Methods - A ROLE FOR SIMULATIVE MENTAL MODELS IN SCIENCE CONCEPT REPRESENTATION: THE CASE OF I

In broad terms, the study was an experiment comparing learning in two conditions, each supporting one of the mental models described in Chapter 2. In both conditions, students made many different speed measurements using a specialized speed measurement tool called a photo-gate timer. The learning activities were the same in each condition except for the way the photo-gate timer was configured. One condition, called the single gate condition, was meant to engage students in the cognitive process of mental simulation hypothesized to be inherent in the motion model and predicted to be useful for understanding instantaneous speed. The

contrasting condition, called the dual gate condition, was meant to engage students in conceptualizing speed as constant over a perceptible length in accordance with the length model. In both conditions, students participated individually to complete a set of essentially identical instructional activities and assessments during two 50 minute periods on separate days. Instruction consisted of a set of guided discovery

investigations with periodic embedded assessment events. I was the sole instructor⁵, and I followed a detailed written protocol. Differences in what students learned were measured by pre- and posttests, and also by a performance assessment. Evidence for differences in cognitive processes during learning and on the performance assessment was obtained from structured notes, student talk, and student drawings.

This chapter is organized into three sections. The first describes the

participants and context in which the study took place. The next gives an overview of the experimental procedures, including instructional design and implementation. The final section describes the instrumentation and procedures for data analysis.

Context and Participants

School Context

The study was conducted in a 400-student charter high school (grades 9 to 12) near a large city on the west coast of the United States. Ethnically, the school was

5 I am an experienced (i.e., National Board Certified) teacher and familiar with this content domain.

predominantly white and Latino, 56% and 29% respectively, with 8% Asian and 3%

African American students. Females made up 51% of the student body, and 6% of the students were socioeconomically disadvantaged (CBEDS, 2010).

The school served an economically diverse district in the midst of several more affluent districts. As a gauge of the school‘s socioeconomic makeup, the percentage of students with at least one parent having earned a college degree, 64%, can be

contrasted with 98% reported for schools in one of the nearby affluent districts

(CBEDS, 2010). On the 2009 state standardized test for mathematics, 30% of students scored proficient or above. The figure for a school in the nearby affluent district was 93% (STAR, 2010).

Ninth graders at the school represented the population of interest. These 98 students were organized into four classes, all of which were taught by the same set of teachers. This group included a single physics teacher and two different math teachers, one for algebra and a second teacher for geometry.

Participant Characteristics

Thirty-six students were recruited for the study from the school's four ninth-grade physics classrooms. Students were 14 or 15 years old. In ethnicity, the sample of 36 students reflected the school‘s demographic characteristics. There were 22 white students (61%), 3 Asian students (8%), and 11 Latino students (31%). In gender, boys slightly overbalanced girls, 20 to 16 (55% male).

These 36 students were drawn disproportionately from the upper end of the achievement distribution within the ninth grade. This disproportion is evidenced by the fact that a high percentage of participants (72%) were identified by their algebra teacher as fit to miss two periods of algebra to participate in the study. Among the entire 98 students, roughly 50% were identified as fit to miss class. This imbalance is corroborated by the observation of higher mean scores on the study‘s specialized pretest for students who participated in the study vs. a larger group who only took the pretest. On a 0 to 1 scale, the mean for the 36 students in the study was .317, while the mean was .183 for the remaining students, t(81) = 3.81, p = .003, d = .696. The

standard deviation of pretest scores for students in the study was also higher, .227 vs.

.169. Scores for students in the study were normally distributed, Kolmogorov-Smirnov (K-S) Test⁶, p = .471.

Recruiting Participants

I began recruiting students for the study the day after the teacher administered the pretest, which, from the teacher‘s standpoint, served as a diagnostic test.

