ses0b o g a p 062209 formatted
Upload: michigan-mathematics-and-science-teacher-leadership-collaborative
Post on 19-Jan-2015
163 views
DESCRIPTION
TRANSCRIPT
Overview of SessionOverview of Session
Introducing OGAP Proportionality
Framework
Developing an understanding of
the structure of proportional
situations
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Goals of the SessionGoals of the Session
• To become familiar with findings from research on
formative assessment
• To understand the meaning of and purpose for
formative assessment
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
OGAP Proportionality OGAP Proportionality FrameworkFramework
Mathematical Topics And
Contexts
Structures of Problems
Other Structures
Evidence in Student Work to Inform Instruction
Proportional Strategies
Transitional Proportional
Strategies
Non-proportionalReasoning
Underlying Issues, Errors,
Misconceptions
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Structures of Proportionality Structures of Proportionality Problems – How the Problems Problems – How the Problems are Builtare Built
• Multiplicative relationships in a problem (Karplus,
Polus, & Stage, 1983; VMP OGAP Pilots, 2006)
• Context (Heller, Post, & Behr, 1985; Karpus, Polus, &
Stage, 1983)
• Types of problems (Lamon, 1993)
• Complexity of the numbers (Harel & Behr, 1993)
• Meaning of quantities as defined by the context and
the units (Silver, 2006 Vermont meeting; VMP OGAP
Pilots, 2006)
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
• When the multiplicative relationships in a proportional
situation are integral, it is easier for students to solve
than when they are non-integral
(Cramer, Post, & Currier, 1993; Karplus,
Polus,
& Stage, 1983; VMP OGAP Pilots, 2006)
A Research FindingA Research Finding
OGAP Proportionality Framework
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Multiplicative RelationshipsMultiplicative Relationships
Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples?
3 boxes
2 bushels 8 bushels
x boxes
=
Integral multiplicative relationship
Non-integral multiplicative relationship
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Multiplicative RelationshipsMultiplicative Relationships
Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples?
3 boxes
x boxes 8 bushels
2 bushels
=
Non-integral multiplicative relationship
Integral multiplicative relationship
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
RelationshipsRelationships
Carrie is packing apples for an orchard’s mail order business. It takes 3 boxes to pack 2 bushels of apples.
How many boxes will she need to pack 7 bushels of apples?
What are the multiplicative relationships in this proportional
situation?
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
A Research FindingA Research Finding
• When the multiplicative relationships in a proportional
situation are both non-integral then students have
more difficulty and often revert back to non-
proportional reasoning and strategies.
(Cramer, Post, & Currier, 1993; Karplus,
Polus, & Stage, 1983; VMP OGAP Pilots,
2006)
OGAP Proportionality Framework
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Structures of Proportionality Structures of Proportionality ProblemsProblems
• Multiplicative relationships in a problem (Karplus,
Polus, & Stage, 1983; VMP OGAP Pilots, 2006)
• Context (Heller, Post, & Behr, 1985; Karpus, Polus, &
Stage, 1983)
• Types of problems (Lamon, 1993)
• Complexity of the numbers (Harel & Behr, 1993)
• Meaning of quantities as defined by the
context and the units (Silver, 2006 Vermont meeting;
VMP OGAP Pilots, 2006)
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Case Study: Multiplicative Case Study: Multiplicative RelationshipsRelationships(VMP Pilot Study, Grade 7 Students, (VMP Pilot Study, Grade 7 Students, n=153)n=153)
• Three similar problems administered across a one week
period
• Main difference between the problems is the
multiplicative relationship within and between figures
PILOT 1: A school is enlarging its playground. The dimensions of the new playground are proportional to the dimensions of the old playground. What is the length of the new playground?
40 ft.
80 ft.120 ft.
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Student Work Analysis Student Work Analysis (n=6 students)(n=6 students)
• Part 1. Solve each problem.
• Identify the multiplicative relationship within and
between the figures.
• Anticipate difficulties that students might have when
solving each problem.
• Part 2. Discussion with a partner:
• Identify the multiplicative or additive relationship
evidenced in the student response (e.g., x 3,
between figures; + 6, within figures).
