Teaching and Learning Fractions with Conceptual Understanding Algebra Forum IV San Jose, CA May 22, 2012 Compiled and Presented by April Cherrington Joan Easterday Region 4 Region 1 Susie W. Hakansson, Ph.D. California Mathematics Project
Description Fractions from a number line approach represents a shift in thinking about fractions, moving beyond part-whole representations to thinking of a fraction as a point on the number line. Included in this session will be the following: rationale, comparing and ordering, and justification.
Outline for Today • (25 minutes) Introduction • (105 minutes) Breakout session (includes Q&A) • (25 minutes) Summary and Reflection
Introduction • CaCCSS-M Task Force • Conceptual understanding • Order problems • Cognitive level • Language issues • Why number line? • Fraction progressions • Standards for Mathematical Practice • Challenges students face • Overview of break out session
Fractions Task Force • Greisy Winicki-Landman, Chair • Nadine Bezuk • April Cherrington • Pat Duckhorn • Joan Easterday • Doreen Heath Lance • Pam Hutchison • Natalie Mejia • Gregorio Ponce • Debbie Stetson • Kathlan Latimer
Demands of CaCCSS-M “… almost all teachers are placing a lower priority on student understanding in recent years, ….” “… the sort of high quality PD that an really affect teachers in their ability to produce students who understand is very, very difficult to do, and very few people have much clue about how to do it.” Scott Farrand
Fraction Sense: Comparing • 8/15 > 1/2 (?) • 7/22 > 1/3 (?) • 6/11 > 7/15 (?) • 7/8 > 8/9 (?) Solve these problems mentally without using algorithms. Justify your thinking.
Cognitive Demand Spectrum Memorization Procedures Without Connections to understanding, meaning, or concepts Procedures With Connectionsto understanding, meaning, or concepts Doing Mathematics Tasks that require engagement with concepts, and stimulate students to make connections to meaning, representation, and other mathematical ideas Tasks that require memorized procedures in routine ways
Why Is English So Hard? • The soldier decided to desert his dessert in the desert. • Upon seeing the tear in the painting, I shed a tear. • After a number of injections, my jaw got number. • A minute is a minute part of a day.
Why Is English So Hard? • There is no egg in eggplant and no ham in hamburger. • How can a slim chance and a fat chance be the same, while a wise man and a wise guy are opposites? • Did you say thirty or thirteen? • Did you say two hundred or two hundredths? • Did you say fifty or sixty?
The Guinevere Effect 9th and 10th graders’ responses • Tom had 5 apples. He ate 2 of them. How many apples were left? • A. 10 B. 7 C. 5 D. 3 (100%) • Guinevere had 5 pomegranates. She ate 2 of them. How many pomegranates were left? • A. 10 (22%) B. 7 (24%) C. 5 (23%) D. 3 (31%)
Key Strategies for English Learners • Access prior knowledge • Frontload language • Build on background knowledge • Extend language • Be aware of multiple meanings of words • Have students Think, Ink, Pair, Share (TIPS)
Teachers learn to amplify and enrich--rather than simplify--the language of the classroom, giving students more opportunities to learn the concepts involved. Aída Walqui, Teacher Quality Initiative
Why Number Line? “Hung-Hsi Wu attempts to bring coherence to the teaching and learning of fractions by beginning with the definition of a fraction as the length on the number line (1998). This approach eliminates the ‘conceptual discontinuity’ (2002) encountered moving from work with whole numbers to fractions; it also brings coherence to the various meanings of fractions and allows for both conceptual work to operations on fractions (2008). Wu asserted that ‘The number line is to fractions what one’s fingers are to whole numbers ...”
Basic Assumptions about the Number Line and Its Use • Using the number line, there are basically two types of tasks: • Given a point on the number line, assign a number to it (its coordinate) • Given a number, place it as a point on the number line
WHY THE Number Line? • It serves as a visual/physical model to represent the counting numbers and constitutes an effective tool to develop estimation techniques, as well as a helping instrument when solving word problems. • It constitutes a unifying and coherent representation for the different sets of numbers (N, Z, Q, R), which the other models cannot do.
WHY THE Number Line? • It is an appropriate model to make sense of each set of numbers as an expansion of other and to build the operations in a coherent mathematical way. • It enables to present the fractions as numbers and to explore the notion of equivalent fractions in a meaningful way.
WHY THE Number Line? • The number line, in some way, looks like a ruler, fostering the use of the metric system and the decimal numbers. • It fosters the discovery of the density property of rational numbers. • It provides an opportunity to consider numbers that are not fractions.
Common Core Standards Mathematics Grades 3, 4 and 5 Number and Operations - Fractions • Grade 4 • Extend understanding of fraction equivalence and ordering • Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. • Understand decimal notation for fractions, and compare decimal fractions. • Grade 3 • Develop understanding of fractions as numbers • Grade 5 • Use equivalent fractions as a strategy to add and subtract fractions. • Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Common Core Standards Mathematics Grades 6 and 7 Number and Operations - Fractions • Grade 6 • Apply and extend previous understandings of multiplication and division to divide fractions by fractions. • Grade 7 • Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
CaCCSS-M: Mathematical Practice • We will focus two of the Standards for Mathematical Practice: • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others
Reason abstractly and quantitatively DO STUDENTS: • Make sense of quantities and their relationships in problem situations? • Decontextualize a problem? • Contextualize a problem? • Create a coherent representation of the problem, consider the units involved, and attend to the meaning of quantities?
Construct viable arguments and critique the reasoning of others DO STUDENTS: • Justify their conclusions, communicate them to others, and respond to arguments of others? • Hear or read arguments of others and decide whether they make sense, and ask useful questions to clarify or improve the argument?
How do I think about a task that opens up opportunities to implement the Standards for Mathematical Practice?
Question What are some of the challenges that students have with fractions?
Overview of Breakout Sessions • Appropriate grade level problem • Twelve (12) cards • Videos of students working with 12 cards • Human Number Line activity • Reflection
Break Out Rooms • Elementary-Oak Grove Room • 2nd Floor South • Middle Grades-SJES Room • Here • High School-Milpitas Room • 2ndFloor North