Recruitment consisted of a short presentation in which I showed the class two pieces of apparatus used in the study (photo-gate timer box, Figure 1, upper center, and marble ramp, Figure 2), and I explained that volunteers would work one-on-one with me for three sessions (later reduced to two sessions) once per week to learn about the speed of a ball rolling on the ramp. I offered no incentives for participation, although in each class the teacher suggested that it was likely to be a fruitful experience and generally encouraged students to participate. I then distributed an information sheet about the study that also consisted of a short survey in which students indicated how eager they were to volunteer. There were 15 students either not present for recruitment or who responded that they did not want to participate in the study.

In the days and weeks following the administration of this survey, I

approached those students who indicated neutral or higher interest in person, saying,

―I saw you responded ____ on the survey, do you think you would like to participate?‖

If they responded affirmatively, I scheduled their participation. This personal contact and scheduling, the final step in recruitment, continued over the length of the study as I worked through the participant pool.

Assigning Participants to Condition

Most students (26 out of the 36), participated in the study during their algebra class, missing two classes in two consecutive weeks. The remaining 10 participated after school. To ensure that participants could miss algebra without detrimental effects, the algebra teacher screened the list of all 9^th grade students and identified suitable candidates. This screening did not include seven students who had tested out of algebra, three of whom were eventually recruited for the study. Based on these

6 The K-S tests the null hypothesis that the distribution is normal.

conditions, candidates for the study were sampled from two strata: those who had screened or tested as fit to miss algebra, and those who had not.

Within each stratum, students were randomly assigned to condition concurrent with scheduling their participation. As a result of two dual gate students‘ eventually cancelling their participation after scheduling, 19 students were assigned to the single-gate condition, and 17 were assigned to dual single-gate. These groups each included 5 students in the not-to-miss-algebra stratum. Random assignment provided for parity between experimental conditions as shown in a multivariate ANOVA on all measured individual difference variables, including ethnicity, gender, pretest score, algebra teacher‘s screening (of fitness to miss algebra), and ramp prediction (a second form of pretest), F(6,27) = .445, p = .842.

Procedures

Outside of pretest and recruitment, students completed the study in two 50 minute periods, one calendar week apart. I guided each student individually through the learning and assessment activities. Throughout the process I followed a detailed instructional protocol, Appendix A, maintaining a natural rapport with students but very often reading aloud from the protocol. The following paragraphs provide a description of the instructional materials and an overview of the procedures detailed by the protocol. All procedures were videotaped, and I took structured notes on a laptop computer.

Instructional Materials

Photo-gate timer. The instruction was focused on speed measurements using a commercially available photo-gate timer (Figure 1). The timer was specifically

designed to display the elapsed time for an object rolling through either one or two photo-gates. The timer started or stopped on signals from a photo-gate when its light beam was interrupted. When using a single photo-gate, the timer started when the beam was first interrupted, and it stopped when the interruption ended (Figure 1, bottom center). With two photo-gates, the timer started when the beam for photo-gate A was interrupted and stopped when the beam for B was interrupted (Figure 1, bottom

right). The time readout was digital, to four significant figures. The measured times were typically small decimal values (e.g., .0233 seconds). The photo-gate was designed to be clamped upright on a track sold by the manufacturer that facilitated proper beam height and alignment for a 1.91 cm marble. However, they functioned adequately for my purposes when placed upside down on a flat surface, forming a tunnel for the marble to roll through (Figure 1, bottom center and bottom right).

Figure 1. Photo-gate timer. With two photo-gates, the timer starts when the beam at A is interrupted and stops when B is interrupted. With one photo-gate, the timer starts when the beam is first interrupted and stops when the interruption ends.

Drawing (top) of photo-gates and timer box courtesy of Grace Shemwell.

Ramp. The bulk of the learning activities were organized around questions of what happens to the speed of a marble rolling down a ramp. I built the ramp for the study from plastic building blocks and parallel fiberglass rods that formed a channel for the marble (Figure 2). The marble rolled down the slight incline after being released either by hand or by starting gates that could be repositioned higher or lower on the ramp as needed. The pitched section of the ramp was 50 cm long and 2cm high,

providing for an incline of about 2 degrees. This incline transitioned smoothly to a flat section 25 centimeters long. The ramp was always oriented such that the marble rolled down the ramp left to right. All speed measurements were taken on the flat section.

