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<ul><li><p>IMPLEMENTING THE "PROFESSIONAL STANDARDS FOR TEACHING MATHEMATICS": TheInfluence of Teachers' Beliefs and Knowledge on Learning EnvironmentsAuthor(s): Cheryl Ann Lubinski and Nancy Nesbitt VaccSource: The Arithmetic Teacher, Vol. 41, No. 8 (APRIL 1994), pp. 476-479Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41196057 .Accessed: 12/06/2014 17:01</p><p>Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp</p><p> .JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.</p><p> .</p><p>National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.</p><p>http://www.jstor.org </p><p>This content downloaded from 188.72.126.88 on Thu, 12 Jun 2014 17:01:56 PMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/action/showPublisher?publisherCode=nctmhttp://www.jstor.org/stable/41196057?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>IMPLEMENTING THE PROFESSIONAL STANDARDS FOR TEACHING MATHEMATICS </p><p>The Influence of Teachers' Beliefs and Knowledge on Learning Environments </p><p>was sitting in his second-grade classroom on the third day of school. </p><p>He'd just finished writing on his paper after his teacher, Ms. Kates, had given the class a problem to solve. When asked what he wrote, he held up his paper and dis- played the following: </p><p>218 + 153 </p><p>371 </p><p>He explained, "Two hundred plus one hun- dred is three hundred. One plus five is six. Eight plus three is eleven. That's one 'ten' and one left over, so you have to add one more to the six 'tens' and you get seven 'tens' and one left over." One could see where Seth had erased a six and written a seven in the tens column. He did not have the traditional marks above the tens col- umn that children are often taught to use. What caused him to think as he did? Is this representation acceptable to the teacher? How will she respond to his explanation? What will she plan to have him do next? All these questions depend on the type of learn- ing environment that is being established in this classroom. </p><p>Prepared by Cheryl Ann Lubinski Illinois State University Normal, IL 61761 Edited by Nancy Nesbitt Vacc University of North Carolina Greensboro, NC 27412 </p><p>Cheryl Lubinski teaches at Illinois State University in Normal, Illinois. Her research interests include teachers ' instructional decision making and math- ematics. </p><p>The Editorial Panel welcomes readers' responses to this article or to any aspect of the Professional Standards for Teaching Mathematics for consider- ation for publication as an article or as a letter in "Readers' Dialogue." </p><p>If we can assume that teachers' instruc- tional practices, in part, create the learning environment, then it is important to know what factors influence a teacher's instruc- tional decisions that are reflected in these practices. Research indicates that teachers' beliefs and teachers' knowledge are related to the instructional decision-making pro- cess (Fennema and Franke 1992; Pajares 1992; Thompson 1992). Thus, what a teacher believes about teaching and learn- ing mathematics and what a teacher knows about the content, methods, and materials available to teach mathematics influence the teacher's instructional decisions. </p><p>Both the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) and the Professional Stan- dards for Teaching Mathematics (NCTM 1991) foster knowledge about teaching </p><p>and learning. These documents offer a vi- sion of what content, methods, and materi- als could be used to create a learning envi- ronment that reflects needed changes in mathematics education. Moreover, the pro- fessional standards document "spells out what teachers need to know to teach toward new goals for mathematics education" (NCTM 1991, vii). Further, this document focuses on learning environments that fos- ter problem solving with understanding. </p><p>The Learning Environment In Ms. Kates' s second-grade classroom on the third day of school, the mathematics learning environment was being estab- lished. The students' desks were arranged </p><p>476 ARITHMETIC TEACHER </p><p>This content downloaded from 188.72.126.88 on Thu, 12 Jun 2014 17:01:56 PMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>in groups of four or five. She prepared the students for mathematics by asking them, "What is your listening position?" They were reminded that they were to face the person who was talking. This position is important to Ms. Kates because she be- lieves that listening to each other's expla- nations about how to solve a problem is important. </p><p>Ms. Kates began the lesson by posing the following problem: </p><p>At the movieJurassic Park we sat in two rows. Out of curiosity we decided to count the unpopped kernels left in the popcorn boxes. The people in row 1 had 153 kernels of popcorn. The people in row 2 had 218 kernels of