S6 | EDUCATION WEEK n November 12, 2014 Making Sense of the Math: The Common Core in Practice > www.edweek.org/go/math-report Approach to Fractions Seen as Key Shift in Standards Number line emphasized By Liana Heitin F or many elementary teachers, fractions have traditionally sprung to mind lessons involving pizzas, pies, and chocolate bars, among other varieties of "wholes" that can be shared. But in what many experts are calling one of the biggest shifts associated with the Common Core State Standards for mathematics, more teachers are now being asked to emphasize fractions as points on a number line, rather than just parts of a whole, to underscore their relationships to integers. As common-core proponents see it, the new standards do a much better job of putting fractions into context, which will help students make connections across other math concepts. "The ultimate underlying principle is you want kids to understand that fractions are numbers," said William G. McCallum, a mathematics-education professor at the University of Arizona, in Tucson, and one of the lead writers of the common standards. "They're new, but they're not in a different galaxy." "I should not have to change what I know about numbers to learn fractions," said Zachary Champagne, an assistant in research at the Florida Center for Research in Science, Technology, Engineering, and Mathematics at Florida State University, in Tallahassee. Fractions instruction in schools has long been seen as a problem area. In 2010, the U.S. Department of Education's Institute of Education Sciences released a report on effective K-8 fractions instruction as part of its What Works Clearinghouse. The report noted that half of 8th graders could not place three fractions in order from least to greatest on the 2004 National Assessment of Educational Progress in math. It also found that fewer than 30 percent of 17-year-olds could convert 0.029 into a fraction. Slow Rollout The recommendations from that report, as well as those of a 2008 federal study by the National Mathematics Advisory Panel that called difficulty with fractions "pervasive," are reflected in the common core. In addition to emphasizing fractions as points on the number line, the common standards differ from previous standards in other important ways: They delay arithmetic with fractions until students have a thorough understanding of what a fraction is; they eliminate explicit instruction on lowest common denominators; they do not differentiate between proper and improper fractions; and they place an early emphasis on decimal equivalents. Under the common core, 1st and 2nd graders learn some basic vocabulary on fractions, including describing parts of shapes as halves and quarters, and 6th graders learn division of fractions. But the bulk of fractions instruction goes on between grades 3 and 5. The 3rd grade rollout of fractions is intended to be slow and steady. The standards require students to view fractions as divided wholes and as numbers on a number line, as well as to reason about a fraction's size. There's no arithmetic with fractions that year. "We're allowing time for students to explore and delve deeply into the meaning of fractions," said Denise M. Walston, the director of mathematics for the Washingtonbased Council of the Great City Schools. In the past, teachers and textbooks have rushed into operations before students really understood the basics of fractions, said Jonathan A. Wray, the instructional facilitator for FRACTIONS BY GRADE LEVEL Under the common-core standards, fractions instruction extends across grades 3 through 6, with operations being taught after students have a firm understanding of where fractions lie on a number line. The standards, shown here by grade, have been edited for length and clarity. secondary mathematics curricular programs in Maryland's Howard County public schools. Mr. Wray, who also helped write the 2010 ies report, said he has seen students make it to middle school still thinking of the numerator and denominator as separate numbers. "They're using algorithms to come up with equivalent fractions, they're using cross-multiplication-that was getting them by, but they didn't understand anything behind it," he said. " Teachers need help with this. This certainly isn't the way most of us learned fractions." JONATHAN A. WRAY 1 b E AD GR 3 Develop understanding of fractions as numbers STANDARDS: * nderstand 1/b as whole U divided into b parts * nderstand a fraction as a U number on the number line * xplain equivalence of E fractions and compare fractions by reasoning about their size Putting fractions on a number line early can also help solidify students' understanding that fractions can and should be compared to whole numbers, experts say. Circular and rectangular representations can still be useful, said Mr. McCallum, especially in the context of learning about quarters and halves. But "it's not so easy to divide a pizza into five equal pieces," he said. In addition, the number line helps ensure students use consistent units. Mr. Wray pointed out that students who are trying to compare fractions with a circular model may end up drawing two circles of significantly different sizes. If they shade one-half of the larger circle and three-fourths of the smaller circle, they could make the argument that onehalf is greater than three-fourths, he said. "The fact that [number lines] are mentioned explicitly as a teaching and learning tool in the common core, I think that has changed the landscape a little bit," said Mr. Wray. Of course, many teachers have been putting fractions on number lines for years, said Ms. Briars, but the common core makes this "much more of a central representation." Howard County (Md.) Public Schools Uncommon Denominators Students first need a good grasp of what a unit fraction is- that is, a fraction with a one in the numerator, the basic unit of measurement for larger fractions-before moving on to operations, according to the standards' authors. "This emphasis on units and unit fractions, and how nonunit fractions are built from unit fractions, they did this very purposefully to help students use the knowledge they have from how whole numbers work," said Diane J. Briars, the president of the Reston, Va.-based National Council of Teachers of Mathematics. "It helps teachers hammer home the point that a denominator is just the label that tells you the size of the partition," she said. A notable absence from the standards, meanwhile, is any mention of "finding the lowest common denominator." Students have traditionally spent large amounts of time practicing reducing fractions to their lowest form, and in many classrooms, an answer was marked wrong if it was not simplified. "The question is, 'Why?' " said Mr. McCallum. "It's not mathematically important." Students do need to compare equivalent fractions, which means they will need to simplify at times. "But it's not an overriding concern," said Mr. McCallum. "And there are situations where it positively is getting in the way of understanding." Mr. Champagne, of Florida State University, offered an exam-http://www.edweek.org/go/math-report http://www.edweek.org/go/math-report

