Teaching Mathematics
How Do Manipulatives Help Students Communicate Their Understanding of Double-Digit Subtraction?
Paper Presentation in a Themed Session Rabab Abi-Hanna, Eileen Fernández
Multi-digit subtraction is difficult for students to learn. This study explores how second-grade students communicate their understanding of double-digit subtraction through the use of manipulatives. Using clinical interview and a variety of manipulatives, we created a venue to help us elicit student’s understanding of double-digit subtraction. Through qualitative analysis we were able to identify differences in students’ understanding that were not apparent from the typical assessments administered in the classroom. Findings suggest that manipulatives helped reveal cognitive constructs and difficulties that the handwritten algorithms were not conveying. For example, students who exhibited an understanding of the subtraction process had not yet developed an understanding of ten and 10 ones interchangeability. These results highlight the role of manipulatives as communication tools that help reveal students’ actual cognitive development. This suggests another approach to teaching: questioning from the perspective of just understanding what students are thinking and not teaching. Recognizing learning differences can come from creating a space to allow students to articulate their understanding. We offer suggestions to assist teachers in recognizing learner differences and use them as a productive resource in lesson planning.
Students’ Mathematics Misconceptions: Detection, Deconstruction and Correction
Paper Presentation in a Themed Session Nahid Golafshani
Misconceptions and errors in mathematical thinking are often confused. Although they are generally similar in that they make students achieve incorrect solutions, they are actually very different notions. An error can range from being a miscalculation to an incorrect misunderstanding of the problem being asked. On the other hand, a misconception occurs when a mathematical premise or rule is incorrectly generalized. An example of a misconception can be found when students write 0.10 is greater than 0.9. The root of the misconception is related to their prior correct knowledge of 10 is greater than 9. The symbolic decimal does not make sense to some students for different reasons. The significant of this study is to help teachers truly understand student’s thinking in solving problems related to the topics being addressed. As a result, they will be better able to make the mathematics make sense to their students. Understanding students thinking will enable the teachers to understand what level of mathematics the students have already mastered and where to go next with specific concepts. This is increasingly necessary as many of our mathematics teachers in primary and junior grades have little training in mathematics. The data gathered for this study are from the students’ math test results and scratch paper in primary and junior levels. The solution and the discerned steps to arrive at the solution to each test item on the students’ tests are examined to detect possible misunderstandings or misconceptions may take place. The information gives some background knowledge on common stumbling blocks for students which lead to a greater understanding of how teachers can assist students appropriately. Using some of the examples of misconceptions from the students’ work, detection, deconstruction and correction techniques of the misconception are suggested and discussed in this study. The findings suggest that misconceptions are not procedural errors. But they are resided in students’ conceptions and are believed to be correct. The information provided in this study could be of value to the teachers and educators of mathematics programs. However, this study should be extended to include a large population and many more grade levels to determine the common roots and characteristics of different misconceptions in all the strands of mathematics.
Promoting Student Discourse in the Mathematics Classroom to Enhance Relational Understanding
Paper Presentation in a Themed Session Robert Francis Cunningham
An active learning strategy employed a sequence of PowerPoint tableaus to generate student discourse in three sections of linear algebra (n=63). Each tableau presented a conceptual question followed by two sample responses with the possibility of both being correct, only one of them correct, or both incorrect. The questions were suggested by research and focused on common student misconceptions. Students were polled individually on which of the responses they thought were correct, if any, and a tally of responses was recorded. Groups of three were then formed to discuss their reasoning among peers and a final poll was then tallied. Results indicate that after peer discussions many correctly revised their answers and for most tableaus the majority of the students selected the correct answer on the final poll. Surveys conducted at the end of the semester revealed that most (90%) agreed or strongly agreed that the tableaus and associated discussions had a positive impact on their understanding and (65%) supported its continued use in the course. Open ended questions on the survey suggest that the perception of the strategy held by high scoring students differed from those held by low scoring students and suggestions for improved implementation of the active learning strategy were offered. However, more research is warranted given that the strategy promoted an unusually high level of student engagement and discourse with the potential to improve relational understanding of challenging mathematical topics.
