Unformatted text preview: Thinking, Doing, and Talking Mathematically: Planning Instruction for Diverse Learners David J. Chard
University of Oregon
College of Education Alexander ATK/ 704 DEF/ 304 Predicting Risk of Heart Attack
Predicting Risk of Heart Attack Researchers have reported that ‘waist
tohip’ ratio is a better way to predict heart attack than body mass index. A ratio that exceeds .85 puts a woman at risk of heart attack. If a woman’s hip measurement is 94 cm, what waist measurement would put her at risk of heart attack? Students with Learning Difficulties
•More than 60% of struggling learners evidence difficulties in mathematics (Light & DeFries, 1995). •Struggling learners at the elementary level have persistent difficulties at the secondary level, because the curriculum is increasingly sophisticated and abstract. hat Does Research Say Are Effective Instructiona
hat
Practices For Struggling Students? Explicit teacher modeling. Student verbal rehearsal of strategy steps during problem solving. Using physical or visual representations (or models) to solve problems is beneficial. Student achievement data as well as suggestions to improve teaching practices.
Fuchs & Fuchs (2001); Gersten, Chard, & Baker (in review) hat Does Research Say Are Effective Instructiona
Practices For Struggling Students? Cross age tutoring can be beneficial only when tutors are welltrained. Goal setting is insufficient to promote mathematics competence Providing students with elaborative feedback as well as feedback on their effort is effective (and often underutilized). Fuchs & Fuchs (2001); Gersten, Chard, & Baker (in review) Mathematical Proficiency
Mathematical
1. Conceptual understanding – comprehension of mathematical concepts, operations, and relations 2. Procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately 3. Strategic competence – ability to formulate, represent, and solve mathematical problems 4. Adaptive reasoning – capacity for logical thought, reflection, explanation, and justification 5. Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.
(U. S. National Research Council, 2001, p. 5) Common Difficulty Areas for Struggling Learners Memory and Conceptual
Difficulties Background Knowledge
Deficits Linguistic and Vocabulary
Difficulties Strategy Knowledge
and Use Memory and Conceptual Difficulties
Students experience problems:
•Remembering key principles;
•Understanding critical features of a concept;
•Because they attend to irrelevant features of a concept or problem. Addressing Diverse Learners Through Core Instruction
Thoroughly develop concepts,
principles, and strategies using
multiple representations. Memory and Conceptual
Difficulties Gradually develop knowledge
and skills that move from
simple to complex.
Include nonexamples to
teach students to focus on
relevant features.
Include a planful system of
review. Big Idea Number
Plan and design instruction that:
• Develops student understanding from concrete to conceptual,
• Scaffolds support from teacher peer independent application. Sequencing Skills and Strategies
Adding w/ manipulatives/fingers
Adding w/ semiconcrete objects Concrete/
conceptual Adding using a number line
Min strategy
Missing addend addition Semiconcrete/
representational Addition number family facts
Mental addition (+1, +2, +0)
Addition fact memorization Abstract Rational Numbers Rational Numbers What rational number represents the filledspaces?
What rational number represents the empty spaces?
