David J. Chard November 19, 2004 University of Oregon dchard@uoregon
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Transcript of David J. Chard November 19, 2004 University of Oregon dchard@uoregon
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David J. ChardNovember 19, 2004
University of [email protected]
David J. ChardNovember 19, 2004
University of [email protected]
Preventing Mathematics Difficultiesin Young Children
Preventing Mathematics Difficultiesin Young Children
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Objectives
• Briefly review the research literature on effective mathematics instruction for students with learning disabilities and low achievement.
• Define the big idea of number and operations.• Describe instructional scaffolding that promotes
access to number and operations.• Examine models of early number development.• Discuss trends in assessment and instruction that are
currently under study.
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1-5%
5-10%
80-90%
Intensive, Individual Interventions•Individual Students•Assessment-based•High Intensity
Targeted Group Interventions•Some students (at-risk)•High efficiency•Rapid response
Universal Interventions•All students•Preventive, proactive
Building a Culture of Competence
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3 7
1 8
8 3 7 5 3 4
4 2 1 7 6 2
3 6 9 5
6 85 2
Quantity Discrimination Missing Number
3 4 _
1 2 _
7 8 _ 5 6 _ 3 4 _
4 5 6 2 3 _ 6 7 _
2 3 _ 7 8 _
6 7 _5 6 _
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(MLSC, p. 5, 2002)
Mathematics proficiency includes:Mathematics proficiency includes:
Productive disposition: habitual inclination to see mathematicsas sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy
Productive disposition: habitual inclination to see mathematicsas sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy
Adaptive reasoning: capacity for logical thought, reflection,explanation, and justification
Adaptive reasoning: capacity for logical thought, reflection,explanation, and justification
Strategic competence: ability to formulate, represent, and solvemathematical problems
Strategic competence: ability to formulate, represent, and solvemathematical problems
Procedural fluency: skill in carrying out procedures flexibly,accurately, efficiently, and appropriately
Procedural fluency: skill in carrying out procedures flexibly,accurately, efficiently, and appropriately
Conceptual understanding: comprehension of mathematical concepts, operations, and relations
Conceptual understanding: comprehension of mathematical concepts, operations, and relations
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Initial Comments about Mathematics ResearchInitial Comments about Mathematics Research
• The knowledge base on documented effective
instructional practices in mathematics is less
developed than reading.
• The knowledge base on documented effective
instructional practices in mathematics is less
developed than reading.
• Efforts to study mathematics and mathematics
disabilities has enjoyed increased interests recently.
• Efforts to study mathematics and mathematics
disabilities has enjoyed increased interests recently.
• Mathematics instruction has been a concern to U.S.
educators since the 1950s, however, research has
lagged behind that of reading.
• Mathematics instruction has been a concern to U.S.
educators since the 1950s, however, research has
lagged behind that of reading.
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Identifying High Quality Instructional ResearchIdentifying High Quality Instructional Research
110 studies High quality researchdesigns?
956 studiesExperimental study?
8,727 papers Elementary or secondary study in mathematics?
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RESULTSFeedback to Teachers on
Student Performance
1. Seems much more effective for special educators than general educators, though there is less research for general educators.
2. May be that general education curriculum is often too hard.
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Overview of Findings
• Teacher modeling and student verbal rehearsal remains phenomenally promising and tends to be effective.
• Feedback on effort is underutilized and the effects are underestimated.
• Cross-age tutoring seems to hold a lot of promise as long as tutors are well trained.
• Teaching students how to use visuals to solve problems is beneficial.
• Suggesting multiple representations would be good.
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Translating Research to Practice
At your table respond to the following questions:
1. How does the research support/confirm what I’m already doing in my classroom in mathematics?
2. How might I apply what I’ve learned to change what I’m doing in mathematics instruction?
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Big Ideas
Conspicuous StrategiesIntegration
Review/Reteaching
Scaffolding
Architecture for Learning
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Big Idea – Number Operations
Plan and design instruction that:Plan and design instruction that:
• Develops student understanding from concrete to conceptual,• Develops student understanding from concrete to conceptual,
• Applies the research in effective math instruction, and• Applies the research in effective math instruction, and
• Scaffolds support from teacher peer independent application. • Scaffolds support from teacher peer independent application.
