1. Introduction

Graphic representation, a method used to show and represent values, increases, decreases, comparisons to either make predictions or show a report of how a certain situation was yesterday and how it is today, is an integral part of all scientific subjects. Scientific graphs visually communicate data and information about variables and their relationships and are often used in the analysis of data to determine patterns and relationships [11, 21]. Be that as it may, the specific purpose and usage of graphs may differ tremendously, even in subjects as closely related as mathematics and physics [19]. Graph comprehension is thus subject specific, that is, it depends on the discipline characteristics of different subjects [6].

2.1.1 Students' difficulties with kinematics graphs

DOI: http://dx.doi.org/10.5772/intechopen.88235

concepts and connecting graphs to the real world.

connect the physical concepts to the different features of the graph.

the narrative description of the problem with the graph correctly.

on a velocity-time graph means constant acceleration.

placement over an interval.

167

McDermott et al. [13] investigated difficulties students experience with graphs as used in kinematics, and in their findings, state that the students seem to lack the ability to abstract information from the graphs. This cannot just be due to inadequate mathematics preparation, because often, students that are able to construct and interpret graphs in mathematics cannot do the same for graphs in physics. The difficulties they experience are rather because of an inability to make connections between graphical representations and physical ideas. The difficulties found by McDermott et al. [13] were divided into two groups: connecting graphs to physics

Discipline,Task and Reader Characteristics of Introductory Physics Students' Graph…

Concerning the difficulties students experience in connecting graphs to physical concepts, McDermott et al. [13] found that students often do not know whether to use the value of the graph or the gradient of the graph to subtract the information from. This is referred to as the (function) value/gradient confusion. Students are also confused between changes in the value of the graph and changes in the gradient. Changes in value are easier to see than changes in gradient. As mentioned earlier, students see a constant graph as a graph with a constant gradient (linear graph). When constructing one graph from another, students find it difficult to ignore the form of the original graph. Many students do not have the ability to differentiate between displacement-time, velocity-time, and acceleration-time graphs. This can be due to the confusion between graph value and gradient and/or the inability to

It was also found that students are not able to match the narrative information of the problem with the relevant features of the graph (McDermott, et-al. [13]). In the example used by McDermott et al. [13], the students had to determine the acceleration from a velocity-time graph over certain intervals. Many of them only used the coordinates of one of the endpoints of the line sector ð Þ y=x instead of the change over the interval ð Þ Δy=Δx , despite the fact that they referred to the acceleration as change in velocity divided by the change in time. They were also asked to determine the acceleration of a part of the movement that was not included on the graph. Most of those who determined the acceleration on the given interval wrongly calculated acceleration for the part not given on the graph. This shows that they did not match

In physics, students have to determine the area under graphs before they have done integration in mathematics. Although they have calculated areas of many twodimensional figures, the idea that the area under a graph can be used to determine a physical quantity is very new and strange to them (McDermott et al. [13]). The fact that, for example, the area under a velocity-time is displacement is memorised and used. They do not realise that the area under the graph represents the functional relation f xð ÞΔx and that, for example, the area under a velocity-time graph is Δs ¼ vΔt. They further do not associate a positive area with displacement in the positive direction and a negative area with displacement in the negative direction. When asked to determine the position at a certain instant from a velocity-time graph, students found it hard to understand that they have to determine the dis-

Problems which can be solved by simple recall can be done with ease by most students (McDermott et al. [13]). Students find it hard to solve problems where the detailed interpretation of a graph is needed. To be able to use graphical interpretation to solve problems requires more than just memorization, for example, the gradient of the velocity-time graph is the acceleration and that a constant gradient

According to Redish and Gupta [19], it is important that physical meaning of mathematical symbols is attached when applying mathematical knowledge in physics. Meredith and Marrongelle [14] further explain this by stating that we interpret mathematical concepts in the context of physics; hence according to Redish [18], the blending of the mathematics symbols, structures and rules with physics concepts, principles and laws is significant to students. This is because the blending will help students to solve kinematics/physics equations and interpret graphs. Woolnough [25] discovered that students tend to interpret slope as a mathematical quantity and that it cannot be associated with units as in physics graphs.

The researchers are therefore investigating in the empirical study why participants' performances on similar tasks in mathematics and physics graphs yielded different responses. There are few investigations on students' application of mathematics knowledge in physics [12, 25], whilst most studies focussed on problemsolving (e.g., [5, 18]) and specific aspect interpretation like slope of graphs [17, 25]. The researchers also found out that less study has been conducted in the four qualitative and quantitative constructs' tasks on the effect of discipline, task and reader characteristics in the mathematics and physics contexts and hence this study.

