The Effect of Experiential Learning on Students’ Conceptual Understanding of Functions in Algebra 1

conceptualization. Closely following the reflective phase is the abstract conceptualization phase. This portion of the EL cycle was a time when students began to develop EFFECT OF EL ON CONCEPTUAL UNDERSTANDING IN ALGEBRA 1 29 theories that supported what they have observed. This phase of the cycle started in this study when the teacher called on student groups to present their observations from the previous phase. As students presented, the teacher used that time to clarify any misconceptions that may have surfaced. One of the core principles of social constructivism is the importance of individuals to create or construct knowledge together as they collaborate with each other and their environment (Kim, 2001; Powell & Kalina, 2009). During the abstract conceptualization phase, students collaborated and discussed with their partners about their observation during the experiment. Additionally, the students also discussed with the teacher in a whole group discussion concerning their observations. Additionally, NCTM (2014) recommends that teachers “facilitate meaningful mathematical discourse” (p. 10) to build a “shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments” (p. 10). This constructivist strategy from NCTM was used by the teacher to build understanding and help students move on to the next phase. As students moved through RO, they were developing ideas and theories concerning what they had experienced or observed. The presentations and discourse that occurred at the end of each lesson were to solidify or correct any conceptions or misconceptions that had happened. Active experimentation. This phase of the cycle prompted students to apply their newly developed ideas and theories to new problems or experiences. In this study, students applied their newly developed concepts and theories to data collected by other groups during the concrete experience phase. Active experimentation occurred in the linear task as students applied what they had learned to develop a model that allowed a successful bungee jump off the atrium balcony. In the exponential task, active experimentation occurred when students determined EFFECT OF EL ON CONCEPTUAL UNDERSTANDING IN ALGEBRA 1 30 whose cup was the best by using what they had learned about their coffee mug and then critiquing the work of other peer groups. Traditional mathematics instruction. The TMI treatment procedures and processes were based on what Boaler (2000), Chapko and Buchko (2004), and O’Callaghan (1998) have described as traditional algebra classrooms which emphasize teacher lecture, rote memorization, review of processes, and procedural practice with additional problems. Participants received direct instruction from the teacher at which time students took notes during lecture, reviewed the process presented to them, and in turn used these processes to practice additional problems. Instruments and Measures Since this study was a quasi-experimental 2X2 (2 groups X 2 treatments) repeated measures design, two posttests were administered to all participants in addition to the pretest. EFFECT OF EL ON CONCEPTUAL UNDERSTANDING IN ALGEBRA 1 31 Group Pretest Phase I Treatment Posttest Phase II Treatment Posttest Conceptual Understanding (Exponential & Linear) Experimental Experiential Learning Theory(EL) Bungee Bob’s Conceptual Understanding (Linear) Control Traditional (TMI) Cool Coffee Conceptual Understanding (Exponential) A (2 classes) Modeling Exp. Interpreting. Exp. Translating. Exp. Total Exp. Modeling Lin. Interpreting. Lin. Translating. Lin. Total Lin. Modeling Exp. Interpreting. Exp. Translating. Exp. Total Exp. Modeling Lin. Interpreting. Lin. Translating. Lin. Total Lin. B (2 classes) Modeling Exp. Interpreting. Exp. Translating. Exp. Total Exp. Control Traditional (TMI) Barbie Bungee Modeling Lin. Interpreting. Lin. Translating. Lin. Total Lin. Experimental Experiential Learning Theory (EL) Thermal Throwdown Modeling Exp. Interpreting. Exp. Translating. Exp. Total Exp. Modeling Lin. Interpreting. Lin. Translating. Lin. Total Lin. Figure 5. Summary of the pretest, treatments, and related posttests. Conceptual understanding. The researcher constructed two instruments for assessing participants' conceptual understanding in mathematics. One instrument assessed conceptual understanding of linear functions (Appendix C) and one assessed conceptual understanding of exponential functions (Appendix D). Since this was a quasi-experimental study without random assignment to treatment groups, the two instruments were used together as a pretest to establish that the two groups are equated. Each instrument was then used as a post-test during the appropriate treatment phase (Phase I, Linear and Phase II, Exponential). Both instruments were EFFECT OF EL ON CONCEPTUAL UNDERSTANDING IN ALGEBRA 1 32 constructed from items taken from the State of Texas Assessments of Academic Readiness (STAAR), as