Reform! Research on learning shows: the entire standards-based Some - PowerPoint PPT Presentation

Signs of Change Making Sense of Math Makes Sense • New standards • Focus, Coherence, Rigor Randall I. Charles • College and career readiness Carmel, CA • Higher expectations rcharles155@icloud.com • New curricula • New assessments Brooklyn • etc March 2020 1 Reform! Research on learning shows: “…the entire standards-based Some benefits of understanding mathematics: • Better retention of knowledge reform movement [is] one whose • Better fluency with procedures Better transfer of knowledge • focus [is] more depth of • Better ability to apply/solve problems • Positive dispositions toward and attitudes about mathematics understanding.” • Helpful beliefs about learning mathematics • More likely to enroll in future mathematics classes (McLaughlin, Shepard, Day, 1995) • Better performance on assessments

Many Traditional Expectations Session Overview • Part 1: Essential Understandings & Big Ideas • Part 2: Teaching for Understanding (and Procedural Making Sense of Fluency) • Problem-based learning Math • Computation procedures Understanding • Part 3: Quantitative Reasoning (Word Problems) Instruction/Assessment Task NY-4.NF.2 Each of these fractions refers to the same whole. Place Compare two fractions with different numerators and different denominators. these fractions in order from least to greatest. • Recognize that comparisons are valid only when the 4/3 4/8 4/5 4/12 4/4 two fractions refer to the same whole. • Record the results of comparisons with symbols >, =, and <, and justify the conclusions.

Part 1 NY-4.NF.2 Compare two fractions with different numerators and different denominators. Essential Understandings and Big • Recognize that comparisons are valid only when the Ideas two fractions refer to the same whole. • Record the results of comparisons with symbols >, =, and <, and justify the conclusions. 9 Research says Skill-Focused Standards • NY-3.NBT.3 Multiply one-digit whole numbers by Many teachers obviously would like their students to multiples of 10 in the range 10-90 using strategies based understand the mathematics they study but, when on place value and properties of operations. asked to specify the goal for a particular lesson, most • NY-4.NF.2 Compare two fractions with different U.S. teachers… talked about skill proficiency; few numerators and different denominators. mentioned understanding. • NY-5.MD.1 Convert among different-sized standard measurement units within a given measurement system (TIMSS, Hiebert and Stigler, 2000) when the conversion factor is given. Use these conversions in solving multi-step, real world problems.

Takeaways 
 Standards – “Understand” Pitfalls to Avoid • NY-1.OA.4 Understand subtraction as an unknown- • Avoid a mindset that, “Today I am teaching a skill; addend problem within 20. tomorrow I am teaching understanding.” • NY-2.NBT.1 Understand that the three digits of a three-digit number represent the number of • Avoid interpreting every content standard only as a hundreds, tens, and ones. statement of skills students should acquire. • NY-7.NS.1.c Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (– q ). Takeaway • For every lesson, make explicit what you want “We understand something if we students to KNOW and, as appropriate, what you see how it is related or connected want them to be able to DO. to other things we know.” (Do You UNDERSTAND? Do you know HOW?) J. Hiebert , Signposts for Teaching Mathematics through Problem Solving In F. Lester & R. Charles, Teaching Mathematics Through Problem Solving, Grades PreK-6. NCTM: Reston, VA, 2003.

A web of ideas! Understanding • Connections • Connections • Connections Essential Understandings What are Essential • The size of a fraction is relative to the size of the whole. • On the number line, a fraction to the right of another is the Understandings and Big greater fraction. Ideas? • If two fractions have the same numerator, the one with the greater denominator is the lesser fraction. • If two fractions have the same denominator, the one with the greater numerator is the greater fraction. What do you want students to • Many fractions can be compared using number sense and how each compares to ½ and 1. KNOW. • Etc.