### 2. Theoretical background

#### 2.1 Graph comprehension

According to Okan et al. [15], graph literacy is a necessary skill for decisionmaking, and it has often been neglected. Szyjka [24] citing Fry [7] defined "graphs as two-dimensional representations of points, lines and spaces, where data are displayed through represented words and numbers." A student can show comprehension of graph by being able to read and interpret it, that is, derive its meaning [6, 8]. According to Dori and Sasson [3] and Friel et al. [6], by working with graphs, students acquire graph sense and graphical thinking skills, and they are also able to comprehend the nature of graphs presented to them and are able to give variables and their relationships meaning [11]. Students acquire graph sense by working with graphs, and they gain graphical thinking skills and are also able to comprehend the nature of graphs as well as give variables and their relationships a meaning [11].

Scott [20] reported variation in students' performance in questions set on different levels in a questionnaire with corresponding mathematics and chemistry questions. He conducted a study on the participants' use of mathematics on the mole concept. No significant difference occurred in the participants' responses to the easier questions; however, the more difficult questions yielded a significant difference with better performances in the mathematics questions than the chemistry ones. He argued that algorithmic approaches in mathematics contribute to students' difficulties with calculations in chemistry.

Stahley [22] reported that even though students may have a correct idea or procedure to comprehend and illustrate discipline, task and reader characteristics of a graph [6], their confidence in taking such a decision is lacking. Some of them may understand the concept but lack the principles, and they seem unable to demonstrate the procedure. In physics graphs, physical contexts embed both algebraic and graphical representations [9].

Discipline,Task and Reader Characteristics of Introductory Physics Students' Graph… DOI: http://dx.doi.org/10.5772/intechopen.88235

#### 2.1.1 Students' difficulties with kinematics graphs

of graphs may differ tremendously, even in subjects as closely related as mathematics and physics [19]. Graph comprehension is thus subject specific, that is, it

According to Redish and Gupta [19], it is important that physical meaning of mathematical symbols is attached when applying mathematical knowledge in physics. Meredith and Marrongelle [14] further explain this by stating that we interpret mathematical concepts in the context of physics; hence according to Redish [18], the blending of the mathematics symbols, structures and rules with physics concepts, principles and laws is significant to students. This is because the blending will

Woolnough [25] discovered that students tend to interpret slope as a mathematical

The researchers are therefore investigating in the empirical study why participants' performances on similar tasks in mathematics and physics graphs yielded different responses. There are few investigations on students' application of mathematics knowledge in physics [12, 25], whilst most studies focussed on problemsolving (e.g., [5, 18]) and specific aspect interpretation like slope of graphs [17, 25]. The researchers also found out that less study has been conducted in the four qualitative and quantitative constructs' tasks on the effect of discipline, task and reader characteristics in the mathematics and physics contexts and hence

According to Okan et al. [15], graph literacy is a necessary skill for decisionmaking, and it has often been neglected. Szyjka [24] citing Fry [7] defined "graphs as two-dimensional representations of points, lines and spaces, where data are displayed through represented words and numbers." A student can show comprehension of graph by being able to read and interpret it, that is, derive its meaning [6, 8]. According to Dori and Sasson [3] and Friel et al. [6], by working with graphs, students acquire graph sense and graphical thinking skills, and they are also able to comprehend the nature of graphs presented to them and are able to give variables and their relationships meaning [11]. Students acquire graph sense by working with graphs, and they gain graphical thinking skills and are also able to comprehend the nature of graphs as well as give variables and their relationships a meaning [11]. Scott [20] reported variation in students' performance in questions set on different levels in a questionnaire with corresponding mathematics and chemistry questions. He conducted a study on the participants' use of mathematics on the mole concept. No significant difference occurred in the participants' responses to the easier questions; however, the more difficult questions yielded a significant difference with better performances in the mathematics questions than the chemistry ones. He argued that algorithmic approaches in mathematics contribute

Stahley [22] reported that even though students may have a correct idea or procedure to comprehend and illustrate discipline, task and reader characteristics of a graph [6], their confidence in taking such a decision is lacking. Some of them may understand the concept but lack the principles, and they seem unable to demonstrate the procedure. In physics graphs, physical contexts embed both algebraic and

depends on the discipline characteristics of different subjects [6].

