Spiral teaching sequence and concept maps for facilitating conceptual reasoning of acceleration

The students participating in this study included three teams from two schools, where each team’s students were divided into three groups. There was a total of 402 senior high school students (Grade 12; age 17-18 years), where team A was made up of three classes from one school, while Teams B and C came from another school, as shown in Table 1. All of the students were in the science stream and had completed studying the senior high acceleration content. Different physics teachers taught each class.

Table 1. Number of students per team for the three groups

	Team A *(PR=95%)**	*Team B(PR=88%)**	*Team C(PR=88%)**
Group 1	42	31	29
Group 2	43	26	26
Group 3	148	27	30

*PR (percentile ratio) is based on the high school entrance examination scores

This study’s teaching intervention design included a series of instructional guidance and formative assessment activities, echoing the notion presented in the literature (Langbeheim et al., 2013; Sağlam, 2010). The first and second groups’ instructional procedure adopted a spiral design, which let the students repeatedly review the acceleration concepts. Each group’s instructional procedure and content differed. The instructional design procedure is listed in Table 2 below. On the left of Table 2, the order and time of introducing the teaching materials to the students are listed, while on the right, the teaching materials introduced to each group are indicated with a “•” showing which particular forms of teaching scaffolding each group received.

Table 2. The teaching scaffolding for each of the three groups

Teaching scaffolding	Duration	Group 1	Group 2	Group 3
Preview	5 min	•*	•
Concept map		•
Test Ⅰ	15 min	•	•	•
Review	10 min	•	•
Test II	15 min	•	•	•

*The symbol “•” indicates the adopted teaching scaffolding of each group

With a focus on reasoning the concept of acceleration, the intervention teaching implemented in this study included four kinds of instructional tools/strategies: 1) preview of the acceleration concepts, 2) a concept map, 3) conceptual questions, and 4) review of the solutions with a focus on the appropriate derivation routes and the prevalent difficulties that the students encountered. They are explained as follows.

1) Preview: Introducing the preview of the acceleration concepts helps the students to recall the previously taught acceleration knowledge, and lets them better understand the scientific language. The preview for Group 1 in this study introduced the five formulas of acceleration and integrated the deduction of the five routes in the concept map. On the other hand, Group 2’s preview only listed and explained the five formula routes, but did not introduce the concept map.

Acceleration derivation is introduced as follows:

Average acceleration can be derived by the change in velocity,;
The change of velocity can include magnitude and direction; acceleration can be divided into tangential (a_t) and normal (a_c) components;，where ;
The cause of acceleration is total force. According to Newton’s Second Law , instantaneous acceleration can be derived from the instant total force, or vice versa.
Newton’s Law is valid only when observers are limited to the inertial frames of reference. The concept of velocity is relative, but acceleration is absolute.
For an object moving with constant acceleration in one dimension, we can express acceleration (a), displacement (S) and time (t) as .

2) A concept map: The concept map presents the ways of reasoning related to the concept of acceleration from the two topics of kinematics and Newton’s laws (shown in Figure 1), combined into one formula diagram made up of five routes. When deducing concepts of acceleration, sometimes multiple routes need to be combined, and connection between kinematics and Newton’s Laws may be required. The aim of the content design was to address the weakness of the traditional teaching materials in which the introduction to acceleration often appears to be fragmentary and unrelated, and which also fails to explicitly elucidate the application limitations of each derivation.

For some rather simple questions, a single route, i.e., Routes 1~5 respectively, is enough to solve the problem (Figure 1). However, connections of multiple routes may be essential to successfully solve some complicated questions. Possible links of multiple routes to solve questions regarding acceleration are depicted in Figure 2. For example, Route 6 shows that students need to first select the inertial frame (avoiding observing at an accelerating frame) to observe the variation of velocity (Route 6a), and then determine the magnitude and direction of acceleration (Route 6b); Route 7 starts from drawing a free-force-diagram, determining the total force (Route 7a), then evaluating the tangential and radial components of acceleration respectively (Route 7b); Route 8 first derives acceleration from the object’s displacement (Route 8a), then determines the total force from the acceleration (Route 8b). Questions which require multiple routes to derive the solution are normally more challenging to students than those requiring a single route. Explicitly drawing concept maps as scaffolding to help students deal with the complicated task is suggested by the literature (Chang, 2011; Lindstrøm & Sharma, 2009).