A fraction is relative to the size of the whole. You cannot teach for understanding unless you know what understandings to teach. Essential Understandings NY-4.NBT. 4. Use place value understanding to round • Rounding is a process of finding the closest multiple of 10, 100, 1,000, etc. that a given whole number is closest to. multi-digit whole numbers to any place. • Rounding is based on knowing the halfway point between two consecutive multiples of 10 or100 or 1,000 etc. Round 41,915 to the nearest thousand • Most whole numbers can be rounded to more than one place value. The context can determine the most appropriate place value for rounding. • On the number line, numbers to the left of the halfway point between two consecutive multiples of 10 or 100 or 1,000 etc. are closer to the lower multiple; numbers to the right are closer to the greater multiple. • A convention in mathematics is to agree that numbers exactly halfway between two consecutive multiples of 10 or 100 or 1,000 are closer to the greater multiple.

What is a “ big idea ” ? These ALL connect to the same big idea! A Big Idea is a statement of an idea “Change the way it looks but don’t change the central to the learning of value. ” mathematics, one that links numerous essential understandings into a coherent whole. “Equivalence” 
 Big Idea • Big Idea: Basic Facts and Algorithms for Big Idea: A given number, measure, or expression can be represented using symbols in more than one way where each has the Rational Numbers same value. Most basic facts and algorithms for operations with rational numbers, both mental math and paper and pencil, can be carried out using equivalence to transform calculations into simpler ones.

Big Idea Research Effective Teachers ✓ Have their own mathematics content knowledge anchored on big ideas and essential understandings. “Change it to a simpler problem or ✓ Use big ideas as the glue for teaching, learning, and problems” assessment (connections). Ma, Liping. Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum, 1999. Stigler, James. “The Teaching Gap: Reflections on Mathematics Teaching and How to Improve It.” Paper presented at the Pearson National Educational Leadership Conference, Washington, D.C., March 2004. Big Ideas and Essential Understandings Key Takeaways • For every lesson, make explicit what you want NCTM, “Essential Understandings” Series. • students to KNOW and, as appropriate, what you want them to be able to DO. • R. Charles (2005). “Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics.” Journal of Mathematics Education Leadership, 8 (1), pp.9-24. (Do You UNDERSTAND? Do you know HOW?) • Take every opportunity to make connections using • D. Clements & J. Sarama (Eds) (2004). Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education. Lawrence Erlbaum: Mahwah, NJ. “big ideas.” • Christopher Cross and others (2009). Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. Washington, D.C.: National Research Council.

Part 2 Understanding Teaching for Understanding Student-Made Connections Previously learned math ideas to new math ideas “Researchers have identified a three-phase classroom activity structure [for high-quality mathematics instruction]. In well executed lessons…the teacher poses a problem and ensures that all students understand the context and expectations, students develop strategies and solutions (typically in collaboration with each other), and, through reflection and sharing, the teacher and students work together to explicate the mathematical concepts “Problem-Based Learning” underlying the lesson’s problem.” Munter, Charles. “Developing Visions of High-Quality Mathematics Instruction.” Journal for Research in Mathematics Education , Volume 45, Number 5, November PBL 2014, pp. 584-635.

“First Instruction” PBL: Three-Phase Instructional Model Phase 1: Teacher poses a problem and ensures that all students understand the context and expectations, New Model Old Model Step 1: Problem-Based Learning Phase 2: Students develop strategies and solutions Step 1: Show and Tell (typically in collaboration with each other), and, Phase 1 Phase 2 Phase 3: Through reflection and sharing, the teacher and Phase 3 students work together to explicate the mathematical Step 2: Try it concepts underlying the lesson’s problem. Step 2: Try it Step 3: Practice and Problem Solving Step 3: Practice and Problem Solving Problem-Based Learning Solve and Share Suppose you were going to plant a rectangular garden that covered 60 square feet. The side lengths are Research shows that understanding develops whole numbers. What might it look like? during the process of solving problems in which important math concepts and skills are Draw all possible outlines of your garden. embedded. (Lester & Charles, 2003). Write and explain why you think you found them all. [Extension: 11 square feet]