popcorn left. How much unpopped popcorn did the people in both rows have left? </p><p>The answer posed a real problem for all the students because no student seemed to have a ready solution. They all began to work on solving the problem. They had work mats on which to represent their solutions. Some students left their desks to get base-ten blocks, which were one of many materials available for their use. </p><p>As they were working, Ms. Kates made the comment, "If you're finished, try it another way." Ms. Kates believes that it is important for students to be able to solve problems in a variety of ways. This phi- losophy was evident in the students' work. One student wrote 153+ 218= 368+ 3 = 371 and explained, "I added 400' and '200.' Then I added '50' and 48.' Then I added '3.' " Another student represented the situation as 153 + 218 = 360 + 11 = 371 and said, "I added 400' and '200.' Then I added '50' and '10.' Then I added '8' and '3.' "A student who was using the base-ten blocks first put the hundreds to- gether, then the tens, and then the ones. She noticed that she had ten Is and one 1 left over, so she said, "Seventy is ten more than '60,' so we have one left over, '371.' " It was obvious by listening to the teacher and the children that alternative representations were encouraged and accepted. </p><p>As Ms. Kates went around the room and asked individual children to tell her about their solution strategies, she noticed that some students were finished with the prob- lem. She had to make a decision about what they should do next. Some of the more mature thinkers were asked, "If '37 ker- nels of popcorn were left, how many more would they need to get to '500' kernels of </p><p>popcorn?" This situation kept the more capable students busy until Ms. Kates felt that all the students had found a solution to the original problem. </p><p>After the children had time to work the initial problem individually, Ms. Kates asked some of them to come to the front of the room and explain their thinking to the class. She specifically chose children with different methods of solving the problem so that the class would hear a variety of solution strategies. When Adam explained that he added 100 and 200 to get 300, then 300 and 53 to get 353, and then 353 and 18 to get 37 1 , Ms. Kates asked, "Why did you take off the '53' and the '18?' " and "How did you add '353' and '18?' " The answers to these questions were important to Ms. Kates because she believes that understand- ing students' thinking helps her to plan better for instruction. </p><p>We can learn by listening to </p><p>students. </p><p>The students spent the entire mathemat- ics period solving the problem and explain- ing their strategies. Second graders are not usually doing three-digit-addition problems with regrouping during the first week of school. However, it was evident that the majority of students in this second-grade class could solve such problems with un- derstanding. </p><p>Teacher Beliefs A teacher' s beliefs about students' abilities greatly influence the decisions the teacher makes about the learning environment. If teachers believe that children should solve a variety of problems at an early age, they will make different decisions than will teachers who believe that children should know basic facts before solving word prob- lems. Teachers who believe that the con- tent of the mathematics in their classroom is guided by the textbook make different decisions than do teachers who believe that the content of the mathematics is guided by children's interests and abilities. </p><p>Should fractions be discussed in kinder- garten? Should multiplication and division situations be presented to five-and six-year olds? Should a teacher model a method to solve a problem or allow students to solve a problem in any way they choose? Does a teacher perceive her or his role to be to develop children's thinking or to tell chil- dren how to think? The answers to these questions are related to teachers' beliefs. </p><p>Ms. Kates could have used two-digit numbers or numbers that when added did not involve regrouping. This decision would have more closely represented the content of the textbook. However, she believes that children need challenging problems and that given sufficient time, most of her stu- dents can solve the problems she poses. </p><p>Ms. Kates does not believe that using manipulatives assures students' understand- ing. She believes that the choice of a ma- nipulative is a decision that the student needs to make. Her beliefs about content and materials have been influenced, in part, by her knowledge of children's thinking. This knowledge influences both her plan- ning decisions and her teaching method. </p><p>Knowledge of Children's Thinking Many of the vignettes presented in the professional teaching standards give a glimpse of ways in which teachers