can modify or develop instructional practices to facilitate growth among all students.

Paper-and-pencil tasks that require students to show their work or describe their procedures will also give teachers insights into students’ problem-solving tactics.  Teachers can get valuable information about how to help students improve by analyzing how students complete tasks and looking for patterns related to outcomes.  Did successful students use a different approach than less successful students?

* Full study at:     http://educationnorthwest.org/webfm_send/750

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1 Assessment Strategies to Inform Science and Mathematics Instruction, It’s Just Good Teaching. Northwest Regional Educational Laboratory, June 1997

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Below are two examples of Performance Assessments I would use for my 7th Grade Math Class.  I listed the Standard, Benchmark, etc., and included a short checklist that will help students see what is expected. 

Minnesota Mathematics Standard

Grade 7              Strand:      Geometry & Measurement

Standard:    Use reasoning with proportions and ratios to determine measurements, justify formulas and solve real-world and mathematical problems involving circles and related geometric figures.

 No.    7.3.1.1

 Benchmark:      Demonstrate an understanding of the proportional relationship between the diameter and circumference of a circle and that the unit rate (constant of proportionality) is Pi.  Calculate the circumference and area of circles and sectors of circles to solve problems in various contexts.

                                                                                 

Performance Assessment Type 1

Building a Landscaped Garden

Your family has decided to create a beautiful landscaped garden in your backyard.  They have asked you to help with ideas and design for the garden.  Your Dad gives you the yard’s dimensions:  75 feet long by 50 feet wide.  A scalloped wood fence runs along the perimeter of the backyard so the design will need to leave room around the perimeter to allow easy access.

Your Mom has a few ideas that she runs by you and asks if you have any comments.  Her idea is to form a circular area that will house a number of native plant species.  She thinks she wants a short walkway from the circular garden to another section that will contain an arbor and wooden bench.  You like her idea of the circular garden but suggest that it should also contain seating.  She asks you how big the circular garden should be but then realizes she doesn’t know how to figure out its dimensions.

You tell your Mom not to worry and that you can explain how to figure it out.  You start by throwing out some measures for circumference:  25 feet, 35 feet, and 40 feet. 

Your Mom asks “what is circumference?”  You answer by explaining ______________________________________________________________________________________________________________.

Your Mom likes your thinking but wants the circumference to be bigger.  You both agree on 50 feet.  Your Mom then asks, “How much planting soil will we need for this area?”  Do you know how to figure that out too?”  No problem you say.  You tell her that the amount of planting soil needed will be the area of the circular garden. You explain that the area of the circular garden is _____________________________________________________________.

DIRECTIONS TO COMPLETE THE GARDEN

Complete the story by:

   1.  Describing the next steps to finish planning the garden.

   2.   Creating a diagram to show the backyard’s dimensions and where the circular garden will be placed.  Identify the circumference, diameter, radius, and Area of the garden.  Remember to answer the question of how much planting soil is needed.

   3.   Label all your work and remember to include your name and class number.

Performance Assessment 1 - Checklist

Learning Target:   

Use reasoning with proportions and ratios to determine measurements, justify formulas and solve real-world and mathematical problems involving circles and related geometric figures.

Specific Learning Outcome:      

Interpret data, organize information, and describe key vocabulary by demonstrating an understanding of the proportional relationship between the diameter and  circumference of a circle and that the unit rate (constant of proportionality) is Pi. 

Student	Interpretation Understands what’s being asked	Organization Organized required information and labeled properly	Writing Uses complete sentences, proper punctuation, etc.	Understands key vocabulary (meaning) Circumference, Diameter, Radius, Area

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Performance Assessment Type 2

Finding Pi

Introduction:

Pi is a number.  In fact, it is a ratio.  Let's find Pi.

Instructions:

1     1.  Take the piece of string you have at your table and measure it with your ruler.  Measure the length using  centimeters (cm).  Record your measurement.

2.  Now, make a circle with your string.  Try to make it as round as possible so that no part of the string is jutting out or it looks more like an oval than a circle.  We want a circle.

3.  Once your circle is made, carefully measure its diameter using your ruler.  Again, use centimeters (cm) for your measure.  Record your measurement.

4.  Record the measure of your circle’s radius. 

5.  Record the measure of your circle’s circumference.

6.  Use the circumference and diameter measures to perform the following division:  Divide your circle’s circumference by its diameter.  Record your result to the nearest thousandth.  Do this twice to check that your result is the same each time.

7.  Raise your hand when you are finished.  Compare your result with other groups in class.  What did you find?

8.  Find the Area of your circle and record your result.

Result	Length of String	Diameter of Circle	Radius	Circumference Diameter	Area of Circle
1st Time
2nd Time

 Performance Assessment 2 - Checklist

 Learning Target:   

Use reasoning with proportions and ratios to determine measurements, justify formulas and solve real-world and mathematical problems involving circles and related geometric figures.

Specific Learning Outcome:      

Interpret data, organize information, and describe key vocabulary by demonstrating an understanding of the proportional relationship between the diameter and  circumference of a circle and that the unit rate (constant of proportionality) is Pi. 

Student	Interpretation Understands what’s being asked	Organization Organized required information and labeled properly	Understands key vocabulary (meaning) and mathematical concepts Circumference, Diameter, Radius, Area

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Reflection

In creating my Performance Assessments, I wanted to observe students using their reasoning and critical thinking skills while working with the data.  As I learned, performance assessments focus on the process of solving problems or completing tasks.  My primary emphasis with these assessments is to see what students can do and how they reach their solutions.