• Place your analysis in the cell that corresponds with
the student number and pilot number in the table on
page 3.
• Complete Discussion Questions on page 3.
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Multiplicative Relationships Multiplicative Relationships Study: Discussion QuestionsStudy: Discussion Questions
• What did you see that you expected?
• What surprised you?
• What are the implications for instruction and
assessment?
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
OGAP Study FindingsOGAP Study Findings(2006 Pilot, n=153)(2006 Pilot, n=153)
Multiplicative Relationships within and between figures
Percent of Correct Responses
Pilot 1 Both integral 80%
Pilot 2 One integral, one non-integral 65%
Pilot 3 Both non-integral 35.5%
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Structures of Proportionality Structures of Proportionality ProblemsProblems
• Multiplicative relationships in a problem (Karplus, Polus,
& Stage, 1983; VMP OGAP Pilots, 2006)
• Context (Heller, Post, & Behr, 1985; Karpus, Polus, &
Stage, 1983)
• Types of problems (Lamon, 1993)
• Complexity of the numbers (Harel & Behr, 1993)
• Meaning of quantities as defined by the
context and the units (Silver, 2006 Vermont meeting;
VMP OGAP Pilots, 2006)
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Context MattersContext Matters
• More familiar contexts tend to be easier for students
than unfamiliar contexts. (Cramer, Post, & Currier,
1993)
• How proportionality shows up in different contexts
impacts difficulty. (Harel, & Behr, 1993)
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Context MattersContext Matters
• Which contexts might be more familiar to students?
• How does proportionality show up in these different
contexts?
The scale factor relating two similar rectangles is 1.5. One side of the larger rectangle is 18 inches. How long is the corresponding side of the smaller rectangle?
Nate’s shower uses 4 gallons of water per minute. How much water does Nate use when he takes a 15 minute shower?
A 20-ounce box of Toasty Oats costs $3.00. A 15-ounce box of Toasty Oats costs $2.10. Which box costs less per ounce? Explain your reasoning.
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Structures of Proportionality Structures of Proportionality ProblemsProblems
• Multiplicative relationships in a problem (Karplus,
Polus, & Stage, 1983; VMP OGAP Pilots, 2006)
• Context (Heller, Post, & Behr, 1985; Karpus, Polus, &
Stage, 1983)
• Types of problems (Lamon, 1993)
• Complexity of the numbers (Harel & Behr, 1993)
• Meaning of quantities as defined by the
context and the units (Silver, 2006 Vermont meeting;
VMP OGAP Pilots, 2006)
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
• Ratio
• Rate
• Rate and ratio comparisons
• Missing value
• Scale factor
• Qualitative questions
• Non- proportional
Types of ProblemsTypes of Problems
OGAP Proportionality Framework
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
• Ratio – is a comparison of any two like quantities
(same unit).
The ratio of boys to girls is 1:2. The ratio of people with brown eyes to blue eyes is 1:4.
• Rate – A rate is a special ratio. Its denominator is always 1.
$5.00 per hour
$3.00 per pound
25 horses per acre
Types of ProblemsTypes of Problems
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
• Relationships - Part : Part or Part : Whole
• Referents - Implied or Explicit
Dana and Jamie ran for student council president at Midvale Middle School. The data below represents the voting results for grade 7.
John says that the ratio of the 7th grade boys who voted for Jamie to the 7th grade students who voted for Jamie is about 1:2. Mary disagreed. Mary says it is about 1:3. Who is correct? Explain your answer.
Types of Problems: Ratio Types of Problems: Ratio
7th Grade Votes
Jamie Dana
Boys 24 40
Girls 49 20
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Types of Problems: Ratio Missing Types of Problems: Ratio Missing ValueValue
• Types of Problems: Ratio Missing Value
• Relationships - Part : Part or Part : Whole
• Referents - Implied or Explicit
There are red and blue marbles in a bag. The ratio of red marbles to blue marbles is 1:2. If there are 10 blue marbles in the bag, how many red marbles are in the bag?
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Types of Problems: Rate Types of Problems: Rate Missing ValueMissing Value
• What are the meanings of the quantities in this
problem?