commu- nicate with children about their thinking to develop their students' thought processes further. Having knowledge of children's thought processes allows a teacher to plan instruction on the basis ofthat knowledge. </p><p>Knowledge of children's thinking can be acquired by listening to students' expla- nations and observing what they do. Did they need to use manipulatives to solve and explain the problem? Did they represent their thinking using paper and pencil? Did they draw pictures or write number sen- tences? The answers to these questions give teachers useful information on stu- dents' thinking. </p><p>For example, some of Ms. Kates' s stu- dents used base-ten blocks to solve the popcorn problem. Others wrote number sentences and thought about grouping tens and ones, which indicates a more mature thought process. For some students the numbers were too large. As a teacher ob- serves and questions children, the instruc- tion can be adjusted to fit their level of </p><p>APRIL 1 994 477 </p><p>This content downloaded from 188.72.126.88 on Thu, 12 Jun 2014 17:01:56 PMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>thinking. This idea of ongoing assessment to modify instruction while teaching is important. </p><p>In Ms. Kates' s class, Elizabeth sat and looked at the numbers 218 and 153. The teacher began questioning her about what she was going to do. Knowing that Eliza- beth has difficulty with large numbers, the teacher modified the problem by selecting two-digit numbers for Elizabeth so that she was successful at solving a similar prob- lem. After Elizabeth had explained her thinking to the teacher about the modified problem, the teacher posed the original problem to her again. This time Elizabeth appeared more eager to solve the problem. However, she was still not successful at finding a solution by the end of the class period. This knowledge allows Ms. Kates to prepare modified problems for students similar to Elizabeth on the basis of their thinking. </p><p>Although knowledge of students' think- ing is important, teachers' knowledge of mathematics content and pedagogy is also critical to the culture of the learning envi- ronment. Knowledge of content and peda- gogy in conjunction with knowledge of students' thinking allows a teacher to plan worthwhile mathematical tasks. </p><p>Teachers7 Knowledge and Worthwhile Mathematical Tasks Greater knowledge of the content and pedagogy allows a teacher more flexibility in decision making. Teachers who have knowledge of various problem types and the different strategies children naturally use to solve each of these types have the ability to plan instruction on the basis of students' thinking. For example, consider the following: </p><p>a) We have 24 cookies and 6 plates. If you put the same number of cookies on each plate, how many cookies are on each plate? </p><p>b) We have 24 cookies and each child gets 6. How many children get cookies? </p><p>Each situation can be symbolically repre- sented as 24 -- 6. However, each situation represents a different type of division prob- lem. The teacher who knows that children generate different solution strategies when solving partitive-division problems - </p><p>knowing the number of groups, as in prob- </p><p>Our beliefs about mathematics affect </p><p>the problems we select. </p><p>lem (a) - and measurement-division prob- lems - knowing how many are in each group as in problem (b) - will be able to adjust instruction accordingly if children are having difficulty solving these prob- lems. Further, a teacher who knows the strategies children will naturally use to solve each of these problem types can plan instruction that will develop students' think- ing rather than tell the students how to think. </p><p>When some of the students in Ms. Kates' s room finished the popcorn problem quickly and were able to explain their thinking, they were given a more difficult problem. </p><p>Knowing why this second problem was more difficult is based on knowledge of the content and how students do think about solving problems involving a missing ad- dend. </p><p>How do teachers obtain access to knowl- edge about content and pedagogy? How do they learn about students' thinking? How can appropriate manipulatives be made available to students to assist them in solv- ing problems? Answers to these questions, in part, can be found in both the curriculum and evaluation standards and the profes- sional teaching standards. However, much of this knowledge can be obtained by lis- tening and watching students solve prob- lems. As teachers acquire knowledge in- volving mathematics, they are better able to plan worthwhile mathematical tasks. </p><p>What makes a task worthwhile? Stan- dard 1 in the Professional Standards for Teaching Mathematics (NCTM 1991) lists several points that teachers should con- sider when planning tas...</p></li></ul>