Education Week - November 12, 2014 - Special Report - S2

Education Week - November 12, 2014 - Special Report - S3

Education Week - November 12, 2014 - Special Report - S4

Education Week - November 12, 2014 - Special Report - S5

Education Week - November 12, 2014 - Special Report - S6

Education Week - November 12, 2014 - Special Report - S7

Education Week - November 12, 2014 - Special Report - S8

Education Week - November 12, 2014 - Special Report - S9

Education Week - November 12, 2014 - Special Report - S10

Education Week - November 12, 2014 - Special Report - S11

Education Week - November 12, 2014 - Special Report - S12

Education Week - November 12, 2014 - Special Report - S13

Education Week - November 12, 2014 - Special Report - S14

Education Week - November 12, 2014 - Special Report - S15

Education Week - November 12, 2014 - Special Report - S16

Education Week - November 12, 2014 - Special Report - S17

Education Week - November 12, 2014 - Special Report - S18

Education Week - November 12, 2014 - Special Report - S19

Education Week - November 12, 2014 - Special Report - S20

Education Week - November 12, 2014 - Special Report - S21

Education Week - November 12, 2014 - Special Report - S22

Education Week - November 12, 2014 - Special Report - S23

Education Week - November 12, 2014 - Special Report - S24

Education Week - November 12, 2014 - Special Report - S25

Education Week - November 12, 2014 - Special Report - S26

Education Week - November 12, 2014 - Special Report - S27

Education Week - November 12, 2014 - Special Report - S28

http://ew.edweek.org/nxtbooks/epe/ew_sr_03072018

http://ew.edweek.org/nxtbooks/epe/ew_02282018

http://ew.edweek.org/nxtbooks/epe/ew_02212018

http://ew.edweek.org/nxtbooks/epe/ew_02142018

http://ew.edweek.org/nxtbooks/epe/ew_02072018

http://ew.edweek.org/nxtbooks/epe/ew_01242018

http://ew.edweek.org/nxtbooks/epe/ew_01172018

http://ew.edweek.org/nxtbooks/epe/ew_01102018

http://ew.edweek.org/nxtbooks/epe/ew_12132017

http://ew.edweek.org/nxtbooks/epe/ew_11292017

http://ew.edweek.org/nxtbooks/epe/ew_11152017

http://ew.edweek.org/nxtbooks/epe/ew_11082017

http://ew.edweek.org/nxtbooks/epe/ew_11012017

http://ew.edweek.org/nxtbooks/epe/ew_10252017

http://ew.edweek.org/nxtbooks/epe/ew_10182017

http://ew.edweek.org/nxtbooks/epe/ew_10112017

http://ew.edweek.org/nxtbooks/epe/ew_10042017

http://ew.edweek.org/nxtbooks/epe/ew_09272017

http://ew.edweek.org/nxtbooks/epe/ew_09202017

http://ew.edweek.org/nxtbooks/epe/ew_09132017

http://ew.edweek.org/nxtbooks/epe/ew_09062017

http://ew.edweek.org/nxtbooks/epe/ew_08302017

http://ew.edweek.org/nxtbooks/epe/ew_08232017

http://ew.edweek.org/nxtbooks/epe/ew_07192017