Theorizing STEM Education in the 21st Century

help students to solve kinematics/physics equations and interpret graphs.

quantity and that it cannot be associated with units as in physics graphs.

this study.

2. Theoretical background

to students' difficulties with calculations in chemistry.

2.1 Graph comprehension

graphical representations [9].

166

McDermott et al. [13] investigated difficulties students experience with graphs as used in kinematics, and in their findings, state that the students seem to lack the ability to abstract information from the graphs. This cannot just be due to inadequate mathematics preparation, because often, students that are able to construct and interpret graphs in mathematics cannot do the same for graphs in physics. The difficulties they experience are rather because of an inability to make connections between graphical representations and physical ideas. The difficulties found by McDermott et al. [13] were divided into two groups: connecting graphs to physics concepts and connecting graphs to the real world.

Concerning the difficulties students experience in connecting graphs to physical concepts, McDermott et al. [13] found that students often do not know whether to use the value of the graph or the gradient of the graph to subtract the information from. This is referred to as the (function) value/gradient confusion. Students are also confused between changes in the value of the graph and changes in the gradient. Changes in value are easier to see than changes in gradient. As mentioned earlier, students see a constant graph as a graph with a constant gradient (linear graph). When constructing one graph from another, students find it difficult to ignore the form of the original graph. Many students do not have the ability to differentiate between displacement-time, velocity-time, and acceleration-time graphs. This can be due to the confusion between graph value and gradient and/or the inability to connect the physical concepts to the different features of the graph.

It was also found that students are not able to match the narrative information of the problem with the relevant features of the graph (McDermott, et-al. [13]). In the example used by McDermott et al. [13], the students had to determine the acceleration from a velocity-time graph over certain intervals. Many of them only used the coordinates of one of the endpoints of the line sector ð Þ y=x instead of the change over the interval ð Þ Δy=Δx , despite the fact that they referred to the acceleration as change in velocity divided by the change in time. They were also asked to determine the acceleration of a part of the movement that was not included on the graph. Most of those who determined the acceleration on the given interval wrongly calculated acceleration for the part not given on the graph. This shows that they did not match the narrative description of the problem with the graph correctly.

In physics, students have to determine the area under graphs before they have done integration in mathematics. Although they have calculated areas of many twodimensional figures, the idea that the area under a graph can be used to determine a physical quantity is very new and strange to them (McDermott et al. [13]). The fact that, for example, the area under a velocity-time is displacement is memorised and used. They do not realise that the area under the graph represents the functional relation f xð ÞΔx and that, for example, the area under a velocity-time graph is Δs ¼ vΔt. They further do not associate a positive area with displacement in the positive direction and a negative area with displacement in the negative direction. When asked to determine the position at a certain instant from a velocity-time graph, students found it hard to understand that they have to determine the displacement over an interval.

Problems which can be solved by simple recall can be done with ease by most students (McDermott et al. [13]). Students find it hard to solve problems where the detailed interpretation of a graph is needed. To be able to use graphical interpretation to solve problems requires more than just memorization, for example, the gradient of the velocity-time graph is the acceleration and that a constant gradient on a velocity-time graph means constant acceleration.

To determine to what extent students connect kinematics graphs with the real world in the study of McDermott et al. [13], balls were released to roll down different inclines, and the students had to register the instant a ball passes a certain point. From that information, displacement-time, velocity-time and acceleration-time graphs had to be drawn. When constructing the displacementtime graph, many students indicated the displacement per time interval instead of the displacement at a certain instant, drawing discontinuous graphs. Others indicated the displacement at certain instances correctly but did not connect the dots to indicate continuous motion. The students also struggled to separate the actual path of the ball from the form of the graph. In one of the movements, the ball rolls up an incline and down again. Many students did not represent the velocity as negative, indicating the ball was rolling in the opposite direction. When drawing the acceleration-time graphs, most students did not realise that a positive (negative) acceleration does not necessarily means speeding up (down) the ball. They did not realise that when the ball was rolling up and down the incline, the acceleration was in the same direction. Many students also drew the displacementtime, the velocity-time and the acceleration-time graphs with similar shapes. They found it hard to accept that the same motion can be represented by graphs with different shapes.

3.2 Research questions

(as shown in Table 1).

Cohen's effect sizes

interpreted as follows:

169

comprehension of kinematics graphs?