• What is the meaning of the answer?
Leslie drove at an average speed of 55 mph for 4 hours. How far did Leslie drive?
Start 1 hour 2 hours 3 hours 4 hours
55 miles
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Types of Problems: Rate Types of Problems: Rate ComparisonComparison
• What is the general structure of rate comparison
problems?
A 20-ounce box of Toasty Oats costs $3.00. A 15-ounce box of Toasty Oats costs $2.10. Which box costs less per ounce? Explain your reasoning.
Big Horn Ranch raises 100 horses on 150 acres of pasture. Jefferson Ranch raises 75 horses on 125 acres of pasture. Which ranch has more acres of pasture per horse? Explain your answer using words, pictures, or diagrams.
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Case Study - Meaning of the Case Study - Meaning of the QuantitiesQuantities
• In Part I of this case study, you will analyze 4 student
solutions to Ranch problem. The solutions represent
the kinds of “quantity interpretation” errors that
students make when they solve rate comparison
Big Horn Ranch raises 100 horses on 150 acres of pasture. Jefferson Ranch raises 75 horses on 125 acres of pasture. Which ranch has more acres of pasture per horse? Explain your answer using words, pictures, or diagrams.
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Case Study - Meaning of the Case Study - Meaning of the QuantitiesQuantities
• In pairs, analyze the student solutions and then
respond to the following.
• What is the evidence that the student may not be
interpreting the meaning of the quantities in the
problem?
• Suggest some questions you might ask each student or
activities you might do to help them understand the
meaning of the quantities in the problem and the
solution.
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Case Study - Meaning of the Case Study - Meaning of the QuantitiesQuantities
• What evidence is there
of the student’s
understanding of both
the meaning of the
quantities in the
problem
and in the solution?
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Types of Problems: Missing Types of Problems: Missing ValueValue
Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples?
What is the general structure of a missing
value problem?
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
A Research FindingA Research Finding(Harel, & Behr,1993)(Harel, & Behr,1993)
• The location of the missing value may affect
performance.
Carrie is packing apples for an orchard’s mail order business. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 7 bushels of apples?
Carrie is packing apples for an orchard’s mail order business. It takes 3 boxes to pack 2 bushels of apples. She needs 7 bushels of apples packed. How many boxes will she need?
OGAP Proportionality Framework
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Research ApplicationsResearch Applications
Paul’s dog eats 15 pounds of food in 18 days.How long will it take Paul’s dog to eat 45 pound bag of food? Explain your thinking.
Change this problem to make it easier, and then
harder.
OGAP Proportionality Framework
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Structures of The ProblemsStructures of The Problems
What type of problem is this similarity problem?
Old Playground
90 ft.
630 ft.
New Playground
110 ft.
A school is enlarging its playground. The dimensions of the new playground are proportional to the old playground. What is the measurement of the missing length of the new playground? Show your work.
What type of problem is this similarity problem?
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Structures of The ProblemsStructures of The Problems
OGAP Proportionality Framework
The dimension of 4 rectangles are given below. Which two rectangles are similar?
2” x 8”4” x 10”6” x 12”6” x 15”
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Structures of The ProblemsStructures of The Problems
What is the general structure of scale factor
problems?
OGAP Proportionality Framework
Jack built a scale model of the John Hancock Center. His model was 2.25 feet tall. The John Hancock Center in Chicago is 1476 feet tall.
How many feet of the real building does one foot on the scale model represent? Be sure to show all of your work.
The scale factor relating two similar rectangles is 1.5. One side of the largerrectangle is 18 inches. How long is the corresponding side of the smaller rectangle?
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Structures of the ProblemsStructures of the Problems
The scale factor relating two similar rectangles is 1.5. One side of the larger rectangle is 18 inches. How long is the corresponding side of the smaller rectangle?
If a student was unable to solve this problem successfully, what variables would you change to
make it more accessible? Why?
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
• Students should interact with qualitative predictive and
comparison questions as they are developing their
proportional reasoning….
