Introduction

cooperative learning structure. A processfollowed by a group to promote learning amonggroup members. Examples of cooperative learningstructures include the following:– numbered heads. Students work in groups offour, with each student assigned a numberfrom 1 to 4. Students work together in theirgroups to solve a problem. The teacher thencalls out a number from 1 to 4 and asks groupmembers with that number to explain howtheir group solved the problem.– pairs check. Students work in pairs. One studentsolves a problem while the other studentobserves and coaches; then the studentsswitch roles.– think-pair-share. The teacher poses a problem.Students think about their response individuallyfor a given amount of time and then sharetheir ideas with a partner in attempting toreach a solution to the problem.counting. The process of matching a number inan ordered sequence with every element of a set.The last number assigned is the cardinal numberof the set.counting all. A strategy for addition in whichthe student counts every item in two or more setsto find the total. See also counting on.counting back. Counting from a larger to asmaller number. The first number counted is thetotal number in the set (cardinal number), andeach subsequent number is less than that quantity.If a student counts back by 1’s from 10 to 1, thesequence of numbers is 10, 9, 8, 7, 6, 5, 4, 3, 2, 1.Young students often use counting back as a strategyfor subtraction (e.g., to find 22 – 4, the studentcounts, “21, 20, 19, 18”).counting on. A strategy for addition in whichthe student starts with the number of the knownquantity, and then continues counting theitems in the unknown quantity. To be efficient,students should count on from the larger addend.For example, to find 2+7, they should beginwith 7 and then count “8” and “9”.decomposition of numbers. The taking apartof numbers. For example, the number 13 is usuallytaken apart as 10 and 3 but can be taken apart as6 and 7, or 6 and 6 and 1, and so forth. Studentswho can decompose numbers in many differentways develop computational fluency and havemany strategies available for solving arithmeticquestions mentally. See also composition ofnumbers and recomposition of numbers.denominator. In common fractions, the numberwritten below the line. It represents the number ofequal parts into which a whole or a set is divided.derived fact. A basic fact to which the studentfinds the answer by using a known fact. Forexample, a student who does not know theanswer to 6 x 7 might know that 3 x 7 is 21,and will then double 21 to get 42.design down. See backwards design.developmental level. The degree to whichphysical, intellectual, emotional, social, andmoral maturation has occurred. Instructionalmaterial that is beyond a student’s developmentallevel is difficult to comprehend, and might belearned by rote, without understanding. Contentthat is below the student’s level of developmentoften fails to stimulate interest. See also zone ofproximal development.developmentally appropriate. Suitable toa student’s level of maturation and cognitivedevelopment. Students need to encounter conceptsthat are presented at an appropriate time in theirdevelopment and with a developmentally appropriateapproach. The mathematics should bechallenging but presented in a manner that makesit attainable for students at a given age and levelof ability. See also zone of proximal development.Glossary 91