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

Standards and andExpectations System/PhysicalEnvironment TeacherKnowledgeandPractice Instructional Tools and Materials EnhancedStudent Learning and Achievement Powerful Core Mathematics Instruction

Connections for Effective Mathematics Teaching Teacher Tools, Practices, Programs Student Needs Fundamental Mathematics For Teaching

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

of Part-Whole Relations Rates Quotient 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

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 Tools, Practices, Programs Student Needs Fundamental Mathematics For Teaching

Common Difficulty Areas for Diverse Learners (Student Needs) (Baker, Simmons, & Kame’enui, 1995) Memory and Conceptual Difficulties Linguistic and Vocabulary Difficulties Background Knowledge Deficits Strategy Knowledge and Use

Addressing Diverse Learners Through Effective Core Instruction Memory and Conceptual Difficulties Thoroughly develop concepts, principles, and strategies. 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 Equivalent fractions Definition ExamplesNon-Examples Model Two or more fractions that represent the same point on the number line 2/3 ~ 4/6 8/4 ~ 2/1 2/5 ~ 4/5 3/2 ~ 3/4

Sequenced Examples Combining Like Terms A.2X + 4X =Identify like terms. 6XCombine coefficients = 6 B.3c – 2d + 2c + 5Identify like terms. 5c – 2d + 5Combine coefficients = 5 C.2k + 4m – 2m + k Identify like terms. 3k + 2mCombine coefficients = 3 4 – 2 = 2 D. 7f + 2g – 2No like terms to combine.

Addressing Diverse Learners Through Effective Core Instruction Background Knowledge Deficits Identify and pre-teach prerequisite knowledge. Assess background knowledge. Differentiate practice and scaffolding.

Numerical ExpressionAlgebraic Expression 7 + 3n + 3 variableconstant Here is how you read algebraic expressions sum n + 3 “n plus 3” product 3 x n or 3n “3 times n” difference n – 3 “n less 3” quotient n÷3 “n divided by 3” Build From Previous Knowledge

Addressing Diverse Learners Through Effective Core Instruction Linguistic and Vocabulary Difficulties Define and use mathematical symbols with precision and in multiple contexts. 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)

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. Sample Text 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.

Translate the expressions Foster Students’ Ability to Translate Mathematics ExpressionEnglish Expression 3x + 2 Sara had some cds; Daisy had three more than twice as many as Sara Two more than the product of 3 and x Sara has x cds; Daisy has 2x + 3 cds

2(3x-1) 49 less than the value of some number is 65. What is the number? Mathematics ExpressionEnglish Expression 1/2b = 12 If you divide a number by 5, the result is 30. What is the number? Mathematics ExpressionEnglish Expression

Addressing Diverse Learners Through Effective Core Instruction Strategy Knowledge and Use Model problem-solving strategies through think- alouds. Teach the “how”, “why”, and “when” of strategy application. Have students verbally rehearse the steps to solving problems. Provide interactive problem- solving opportunities.

Verbally rehearse the steps.

RtI: Multi-Tiered Model

Focus Interventionist Grouping Assessment Program Setting Time

A Closer Look at Tier II

Focus Interventionist Grouping Assessment Program Setting Time

Research and Practice Relationship Research – practice = irrelevant Practice – research = irresponsible Research + practice = relevant and responsible Dr. Kame’enui, 2009

Explicit and Systematic Instruction Elementary and Middle School Settings Level of evidence: Strong IES Practice Guide Recommendation 3 Secondary School Settings Level of evidence: Promising Teacher provides clear models Students receive extensive and scaffolded practice and review Teacher and students verbalize and think aloud Teachers provides corrective feedback

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 Middle School Settings Level of evidence: Moderate IES Practice Guide Recommendation 5 Secondary School Settings Level of evidence: Promising Model mathematical concepts using visual representations Relationship is explicitly taught

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 Middle School Settings Level of evidence: Moderate Effect size: 0.82 (p<.001) Secondary School Settings Level of evidence: Possible Sequence of examples is systematic Variation in range of examples Example: Concrete-Representational- Abstract

⅓ ÷ ⅘ 0.54 × ÷ ⅖ ¾ = 0.75 ⅞ − ⅕

Abstract Representation (Formal) Abstract Representation (Formal) Visual Representations (Pre-formal) Visual Representations (Pre-formal) Concrete Models (Informal) 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 ConcreteVisual Representations

Best Evidence Synthesis: Effective Programs in Middle and High School Mathematics Slavin, Lake, & Groff (2009) Best evidence syntheses are similar to meta- analyses, 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