What is the relationship between the filled and empty spaces? Presenting Rational Numbers Conceptually
Definition A rule of correspondence between two sets such that there is a unique element in the second set assigned to each element in the first set Synonyms
rule of correspondence linear function
y = x + 4 x + 4 f(x) = 2/3x 3y + 5x Examples Counter Examples Introduction to the Concept of Linear Functions Input
2 Rule
y = x+4 Output
6 Functions with increasingly complex operations y=x y = 3x+12 f(x) = 2.3x7 Functions to Ordered Pairs Ordered Pairs to Functions x x 3 4 5 6 7 10 y 7 9 11 13 15 ? 2 3 4 5 10 y y = 3x 1
? ? ? ? ?? y is 2 times x plus 1
y = 2x + 1
y = 2(10) + 1
y = 20 + 1 = 21 Primary
Primary
Concept
Development
Practice
Opportunities
Key
Vocabulary
Problem
Solving
Strategy Intermediate Secondary Background Knowledge Deficits
Students experience problems:
•With a lack of early number sense;
•Due to inadequate instruction in key concepts, skills, and strategies;
•Due to a lack of fluency with key skills. For many students struggling with mathematics, mastery of key procedures is dependent on having adequate practice to build fluency. Addressing Diverse Learners Through Core Instruction
Identify and preteach
prerequisite knowledge. Background Knowledge
Deficits Assess background knowledge. Differentiate practice and
scaffolding. Number Families 4 3 7 4+3=7 74=3 3+4=7 73=4 Fact Memorization 5
+2
1 + 8 = 4
+4
5 + 2 = 3
+6
4 + 3 = 2
+7
6 + 0 = 13 +10 +3 5=
3 2 “Manipulative Mode” 13 +10 +3 5=
3 2 13 +10 +3 5=
3 2 13 +10 +3 5=
3 2 13 +10 +3 5=
3 2 13 +10 +3 5=
3 2 Linguistic and Vocabulary Difficulties
Students experience problems:
•Distinguishing important symbols;
•With foundation and domain specific vocabulary;
•With independent word recognition. A Plan for Vocabulary in A Plan for Vocabulary in Mathematics
1. Assess students’ current knowledge. 2. Teach new vocabulary directly before and during reading of domain specific texts. 3. Focus on a small number of critical words. 4. Provide multiple exposures (e.g., conversation, texts, graphic organizers). 5. Engage students in opportunities to practice using new vocabulary in meaningful contexts.
(Baker, Gersten, & Marks, 1998; Bauman, Kame’enui, & Ash, 2003;
Beck & McKeown, 1999; Nagy & Anderson, 1991; Templeton, 1997) Check Your Vocabulary Knowledge
1. 1, 2/3, .35, 0, 14, and 32/100 are _____________.
2. In the number 3/8, the 8 is called the ____________.
3. In the number .50, the _____________ is 5.
4. ¾ and 9/12 are examples of ____________ fractions.
numerator equivalent denominator rational A Plan for Vocabulary in A Plan for Vocabulary in Mathematics
1. Assess students’ current knowledge. 2. Teach new vocabulary directly before and during reading of domain specific texts. 3. Focus on a small number of critical words. 4. Provide multiple exposures (e.g., conversation, texts, graphic organizers). 5. Engage students in opportunities to practice using new vocabulary in meaningful contexts.
(Baker, Gersten, & Marks, 1998; Bauman, Kame’enui, & Ash, 2003;
Beck & McKeown, 1999; Nagy & Anderson, 1991; Templeton, 1997) Recommended Procedures for Recommended Procedures for Vocabulary Instruction Modeling when difficult/impossible to use language to define word (e.g., triangular prism) Synonyms when new vocabulary equates to a familiar word (e.g., sphere) Definitions when more words are needed to define the vocabulary word (e.g., equivalent fractions) Probability Experiment
Odds
Theoretical probability
Tree diagram
These words will not be
Simulation
learned incidentally Experimental probability
or through context. Marzano, Kendall, & Gaddy (1999) A Plan for Vocabulary in A Plan for Vocabulary in Mathematics
1. Assess students’ current knowledge. 2. Teach new vocabulary directly before and during reading of domain specific texts. 3. Focus on a small number of critical words. 4. Provide multiple exposures (e.g., conversation, texts, graphic organizers). 5. Engage students in opportunities to practice using new vocabulary in meaningful contexts.
(Baker, Gersten, & Marks, 1998; Bauman, Kame’enui, & Ash, 2003;
Beck & McKeown, 1999; Nagy & Anderson, 1991; Templeton, 1997) Selection Criteria
Selection Criteria
for Instructional Vocabulary
Tier 1 Description Tier 2 Tier 3 Basic words that Words that appear Uncommon many children frequently in texts words understand which students associated with before entering school Math examples need for conceptual clock, count, square subtrahend, capacity, measure asymptote understanding
perimeter, a specific domain (Beck, McKeown, Kucan, 2002) Tier 3
Tier 3
Uncommon words associated with a specific domain subtrahend, asymptote, symmetry, hypotenuse Teaching children subject matter words (Tier 3)
can double their comprehension
of subject matter texts.