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Number and Operations
1. Understand numbers, ways of representing numbers, relationship among numbers, and number systems
2. Understand meanings of operations and how they relate to one another.
3. Compute fluently and make reasonable estimates.
Standard
Instructional programs from pre-Kindergarten through grade 12 should enable all students to:
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Number and OperationsPre K-2 Expectation:•Count with understanding and recognize “how many” in sets of objects
•Use multiple models to develop initial understanding of place value and the base-ten number system
•Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections
•Develop a sense of whole numbers and represent them in flexible ways, including relating, composing, and decomposing numbers
•Connect number words and numerals to the quantities they represent, using physical models and representations; understand and represent commonly used fractions, such as 1/4, 1/3, and1/2.
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Insight #1
Change in Magnitude of
One
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Insight #1
Change in Magnitude of
One
Insight #2
Change in Magnitude of
Arrays
Demonstratedby humans andnon-human primates
Language
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Early Number Concepts and Number Sense
Number sense refers to a child’s fluidity and flexibility with numbers, the sense of what numbers mean, and an ability to perform mental mathematics and to look at the world and make comparisons (Case, 1998; Gersten & Chard, 1999).
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BIG IDEA #1: Counting
To make counting meaningful, it is important to promote knowledge and awareness of patterns, classification, cardinality principle, and conservation of number.
Reys, Lindquist, Lambdin, Smith, & Suydam (2003); Van de Walle (2004)
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Summary: Counting• Informal Strategies (Gersten & Chard, 1999)
• Parents can help children develop early number sense using various activities such as asking them to:
• ascend and count four steps and then count and descend two steps
• count forks and knives when setting the table.
• Formal Strategies• Rational counting (Stein, Silbert, Carnine, 1997) • “Min” strategy: Starting with the larger number and
counting on when trying to find the answer to either 3+8 or 8+3, whether using one’s fingers, manipulatives, or stick figures (Gersten & Chard, 1999).
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Counting Sets Activity
Have children count several sets where the number of objects is the same but the objects are very different in size.
Discuss how they are alike or different. “Were you surprised that they were the same amount? Why or why not?
(Van de Walle, 2004)
What insight are wehoping all children will
develop?
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Count and Rearrange Activity
Have children count a set. Then rearrange the set and ask, “ How many now?”
If they see no need to count over, you can infer that they have connected the cardinality to the set regardless of its arrangement.
If they choose to count again, discuss why they think the answer is the same.
(Van de Walle, 2004)
What insight are wehoping all children will
develop?
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Learning Patterns Activity
To introduce the patterns, provide each student with about 10 counters and a piece of construction paper as a mat. Hold up a dot plate for about 3 seconds.
“Make the pattern you saw using the counters on the mat. How many dots did you see? How did you see them?
Discuss the configuration of the patterns and how many dots. Do this with a few new patterns each day.
(Van de Walle, 2004)
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Ten-Frame Flash Activity
• Flash ten-frame cards to the class or group, and see how fast the children can tell how many dots are shown.
• Variations: Saying the number of spaces on the card instead
of the number of dots Saying one more than the number of dots (or two
more, and also less than) Saying the “ten fact” – for example, “Six and
four make ten”
(Van de Walle, 2004)
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Part-Part-Whole: Two out of Three Activity
Make lists of three numbers, two of which total the whole. Here is an example list for the number 5.
2-3-45-0-21-3-23-1-42-2-34-3-1
1. Write the list on the board or overhead. 2. Have children take turns selecting the two numbers that make the
whole. 3. Challenge children to justify their answers.
(Van de Walle, 2004)
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Summary: Teaching Number Sense
• Math instruction that emphasizes memorization and repeated drill is limited (Gersten & Chard, 1999).
• Strategies that support mathematical understanding are more generalizable.
