Standard 2.5 Mathematical Problem Solving

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Standard 2.5Mathematical Problem Solving
Communication

• Open-Ended Questions in Math

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Todays Agenda

• Analyze past PSSA results for open-ended items
• What is an open-ended item and how is it
  addressed on the PSSA
• 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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5th, 8th and 11th Grade 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 score for 5th,
  8th and 11th grades and 14 for the 3rd grade
  PSSA
• Responses to open-ended items are graded using a
  rubric (0 to 5 points)

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Open-ended Results From 2001-2002
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Grade 5
7
Grade 8
8
Grade 11
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What Is an Open-Ended Item?
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What is Problem Solving?

• Reasoning, communication and making connections
• Why?
• How do you know?
• Convince me
• Explain that

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Sample 5th Grade Open-ended Items

• Rename a number in a variety of ways
• Sort data using Venn diagrams
• Develop a tree diagram ands list the elements
• Draw geometric shapes
• Draw and label the parts of a right triangle

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Sample 5th Grade Open-ended Items

• Display data using tables, charts, lists, tallies
  or graphs
• List all possible combinations
• Continue a pattern
• Do simple transformations of figures
• See the 5th grade standards for more
  possibilities

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Explanations

• At the 3rd and 5th grade levels, open-ended items
  may
• Require no explanation
• Require an explanation that focuses on
  mathematical concept knowledge
• Require an explanation that tells why each step
  was taken
• Required by all 8th 11th grade open-ended
  items

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Open-ended Items

• If an open-ended item is straight problem solving
  (computational type) it will have the following
  directions
• Show each step of your math work
• Explain WHY you chose each step

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Scratch Paper

• If students use scratch paper for open-ended
  items, make sure they transfer ALL of their work
  to the test booklet.
• It may be helpful to have students NOT use
  scratch paper for the open-ended items.

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Labels

• Answers must be labeled correctly
• Especially money and measurement
• Ex) 57 cents can be written as 0.57 or 57?
• NOT as 0.57? or 0.57?
• Perimeter in, cm, etc
• Area sq in, cm2, etc

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Calculator Usage

• Calculators can be used for all open-ended items
  on the 5th, 8th and 11th grade PSSA
• Students need to be familiar with using the
  specific calculator before the test!
• Calculators are NOT permitted
• on the 3rd grade PSSA

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Sample 5th Grade

• Open-ended Question

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Sample 5th Grade Question

• Some 5th graders competed in a field-day race.
  The 5th grade results for the 50-meter race are
  shown in the chart.
• (Standards 2.1, 2.5, 2.6)

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Sample 5th Grade Item
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Sample 5th Grade Item

• List the runners from fastest to slowest.
  REMEMBER the runner completing the race in the
  least amount of time is the winner.
• CALCULATE the average time for the 5th grade.
  Which runners time was closest to the average
  for 5th grade? CALCULATE by how much time this
  students time differed from the average.

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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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Solving Open-ended Items

• Tips from Teachers

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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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Tips for Solving

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

work
explanation
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The Work (a students response)

• A)
• 1) Fastest 1. Tia 7.68 sec
• 2. Rose 7.86 sec
• 3. Andrew 8.05 sec
• 4. Matt 8.28 sec
• 5. Donna 8.72 sec
• Slowest 6. José 8.98 sec

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The Work (a students response)

• 2) 7.68 7.86 8.05 8.28 8.72
  8.98 49.57 sec
• 3) 49.57 6 8.26 sec (average)
• 4) Matts time 8.28 sec is the closest
• 5) 8.28 8.26 .02 sec

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Explanation Tips

• Make sure ALL steps are explained in words
• 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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Explanation Tips

• Encourage students to EXPLAIN their work - not
  DESCRIBE it
• Description
• I divided 49.57 by six which equals 8.26
• Explanation
• I divided the total time by the number of
  runners TO FIND the average time

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Explanation Tips

• Explain which operation was performed and what
  each number means
• Start each step with I added
• I subtracted
• I multiplied, etc
• Then explain what each number represents
• I divided the total time by the number of
  runners

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Explanation Tips

• 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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The Explanation(a students response)

• I ordered the runners from fastest to slowest.
  Tia won the race. The runners up in order were
  Rose, Andrew, Matt, Donna and last was José.
• I added all the runners times together TO FIND
  the total time.

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The Explanation (a students response)

• 3) I divided the total time by the number of
  runners TO GET the average time for 5th grade.
• 4) I compared the runners time to the average
  and I see that Matts time was the closest to the
  average.
• 5) I subtracted the average time from Matts
  time TO FIGURE OUT how much he differed from the
  average.

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The Final Product

• work
• A) 1) Fastest 1. Tia 7.68 sec
• 2. Rose 7.86 sec
• 3. Andrew 8.05 sec
• 4. Matt 8.28 sec
• 5. Donna 8.72 sec
• Slowest 6. José 8.98 sec
• B) 2) 7.68 7.86 8.05 8.28 8.72 8.98
  49.57 sec
• 3) 49.57 6 8.26 sec (average)
• 4) Matts time 8.28 sec is the closest
• 5) 8.28 8.26 .02 sec

• explanation
• 1) I ordered the runners from fastest to
  slowest. Tia won the race. The runners up in
  order were Rose, Andrew, Matt, Donna and last was
  José.
• 2) I added all the runners times together TO
  FIND the total time.
• 3) I divided the total time by the number of
  runners TO GET the average time for 5th grade.
• 4) I compared the runners time to the average
  and I see that Matts time was the closest to the
  average.
• 5) I subtracted the average time from Matts
  time TO FIGURE OUT how much he differed from the
  average time.

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Open-ended Items

• In Your Math Classroom

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Where to Find Open-Ended Problems

• Mathematics Assessment handbook Released Items
  Handbook
• www.pde.state.pa.us
• Click on K-12 Schools
• Click on Assessment Testing
• Click on Assessment Handbooks
• Mathematics Instructional Rubrics (Primary thru
  High School)
• www.pde.state.pa.us
• Click on K-12 Schools
• Click on Curriculum Instruction
• Click on Mathematics
• Click on Whats New

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Where to Find Open-Ended Problems

• Open-ended problems created at the 2002 Math
  Governors Institutes
• www.pde.state.pa.us
• Click on K-12 Schools
• Click on Curriculum Instruction
• Click on Mathematics
• Click on Professional Education
• PSSA Practice Tests (5th, 8th 11th)

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Where to Find Open-Ended Items

• Text resource books
• Multi-step word problems from your textbook
• Teacher-made questions
• Internet resources

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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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Hints on 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
  (or part) 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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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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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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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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Tips for Rubric Use

• Copy the rubric on the back of an open-ended item
  and circle or highlight where their score falls
• Have students correct lower scoring papers to
  make them a 5
• Have students grade their own solutions or
  another classmates solution using the rubric

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The Importance of

• Mathematics Vocabulary

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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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Math Vocabulary Development

• 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 26 in the
  2000-2001 math handbook (for grade 5)

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Math Vocabulary Development

• Math vocabulary bulletin board - add new words to
  it year-round
• Math Word Wall
• Student developed math vocabulary notebook
• Math word-of-the-day

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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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Good luck to you and your students!!