Figure 2. Ramp used for most of the learning activities.

The marble rolled downhill left to right. All speed

measurements were made on the flat section at the bottom, here bordered by plain white paper. This plain border could be replaced with a 1 cm measurement scale to facilitate measuring speed with two photo-gates.

Photo-gate simulator. A third important learning tool was a cardboard simulator which displayed the interactions between the marble and the photo-gate beam(s) that provided the displacement and elapsed time needed for the speed measurement (Figure 3). It simulated the motion of the marble on the flat section of the ramp. A coin representing the marble could be pulled slowly along the track by a thread at the right (here replaced by a piece of yarn). The coin and track were overlaid with wax paper on which could be taped one or two photo-gates (plastic transparency cutouts) at whatever position(s) the student chose. A dot on each photo-gate

Flat section

represented the light beam. As I pulled the string and advanced the marble through one or two photo-gates, the student could track the interruption of the beam(s) by the marble that triggered the start and stop of the timer.

Video. A small component of the instruction in both conditions was to show students a short video simulating a ―standard‖ lecture on the relationship between average and instantaneous speed. I made the video by training a video camera on a computer screen displaying a sequence of power point slides while I read from the voiceover script. The purpose of the video was to compare students in the two experimental conditions on recognition of didactically presented information, as measured by a coordinated question on the posttest. The video‘s script and images are provided in Appendix B. The screen showed several depictions of accelerated objects, while a narrator explained them. The video defined constant acceleration and

explained the relationship between average velocity for a time interval and

instantaneous velocities at the beginning and end of the interval. The video lasted just over 3 minutes.

Other equipment. The marble used for the experiment was a steel ball bearing 1.91 cm in diameter. For the sake of numerical simplicity, I rounded the diameter measurement to 2 cm. Two different length measurements were provided to measure the displacement of the marble. The first was a set of two paper strips marked in 1 cm increments which ran the length of the flat section of the ramp shown in Figure 2.

They facilitated the measurement of the spacing between two photo-gates. Of course, Figure 3. Photo-gate simulator. The string pulls the coin, representing the marble, slowly to the right. The photo-gates are represented by rectangular transparency cutouts. The dots are the photo-gate beams. The student watches the movement of the coin and says aloud when the timer starts and when it stops.

these measurement scales were of no use to the single-gate students, who needed to use the diameter of the marble as the displacement. I used the marble pad shown in Figure 4 as an unobtrusive way to present this information. The marble sat on a plastic ring adjacent to the full scale figure showing the diameter to be 2 cm. Both the pad and the centimeter strips were visible to each student, regardless of condition, but only during the later, quantitative measurements with the photo-gates. For earlier,

qualitative measurements, both pad and centimeter strips were obscured or put away.

Figure 4. Marble and pad. When not in use, the marble sat on a special pad showing that its diameter was 2 cm.

The first page of the experimental protocol (Appendix A) lists a range of other equipment used for the instruction. The list includes a calculator, paper and pencil, video viewer (laptop computer), masking tape, tweezers, etc.

Overview of Instructional Protocol: Rationale

Purpose of instruction. From my point of view as researcher, the learning activities detailed by the protocol were intended to provide multiple instructional events incorporating the complementary activities of photo-gate use and photo-gate explication. Photo-gate use required students to deploy the photo-gate(s) to measure speed, generally as a metric value in centimeters per second, but sometimes as a relative magnitude. Photo-gate explication required students to describe how the photo-gate or photo-gates could be used to measure the change in position of the marble associated with the elapsed time on the timer. I use the term explication rather than explanation because the emphasis was on describing displacement in explicit terms (i.e., describing precisely the displacement of the marble associated with elapsed

time) and less on the interpretive process of helping another person understand the idea of displacement. Explication took two different forms, a verbal response to an experimenter prompt, and a drawing task. Of course, the two instructional events, use and explication, were meant to incorporate different cognitive processes in the single gate condition than in the dual gate condition. The study‘s hypothesis (Chapter 2) provides a detailed discussion of these intended differences.