• Teaching number sense shifts the focus from computation to mathematical understanding and better helps students with learning difficulties.
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Scaffolding
TeacherSupport
StudentIndependence
Time
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Instructional Scaffolding includes:Instructional Scaffolding includes:
Carefully selecting and sequencing of examples.Carefully selecting and sequencing of examples.
Pre-teaching prerequisite knowledge.Pre-teaching prerequisite knowledge.
Providing specific feedback on students’ efforts.Providing specific feedback on students’ efforts.
Offering ample opportunities for students’ to discuss their approaches to problem solving.Offering ample opportunities for students’ to discuss their approaches to problem solving.
Sequencing instruction to avoid confusion of similar concepts.Sequencing instruction to avoid confusion of similar concepts.
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Adding w/ manipulatives/fingers
Adding w/ semi-concrete objects (lines or dots)
Adding using a number line
Min strategy
Missing addend addition
Addition number family facts
Mental addition (+1, +2, +0)
Addition fact memorization
Adding w/ manipulatives/fingers
Adding w/ semi-concrete objects (lines or dots)
Adding using a number line
Min strategy
Missing addend addition
Addition number family facts
Mental addition (+1, +2, +0)
Addition fact memorization
Concrete/conceptual
AbstractAbstract
Semi-concrete/representational
Sequencing Skills and StrategiesSequencing Skills and Strategies
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Teach prerequisite skills thoroughly.
6 + 3 = ?
What are the prerequisite skills students
need to learn in Pre-K and K before
learning to add
single digit numbers?
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Adding with concrete models Adding with concrete models
Min strategy with addition
(counting up from the larger addend)
Min strategy with addition
(counting up from the larger addend)
Addition number familiesAddition number families
Mental addition (+1; +2, +0)
Addition fact memorization
Mental addition (+1; +2, +0)
Addition fact memorization
Mental subtraction
Subtraction fact memorization
Mental subtraction
Subtraction fact memorization
Subtraction number familiesSubtraction number families
Min strategy with subtraction
(counting up from the
subtrahend to the minuend)
Min strategy with subtraction
(counting up from the
subtrahend to the minuend)
Subtracting with semi-concrete
or number line
Subtracting with semi-concrete
or number line
Adding with semi-concrete objects
Or adding using a number line
Adding with semi-concrete objects
Or adding using a number line
Subtracting with concrete modelsSubtracting with concrete models
Missing addend additionMissing addend addition
Oral counting; number vocabulary
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Scaffold the Instruction
TeacherSupport
StudentIndependence
Time
model guide strategically integrate monitored independent independentmodel guide strategically integrate monitored independent independent
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Acquisition of Mathematical Knowledge
Work Among Peers: Instructional InteractionsWork Among Peers: Instructional Interactions
Teacher Support
Peer Support
Independence
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Lesson Planning: Addition with Manipulatives Lesson Planning: Addition with Manipulatives
Day 1 & 2 problemsDay 1 & 2 problemsDay 3 & 4 problemsDay 3 & 4 problems
Day 9 until accurateDay 9 until accurate
Day 7 & 8 problemsDay 7 & 8 problems
Day 5 & 6 problemsDay 5 & 6 problems
Until fluentUntil fluent
ModelModel GuideStrategyGuide
StrategyStrategy
IntegrationStrategy
IntegrationTeacher
MonitoredIndependent
TeacherMonitored
Independent
Independent(no teacher
monitoring)
Independent(no teacher
monitoring)
ScaffoldingScaffolding
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Introduction to the Concept of AdditionIntroduction to the Concept of Addition
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Addition of Semi-concrete Representational ModelsAddition of Semi-concrete Representational Models
5 + 3 = ? 5 + 3 = ?
5 + 3 = 8=
5 + 3 = 8=
= =++
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The Min StrategyThe Min Strategy
5 + 3 = ? 5 + 3 = ?
5 + 3 = ? 5 + 3 = ?
