Standard 2.5 Mathematical Problem Solving & Communication

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Standard 2.5Mathematical Problem Solving
Communication
Open-Ended Questions in Math
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Todays Agenda

• What is an open-ended item and how is it
  addressed on the PSSA
• Analyze past PSSA results for open-ended items
• Learn tips on solving open-ended items
• Learn where to find open-ended items or how to
  create your own
• Learn ways to use open-ended items in your
  classroom/curriculum

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What Is an Open-Ended Item?

• An open-ended math item asks students to solve a
  multi-step problem and explain WHY they chose
  each step.

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PSSA

• Each student has 4 open ended items on the PSSA
• 3 on the common form
• 1 on the matrix form
  

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PSSA

• Each open-ended item has a maximum value of 5
  points.
• Therefore, open-ended items account for
  approximately 18 of the students PSSA score.
• Responses to open-ended items are graded using a
  rubric.

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The PSSA Rubric

• To earn all 5 points, students need to show each
  step of their work, and explain WHY they chose
  each of their steps.
• They can earn at least 1 point by showing a
  correct step toward solving the problem.

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Calculator Usage

• Calculators can be used for all open-ended items
  on the PSSA
• Students need to be familiar with using the
  specific calculator before the test!

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Open-ended Results From 2000-2001
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Grade 5
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Grade 8
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Grade 11
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Sample 8th Grade Question

• Four members of the Johnson family took a trip
  from Pittsburgh to Harrisburg, a distance of 221
  miles. It took them 4 hours and 15 minutes to
  make the trip. The car required 13 gallons of
  gasoline at 1.25 per gallon. The turnpike toll
  was 6.50, and they spent 12.84 for food. What
  was the average cost per mile based on the total
  expenses of gas, food and tolls for this trip?
• (Standards 2.2 2.5)

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The Directions

• For full credit, you must do the following
• Show OR describe each step of your work, even if
  you did it in your head (mental math) or used a
  calculator,
• Write an explanation stating the mathematical
  reason(s) why you chose each of your steps.

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Tips for Solving from Teachers

• Make 2 columns
• Put work on the left
• Put explanation on the right

work
explanation
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Solving Tips from Teachers

• Show and number each step of the work
• Even/especially if the work was done in the
  students head or calculator.

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The Work

• 1) 1.25 X 13 gal 16.25
• 2) 16.25 6.50 12.84 35.59
• 3) 35.59 ? 221 mi ? 0.1610407
• 4) 0.16 per mile

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Explanation Tips

• Encourage students to EXPLAIN their work - not
  DESCRIBE it
• Description
• I multiplied 1.25 and 13 and got 16.25
• Explanation
• I multiplied the price of gas per gallon and
  the number of gallons to get the price for the
  gas used.

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Explanation Tips from Teachers

• Make sure ALL steps are explained in words.
• Encourage students not to use numbers in their
  explanations this will stop them from
  describing their work.

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Explanation Tips from Teachers

• Use magic words in the explanation.
• These are words that gear students to explain
  their work rather than describe it.
• Article on Magic Words can be found in the
  math assessment handbook

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What are the Magic Words?

• To find
• To get
• To see
• To figure out
• To show
• To determine
• Because
• Since
• Therefore.

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Explanation Tips from Teachers

• Encourage your students
• NOT to use the words
• to find THE ANSWER
• The words the answer do not explain what the
  answer represents.

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The Explanation

• 1) I multiplied the price of gas and the number
  of gallons TO GET the total cost of gas.
• 2) I added the cost of gas, food and tolls
  together TO FIND the total cost of the trip.
• 3) I divided the total cost of the trip by the
  number of miles and I FOUND the cost per mile.
• 4) SINCE I had many decimal places, I rounded to
  the hundredth BECAUSE I wanted money.