4. Research design and methodology

DOI: http://dx.doi.org/10.5772/intechopen.88235

The following research questions were addressed in the empirical study:

Discipline,Task and Reader Characteristics of Introductory Physics Students' Graph…

• What characteristics of graphic tasks (reading coordinates, connecting representations and interpreting the area under and the slope of a graph) hamper the participants' performances in mathematics and kinematics?

• What is the role of discipline and reader characteristics on the participants'

In order to address the research questions, a questionnaire was designed, consisting of two sections, one section focusing on kinematics graphs and the other one focusing on corresponding graphs in mathematics. The kinematics questions were designed using Beichner's Test of Understanding Graphs in Kinematics (TUG-K) model (1994). The questions were based on the reading of coordinates, connection of representations, understanding and calculating the area under a graph and the gradient of a graph. Mathematics section was comprised of linear functions and graphs, the required skills and knowledge to solve kinematics graphs and equations. Validation of the content of questionnaires was done by two academics in the same research field. The questionnaires were further piloted using 30 first year physics students enrolled at Central University of Technology, Free State (CUT). Thereafter, changes necessary in the questionnaires were then effected. The final questionnaire showed a reliability with the Cronbach's alpha coefficients of

0.69 in the kinematics section and 0.75 in the section on mathematics.

The pairs of kinematics and mathematics questions are attached as Appendices. The corresponding mathematics and kinematics questions were not identical in order to prevent similarities in students' answers based on recognition of graphs in questions in the two sections. Discipline characteristics further necessitated differences. For example, in kinematics graphs, the independent variable, time, can only have positive values, whilst positive and negative x-values can be used in mathematics graphs. Still, care was taken that the corresponding mathematics and physics tasks in the questionnaire require the same judgement and similar visual decoding

The results of the questionnaire were statistically analysed using effect sizes, because no random sampling (only available sampling) was done. Effect sizes yield important results in any empirical study and can be used to give the practical significance of such results [10]. In this study, comparison between differences in proportions for mathematics and physics successes were interpreted according to

w ¼

freedom is retained from the McNemar test [23]. This effect size determines whether there is a practically significant difference between the proportion of students who succeeded in answering the mathematics correctly and the proportion of students who succeeded in answering the physics correctly. The w-values are

ffiffiffiffi χ2 n r

where n is the total number of participants and the χ2-value with one degree of

(1)

According to a study done by Beichner [1] in which he used the Test of Understanding Graphs in Kinematics, similar difficulties and misconceptions were found. It was found that students struggled to determine gradient in the correct way especially if the graph did not run through the origin. Students considered the graph as a picture of the path followed by the object and not as an abstract mathematical representation of the movement. When answering the questions, the students did not distinguish between the variables' displacement, velocity and acceleration. As indicated above, they believe that the displacement-time, the velocity-time and the acceleration-time graphs have to look similar. Beichner [1] also found that the students did not recognise the meaning of the area under the different graphs. In the answering of many of the questions, the confusion between the graph value, the gradient and the area under the graph was clear.

Some of these misconceptions are caused by the fact that students do not connect what they learn in physics with their everyday experiences Brungardt and Zollman [2]. The difficulty students have with negative velocity can, in part, be because a speedometer only indicates positive speed. Students may associate the word "negative" with decreasing or lesser quantity. This then means that vocabulary also causes problems for the students. They use the word "constant" to refer to a linear graph with a constant gradient, whilstthe words "up" and "down" are sometimes used to indicate an increase or decrease of magnitude or to indicate direction.

#### 3. Aim and research questions

#### 3.1 Research aim

The aim of the research is to investigate how discipline, task and reader characteristics influence physics students' graph comprehension in the corresponding mathematics and kinematics questions. The participants were 152 willing first year physics students enrolled at the Central University of Technology, Free State (CUT) in South Africa.

