6. Results

#### 6.1 Results: reader characteristics

The number of students and the average percentages obtained by each group are given in Table 2 for gender, Table 3 for the last school year and Table 4 for the faculty in which they are enrolled.

The effect sizes for differences between groups are medium (≥0.5) for gender, small for last school year and insignificant for faculty. In all cases, the effect size values were larger for mathematics than physics.

#### 6.2 Results: task characteristics

Table 5 summarises the average percentages correctly obtained by the participants as well as the results of the McNemar test for each question pair (refer to Appendix). The questions are categorised in constructs according to the tasks to be


performed, that is, reading coordinates, connecting representations, area (qualitative and quantitative) and slope (qualitative and quantitative). In Table 5, the label "M" is used for the mathematics questions, whilst "P" indicates physics (kinematics) questions. The percentages of participants who had the specific question correct are given in Table 5 as well as the w-values calculated from the McNemar test, indicating the effect size of differences in responses. Medium effect sizes (0:3≤ w ≤0:5) are marked with a single star (\*) and large effect sizes (w . 0:5) with a double star (\*\*). Large effect sizes imply that the pair of questions were answered significantly different, that is, either the mathematics question correct and the physics incorrect or vice versa. The w-values that are not marked show a small effect size (w , 0:3), that is, the pair of questions were answered similarly,

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

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

Four additional physics questions aided in the interpretation of the results of Table 2. These questions are incorporated in the Appendix, and participants' per-

that is, either both correct or both incorrect.

Results of the effect sizes for paired mathematics and physics questions.

formances are given in Table 6.

Table 4.

Table 5.

173

Performances by faculties.

Table 2. Gender performances.


Table 3. Performance by last schooling attended.

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


#### Table 4.

Requirements for successful execution of the qualitative and quantitative tasks

For all tasks, the discipline characteristics of the question thus determine what visual decoding and judgement tasks have to be done. Inability to perform the correct contextual tasks is expected to prohibit execution of correct visual decoding

The number of students and the average percentages obtained by each group are given in Table 2 for gender, Table 3 for the last school year and Table 4 for the

The effect sizes for differences between groups are medium (≥0.5) for gender, small for last school year and insignificant for faculty. In all cases, the effect size

Table 5 summarises the average percentages correctly obtained by the participants as well as the results of the McNemar test for each question pair (refer to Appendix). The questions are categorised in constructs according to the tasks to be

on slopes (tasks 4a and 4b) are similar to those for area. From mathematics, participants should know the meaning and formula for calculating the slope of a graph and be able to attach the kinematics meaning to it, that is, v ¼ ds=dt = gradient of s-t graph at time t. Thereafter, the visual decoding and judgement required

by the different questions can be performed.

Theorizing STEM Education in the 21st Century

6.1 Results: reader characteristics

faculty in which they are enrolled.

6.2 Results: task characteristics

values were larger for mathematics than physics.

and judgement.

6. Results

Table 2.

Table 3.

172

Performance by last schooling attended.

Gender performances.

Performances by faculties.


#### Table 5.

Results of the effect sizes for paired mathematics and physics questions.

performed, that is, reading coordinates, connecting representations, area (qualitative and quantitative) and slope (qualitative and quantitative). In Table 5, the label "M" is used for the mathematics questions, whilst "P" indicates physics (kinematics) questions. The percentages of participants who had the specific question correct are given in Table 5 as well as the w-values calculated from the McNemar test, indicating the effect size of differences in responses. Medium effect sizes (0:3≤ w ≤0:5) are marked with a single star (\*) and large effect sizes (w . 0:5) with a double star (\*\*). Large effect sizes imply that the pair of questions were answered significantly different, that is, either the mathematics question correct and the physics incorrect or vice versa. The w-values that are not marked show a small effect size (w , 0:3), that is, the pair of questions were answered similarly, that is, either both correct or both incorrect.

Four additional physics questions aided in the interpretation of the results of Table 2. These questions are incorporated in the Appendix, and participants' performances are given in Table 6.


mathematics and physics questions, indicating that participants who managed the

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

The average percentages in the four mathematics questions on comparison or calculations of the area under graphs (M\_A1, M\_A2, M\_A3 and M\_A4) ranged from 56.9 to 72.4%. Higher percentages were obtained in the qualitative than the quantitative questions in the corresponding physics questions on area under kinematics graphs (P\_A1, P\_A2, P\_A3 and P\_A4), where the participants obtained low percentages (≤40%), indicating that they did not apply their existing mathematics

The w-values for the corresponding pairs of questions on area were all medium to large, confirming inconsistencies in the students' responses. Students who were successful in the mathematics contexts generally failed to transfer their mathematics knowledge to the kinematics context. Practically significant differences in answers were obtained when comparing qualitative questions M\_A2 (largest area under graph) and P\_A2 (largest displacement from v-t graph), as well as quantitative questions M\_A4 (calculation of area under section of x-y graph) and P\_A4

Possible reasons for the poor performances in the physics questions on the area were investigated by additional qualitative item P\_A5 and quantitative item P\_A6. In P\_A5, the participants were asked whether displacement can be obtained from the area or slope of velocity-time or acceleration-time graphs. Only half of the students (53.5%) knew that the option "area under a velocity-time graph" is the way to determine displacement. Approximately a quarter of the participants chose the incorrect option "gradient of a velocity-time graph," showing area-slope confusion. The slope-area confusion was confirmed in the additional question P\_A6 that assessed the participants' understanding of what task should be performed and how it should be performed to determine the displacement in a straight-line velocitytime graph over an interval starting at the origin. Only 51.7% had P\_A6 correct, and a large number of students (30%) indicated that they would calculate the slope making the same slope-area mistake as in P\_A5. Both these additional questions indicate that a lack of physics conceptual knowledge contributed to participants'