5 + 3 = 8 5 + 3 = 8
88
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Missing Addend AdditionMissing Addend Addition
4 + ? = 6 4 + ? = 6
4 + 2 = 6=
4 + 2 = 6=
4 + ? = 6 =
4 + ? = 6 =
Early AlgebraicReasoning
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1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
Number Line Familiarity
“What is one more than 3?”
“What is 2 less than 8?”
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Mental Math Mental Math
1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10 5 + 1 =5 + 1 =
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Mental Math Mental Math
1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10 5 + 1 =5 + 1 =
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Mental Math Mental Math
5 + 2 =5 + 2 =1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
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Mental Math Mental Math
5 + 2 =5 + 2 =1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
“In the story of Aunt Flossie’s Hats, how many hats did Aunt Flossie have?”
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4 34 3
Number Families Number Families
77
4 + 3 = 74 + 3 = 7
3 + 4 = 73 + 4 = 7 7 - 3 = 47 - 3 = 4
7 - 4 = 37 - 4 = 3
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Fact Memorization Fact Memorization
5 + 2 = 6 + 0 =
1 + 8 = 4 + 3 =
5+2
2+7
4+4
3+6
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Lesson Planning: Addition with Semi-Concrete ModelsLesson Planning: Addition with Semi-Concrete Models
Day 1 & 2 problemsDay 1 & 2 problemsDay 3 & 4 problemsDay 3 & 4 problems
Day 9 until accurateDay 9 until accurate
Day 7 & 8 problemsDay 7 & 8 problems
Day 5 & 6 problemsDay 5 & 6 problems
Until fluentUntil fluent
3 3
3 new, 2 conc3 new, 2 conc
11
33
22
44
11
22
4 -5 daily4 -5 daily
4 new, 3 conc4 new, 3 conc
4 new, 2 conc4 new, 2 conc
2 new, 2 conc2 new, 2 conc
ModelModel GuideStrategyGuide
StrategyStrategy
IntegrationStrategy
IntegrationTeacher
MonitoredIndependent
TeacherMonitored
Independent
Independent(no teacher
monitoring)
Independent(no teacher
monitoring)
ScaffoldingScaffolding
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Relational understanding of basic place value requires an integration of new and difficult-to-construct concepts of grouping by tens (the base-ten concept) with procedural knowledge of how groups are recorded in our place-value scheme, how numbers are written, and how they are spoken.
(Van de Walle, 2004)
Place Value
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Counting•By ones•By groups and singles•By tens and ones
Oral NamesStandard: Thirty-twoBase-Ten: Three tens and two
Written Names
32
Standard and equivalent groupings meaningfully used to represent quantities
Base-Ten Concepts
Van de Walle (p. 181, 2004)
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Developing Whole Number Computation:Algorithms
Algorithms are steps used to solve a math problem.
When the number of digits or complexity of computation increases, the need for a traditional algorithm increases.
Traditional algorithms require an understanding of regrouping.
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Working With Models1. Display the problem at the top of the place value mat:
27+54 2. Think aloud as you attempt to answer the
question.
Teacher: You are going to learn a method of adding that most “big people” learned when they were in school. Here’s one rule: You begin in the ones column. This is the way people came up with a long time ago, and it worked for them. Okay, I will start with the ones place. We have seven ones and 4 ones on our two ten-frames in the ones place. I am going to fill the first ten-frames by moving some ones in the second ten-frame. I filled up 10. There’s 11. That’s 10 and one. Can I make a trade? Yes, because now I have a ten and an extra. So, I am going to trade the ten ones for a 10. I have one left in the ones column, which is not enough to trade tens. So the answer is 8 tens and a one – 81.
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Working With Models3. Next, have students work through similar problems
with you prompting them (ask leading questions) as needed.
4. Peer or partner group work or independent work with you monitoring it.
5. Finally, students should be able to do it independently and provide explanations.
Have students explain what they did and why. Let students use overhead models or magnetic pieces to help with their explanations.
Van de Walle (2004)
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In your group…
• Describe two visual (concrete or semi-concrete) models that would teach number and number operations (e.g., teach the concept of 4 and 9-4=5).
