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Fractions / Decimals / Percents: Success for ALL Students Heather Lindfors-Navarro February 2-3 2013 GSDMC Annual Conference TODOS: Mathematics for All The mission of TODOS: Mathematics for ALL is to advocate for an equitable and high quality mathematics education for all students— in particular, Hispanic/Latino students— by increasing the equity awareness of educators and their ability to foster students’ proficiency in rigorous and coherent mathematics. Diving Deeper Into the Common Core State Standards How the Mathematical Practices support a language acquisition in a classroom setting that is Culturally responsive. What I have learned about introducing concepts related to fractions, decimals, and percents so that ALL of my students can experience success. I will share with you: the importance of developing the ideas in relation to benchmarks,and relationships to the concept of one‐half, The importance of connecting these mathematical ideas with students’ foundational knowledge Questions that help guide teachers in facilitating lessons that involve rigorous mathematics. standards for mathematical practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critiques the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Instructional practices that support language acquisition MP Alignment Pedagogy Description Culturally Responsive Teaching (CRT) Instruction that builds on the languages and cultures that children bring from their home and community (Slaving & Cheung, 2005). Cooperative Learning Instruction that involves the use of small groups as a means to optimize students’ own and each other’s learning. MP 3 Instructional Conversation Extended instructional discourse between the teacher and students. MPs 2,3, 8 Instruction allowing student to articulate their thinking which in turn Cognitively Guided provides teachers with a better understanding of how children learn Instruction (CGI) math. MPs 1-8 Instruction utilizing technology to help connect learning in the Technologyclassroom to real-life situations and allows students to access MPs 4,5 Enriched Instruction information their native language. Almaguer, I., Diaz, Z., Esquiredo, J., Ramirez, O. in (2011). Teaching for Excellence and Equity in Mathematics. Vol. 3 No.1, Developing Mathematics Literacy for Bilingual Learners: A framework for Effective Learning. Culturally Responsive Mathematics teaching (CRMT) CRMT Evidence Reflective Question for Teachers How does my MP lesson... Alignment Cognitive demand ...enable students to closely explore and analyze concepts, procedures, and reasoning strategies? MPs 1,2,3 Depth of knowledge & student understanding ...make student thinking/understanding visible and deep? MPs 1,2,3 Mathematical discourse ...create opportunities to discuss mathematics in meaningful and rigorous ways? MPs 3,6 Power and participation ...distribute mathematics knowledge authority, value student contributions, and address status differences among students? Academic language support (for ELLs) ...provide academic language support for English Language Learners? Cultural/communitybased funds of knowledge/social justice ...(a.) help students connect math with relevant/authentic situations in their lives? ...(b.)support students’ use of mathematics to understand, critique, and change an important equity or social justice issue in their lives? • MPs 3, 6 Aguirre, J. (2012). Noticias de TODOS. Developing Culturally Responsive Mathematics Teachers. problem solving Rich mathematical problems that promote reasoning Opportunities for student groupings to maximize learning Problems demanding justification and communicating mathematically • Leiva, M. (2006). National Mathematics Panel Testimony. Teacher questions What are the prerequisite concepts and skills? What is the required Mathematics AND English vocabulary? What do the words, phrases, and sentences mean? Are these part of our common experiences? If not, how can we change the problem or use a teachable moment to teach and reach all students? CCSS Fraction progression Grade 1/2 : Describe partitions of shapes into equal shares Grade 3: Partitioning a whole into equal parts, unit fractions, preliminary reasoning about equivalent fractions, add decimal fractions by converting them to fractions Grade 4: Includes wholes that are collections of objects, equivalent fractions, comparisons, addition, subtraction(common denominators) and finite decimals, multiplication of a whole number and a unit fraction, decimal fractions (fractions w/ denominator of 10 or 100) Grade 5: Adding and subtractions fractions w/ unlike denominators, multiplying fractions by whole numbers and fractions, dividing whole numbers by fractions, scaling Grade 6: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. Recognizing and describing ratios, rates, and proportional relationship from other types of situations. (CCProgression, 2011) Meaning of per cent The word percent means “per 100” (cent is an abbreviation of the Latin centum “hundred”). If 35 milliliters out of every 100 milliliters in a juice mixture are orange juice, then the juice mixture is 35% orange juice (by volume). If a juice mixture is viewed as made of 100 equal parts, of which 35 are orange juice, then the juice mixture is 35% orange juice. Foundational lessons Introduce fractions/decimals/p ercents at the same time. Initial diagnostic Name That Portion, Fractions, Percents, and Decimals. (2004). Investigations in