The purpose of the instruction in terms of instructional events and cognitive processes was not known to the students. From their point of view, the activity was focused on learning how to measure the speed of an object using the photo-gate(s), and on investigating the effect of releasing a marble at different points along an incline on the speed of the marble at the bottom of that incline.

Speed (not changes in speed) as the focal concept. The instructional events using the photo-gates were designed to help students in both conditions conceptualize speed as a coordinated relationship between a linear displacement and its associated time interval. The instruction avoided use of the photo-gates to help students

conceptualize changes in speed. This distinction between speed and changes in speed is important because it implies that differences in scores on pre- and posttests, which measured understanding of changes in speed, can be interpreted as reflecting a degree of transfer beyond what was taught directly.

The focus on speed (and not changes in speed) occurred in both photo-gate use and explication. In photo-gate use, all speed measurements were made at the bottom of the ramp as opposed to on the incline. Thus, the measured speed was framed as ―the speed‖ that resulted from something that happened higher up on the ramp. This is not to say that students never reasoned about changes in speed of the marble as it rolled on the ramp. For instance, I asked them what would happen to the speed at the bottom if the marble were to be released from a higher or lower point on the ramp. However, the actual measurement, which was the only part of the instruction that differed by

experimental condition, only concerned ―the speed‖ in the flat section at the bottom of the ramp. Similarly, in explication of photo-gate use, there was never any discussion of changes in the speed of the marble as it rolled through the photo-gates.

The performance assessment (described in detail in the instrumentation section of this chapter) was an exception to the rule that the photo-gate(s) were used to

measure speed and not changes in speed. In this activity, students used the photo-gate(s) in different positions to measure changes in the speed of a marble on the main incline of a curved roller coaster track. However, in contrast to the learning phases of the experimental protocol, I gave students no instructor support during or after the performance assessment. Nevertheless, the fact that the performance assessment involved direct consideration of changes in speed opens up the possibility that posttest scores may not have reflected learning that occurred before the performance

assessment (and therefore transfer of learning), but during the performance assessment (and therefore not transfer). To provide a check against this possibility, the

performance assessment incorporated a prediction element which was independent of photo-gate use.

Guided discovery as the teaching paradigm. Except for a three-minute video shown near the end of the second day‘s activities, the instruction did not involve teaching by explaining or telling. Instead, I as instructor posed questions and provided students the means of answering them. The bulk of this pattern consisted of

investigations of the behavior of a marble on a ramp using a predict-observe-explain cycle in which photo-gate measurements at the bottom of the ramp provided the necessary evidence. A second important component of this pattern was helping

students conceptualize, by means of a system of graded hints involving the photo-gate simulator, how either one or two photo-gates could be used to measure speed in centimeters per second.

The rationale for the guided discovery paradigm was two-fold. First, it was meant to provide for an active, constructive, and interactive instructional environment that would be efficacious for learning (Chi, 2009). Second, by making the students the primary actors in the learning process, the instructor‘s role was constrained to that of guide, thus enabling more uniform implementation of instruction across participants.

Naturally, not all instruction was by guided discovery. At times I would simply show or tell things to students. However, these instances were written into the

experimental protocol and confined to elements of simple declarative or procedural knowledge, such as helping students correct mathematical calculations or showing students the functional parts of the photo-gates and timer box.

Contingencies. Contingencies were built into the protocol for student mistakes and misunderstandings. For instance, at step 1.2.6 in the protocol, I asked students to do a simple speed calculation given distance in centimeters and elapsed time in seconds. Some students incorrectly divided time by distance. Following the next step in the protocol, I modeled the correct calculation once, and then asked the student to

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