I would set students up in groups of two for each assignment, although both can be done individually as well.  Performance Assessment 1 asks students to integrate writing into their solution.  The expectation here is that students will be explaining their solution and drawing conclusions from the data while thinking through the process.

Performance Assessment 2 requires students to sequence their thinking to affect their final answer.  It also provides a compare / contrast discussion when speaking with other student groups.  This is a critical component because students see and hear what their peers came up with, reinforcing the learning target.

Both assessments can be given either at the start of a unit or within a few days of starting.  Obviously, students need some background knowledge in dealing with geometry and understanding the vocabulary terms like diameter, radius, and Pi, so these assessments are not to be initiated without some prior knowledge or instruction.

Going forward, I will definitely use these two assessments in my classroom.  My goal will be to help students understand the "why" in their learning and give me a good perspective of my instruction techniques.  I believe performance assessments are a good tool in providing a road map to help students meet their learning targets.

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         3 – S Process                                 Assignments Matter                              Chapter 1

S	What statements can you make about the reading?  What did you learn that others need to know?      The statements I can make about the reading are that the type of assignments matter for student success.  As the reading points out, “well crafted assignments hold the potential to make learning and teaching more focused and relevant …”      I learned a couple of things.  First, as Robert Marzano points out, students should be held to “…high expectations …” that lead to “… establishing challenging goals …” (2003, p.35).  This means the rigor of your class assignments must be present to align with your high expectations.  Second, “task predicts performance,” (City, Elmore, Fieman & Teitel, 2009, p. 30).  There is a strong correlation to the type of assignments as excellent vehicles to prepare students for the routines and demands of college work.
S	What strategies will you try?  How will you apply your learning?      One strategy I will try include is showing students the “why” behind the learning.  It’s obvious we can get to the ‘what’ and ‘how’ but teaching to the ‘why’ is critical to evidence outcome.  The example used in the reading is Ms. H’s English class.  She used explicit / intentional teaching after assigning her class to write a critical review of a novel.  But before she let them begin, she taught the students the step-by-step skills needed to produce a critical review.      I will also remember to hold my students to high expectations and ensure the rigor of their learning is at a high level.  I will apply this in the classroom by making sure my instruction is clear, explicit, and intentional, so as to benefit the students learning outcomes.  As the reading points out, “… because assignments are evidence of outcomes as well as the production of the outcomes.”
S	What is the source?  Where did you get the information? The source for this 3 – S Process is Chapter 1 of our text: Assignments Matter: Making the Connections That Help Students Meet Standards by Eleanor Dougherty.

3 – S Process              Assignments Matter                   Chapter 5

S	What statements can you make about the reading?  What did you learn that others need to know?     The statements I can make about the reading are that there is a method for building on student learning by sequencing assignments.  Assignments are “… the stuff of units and courses; they deliver on the big ideas, themes, concepts, topics, or issues that constitute a course.”    I learned that a well sequenced course connects content and skills from assignment to assignment and unit to unit.  I learned that the best student learning experiences occur along a progression of complexity in assignments, which allows coursework to prepare students for the next grade level.    I also learned about the 5 principles that help guide unit & course design: 1) Thoughtful engagement, 2) Literacy as common practice, 3) Value, 4) Manageability, and 5) Pressure gently applied.
S	What strategies will you try?  How will you apply your learning?      I like the strategy of thinking about ways to sequence assignments and putting that sequencing in the design of a unit.  I will use the 5 Principles to guide my unit design, instruction, and student assignments so that I build a learning progression.      The trick will be to sequence / scaffold my assignments in accordance with the right learning progression in math.  Since math content is generally taught in a progression anyway, this may be easier than it appears.  I believe the key will be defining the students’ products and activities that best suit the increased learning they will need to continue on.
S	What is the source?  Where did you get the information? The source for this 3 – S Process is Chapter  5 of our text: Assignments Matter: Making the Connections That Help Students Meet Standards by Eleanor Dougherty.

3 – S Process             Assignments Matter                 Chapter 8

S	What statements can you make about the reading?  What did you learn that others need to know?      The statements I can make about the reading are 1) that data analysis is integrated into education, 2) assignments should be collected, stored, reviewed, and assessed and, 3) the data culled from this process helps inform teachers about their instruction and aids in curriculum design.      I learned that in any school there should be a “file drawer” of all student work (i.e., assignments) across grade levels.  Teachers and other school staff should collectively assess their work and discuss the students’ learning outcomes.  This process only seeks to help better target instruction, target standards, and guide future curriculum approaches.
S	What strategies will you try?  How will you apply your learning?      I like two specific strategies outlined in the reading.  The first is “calibrating alignment to standards.”  This requires selecting one or two standards in an area to calibrate, and then create your collection of assignments for these standards.  After you have the collection, either in a PD session or, asking a peer group of teachers separately, to match each assignment to the corresponding standard.  This presents a somewhat unbiased point of view.       The second strategy I will try is to employ a strong reteaching practice for students when I see a lack of understanding, either through instruction or after analyzing assignments.  I learned that reteaching should simultaneously repair and ratchet up demands, gently but firmly moving students along a learning progression toward achievement.
S	What is the source?  Where did you get the information? The source for this 3 – S Process is Chapter  8 of our text: Assignments Matter: Making the Connections That Help Students Meet Standards by Eleanor Dougherty.