Recent research contends that good science teaching depends on teachers having a strong command of the science content and a strong conceptual understanding of that content (Antink-Meyer & Meyer, 2016; Moodley & Gaigher, 2019; Namdar, 2018; Potvin & Cyr, 2017). In line with the aforementioned research stream on teachers’ conceptual understanding of science content, a number of scholars have studied both prospective teachers’ (Çam et al., 2015; Diamond et al., 2014; Harrell & Subramaniam, 2014a, 2014b, 2015; Kiray et al., 2015; Riegle-Crumb et al., 2015; Subramaniam & Esprivalo Harrell, 2013; Subramaniam et al., 2019) and in-service teachers’ (Bergqvist & Rundgren, 2017; Heller et al., 2012) conceptual understanding of a number of science topics. The study presented in this article continues this stream of research and focuses on prospective elementary teachers’ conceptual understanding of average speed.

This exploratory and descriptive qualitative study aimed to investigate the stability of prospective elementary teachers’ conceptual understanding of average speed. Stability here refers to continued and repeated use of scientifically accepted conceptions of average speed and alternative conceptions of average speed that prospective elementary teachers draw on to define, explain, and use when they teach the concept of average speed. Alternative conceptions refer to ideas or notions that counter the scientifically accepted understanding of a phenomenon (Butler et al., 2015; Chi, 2008; Vosniadou & Skopeliti, 2017) and, in the case of this study, the alternative conceptions of average speed. The research question that guided the study was, “What are the stable and persistent conceptions of average speed held by prospective elementary teachers in their descriptions of teaching average speed?”

The intent of this investigation was twofold. First and foremost, the investigation on the concept of average speed was conducted in response to past prospective elementary teachers’ lesson plans on average speed. As educators of science teachers, we had noticed that prospective teachers—when asked to construct lesson plans on the concept of average speed—exhibited the following characteristics in their lesson plans:

1. The objective of the lesson was predominantly centered on stating and recalling the formula for calculating speed, not average speed.
2. Average speed as a complex property concept similar to concepts such as area, acceleration, and others lacked any (a) association with numerical values, (b) characterization with magnitude, and (c) association with units.
3. Instructional strategies to teach the concept of average speed centered on activities that predominantly involved (a) using a toy car that was pushed down a ramp or pushed across the ground, (b) recording the distance travelled by the toy car down the ramp or across the ground, (c) recording the time taken by the toy car to move down the ramp or across the ground, (d) calculating and recording the speed, and (e) repeating steps (a), (b), (c), and (d) two more times, then calculating the average of the three recorded speeds.

In general, the teaching and learning of a complex concept such as average speed was a hands-on activity rather than a cognitive task to develop conceptual understanding of average speed. By directly addressing this deficiency, we wanted to identify prospective elementary teachers’ alternative conceptions about average speed and simultaneously develop a framework that would enable prospective elementary teachers to address and reconstruct their alternative conceptions as a more scientifically appropriate understanding of average speed.

The second goal for this investigation was to build the research base on average speed, a common physics topic taught in both primary and secondary classrooms that is not extensively investigated and understood from prospective elementary teachers’ perspective.

Conceptual understanding of average speed

The literature about the conceptual understanding and alternative conceptions of average speed is limited. According to the literature, the concept of average speed is described as a scalar quantity defined as the rate of movement of an object expressed as the total distance traveled by the object divided by the total time taken or the elapsed time; average speed is differentiated from speed, which is the rate of movement of an object expressed as how far the object travels divided by a set time (Hewitt, 2015; Peacock et al., 2017). Thus, the basic concepts of average speed include a moving object, a frame of reference (point A to point B), total distance travelled by the moving object (point A to point B), and the total time taken to travel the distance (point A to point B; Hewitt, 2015; Peacock et al., 2017). Accordingly, average speed is a complex property concept with a definite symbol, definite formula (descriptive and symbolic), and definite units. Apart from the aforementioned unique concepts that are associated with the conceptual understanding of average speed, average speed is easily distinguished from other complex concepts such as velocity, average velocity, and acceleration based on the understanding that average speed is a scalar quantity while velocity and acceleration are vector quantities with their own descriptive and symbolic formulae and units.

