David_20Chard_20SMEC_20Plenary

David_20Chard_20SMEC_20Plenary - Thinking Doing and Talking...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 non-examples 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) = 2.3x-7 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 7-4=3 3+4=7 7-3=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 ...
View Full Document

This note was uploaded on 11/11/2011 for the course MATH 110 taught by Professor Staff during the Winter '08 term at BYU.

Ask a homework question - tutors are online