Discipline,Task and Reader Characteristics of Introductory Physics Students' Graph… DOI: http://dx.doi.org/10.5772/intechopen.88235

#### 3.2 Research questions

To determine to what extent students connect kinematics graphs with the real

world in the study of McDermott et al. [13], balls were released to roll down different inclines, and the students had to register the instant a ball passes a certain point. From that information, displacement-time, velocity-time and acceleration-time graphs had to be drawn. When constructing the displacementtime graph, many students indicated the displacement per time interval instead of the displacement at a certain instant, drawing discontinuous graphs. Others indicated the displacement at certain instances correctly but did not connect the dots to indicate continuous motion. The students also struggled to separate the actual path of the ball from the form of the graph. In one of the movements, the ball rolls up an incline and down again. Many students did not represent the velocity as negative, indicating the ball was rolling in the opposite direction. When drawing the acceleration-time graphs, most students did not realise that a positive (negative) acceleration does not necessarily means speeding up (down) the ball. They did not realise that when the ball was rolling up and down the incline, the acceleration was in the same direction. Many students also drew the displacementtime, the velocity-time and the acceleration-time graphs with similar shapes. They found it hard to accept that the same motion can be represented by graphs

Theorizing STEM Education in the 21st Century

According to a study done by Beichner [1] in which he used the Test of Understanding Graphs in Kinematics, similar difficulties and misconceptions were found. It was found that students struggled to determine gradient in the correct way especially if the graph did not run through the origin. Students considered the graph as a picture of the path followed by the object and not as an abstract mathematical representation of the movement. When answering the questions, the students did not distinguish between the variables' displacement, velocity and acceleration. As indicated above, they believe that the displacement-time, the velocity-time and the acceleration-time graphs have to look similar. Beichner [1] also found that the students did not recognise the meaning of the area under the different graphs. In the answering of many of the questions, the confusion between the graph value, the gradient and the area under the graph

Some of these misconceptions are caused by the fact that students do not connect what they learn in physics with their everyday experiences Brungardt and Zollman [2]. The difficulty students have with negative velocity can, in part, be because a speedometer only indicates positive speed. Students may associate the word "negative" with decreasing or lesser quantity. This then means that

vocabulary also causes problems for the students. They use the word "constant" to refer to a linear graph with a constant gradient, whilstthe words "up" and "down"

The aim of the research is to investigate how discipline, task and reader characteristics influence physics students' graph comprehension in the corresponding mathematics and kinematics questions. The participants were 152 willing first year physics students enrolled at the Central University of Technology, Free State (CUT)

are sometimes used to indicate an increase or decrease of magnitude or to

with different shapes.

was clear.

indicate direction.

3.1 Research aim

in South Africa.

168

3. Aim and research questions

The following research questions were addressed in the empirical study:


### 4. Research design and methodology

In order to address the research questions, a questionnaire was designed, consisting of two sections, one section focusing on kinematics graphs and the other one focusing on corresponding graphs in mathematics. The kinematics questions were designed using Beichner's Test of Understanding Graphs in Kinematics (TUG-K) model (1994). The questions were based on the reading of coordinates, connection of representations, understanding and calculating the area under a graph and the gradient of a graph. Mathematics section was comprised of linear functions and graphs, the required skills and knowledge to solve kinematics graphs and equations. Validation of the content of questionnaires was done by two academics in the same research field. The questionnaires were further piloted using 30 first year physics students enrolled at Central University of Technology, Free State (CUT). Thereafter, changes necessary in the questionnaires were then effected. The final questionnaire showed a reliability with the Cronbach's alpha coefficients of 0.69 in the kinematics section and 0.75 in the section on mathematics.

The pairs of kinematics and mathematics questions are attached as Appendices. The corresponding mathematics and kinematics questions were not identical in order to prevent similarities in students' answers based on recognition of graphs in questions in the two sections. Discipline characteristics further necessitated differences. For example, in kinematics graphs, the independent variable, time, can only have positive values, whilst positive and negative x-values can be used in mathematics graphs. Still, care was taken that the corresponding mathematics and physics tasks in the questionnaire require the same judgement and similar visual decoding (as shown in Table 1).

The results of the questionnaire were statistically analysed using effect sizes, because no random sampling (only available sampling) was done. Effect sizes yield important results in any empirical study and can be used to give the practical significance of such results [10]. In this study, comparison between differences in proportions for mathematics and physics successes were interpreted according to Cohen's effect sizes

$$w = \sqrt{\frac{\chi^2}{n}}\tag{1}$$

where n is the total number of participants and the χ2-value with one degree of freedom is retained from the McNemar test [23]. This effect size determines whether there is a practically significant difference between the proportion of students who succeeded in answering the mathematics correctly and the proportion of students who succeeded in answering the physics correctly. The w-values are interpreted as follows:


A w-value of > 0.5 indicated a practically significant difference between the two aspects considered. For this study, a small effect size indicates that the mathematics and physics questions were answered similarly, either both correct or both incorrect. A large effect size means that the mathematics and physics questions were answered differently, either the mathematics correctly and the physics incorrectly or vice versa.