Figure 1. The concept map depicts single routes of reasoning acceleration

Figure 2. The concept map depicts multiple routes of reasoning acceleration

3) Conceptual questions: The design of the two tests of conceptual questions on acceleration emphasized concept clarification, and reduced the amount of complicated calculation. Moreover, in line with previous research, the level of difficulty of the test content suited the background of the students and was designed to highlight the acceleration difficulties identified in the literature, corresponding to the design idea of the acceleration concept map. After the students completed Test I, the teacher analyzed the deduction of questions in the concept map and integrated the review of the solutions. This was followed by Test II, to see the results. This adheres to the notion of formative assessment (Beatty et al., 2006; Sağlam, 2010). The format of the conceptual questions was single-answer multiple-choice.

4) Review: The review of the solutions involved discussion of the appropriate derivation routes and the prevalent difficulties the students encountered for each question. These explanations not only involved correct reasoning and solutions, but also included discussion of the commonly seen mistakes made by previous students who had taken the test, and the possible reasons for these mistakes. Each question’s choices were designed with reference to students’ acceleration difficulties identified in the literature. Moreover, the explanations could also link to the concept map in order to clarify the acceleration concepts and the multiple routes integrated into the concept map to carry out inferencing, reflecting the assertion of using concept maps to reduce cognitive load (Lindstrøm & Sharma, 2009).

After the completion of the course, 55 students from Group 1 and Group 2 of Team C completed the questionnaire. The survey design included three parts to allow for multi-dimensional points of view and in-depth understanding. The first part consisted of eight close-ended items using a 5-point Likert scale for the qualitative classification of the students’ level of satisfaction including cognition, affect, and metacognition. The Cronbach's α exceeded .85, indicating the reliability of the questionnaire. The second part required the students to list in order the four kinds of teaching scaffolding from the most to the least effective, while the third part consisted of open-ended questions; the students were invited to write down what they had learned as well as their teaching design suggestions, and to point out the strengths and weaknesses of the course. Finally, according to the previous quantitative and qualitative analyses, two students and one teacher were interviewed, with a focus on those questions requiring further investigation.

Both qualitative and quantitative analyses were performed, where the quantitative analysis was performed on the results of the conceptual questions and the first and second parts of the questionnaires, whereas the qualitative analysis included the third part of the questionnaire and the interviews. The data analysis was performed as follows: Regarding students’ performance on the conceptual questions, in order to account for the differences in students’ pre-instruction physics abilities between groups, students’ performance on the conceptual questions was modified using their scores on the physics midterm examination at school. In particular, one group was selected as the benchmark group, and the average midterm score of each group was divided by the average midterm score of this benchmark group in order to obtain a correction factor for each group. Students’ average scores on the conceptual questions were then multiplied by the corresponding correction factor in each group before any between-group comparison was made. Then, the correct ratio (%) of the conceptual questions was used to analyze the effect of the teaching intervention on each group in order to more deeply probe the influence of the teaching intervention. Using effect size can represent the degree of effect of an experiment (Savinainen & Scott, 2002). The formula for Cohen’s d effect size is the difference between the average value of the two groups divided by the two groups’ combined standard deviation, where the higher the value, the greater the result of the experiment (Cohen, 1988).

Following this, the concept questions were divided into kinematics and Newton’s Law questions for comparative analysis. Comparative analysis was also applied to divide the concept questions into those which required integration of one or two concepts.