http://ew.edweek.org/nxtbooks/epe/ew_06212017

http://ew.edweek.org/nxtbooks/epe/ew_06142017

http://ew.edweek.org/nxtbooks/epe/ew_06072017

http://ew.edweek.org/nxtbooks/epe/ew_05312017

http://ew.edweek.org/nxtbooks/epe/ew_05242017

http://ew.edweek.org/nxtbooks/epe/ew_05172017

http://ew.edweek.org/nxtbooks/epe/ew_05102017

http://ew.edweek.org/nxtbooks/epe/ew_04262017specialreport

http://ew.edweek.org/nxtbooks/epe/ew_04262017

http://ew.edweek.org/nxtbooks/epe/ew_04192017

http://ew.edweek.org/nxtbooks/epe/ew_04052017

http://ew.edweek.org/nxtbooks/epe/ew_03292017

http://ew.edweek.org/nxtbooks/epe/ew_03222017

http://ew.edweek.org/nxtbooks/epe/ew_03082017

http://ew.edweek.org/nxtbooks/epe/ew_03012017

http://ew.edweek.org/nxtbooks/epe/ew_02222017

http://ew.edweek.org/nxtbooks/epe/ew_02152017

http://ew.edweek.org/nxtbooks/epe/ew_02082017

http://ew.edweek.org/nxtbooks/epe/ew_01252017

http://ew.edweek.org/nxtbooks/epe/ew_01252017specialreport

http://ew.edweek.org/nxtbooks/epe/ew_01182017

http://ew.edweek.org/nxtbooks/epe/ew_01042017

http://ew.edweek.org/nxtbooks/epe/ew_12142016

http://ew.edweek.org/nxtbooks/epe/ew_12142016v2

http://ew.edweek.org/nxtbooks/epe/ew_11302016

http://ew.edweek.org/nxtbooks/epe/ew_11162016

http://ew.edweek.org/nxtbooks/epe/ew_11092016

http://ew.edweek.org/nxtbooks/epe/ew_11022016

http://ew.edweek.org/nxtbooks/epe/ew_10262016

http://ew.edweek.org/nxtbooks/epe/ew_10192016

http://ew.edweek.org/nxtbooks/epe/ew_10122016

http://ew.edweek.org/nxtbooks/epe/ew_10052016

http://ew.edweek.org/nxtbooks/epe/ew_09282016

http://ew.edweek.org/nxtbooks/epe/ew_09212016

http://ew.edweek.org/nxtbooks/epe/ew_09142016

http://ew.edweek.org/nxtbooks/epe/ew_09072016

http://ew.edweek.org/nxtbooks/epe/ew_08312016

http://ew.edweek.org/nxtbooks/epe/ew_08242016

http://ew.edweek.org/nxtbooks/epe/ew_08032016

http://ew.edweek.org/nxtbooks/epe/ew_08032016

http://ew.edweek.org/nxtbooks/epe/ew_07202016

http://ew.edweek.org/nxtbooks/epe/ew_01112017

http://ew.edweek.org/nxtbooks/epe/ew_06082016

http://ew.edweek.org/nxtbooks/epe/tc_06092016

http://ew.edweek.org/nxtbooks/epe/dc_06022016

http://ew.edweek.org/nxtbooks/epe/ew_06012016

http://ew.edweek.org/nxtbooks/epe/ew_05182016

http://ew.edweek.org/nxtbooks/epe/ew_05112016

http://ew.edweek.org/nxtbooks/epe/ew_sr_05112016

http://ew.edweek.org/nxtbooks/epe/ew_04272016

http://ew.edweek.org/nxtbooks/epe/ew_04202016

http://ew.edweek.org/nxtbooks/epe/ew_04132016

http://ew.edweek.org/nxtbooks/epe/ew_03302016

http://ew.edweek.org/nxtbooks/epe/ew_sr_03302016

http://ew.edweek.org/nxtbooks/epe/ew_03232016

http://ew.edweek.org/nxtbooks/epe/ew_03162016

http://ew.edweek.org/nxtbooks/epe/ew_03092016