well as items from O’Callaghan’s (1998) pre and post assessments. For both the pre and post assessments, linear and exponential questions were chosen which reflect three aspects of conceptual understanding (modeling, evaluating, and translating) in mathematics. When designing the instruments for this study, the decision to use a majority of multiplechoice items was based on the research conducted with year one medical students and the use of distractors by Ali, Carr, and Ruit (2016). In their article, Ali et al. (2016) showed that multiplechoice distractors enhanced the validity and reliability of scores obtained from multiple choice instruments. Herman-Abell and DeBoer (2014), through the use of multiple-choice instruments, assessed student’s conceptual understanding of the fundamental ideas of energy. They found that using “distractor-driven assessment items” (p. 21) provided insight into student understanding and misconceptions. Although these studies were not conducted in mathematics classrooms, they do however show that the use of multiple-choice instruments can assess student conceptual understanding. There is considerable research supporting the claim of multiple-choice items can measure the same thing as constructed response items (Walner & Thissen, 1992; Walstad & Becker, 1994; Bridgeman, 1991; Bennett et al., 1989; Becker & Johnston, 1999), there is also research that disputes these claims (Dufresne, Leonard, & Gerace, 2002; Kuechler & Simkin, 2010; Birenbaum & Tatsuoka, 1987). However, what much of the research does agree upon is the idea of including multiple sources of information to assess student knowledge and understanding reliably. It is for this reason that multiple-choice items and constructed response items were included in the instrument measuring student conceptual understanding. EFFECT OF EL ON CONCEPTUAL UNDERSTANDING IN ALGEBRA 1 33 According to the Texas Education Agency (TEA) (2015) each multiple-choice item was vetted by their panel with the following criteria for distractors: Figure 6. Criteria for distractors on Texas STAAR assessment. Adapted from Texas Education Agency (2015). Participants’ conceptual understanding was assessed three times. Once as a pretest that measured participants’ accuracy on an assessment of exponential and linear functions, once as a posttest for linear functions after treatment phase I and once as a posttest for exponential functions after treatment phase II. Each assessment was designed to examine participants’ accuracy in modeling, interpreting, and translating functions. Pretest: Conceptual understanding exponentials and linear functions. Prior to Phase I, all participants completed a 26-item pretest measuring participants’ conceptual understanding on both linear and exponential functions. Student were given 50 minutes to complete the pretest. Graphing calculators were allowed. In order to ensure that students try their best, the conceptual understanding pretest was graded for accuracy so that pre and post test scores could be compared. Since this material may have been new for some students, a participation daily grade was recorded for all students. The items on the pretest were identical to the items on the posttests which was administered at the end of Phase I (Linear) and Phase II (Exponential). To address priming effects of using a pretest, the pretest was administered three weeks prior to the beginning of Phase I. The questions selected for this instrument from the STAAR and O’Callaghan (1998) assessments, were chosen because of their correlation to the constructs being measured (modeling, interpreting, and translating). For example, modeling is defined in this study as EFFECT OF EL ON CONCEPTUAL UNDERSTANDING IN ALGEBRA 1 34 transitioning from a problem situation to a mathematical situation. Examples would include the fluency of students to transition from a linear or exponential word problems to tables, equations, and graphs (O’Callaghan, 1998). Figure 7 is a multiple choice question from the conceptual understanding pretest that measures students’ conceptual understanding of modeling functions. Modeling was defined in this study as transitioning from a problem situation to a mathematical situation. This question assesses modeling of exponential functions by giving students the problem situation concerning the amount of fertilizer a company has after w weeks and asking them to choose the appropriate symbolic rule(equation), which is a mathematical situation. Figure 7. Question two from exponential assessment. Selected from Texas Education Agency STAAR 2017. The constructed response items included in the instruments were chosen from the conceptual understanding instrument designed by O’Callaghan (1998). Figure 8 is a constructed response question from the conceptual understanding pretest that measures students’ conceptual understanding of modeling functions. This question assesses modeling of linear functions by giving students the problem situation concerning the weigh