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Helping Struggling Learners with Mathematical Reasoning David J. Chard Southern Methodist University Simmons School of Education and Human Development Purposes of the Presentation • Discuss the ingredients needed to create a system of mathematics education that will ensure greater achievement for all students. • Consider the needs of teachers and students to improve mathematics achievement. • Review evidence to identify and provide examples of instructional practices that can be incorporated into core instruction and interventions Powerful Core Mathematics Instruction System/ Physical Environment Standards and Expectations Teacher Knowledge and Practice Instructional Tools and Materials Enhanced Student Learning and Achievement Connections for Effective Mathematics Teaching Teacher Mathematics For Teaching Tools, Practices, Programs Fundamental Mathematics Student Needs http://www.NCTQ.org Understanding Fundamentals of Mathematics • Historical development • Number systems and relationships The Real Numbers Defines Division Defines Addition, Subtraction, and Multiplication Understanding Fundamentals of Mathematics • • • • Historical development Number systems and relationships Key mathematical principles Appropriate models of mathematical concepts and dispelling common misconceptions Modeling Fractions Set Models Number Line Area Models Fraction Equivalence Rates Part-Whole Relations 3 4 Quotient 3 4 of Operations Understanding Fundamentals of Mathematics • • • • Study historical development of mathematics Understand number systems and relationships Know key mathematical principles Prepare appropriate models of mathematical concepts and dispel common misconceptions • Understand that common algorithms are derived from formulas Demystifying Mixed Numbers Definition of fraction addition 2 3 5 Understanding Fundamentals of Mathematics • • • • Study historical development of mathematics Understand number systems and relationships Know key mathematical principles Prepare appropriate models of mathematical concepts and dispel common misconceptions • Understand that common algorithms are derived from formulas • Develop a positive disposition for mathematics and problem solving Connections for Effective Mathematics Teaching Teacher Mathematics For Teaching Tools, Practices, Programs Fundamental Mathematics Student Needs Common Difficulty Areas for Diverse Learners (Student Needs) (Baker, Simmons, & Kame’enui, 1995) Memory and Conceptual Difficulties Background Knowledge Deficits Linguistic and Vocabulary Difficulties Strategy Knowledge and Use Addressing Diverse Learners Through Effective Core Instruction Thoroughly develop concepts, principles, and strategies. Memory and Conceptual Difficulties Gradually develop knowledge and skills while ensuring student success. Include non-examples to teach students to focus on relevant features. Include a planful system of review. Presenting Mathematical Ideas Conceptually Definition Model Two or more fractions that represent the same point on the number line Equivalent fractions 2/3 ~ 4/6 2/5 ~ 4/5 8/4 ~ 2/1 3/2 ~ 3/4 Examples Non-Examples Sequenced Examples Combining Like Terms A. 2X + 4X = Identify like terms. 6X Combine coefficients 2+4=6 B. 3c – 2d + 2c + 5 Identify like terms. 5c – 2d + 5 Combine coefficients 3+2=5 C. 2k + 4m – 2m + k Identify like terms. 3k + 2m Combine coefficients 2+1=3 4–2=2 D. 7f + 2g – 2 No like terms to combine. Addressing Diverse Learners Through Effective Core Instruction Identify and pre-teach prerequisite knowledge. Background Knowledge Deficits Assess background knowledge. Differentiate practice and scaffolding. Build From Previous Knowledge Numerical Expression Algebraic Expression 7+3 n+3 variable constant Here is how you read algebraic expressions sum n+3 product 3 x n or 3n “n plus 3” “3 times n” difference quotient n–3 “n less 3” n÷3 “n divided by 3” Addressing Diverse Learners Through Effective Core Instruction Define and use mathematical symbols with precision and in multiple contexts. Linguistic and Vocabulary Difficulties Describe and develop vocabulary deliberately and thoughtfully. Model and encourage the use of mathematical vocabulary in classroom discourse. Provide students opportunities to discuss mathematics and to receive feedback on their discussions. 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) Sample Text Like terms can be combined because they have the same variable raised to the same power. They may have different coefficients. When you combine like terms, you simplify the expression but you do not change the value of the expression. Equivalent expressions have the same value for all values of the variables. Key Vocabulary Like Terms Coefficients Simplify Equivalent expressions Discuss 1. What is the first step in simplifying the expression 3 + 2(d+3f) - 2f? 2. How many different sets of like terms are in the expression in (1) above? 