http://ew.edweek.org/nxtbooks/epe/ew_02242016

http://ew.edweek.org/nxtbooks/epe/ew_sr_02242016

http://ew.edweek.org/nxtbooks/epe/ew_02172016

http://ew.edweek.org/nxtbooks/epe/ew_02102016

http://ew.edweek.org/nxtbooks/epe/ew_sr_01272016

http://ew.edweek.org/nxtbooks/epe/ew_01272016

http://ew.edweek.org/nxtbooks/epe/ew_01202016

http://ew.edweek.org/nxtbooks/epe/ew_sr_01132016

http://ew.edweek.org/nxtbooks/epe/ew_01132016

http://ew.edweek.org/nxtbooks/epe/qc_01072016

http://ew.edweek.org/nxtbooks/epe/ew_01062016

http://ew.edweek.org/nxtbooks/epe/ew_12092015

http://ew.edweek.org/nxtbooks/epe/ew_12022015

http://ew.edweek.org/nxtbooks/epe/ew_sr_11112015

http://ew.edweek.org/nxtbooks/epe/ew_11112015

http://ew.edweek.org/nxtbooks/epe/ew_11042015

http://ew.edweek.org/nxtbooks/epe/ew_10282015

http://ew.edweek.org/nxtbooks/epe/ew_10212015

http://ew.edweek.org/nxtbooks/epe/ew_sr_10212015

http://ew.edweek.org/nxtbooks/epe/ew_10142015

http://ew.edweek.org/nxtbooks/epe/ew_10072015

http://ew.edweek.org/nxtbooks/epe/ew_09302015

http://ew.edweek.org/nxtbooks/epe/ew_sr_09302015

http://ew.edweek.org/nxtbooks/epe/ew_09232015

http://ew.edweek.org/nxtbooks/epe/ew_09162015

http://ew.edweek.org/nxtbooks/epe/ew_09092015

http://ew.edweek.org/nxtbooks/epe/ew_08262015

http://ew.edweek.org/nxtbooks/epe/ew_08192015

http://ew.edweek.org/nxtbooks/epe/ew_08052015

http://ew.edweek.org/nxtbooks/epe/ew_07082015

http://ew.edweek.org/nxtbooks/epe/ew_06102015

http://ew.edweek.org/nxtbooks/epe/tc_06112015

http://ew.edweek.org/nxtbooks/epe/dc_06042015

http://ew.edweek.org/nxtbooks/epe/ew_06032015

http://ew.edweek.org/nxtbooks/epe/ew_05202015

http://ew.edweek.org/nxtbooks/epe/ew_sr_05132015

http://ew.edweek.org/nxtbooks/epe/ew_05132015

http://ew.edweek.org/nxtbooks/epe/ew_05062015

http://ew.edweek.org/nxtbooks/epe/ew_04222015

http://ew.edweek.org/nxtbooks/epe/ew_sr_04152015

http://ew.edweek.org/nxtbooks/epe/ew_04152015

http://ew.edweek.org/nxtbooks/epe/ew_04012015

http://ew.edweek.org/nxtbooks/epe/ew_03252015

http://ew.edweek.org/nxtbooks/epe/ew_sr_03182015

http://ew.edweek.org/nxtbooks/epe/ew_03182015

http://ew.edweek.org/nxtbooks/epe/ew_03042015

http://ew.edweek.org/nxtbooks/epe/ew_02252015

http://ew.edweek.org/nxtbooks/epe/ew_sr_02252015

http://ew.edweek.org/nxtbooks/epe/ew_sr_02182015

http://ew.edweek.org/nxtbooks/epe/ew_02182015

http://ew.edweek.org/nxtbooks/epe/ew_02042015

http://ew.edweek.org/nxtbooks/epe/ew_01282015

http://ew.edweek.org/nxtbooks/epe/ew_01212015

http://ew.edweek.org/nxtbooks/epe/ew_sr_01212015

http://ew.edweek.org/nxtbooks/epe/ew_01142015

http://ew.edweek.org/nxtbooks/epe/qc_01082015

http://ew.edweek.org/nxtbooks/epe/ew_01072015