On the other hand, the available literature on the alternative conceptions of average speed states that the alterative conceptions hinder or facilitate the teaching and learning of average speed (Borghi et al., 1993; Reed & Jazo, 2002; Palic Sadoglu & Durukan, 2018; Yıldız, 2014, 2016). Specifically, these studies identify the following three alternative conceptions as being predominant: (1) the interchangeable use of the terms speed and velocity in the conceptual understanding of average speed; (2) the interchangeable use of the terms average speed and average velocity to indicate the same quantity, the rate at which distance is covered by a moving object within a time interval; and (3) the lack of understanding that speed is a scalar quantity and, in contrast, velocity is a vector quantity.

Method

The study involved 84 prospective teachers from four sections of an elementary teacher education program. A pretest questionnaire was administered to all 84 prospective primary teachers on the first day of a 15-week elementary science methods course. The purpose of the pretest questionnaire was to elicit all 84 participants’ conceptions of average speed. The pretest questionnaire consisted of two questions and elicited participants’ conceptual understanding of average speed. The same test was administered at the end of the 15-week course. There was no intervention after the administration of the pretest because the main aim of the study was to investigate the stability of prospective elementary teachers’ conceptual understanding of average speed. A sample of 55 participant responses were collected during the administration of the posttest. The prompts used in the pre- and posttests were as follows:

1. Define average speed.
2. Describe how you intend to explain average speed to your students when teaching average speed in your future science classroom.

Data analysis

A two-tier coding scheme (Stroupe, 2017) was adopted to analyze the data. This coding scheme consisted of two primary categories: participants’ definition of average speed and the features of participants’ explanations of average speed.

Coding category 1

The first coding scheme aimed to characterize each participant’s definition of average speed as either accurate or inaccurate in the pre- and posttest responses to the prompt “Define average speed.” Each participant’s definition was compared to the scientifically accepted definition for average speed for accuracy and inaccuracy. By doing so, responses to the prompt “Define average speed” were categorized as an accurate definition (17 pretest responses and 16 posttest responses) or an inaccurate definition (67 pretest responses and 39 posttest responses). This coding also enabled the categorization of responses to the prompt “Describe how you intend to explain average speed to your students when teaching average speed in your future science classroom” to be placed within the categories of accurate and inaccurate definitions for average speed.

Coding category 2

The second coding scheme aimed to characterize the features of explanations of average speed by mapping the associated and unassociated concepts used in each participant’s response to the prompt “Describe how you intend to explain average speed to your students when teaching average speed in your future science classroom.” Each associated and unassociated concept was coded using the concepts present within the scientifically accepted understanding and explanation of average speed for alignment and for accuracy or inaccuracy. These codes were used to develop themes and tallied to determine frequencies and percentages (see Table 1)

A final step in the analysis process involved a Pearson’s chi-square test on the pre- and posttest frequencies of the themes with student understanding of average speed. A chi-square test is used to compare frequencies that occur in different groups (Gay et al., 2009). The assumptions of the test include the following: The data in each cell represent frequencies; there are two categories that are mutually exclusive; the groups are independent, and only one subject is represented in a cell; and the value of each cell is greater than five or more frequencies for more than 80% of the cells. The chi-square test found no statistical significance between the frequencies of accurate and inaccurate explanations between pretest and posttest responses. Although the frequency of accurate explanations increased slightly from pretest to posttest, the difference was not significant, χ²(1, N = 139) = 1.44, p = 0.23. The chi-square results confirmed the null hypothesis, namely that the proportion of accurate and inaccurate explanations does not vary between the pre- and posttest intervention. A two-tier coding scheme (Coding Category 1 and Coding Category 2) used in the study was independently conducted by the four authors. Inter-rater reliability (Kurasaki, 2000) was high, with a consensus of 90% coder agreement by the four authors.

Findings

Analysis of data indicated that 20.3% of definitions of average speed in the pretest and 29.1% of the definitions in the posttest were accurate, while the percentages of inaccurate definitions were 79.8% in the pretest and 70.9% in the posttest (Table 1).

Participants who had both accurate and inaccurate definitions of average speed used two predominant stable and persistent ways to explain average speed to their students: (1) the use of the conceptual understanding of speed to explain the conception of average speed, and (2) the notion of average speed as the average of multiple recordings of an object’s speed taken in three or more trials of the object moving a set distance in a set time (see Table 2).

For participants in this study, whether they held an accurate or inaccurate definition of average speed, their explanation of average speed was conveyed as the measure of how far an object traveled in a set time instead of a measure of the total distance traveled by the object between two points in a given period of time (total time taken). Table 2 provides sample quotes taken from the pre- and posttest data to illustrate the above stable and persistent theme. The other persistent explanation used by participants to convey the understanding of average speed, whether they held an accurate or inaccurate definition, was the notion of conceptualizing average speed as being made up of two separate entities: average and speed. In this case, participants conceptualized average speed as calculating the mean of three or more recordings of an object’s speed. The three or more speeds are calculated from three or more trials of the object moving a certain distance in a certain period of time, with the formula distance/time used to find the speed of the object in each trial. In this case, the concept of average speed was conceptualized as only a numerical value derived from calculating the average, not as the measure of the total distance traveled by the object between two points in a given period of time (total time taken).