3. Explain why 3a + 3b + 3c is already simplified. The purpose of teaching students key mathematical vocabulary is to anchor their understanding in words that they can then use to communicate mathematically. Foster Students’ Ability to Translate Translate the expressions Mathematics Expression 3x + 2 Sara has x cds; Daisy has 2x + 3 cds English Expression Two more than the product of 3 and x Sara had some cds; Daisy had three more than twice as many as Sara Mathematics Expression English Expression 2(3x-1) 49 less than the value of some number is 65. What is the number? Mathematics Expression English Expression 1/2b = 12 If you divide a number by 5, the result is 30. What is the number? Addressing Diverse Learners Through Effective Core Instruction Model problem-solving strategies through thinkalouds. Strategy Knowledge and Use Teach the “how”, “why”, and “when” of strategy application. Have students verbally rehearse the steps to solving problems. Provide interactive problemsolving opportunities. Verbally rehearse the steps. RtI: Multi-Tiered Model RtI:Multi-Tiered Model Focus Program Interventionist Setting Grouping Time Assessment A Closer Look at Tier II A Closer Look at Tier II Focus Program Interventionist Setting Grouping Time Assessment Research and Practice Relationship Research – practice = irrelevant Practice – research = irresponsible Research + practice = relevant and responsible Dr. Kame’enui, 2009 Explicit and Systematic Instruction Teacher provides clear models Students receive extensive and scaffolded practice and review Teacher and students verbalize and think aloud Teachers provides corrective feedback Elementary and Middle School Settings • Level of evidence: Strong • IES Practice Guide Recommendation 3 Secondary School Settings • Level of evidence: Promising Learning Outcomes • Research findings: – Improved proficiency in solving word problems – Improved procedural fluency • Possible outcomes: – Connection between conceptual and procedural understanding – Generalize concepts and procedures – Ability to communicate problem solving and reasoning Visual Representations Model mathematical concepts using visual representations Relationship is explicitly taught Middle School Settings • Level of evidence: Moderate • IES Practice Guide Recommendation 5 Secondary School Settings • Level of evidence: Promising Considerations for Using Visual Representations • Explicitly link the concrete model and visual representation with the abstract mathematics • Use consistent language across representations • Provide ample practice opportunities • Select examples and non-examples carefully Sequence and Range of Examples Sequence of examples is systematic Variation in range of examples Example: Concrete-RepresentationalAbstract Middle School Settings • Level of evidence: Moderate • Effect size: 0.82 (p<.001) Secondary School Settings • Level of evidence: Possible ⅞−⅕ ⅓÷⅘ ¾ = 0.75 0.54 × 0.2 21 ÷ ⅖ Abstract Representation (Formal) Visual Representations (Pre-formal) Concrete Models (Informal) Webb, Boswinkel, Dekker, 2008 Learning Outcomes • Research findings: – Improved achievement in general mathematics, pre-algebra concepts, word problems, operations • Possible outcomes: – Understand meaning of abstract symbols – Develop strategies for translating word problems into abstract numerical statements – Scaffold students’ conceptual understanding Developing Understanding of Fractions Concrete Visual Representations Best Evidence Synthesis: Effective Programs in Middle and High School Mathematics Slavin, Lake, & Groff (2009) • Best evidence syntheses are similar to metaanalyses, pooling effects across studies to determine the contribution of a particular approach. • Very similar to the What Works Clearinghouse model. Findings • Programs and approaches effective with one subgroup, appear to be effective with all subgroups • There is a lack of evidence that particular curricular materials are better than others (e.g., NSF-supported vs. non-NSF supported) • Approaches that produce significantly positive effects are those that fundamentally change what students do every day in the classroom • What new information have you learned? • What remains confusing or unanswered? • How might you apply what you’ve learned? What I Believe We Must Do • Take responsibility for our children’s mathematics development • Determine what we need to learn and invest our time and energy to make it happen • Focus our attention on the whole picture; improvement in one aspect is insufficient • Sustain our effort to improve mathematics instruction for all children • Develop sophisticated measurement systems to track progress and make difficult resource allocations to ensure success Thank You