http://ew.edweek.org/nxtbooks/epe/ew_12102014

http://ew.edweek.org/nxtbooks/epe/ew_12032014

http://ew.edweek.org/nxtbooks/epe/ew_11122014

http://ew.edweek.org/nxtbooks/epe/ew_sr_11122014

http://ew.edweek.org/nxtbooks/epe/ew_11052014

http://ew.edweek.org/nxtbooks/epe/ew_10292014

http://ew.edweek.org/nxtbooks/epe/ew_10222014

http://ew.edweek.org/nxtbooks/epe/ew_sr_10222014

http://ew.edweek.org/nxtbooks/epe/ew_10152014

http://ew.edweek.org/nxtbooks/epe/ew_10082014

http://ew.edweek.org/nxtbooks/epe/ew_sr_10012014

http://ew.edweek.org/nxtbooks/epe/ew_10012014

http://ew.edweek.org/nxtbooks/epe/ew_09242014

http://ew.edweek.org/nxtbooks/epe/ew_09172014

http://ew.edweek.org/nxtbooks/epe/ew_09102014

http://ew.edweek.org/nxtbooks/epe/ew_08272014

http://ew.edweek.org/nxtbooks/epe/ew_08202014

http://ew.edweek.org/nxtbooks/epe/ew_sr_08202014

http://ew.edweek.org/nxtbooks/epe/ew_08062014

http://ew.edweek.org/nxtbooks/epe/ew_07092014

http://ew.edweek.org/nxtbooks/epe/ew_06112014_v2

http://ew.edweek.org/nxtbooks/epe/ew_06112014

http://ew.edweek.org/nxtbooks/epe/dc_06052014

http://ew.edweek.org/nxtbooks/epe/ew_06042014

http://ew.edweek.org/nxtbooks/epe/ew_05212014

http://ew.edweek.org/nxtbooks/epe/ew_05142014

http://ew.edweek.org/nxtbooks/epe/ew_05072014

http://ew.edweek.org/nxtbooks/epe/ew_sr_04232014

http://ew.edweek.org/nxtbooks/epe/ew_04232014

http://ew.edweek.org/nxtbooks/epe/ew_04162014

http://ew.edweek.org/nxtbooks/epe/ew_04022014

http://ew.edweek.org/nxtbooks/epe/ew_03262014

http://ew.edweek.org/nxtbooks/epe/tc_20140313

http://ew.edweek.org/nxtbooks/epe/ew_03122014

http://ew.edweek.org/nxtbooks/epe/ew_sr_03052014

http://ew.edweek.org/nxtbooks/epe/ew_03052014

http://ew.edweek.org/nxtbooks/epe/ew_02262014

http://ew.edweek.org/nxtbooks/epe/ew_sr_02192014

http://ew.edweek.org/nxtbooks/epe/ew_02192014

http://ew.edweek.org/nxtbooks/epe/ew_01292014_v2

http://ew.edweek.org/nxtbooks/epe/ew_02052014

http://ew.edweek.org/nxtbooks/epe/ew_01292014

http://ew.edweek.org/nxtbooks/epe/ew_01222014

http://ew.edweek.org/nxtbooks/epe/ew_01152014

http://ew.edweek.org/nxtbooks/epe/qc_01092014

http://ew.edweek.org/nxtbooks/epe/ew_01082014

http://ew.edweek.org/nxtbooks/epe/ew_12112013

http://ew.edweek.org/nxtbooks/epe/ew_12042013

http://ew.edweek.org/nxtbooks/epe/ew_11132013

http://ew.edweek.org/nxtbooks/epe/ew_11062013

http://ew.edweek.org/nxtbooks/epe/ew_sr_10302013

http://ew.edweek.org/nxtbooks/epe/ew_10302013

http://ew.edweek.org/nxtbooks/epe/ew_10232013

http://ew.edweek.org/nxtbooks/epe/ew_10162013

http://ew.edweek.org/nxtbooks/epe/ew_10092013

http://ew.edweek.org/nxtbooks/epe/ew_sr_10022013

http://ew.edweek.org/nxtbooks/epe/ew_10022013