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The Final Product

• Work
• 1) 1.25 X 13 gal 16.25
• 2) 16.25 12.84 6.50
• 35.59
• 3) 35.59 ? 221mi ? 0.161041
• 4) 0.16 per mile

• Explanation
• 1) I multiplied the price of gas and the number
  of gallons TO GET the total cost of gas.
• 2) I added the cost of gas, food and tolls
  together TO FIND the total cost of the trip.
• 3) I divided the total cost of the trip by the
  number of miles and I FOUND the cost per mile.
• 4) SINCE I had many decimal places, I rounded to
  the hundredth BECAUSE I wanted money.

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Where to Find Open-Ended Problems

• Mathematics assessment handbook
• www.pde.psu.edu/pssa/mathbook.pdf
• Released Items handbook
• www.pde.psu.edu/pssa/mathrelitems.pdf
• (2002 Released Items booklet coming soon)
• PSSA Practice Tests

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Suggested Resources from Teachers

• Measuring Up books
• 1-800-822-1080 or www.patesthelp.com
• PSSA Mathematics Coach books
• 1-800-221-9372 or www.educationaldesign.com
• Continental Press PAM Prep
• 1-800-233-0759
• Exemplars
• www.exemplars.com

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Where to Find Open-Ended Items

• Intermediate Units Performance Assessment
  Lesson Plan Topics
• Text resource books
• Multi-step word problems from your textbook
• Teacher-made questions

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Hints to Creating Your Own Open-Ended Problems

• Make sure problems address a standard at or near
  your grade level.
• Make sure all problems require more than 1 step
  to solve.
• Use your students names, other teachers names
  and your schools name to make the problems more
  interesting to your students.

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Hints to Creating Your Own Open-Ended Problems

• Use the same directions that the PSSA uses at
  your level (show/describe AND explain)
• Use a specific rubric for each of your problems
  that is based on the general rubric in the
  handbook
• Have students use calculators so they can focus
  more on the explanation and less on the number
  crunching.

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Ways to Use Open-Ended Items in Your Classroom

• Put a problem on every test or quiz
• Homework
• Math journal
• Open-ended portfolio..
• DO NOT use only as extra credit

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Practice, Practice, Practice

• Practice should occur the entire year
• Open-ended questions should be addressed about
  once a week
• Incorporate these types of questions into ALL
  grade levels
• Eventually open-ended questions should become an
  integrated part of the math curriculum at ALL
  grade levels

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Curriculum

• Your Goal
• Open-ended items should become a part of your
  math curriculum!
• NOT just an added activity to do if you have time
  at the end of the chapter or on days before
  vacation!
• Your Districts Goal
• To develop a district-wide plan on the the
  implementation of open-ended items in math
  classrooms, kindergarten through 12th grade.

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• Numerous studies reveal that a knowledge of
  mathematics vocabulary directly affects
  achievement in arithmetic, particularly problem
  solving
• Barton, M.L. Heidema, C. (2002) Teaching
  Reading in Mathematics 2nd Ed.

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Tips for Elementary Teachers

• Insist that students use correct mathematical
  vocabulary in their explanations (when
  developmentally appropriate)
• Refer to the Terms to Know in the math
  standards (all grade levels) or page 20 in the
  2000-2001 math handbook (for grade 5)

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Tips for Elementary Teachers

• Have a math vocabulary bulletin board and add new
  words to it year-round
• Have students develop a math vocabulary notebook

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Tips for Secondary Teachers

• Encourage students to use correct math vocabulary
  in their explanations
• Make sure students are familiar with the terms
  used in the standards and PSSA.
• In the 2001-2002 math handbook
• 8th grade page 59, 11th grade page 88
• Review the formula sheet found in the 2001-2002
  math handbook before taking the PSSA
• 8th grade page 61, 11th grade page 90

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Tips for Beginners

• Provide time for students to solve problems
  individually
• Share answers/ideas with partners or in small
  groups
• Discuss as a class

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Conclusion

• Incorporating open-ended questions in your math
  curriculum will not only help improve your PSSA
  scores, but will also help improve your students
  understanding, and the ability to communicate
  that understanding.

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