24 SES 12, Cognitive Development and Mathematics Learning
Algebra has a great significance for students’ future career and high school achievement (Adelman, 2006; Knuth, Stephens, McNeil, & Alibali, 2006). Even though its importance, many of the studies indicate the students’ difficulties and misconceptions on this topic (e.g., Dede & Peker; 2007; Lucariello, Tine, & Ganley, 2014). Some of the scholars suggest that using tasks can be effective to support students’ algebraic thinking and learning (e.g. Lannin, 2005; Yılmaz, 2015).
Based on the definition of Steele (2005), Driscoll (1999) and Kieran (2004) algebraic thinking can be defined as the ability to make generalizations and to understand variables, expressions and meaning of equal sign. In the literature, various types of tasks were used to improve students’ learning and thinking in algebra. Some of the researchers (e.g. Saraswati & Putri, 2016) claimed that using hands on materials such as algebra tiles might be beneficial to reinforce students’ algebraic thinking while some of them (e.g. Walkington, Petrosino & Sherman, 2013) noted that the real life contexts are important to contribute their algebraic thinking.
Furthermore, Leatham and his colleagues (Leatham, Peterson, Stokero & Zoest, 2015) claimed that mathematical tasks can be designed in a way to create instances in a lesson so that teachers have chance to build on students’ mathematics as well as assessing their mathematical understanding. They noted that we need to determine which opportunities in the classroom are more appropriate to elicit students’ thinking and address their misunderstanding. The researchers called the possible learning opportunities as Mathematically Significant Pedagogical Opportunities to Build on Student Thinking (MOST). They described MOST as having sequential and interrelated three components such that it should depend on students’ mathematics, be mathematically significant, and be a pedagogical opportunity.
In this study, we designed mathematical tasks to support students’ algebraic thinking according to MOST Framework and the following criteria: Tasks should 1) encourage students to think about given situation and follow problem solving procedures, 2) allow using manipulatives and hands on aids, 3) lead communication among students (collaborative work), 4) involve in real life contexts, and 5) have potential to provoke students’ misconceptions. The tasks were about finding general rule of given patterns, writing algebraic expressions and setting up and solving equations. In this paper we will specifically discuss the findings about how students’ performance changed during task-assisted instruction as they were trying to find the general term of given patterns in the tasks.
Participants The participants were 28 seventh grade students from the same classroom of a public middle school located in a suburban area in Istanbul which has a school-faculty collaboration with the university. The students mostly come from lower socioeconomic status families. In this study, the tasks were administered by 8 senior pre-service teachers from mathematics education department of the collaboration university. All of the pre-service teachers have similar backgrounds since they took similar pedagogy and mathematics courses during their previous years. Research setting Before the study, researchers discussed the possible MOST instances with the pre-service teachers. During the discussions with pre-service teachers, researchers also suggested the possible interventions and questions to improve students’ algebraic thinking. Furthermore, the researchers shared the instructions for each task to be sure that each of the pre-service teacher apply the tasks in the same way. In the following 7 weeks 8 tasks were applied and each pre-service teacher was responsible from 4 or 3 students. Each task was implemented during 80 minutes. In the implementation process, the students firstly allowed to work individually. After the individual work, the pre-service teachers created a discussion with students according to the instructions. Data collection A 5-item achievement tests to measure students’ algebraic thinking was developed as parallel form tests and applied before and after task implementation process. Task implementation process was videotaped and students’ worksheets were collected. The pre-service teachers were also asked to reflect on each task implementation and this reflection sessions were also videotaped. Data analysis The paired sample t-test was done to compare the students’ pre and post achievement test results. For the qualitative part of the data, the videos from Task 1, Task 2 and Task 8 (the tasks related to finding the rule and the terms of the pattern) were transcribed. In the transcriptions, the verbal expressions of the students were detected to understand how students’ algebraic thinking has changed. In addition, the percentages of the correct answers of the students from their individual work among these tasks were compared to track the improvement of the students’ algebraic thinking.
The results of paired sample t-test showed that there was a significant difference between the pre and post achievement test (p=0.007). One of the items in the test was about finding general term of given pattern. The item was out of 5 points such that the mean score of the item was 2.11 in the pre-test and 3.00 in the post-test. The students’ papers from their individual work were analyzed to understand how students’ algebraic thinking changed throughout the implementation process. The students’ individual work supported the findings from achievement test since the percentage of correct answers in the tasks increased. In Task 1 and Task 2, there were three questions with increasing difficulty and in Task 8 there was one question related to finding the rule and the terms of the pattern. The percentages of their correct answers were 10.71%, 10.71%, 17.85% in the first task and 29.62%, 22.22%, 18.51% in the second task respectively. In Task 8, 28% of the students gave the correct answer their individual work. One of the possible reasons of the decrease in percentages from the second task was the complexity of the questions since the patterns were getting harder from the first question to the last question. The analysis of students’ verbal explanations support changes in students’ algebraic thinking throughout the implementation. For instance, one of the students in Task 8 stated that “We need to write the amount of increase as the coefficient term and the constant square in the pattern as the additional term.” Even though the student was not successful in some of the previous tasks, he understood the rule of pattern conceptually as a result of task-assisted instruction. Acknowledgement This research was supported by The Scientific and Technological Research Council of Turkey (TUBITAK, Grant no: 215K049)
Adelman, C. (2006). The toolbox revisited: Paths to degree completion from high school through college. Washington, DC: US Department of Education. Dede. Y., & Peker. M. (2007). Students’ errors and misunderstanding towards algebra: Pre-Service mathematics teachers’ prediction skills of error and misunderstanding and solution suggestions. Elementary Education Online, 6(1), 35-49. Driscoll, M. (1999). Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10. Heinemann: Portsmounth. Kieran, C. (2004). Algebraic thinking in the early grades: What is it?. The Mathematics Educator, 8(1), 139-151. Knuth, E. J., Stephens, A. C., McNeil, N. M. & Alibali, M. W. (2006). Does understanding equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297-312. Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231-258. Leatham, K. R., Peterson, B. E., Stokero, S. L., & Zoest, L. R. (2015). Conceptualizing Mathematically Significant Pedagogical Opportunities to Build on Student Thinking. Journal for Research in Mathematics Education, 46(1), 88-124. Lucariello, J., Tine, M. T. & Ganley, M. C. (2014). A formative assessment of students’ algebraic misconceptions. Journal of Mathematical Behaviour, 33(1), 30-41. Putri, I. R., & Saraswati, S. (2016). Supporting students’ understanding of linear equations with one variable using algebraic tiles. Journal on Mathematics Education, 7(1), 19-30. Steele, D. F. (2005). Using schemas to develop algebraic thinking. Mathematics Teaching in Middle School, 11(1), 40-46. Yılmaz, N. (2015). Cebir öğretiminde yazma etkinliklerini kullanmanın ortaokul 7. Sınıf öğrencilerinin başarılarına etkisi. Abant İzzet Baysal Üniversitesi Eğitim Fakültesi Dergisi, 15(1), 357-376. Walkington, C., Petrosino, A., & Sherman, M. (2013). Supporting algebraic reasoning through personalized story scenarios: How situational understanding mediates performance, Mathematical Thinking and Learning, 15(2), 89-120.
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