• What misconceptions could students develop based on your models?
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13 - 5 =
+10 +3 -3 -2 “Manipulative Mode”
![Page 58: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/58.jpg)
13 - 5 =
+10 +3 -3 -2
![Page 59: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/59.jpg)
13 - 5 =
+10 +3 -3 -2
![Page 60: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/60.jpg)
13 - 5 =
+10 +3 -3 -2
![Page 61: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/61.jpg)
13 - 5 =
+10 +3 -3 -2
![Page 62: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/62.jpg)
13 - 5 =
+10 +3 -3 -2
![Page 63: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/63.jpg)
13 - 5 =
10 3 3 2
![Page 64: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/64.jpg)
13 - 5 =
10 3 3 2
![Page 65: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/65.jpg)
13 - 5 =
10 3 3 2
![Page 66: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/66.jpg)
13 - 5 =
10 3 3 2
![Page 67: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/67.jpg)
13 - 5 =
10 3 3 2
![Page 68: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/68.jpg)
13 - 5 =
10 3 3 2
![Page 69: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/69.jpg)
13 - 5 =
3 2
![Page 70: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/70.jpg)
13 - 5 =
3 2
![Page 71: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/71.jpg)
13 - 5 =
3 2
![Page 72: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/72.jpg)
13 - 5 =
3 2
![Page 73: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/73.jpg)
13 - 5 =
![Page 74: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/74.jpg)
13 - 5 =
![Page 75: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/75.jpg)
13 - 5 =
![Page 76: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/76.jpg)
13 - 11=
![Page 77: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/77.jpg)
13 - 11=
![Page 78: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/78.jpg)
13 - 11=
![Page 79: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/79.jpg)
Diagnosing and Remediating Errors
• Fact Errors
• Strategy Errors
• Components Errors
• Incorrect Operation or Sign Discrimination
• Random Errors
![Page 80: David J. Chard November 19, 2004 University of Oregon dchard@uoregon](https://reader035.fdocuments.in/reader035/viewer/2022070408/568143ee550346895db07375/html5/thumbnails/80.jpg)
Diagnosing & Remediating Errors
• Fact Errors: 7
3 8 2 - 1 6
3 6 5
Extra Practiceor
Motivation
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Diagnosing & Remediating Errors
• Strategy Errors:
3 8 2 - 1 6 3 7 4
Reteach strategy beginning with teacher model
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Diagnosing & Remediating Errors
• Component Errors: 9
3 8 2 - 1 6 3 8 6
Reteach that component, continue strategy
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Diagnosing & Remediating Errors
• Incorrect Operation:
3 8 2 - 1 6 3 9 8 Precorrect or prompt correct operation.
For random errors - accuracy below 85%: Increase motivation
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Error AnalysisFact? Component? Strategy? Sign?
Fact
Sign
Component
Strategy
2 5x 3 2 5 07 3 7 8 0
2 5x 3 2 5 7
2 5x 3 2 5 0 7 5 1 2 5
2 5 x 3 27 5 5 0
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Why all this error analysis?
• Errors or probable causes of errors imply considerations about– Preskills
– Instruction
– Remediation
• Identifying the type of error allows you to more efficiently address learner needs.
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Basic ConceptsBasic Concepts
Rule RelationshipsRule Relationships
Complex StrategiesComplex Strategies
Facts/AssociationsFacts/Associations Number familiesNumber families
EqualityEquality
Commutative Property of Addition
Commutative Property of Addition
Reasoning/Problem Solving
Reasoning/Problem Solving
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Basic ConceptsBasic Concepts
Rule RelationshipsRule Relationships
Complex StrategiesComplex Strategies
Facts/AssociationsFacts/Associations
ConvergentConvergent
DivergentDivergent
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Future Trends
•System of assessment to :
Detect students at risk for mathematics difficultiesMonitor student progress and fluencyGauge instructional efficacy
•Increased emphasis on early algebraic reasoning
•More thorough teaching on narrower focus
•Professional knowledge, professional knowledge, professional knowledge