http://ew.edweek.org/nxtbooks/epe/ew_09252013

http://ew.edweek.org/nxtbooks/epe/ew_09182013

http://ew.edweek.org/nxtbooks/epe/ew_09112013

http://ew.edweek.org/nxtbooks/epe/ew_08282013

http://ew.edweek.org/nxtbooks/epe/ew_sr_08212013

http://ew.edweek.org/nxtbooks/epe/ew_08212013

http://ew.edweek.org/nxtbooks/epe/ew_08072013

http://ew.edweek.org/nxtbooks/epe/ew_07102013

http://ew.edweek.org/nxtbooks/epe/ew_06122013

http://ew.edweek.org/nxtbooks/epe/dc_06062013

http://ew.edweek.org/nxtbooks/epe/ew_06052013

http://ew.edweek.org/nxtbooks/epe/ew_sr_05222013

http://ew.edweek.org/nxtbooks/epe/ew_05222013

http://ew.edweek.org/nxtbooks/epe/ew_05152013

http://ew.edweek.org/nxtbooks/epe/ew_05082013

http://ew.edweek.org/nxtbooks/epe/ew_sr_04242013

http://ew.edweek.org/nxtbooks/epe/ew_04242013

http://ew.edweek.org/nxtbooks/epe/ew_04172013

http://ew.edweek.org/nxtbooks/epe/ew_04032013

http://ew.edweek.org/nxtbooks/epe/ew_03272013

http://ew.edweek.org/nxtbooks/epe/ew_03132013

http://ew.edweek.org/nxtbooks/epe/tc_20130314

http://ew.edweek.org/nxtbooks/epe/ew_03062013

http://ew.edweek.org/nxtbooks/epe/ew_02272013

http://ew.edweek.org/nxtbooks/epe/ew_sr_02202013

http://ew.edweek.org/nxtbooks/epe/ew_02202013

http://ew.edweek.org/nxtbooks/epe/ew_sr_02062013

http://ew.edweek.org/nxtbooks/epe/ew_02062013

http://ew.edweek.org/nxtbooks/epe/ew_01302013

http://ew.edweek.org/nxtbooks/epe/ew_01232013

http://ew.edweek.org/nxtbooks/epe/ew_01162013

http://ew.edweek.org/nxtbooks/epe/qc_01102013

http://ew.edweek.org/nxtbooks/epe/ew_01092013

http://ew.edweek.org/nxtbooks/epe/ew_12122012

http://ew.edweek.org/nxtbooks/epe/ew_12052012

http://ew.edweek.org/nxtbooks/epe/ew_sr_11142012

http://ew.edweek.org/nxtbooks/epe/ew_11142012

http://ew.edweek.org/nxtbooks/epe/ew_11072012

http://ew.edweek.org/nxtbooks/epe/ew_10312012

http://ew.edweek.org/nxtbooks/epe/ew_sr_10242012

http://ew.edweek.org/nxtbooks/epe/ew_10242012

http://ew.edweek.org/nxtbooks/epe/ew_10172012

http://ew.edweek.org/nxtbooks/epe/ew_10102012

http://ew.edweek.org/nxtbooks/epe/ew_10032012

http://ew.edweek.org/nxtbooks/epe/ew_09262012

http://ew.edweek.org/nxtbooks/epe/ew_09192012

http://ew.edweek.org/nxtbooks/epe/ew_09122012

http://ew.edweek.org/nxtbooks/epe/ew_08292012

http://ew.edweek.org/nxtbooks/epe/ew_sr_08222012

http://ew.edweek.org/nxtbooks/epe/ew_08222012

http://ew.edweek.org/nxtbooks/epe/ew_20120829

http://ew.edweek.org/nxtbooks/epe/ew_sr_08292012

http://ew.edweek.org/nxtbooks/epe/ew_20120822_v2

http://ew.edweek.org/nxtbooks/epe/ew_20120822

http://ew.edweek.org/nxtbooks/epe/ew_test

http://ew.edweek.org/nxtbooks/epe/diplomascount_2012issue34

https://www.nxtbookmedia.com