Number, Data, and Space. TERC. Scott Foresman. Pearson Education, Inc. Background knowledge Name That Portion, Fractions, Percents, and Decimals. (2004). Investigations in Number, Data, and Space. TERC. Scott Foresman. Pearson Education, Inc. Math Pathways and Pitfalls MPP problems require students to analyze fictional students’ thinking, explain the thinking, and argue one’s judgment. Encourages students to take on a role of “expert” and to offer her/his thinking as a way to help the fictional student. Overall this validates student contributions to the mathematical discourse. Kisty, L., Radosavljevic, A. (2010). A Descriptive Analysis of Math Pathways and Pitfalls in a Latina/o Bilingual Classroom. Math Pathways and Pitfalls This seemly small characteristic of MPP (i.e., that of having students analyze fictional students’ thinking) shifts the focus away from a single voice of authority (i.e., the author) to one of community collaboration. It is a vehicle for drawing on students’ knowledge and resources. Defines students as having “voice” which contributes to everyone’s learning Kisty, L., Radosavljevic, A. (2010). A Descriptive Analysis of Math Pathways and Pitfalls in a Latina/o Bilingual Classroom. Key concepts 100% is equivalent to 1 whole One-half is equivalent to 50%, and we can gauge whether an amount is greater than or less than using one-half equivalencies 1% is equivalent to 1 hundredth and 10% is equivalent to 10 hundredths and to 1 tenth Percents are comparisons to 1 or 100% Relate fraction, percent, and decimal amounts using an area model Use benchmarks to compare amounts Different names for the same amount Compare number in different symbolic forms on a number line Use contextual situations Quilting investigation Keisha’s grandma and mom have been making quilts for as long as she can remember. After reading The Keeping Quilt by Patricia Polacco, Keisha asks her mom if she can learn to make quilts too. Keisha’s mom is sewing a quilt that is 8 feet long and 6 feet wide. Keisha is going to help her mom make this quilt into their family’s own “keeping quilt.” Her mom asks her to sew a rectangular piece of the quilt that is 2 feet in length and 3 feet in width. What fraction of the area of the whole rectangular quilt will Keisha sew? Show how you found your solution. This investigation is adapted from “The Area Model of Multiplication of Fractions” by J.K. Tsankova and K. Pjanic in Mathematics Teaching in the Middle School, December 2009/January 2010. Key Instructional practices for ELLs Regular and active participation in the classroom—not only reading and listening but also discussing, explaining, writing, representing, and presenting—is critical to the success of ELLs in mathematics. Research has shown that ELLs can produce explanations, presentations, etc. and participate in classroom discussions as they are learning English. To develop written and oral communication skills, students need to participate in negotiating meaning for mathematical situations and in mathematical practices that require output from students. Application of Common Core State standards for English Language Learners Resources NCTM SmartBrief - At the end of August 2012, NCTM began to email to its members this daily newsletter. It is FREE to ANYONE who wants to receive it. It is a great update on the latest news and trends in education. To subscribe go to: https://www.smartbrief.com/nctm/index.jsp NCTM in TWITTERThe NCTM journals Teaching Children Mathematics (TCM) and Mathematics Teaching in the Middle School (MTMS) now have separate Twitter accounts: https://twitter.com/TCM_at_NCTM https://twitter.com/MTMS_at_NCTM ¿TIENES TAREAS? Tareasplus, http://www.tareasplus.com/, an onlineeducation portal for the Spanish-speaking world, offers thousands of video lessons on mathematics and science subjects from kindergarten through early college. www.todos-math.org. References Aguirre, J. (2012). Noticias de TODOS. Developing Culturally Responsive Mathematics Teachers. Almaguer, I., Diaz, Z., Esquiredo, J., Ramirez, O. (2011). Teaching for Excellence and Equity in Mathematics. Vol. 3 No.1, Developing Mathematics Literacy for Bilingual Learners: A framework for Effective Learning. Core Standards. http://www.corestandards.org/assets/CCSSI_Math%20Stan dards.pdf Core Standards for English Learners. http://www.corestandards.org/assets/application-for-english- learners.pdf Kisty, L., Radosavljevic, A. (2010). A Descriptive Analysis of Math Pathways and Pitfalls in a Latina/o Bilingual Classroom. Leiva, M. (2006). National Mathematics Panel Testimony. Name That Portion, Fractions, Percents, and Decimals. (2004). Investigations in Number, Data, and Space. TERC. Scott Foresman. Pearson Education, Inc. Progressions for the Common Core. (2011). http://commoncoretools.me/wpcontent/uploads/2012/02/ccss_progression_nf_35_2011_08_12.pdf Ramirez, N.G. & Celedon-Pattichis, S. (2012). Second Language Development and Implications for the Mathematics Classroom. In S. Celedón-Pattichis & N. Ramirez (Eds.), Beyond good teaching: Advancing mathematics education for ELLs. (pp. 19-38). Reston, VA: NCTM. Speaker feedback Strongly Disagree 1 Disagree Neutral 2 3 Agree Strongly Agree 4 5 Text your message to this Phone Number: 37607 “ 41632 ___ ___ ___ ___________ ” Session matched Speaker was welltitle and description prepared and Speaker was Other comments, in program book knowledgeable engaging and an suggestions, or effective presenter feedback Example: “41362 545 Great session! ”