2013 Conference Proceedings - University of Nevada, Las Vegas

internal evaluator) were analyzed. We also administered a survey of the nature of mathematicsand mathematics teaching and learning to the teachers. The adaptation of the LMT consisted ofchoosing only those items that pertained to upper elementary mathematics topics. Topics of ratioand proportion, geometry and reasoning, and fundamental concepts of algebra were included.The test had 11 separate items, a few with 3 or 4 parts, and all items were multiple choice. Themodified LMT given toward the beginning of the PD work and at the end of year one had similaritems, however, because of these modifications the scores were used to identify where at leasthalf of the faculty involved had misconceptions. The analysis of the rich problems also helped tocharacterize the knowledge that these mathematics teachers displayed in PD activities.ResultsOne of the problems, as is a trait of the LMT, had teachers analyze an alternate method fordividing two fractions. The student said that he divided the two numerators and the twodenominators in68 ¸ 12=64. The teachers were asked to choose one of the following about whatthis student’s teacher was thinking about the students method (number of teachers selecting ananswer): a) He knew that the method was wrong, even though he happened to get the rightanswer for this problem (3); b) He knew that the student’s answer was actually wrong (0); c) Heknew that the student’s answer was right, but that for many numbers this would produce a messyanswer (4); d) He knew that the student’s method only works for some fractions (6). The correctanswer was selected by only 4 of the 13 teachers. Answers a and d were selected by 9 of thesehigh school mathematics teachers. These two selections imply that the teachers themselves knewthat the student had gotten the right answer and that either they didn’t or couldn’t determine ifthe work presented was equivalent to the standard algorithm for dividing two fractions. Basedon other mathematics problems similar to this question it was clear that these teachers were notused to thinking about alternative methods for solving problems and why various algorithms thatthey used regularly worked or how they were derived. On an exit interview with the internalevaluator near the end of year 2 all of the teachers except one indicated that they believed theirknowledge of mathematics and of the common core state standards had increased. When askedwhy they believe this to be true they referred to the rich mathematical tasks that they solved andhad to demonstrate their reasoning for their solutions. They also indicated that they were moreconfident in being able to implement the common core state standards as a result of theirparticipation in the grant activities as was also supported by our observations and analysis of theProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 120