The predominant descriptive teaching contexts within which these stable and persistent themes were present included the following: (1) the use of verbal and visual examples of average speed (81.0% in the pretest and 80.0% in the posttest), (2) the use of manipulatives (e.g., objects such as cars, marbles, and ramps; 52.4% in the pretest and 69.1% in the posttest), and (3) the use of hands-on activities (26.2% in the pretest and 23.6% in the posttest) to convey the concept of average speed to students (Table 3). Participants’ explanations of average speed—whether they were underpinned by accurate or inaccurate definitions—were located within the aforementioned three identifiable instructional approaches.

Discussion and implications

The findings in this study indicated that whether participants held accurate or inaccurate definitions of average speed, there were two stable and persistent themes in their understandings of the concept: the use of the conceptual understanding of speed to explain average speed and the notion of average speed as the average of multiple recordings of an object’s speed. Additionally, these two themes were predominantly expressed as explanations of average speed using verbal and visual examples, manipulatives, and hands-on activities (teaching contexts). Asking participants to express their explanations of average speed in relation to how they plan to teach average speed in their future classrooms revealed the contextual nature and the dependence on teaching contexts.

The argument arising from these findings is that participants’ definitions of average speed, whether accurate or inaccurate, did not shape their explanations of average speed. Instead, the teaching contexts would determine how participants would convey their explanations of average speed to their students in their future classrooms. Thus, for participants in this study, having an accurate definition of average speed did not mean that they would be able to explain average speed to their future students in a scientifically correct way. The findings also indicated that participants assumed specific teaching contexts (verbal and visual examples, manipulatives, and hands-on activities) as having explanatory powers to convey the understanding of average speed to their future students. That is, participants’ explanations of average speed were inherent within the teaching contexts. It is evident that participants’ understanding of average speed, especially for those participants with inaccurate definitions, was a product of faulty teaching within their K–16 education. Participants with accurate understandings also likely experienced faulty teaching but may have had an accurate definition reinforced through memorization or rote learning.

Participants’ alternative conceptions were associated with the teaching contexts that participants claimed they were using to explain the meaning and understanding of average speed to future students. This study indicates that identifying just the alternative conceptions is not enough; empirical studies need to situate and contextualize the alternative conceptions, especially if they occur with faulty classroom instruction.

The plausible reasons for the stability and persistence of the two alternative conceptions could be that participants in this study were exposed to K–16 instruction that did not identify, label, represent, and distinguish the concept of average speed as a complex concept, with relevant signs, symbols, principles, facts, and definitions. In particular, participants in this study were exposed to K–16 instruction that did not generate examples of the concept of average speed to distinguish it from other concepts, specifically in relation to the concept of speed.

In consideration of the findings of this study, it is apparent that a framework is needed to enable participants to address and reconstruct the two stable and persistent themes. One possible component of such a framework is the inclusion of disciplinary practices (NGSS Lead States, 2013; Stroupe, 2017) specific to teaching a concept such as average speed. Disciplinary practices in this case can involve

• •asking questions to probe prior knowledge about the difference between speed and average speed, including the difference between the respective descriptive and symbolic formulas for speed and average speed;
• •using drawings to show the difference between speed and average speed;
• •planning and carrying out investigations on average speed that use manipulatives;
• •using mathematics to communicate the significance of the numerical components of the symbolic formula for average speed, v_av = Δd/Δt or v_av = s/Δt; and
• •asking students to construct and communicate their own explanations of the descriptive formula for average speed, Total Distance/Total Time Taken.

Science instruction focused and based on the specific disciplinary practices (questioning, comparing and contrasting, drawing, mathematical thinking, and communicating) will promote specific purposes, orderliness, and values of the objects and contexts for teaching the concept of average speed and other physics concepts.

Karthigeyan Subramaniam (Karthigeyan.Subramaniam@unt.edu) is an associate professor of science education, Christopher S. Long is an assistant professor of science education, Pamela Esprivalo Harrell is a professor of science education, and Nazia Khan is a senior lecturer of teacher education and administration, all in the College of Education at the University of North Texas in Denton, Texas.