(Lamon,1993)
A Research FindingA Research Finding
OGAP Proportionality Framework
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Types of Problems: QualitativeTypes of Problems: Qualitative
Kim ran more laps than Bob. Kim ran her laps in less time than Bob ran his laps. Who ran faster?
If Kim ran fewer laps in more time than she did yesterday, would her running speed be:
a) faster; b) slower; c) exactly the same; d) not enough information.
Why do you think researchers suggest these types of
problems as important stepping stones?
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
• Students need to see examples of proportional and
non-proportional situations so they can determine
when it is appropriate to use a multiplicative solution
strategy.
(Cramer, Post, & Currier,
1993)
A Research FindingA Research Finding
OGAP Proportionality Framework
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Solve these problemsSolve these problems(Cramer, Post, & Currier, (Cramer, Post, & Currier, 1993)1993)
Sue and Julie were running equally fast around a track. Sue started first. When she had run 9 laps, Julie had run 3 laps.When Julie completed 15 laps, how many laps had Sue run?
3 U.S. dollars can be exchanged for 2 British pounds.How many pounds for $21 U.S. dollars?
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
A Research FindingA Research FindingA Classic Non-proportional A Classic Non-proportional Example* Example*
Sue and Julie were running equally fast around a track. Sue Started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?
•22 out of 33 undergraduate students treated this as a proportional relationship. (Cramer, Post, & Currier, 1993)
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
A Contrasting Research A Contrasting Research FindingFinding
Three U.S. dollars can be exchanged for 2 British pounds. How many pounds for 21 U.S. dollars?
•Same group – 100% solved it correctly using traditional proportional algorithm.
•No one in the same group could explain why this is a
proportional relationship while the “running laps” is not.
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Case Study - Proportional and Case Study - Proportional and Non-proportional?? (VMP Pilot Non-proportional?? (VMP Pilot Study, ???)Study, ???)
Kim and Bob were running equally fast around a track. Kim started first. When she had run 9 laps, Bob had run 3 laps. When Bob completed 15 laps, how many laps had Kim run?
Do student work sort!
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Vermont Version Grade 6(n= Vermont Version Grade 6(n= 82)82)
Kim and Bob were running equally fast around a track. Kim started first. When she had run 9 laps, Bob had run 3 laps. When Bob completed 15 laps, how many laps had Kim run?
• 39/82 (48%) solved as a proportion
• 33/82 (40%) solved as an additive situation
• 10/82 (12%) non-starters
What are the instructional implications?
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Elements of a Proportional Elements of a Proportional Structure That Affect Structure That Affect PerformancePerformance
• Problem types (comparison, missing value, etc.)
• Mathematical topics/contexts (scaling, similarity, etc.)
• Multiplicative relationships (integral or non-integral)
• Meaning of quantities (ratio relationships and ratio
referents)
• Type of numbers used (integer vs. non-integer)
No wonder proportions are tough to teach and
learn.
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
What Are the Hallmarks ofWhat Are the Hallmarks ofa Proportional Reasoner?a Proportional Reasoner?
• Recognizes the nature of proportional relationships,
• Finds an efficient method based on multiplicative
reasoning to solve problems,
• Represents the quantities in the solution with units that
reflect the meaning of the quantities for the problem
situation.
• Ultimately, a proportional reasoner should not be
deterred by structures, such as context, problem types,
the quantities in the problems. (Cramer, Post, &
Currier, 1993; Silver, 2006)
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Activity: Activity: Administering the OGAP Administering the OGAP Pre-assessmentPre-assessment
• Analyze each of the tasks for:
• Problem types
• Mathematical topics/contexts (scaling, similarity, etc.)
• Multiplicative Relationships (integral or non-integral)
• Ratio Relationships (part:whole or part:part) and
referents (implied or implicit - if applicable)
• Type of numbers used (integer or non-integer)
• Internal Structure (parallel or non-parallel)
Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
General Directions: General Directions: Administering the OGAP Administering the OGAP Pre-assessmentPre-assessment
• Administer the pre-assessment and bring a set of 20 to
25 to our next session
• Calculators are not allowed
• Tips for students
• Time
• Level of teacher assistance
• Do not analyze student work before our next meeting