The effect size for teaching subject matter words is .97
(Stahl & Fairbanks, 1986) Word Identification Strategies
•Teach the meanings of affixes; they carry clues about word meanings (e.g., meter, gram, pent, etc.)
•Teach specific glossary and dictionary skills A Plan for Vocabulary in A Plan for Vocabulary in Mathematics
1. Assess students’ current knowledge. 2. Teach new vocabulary directly before and during reading of domain specific texts. 3. Focus on a small number of critical words. 4. Provide multiple exposures (e.g., conversation, texts, graphic organizers). 5. Engage students in opportunities to practice using new vocabulary in meaningful contexts.
(Baker, Gersten, & Marks, 1998; Bauman, Kame’enui, & Ash, 2003;
Beck & McKeown, 1999; Nagy & Anderson, 1991; Templeton, 1997) C a re fully S e le c te d G ra p h ic O rg a nize rs A Plan for Vocabulary in
Mathematics
Mathematics
1. Assess students’ current knowledge. 2. Teach new vocabulary directly before and during
Teach
reading of domain specific texts. 3.
3. Focus on a small number of critical words. 4. Provide multiple exposures (e.g., conversation,
Provide
texts, graphic organizers).
texts, 5. Engage students in opportunities to practice using
Engage
new vocabulary in meaningful contexts.
new
(Baker, Gersten, & Marks, 1998; Bauman, Kame’enui, & Ash, 2003;
Beck & McKeown, 1999; Nagy & Anderson, 1991; Templeton, 1997) “…students must have a way to participate in the mathematical practices of the classroom community. In a very real sense, students who cannot participate in these practices are no longer members of the community from a mathematical point of view.” Cobb (1999)
(Cobb and Bowers, 1998, p. 9) Extending mathematical knowledge through conversations
Discuss the following ideas about
rational numbers.
1. Describe how you know that
¾ and .75 are equivalent.
2. Explain how you can simplify a
rational number like 6/36. If you multiply ¾ by 1, it does not change its value. That’s why ¾ and .
75 or 75/100 are equivalent. I can convert ¾ to .75 by multiplying by 1 or 25/25. Encourage Interactions Encourage Interactions with Words Questions, Reasons, Examples:
– If two planes are landing on intersecting landing
strips, they must be cautious. Why? – Which one of these things might be symmetrical? Why or why not? A car? A water bottle? A tree? Relating Word
– Would you rather play catch with a sphere or a rectangular prism? Why? A Plan for Vocabulary in A Plan for Vocabulary in Mathematics
1. Assess students’ current knowledge. 2. Teach new vocabulary directly before and during reading of domain specific texts. 3. Focus on a small number of critical words. 4. Provide multiple exposures (e.g., conversation, texts, graphic organizers). 5. Engage students in opportunities to practice using new vocabulary in meaningful contexts.
(Baker, Gersten, & Marks, 1998; Bauman, Kame’enui, & Ash, 2003;
Beck & McKeown, 1999; Nagy & Anderson, 1991; Templeton, 1997) Strategy Knowledge and Use
Students experience problems:
•Remembering steps in a strategy;
•Developing selfquestioning skills;
•Selecting an appropriate strategy to fit a particular problem. You could use the ‘Algebrator” . . . Step 1. Enter the equation into the window. Step 2. Let the Algebrator solve it. Step 3. Stop Thinking!!! . . . What would you be missing? Thank You ...
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This note was uploaded on 11/11/2011 for the course MATH 110 taught by Professor Staff during the Winter '08 term at BYU.
 Winter '08
 Staff
 Math

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