Effect sizes of the reader characteristics on students' performances in the math-

ematics and physics sections of the questionnaire were statistically determined using Cohen's effect sizes [4]. The characteristics evaluated were the participants' gender, their study courses and whether they completed school the previous year or two or more years prior to the study. A gap between school and university physics may prevent knowledge retention and consequently lower performances. The statistical results are interpreted as follows for differences in average percentages in

Discipline,Task and Reader Characteristics of Introductory Physics Students' Graph…

Effect size of 0.8 is large, that is, the difference is of practical significance.

5. Analysis of task characteristics of questions in the two disciplines

Before the empirical results are discussed, the characteristics of the tasks set in the mathematics and kinematics contexts were analysed on the level of the participants. This implies that this analysis may differ for more or less advanced participants. For example, more experienced participants may distinguish characteristic features of graphs by visual decoding only and consequently may not need to

As indicated in Table 1, each task (e.g., reading coordinates, etc.) requires different mathematical and kinematics contextual knowledge, although similar visual decoding and judgement are to be performed in both the contexts. The first task, reading coordinates, is the simplest and requires only contextual knowledge and visual decoding. The other graph tasks require contextual knowledge, visual

It is important to note that the kinematics tasks can only be done if the mathematics contextual knowledge is transferred and integrated with kinematics knowledge. In the first task (reading coordinates), participants should have contextual mathematics knowledge of Cartesian coordinates and integrate it with kinematics knowledge about the variables of position (s), velocity (v), acceleration (a) and time (t). Conventionally, the independent variable t is placed on the x-axis and the dependent (s, v or a) on the y-axis. In the questionnaire items, participants needed to connect the proper dependent variable (function value) to a given independent

The second task (called connecting representations) requires mathematical knowledge of the graphical representation and formula of straight-line, parabolic and hyperbolic functions. In the kinematics questions, participants needed to recognise the mathematical formats and graph forms of the given expressions containing kinematics variables, instead of mathematical symbols. Proper understanding further requires insight that the given kinematics equations and graphs represent functions of time. Without having and integrating this contextual mathematics and kinematics knowledge, the participants will not know which visual

In order to accomplish "area quantitative" and "area qualitative" tasks (tasks 3a and 3b in Table 1) on kinematics, participants must recall the kinematics relation

vdt = area under v-t graph. Only then can the participants perform the

from the area under a line graph. Blending these kinematics and mathematics knowledge elements should result in understanding that displacement in interval dt

vdt. Then they should know from mathematics that the integral is determined

the mathematics and physics sections:

DOI: http://dx.doi.org/10.5772/intechopen.88235

explicitly perform judgement.

decoding and judgement.

variable, using visual decoding.

<sup>s</sup> <sup>¼</sup> <sup>Ð</sup>

is <sup>s</sup> <sup>¼</sup> <sup>Ð</sup>

171

decoding and judgement tasks to perform.

expected visual decoding and judgement tasks.

Effect size of 0.2 shows a small effect.

Effect size of 0.5 is medium but observable effect.


Table 1. Task characteristics of questions. Discipline,Task and Reader Characteristics of Introductory Physics Students' Graph… DOI: http://dx.doi.org/10.5772/intechopen.88235

Effect sizes of the reader characteristics on students' performances in the mathematics and physics sections of the questionnaire were statistically determined using Cohen's effect sizes [4]. The characteristics evaluated were the participants' gender, their study courses and whether they completed school the previous year or two or more years prior to the study. A gap between school and university physics may prevent knowledge retention and consequently lower performances. The statistical results are interpreted as follows for differences in average percentages in the mathematics and physics sections:

Effect size of 0.2 shows a small effect.

• w , 0:3 is a small effect.

• w . 0:5 is a large effect.

or vice versa.

Table 1.

170

Task characteristics of questions.

• 0:3 ≤ w ≤0:5 is a medium effect.

Theorizing STEM Education in the 21st Century

A w-value of > 0.5 indicated a practically significant difference between the two aspects considered. For this study, a small effect size indicates that the mathematics and physics questions were answered similarly, either both correct or both incorrect. A large effect size means that the mathematics and physics questions were answered differently, either the mathematics correctly and the physics incorrectly

Effect size of 0.5 is medium but observable effect.

Effect size of 0.8 is large, that is, the difference is of practical significance.