Mathematics item M\_S1 and physics item P\_S1 required students' judgement of intervals where the slope and the instantaneous velocity (on a position-time graph), respectively, are the highest. In both questions, <50% of the students chose the correct answer. According to the small w-value (0.18), the majority of students were unsure in both the mathematics and physics questions. It seems as if a lack of mathematics knowledge and understanding of the concept of slope is transferred from mathematics to physics. This deduction was confirmed in the additional physics questions P\_S4 and P\_S5, in which the participants had to identify the intervals on a velocity-time graph, where the gradient and acceleration, respectively, are negative. The w-value for these two questions is 0.17, indicating that the participants who did not know where the slope is negative, did not also know where the acceleration of the v-t graph is negative. In both questions, the option chosen by the second-most participants was DE, the interval with both negative function values and negative slope. This shows that students struggle to discriminate

(calculation of change of velocity from an acceleration-time graph).

failure in the kinematics questions on area.

6.2.4 Slope qualitative and quantitative (tasks 4a and 4b)

mathematics tasks could not do the kinematics tasks.

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

6.2.3 Area qualitative and quantitative (tasks 3a and 3b)

knowledge.

175

#### Table 6. Additional physics questions.

For all four tasks, the results (Table 5) show that participants performed better in the mathematics questions than the corresponding physics questions. Comparison of the average percentages and w-values between the tasks shows differences in how participants performed. Their responses thus seem to depend on the characteristics of the tasks, as discussed below:

### 6.2.1 Coordinates (task 1)

In task 1, the reading of coordinate values from the given graphs in the mathematics and physics contexts was assessed (questions M\_C1, P\_C1 and P\_C2). The participants performed well in this task (>80% correct), and the low w-values (0.04 and 0.22) indicate consistency in responses, that is, the majority of participants answered correctly in both pairs of questions. It therefore, seems that the participants effectively transferred their mathematics knowledge about coordinates in a Cartesian plane to the kinematics domain. The lowest average performance (84.2%) obtained in the second kinematics question (P-C2) is probably due to the need to estimate the position (y) value by using the scale, which seems to be more difficult than reading values from intersections of grid lines as is the case in the other questions.

#### 6.2.2 Representations (task 2)

In both sets of mathematics and physics questions on the representation task, five graphs of different forms were given (see Appendix). In the three pairs of questions, the participants had to match a straight-line, hyperbolic and quadratic function to one of the given mathematics graphs and linear motion equations to kinematic graphs.

The vast majority of participants knew that the mathematics function in item M\_R1 is a straight-line graph and chose either the correct one, option 1 (67.6%), or the additional straight-line, option 2 (21.2%). With regard to the hyperbolic and parabolic functions g(x) in item M\_R2 and h(x) in item M\_R3, respectively, more than 70% of participants related each to the correct graphs. In both latter cases, the second largest contingent of participants (about 20%) connected the hyperbolic function to the parabolic graph or vice versa. These participants seem to confuse the representations of hyperbola and parabola in the mathematics contexts.

With regard to the physics items on this task, the largest correct percentage (65.1%) was also obtained for the straight-line representation (P\_R1). The small wvalue of 0.03 indicates transfer of these participants' mathematics knowledge to kinematics. For the hyperbolic and parabolic equations only, small percentages of participants succeeded (about 38 and 29%, respectively). The large w-values (0.58 and 0.44) imply medium to practically significant differences in responses to the

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

mathematics and physics questions, indicating that participants who managed the mathematics tasks could not do the kinematics tasks.

### 6.2.3 Area qualitative and quantitative (tasks 3a and 3b)

The average percentages in the four mathematics questions on comparison or calculations of the area under graphs (M\_A1, M\_A2, M\_A3 and M\_A4) ranged from 56.9 to 72.4%. Higher percentages were obtained in the qualitative than the quantitative questions in the corresponding physics questions on area under kinematics graphs (P\_A1, P\_A2, P\_A3 and P\_A4), where the participants obtained low percentages (≤40%), indicating that they did not apply their existing mathematics knowledge.

The w-values for the corresponding pairs of questions on area were all medium to large, confirming inconsistencies in the students' responses. Students who were successful in the mathematics contexts generally failed to transfer their mathematics knowledge to the kinematics context. Practically significant differences in answers were obtained when comparing qualitative questions M\_A2 (largest area under graph) and P\_A2 (largest displacement from v-t graph), as well as quantitative questions M\_A4 (calculation of area under section of x-y graph) and P\_A4 (calculation of change of velocity from an acceleration-time graph).

Possible reasons for the poor performances in the physics questions on the area were investigated by additional qualitative item P\_A5 and quantitative item P\_A6. In P\_A5, the participants were asked whether displacement can be obtained from the area or slope of velocity-time or acceleration-time graphs. Only half of the students (53.5%) knew that the option "area under a velocity-time graph" is the way to determine displacement. Approximately a quarter of the participants chose the incorrect option "gradient of a velocity-time graph," showing area-slope confusion. The slope-area confusion was confirmed in the additional question P\_A6 that assessed the participants' understanding of what task should be performed and how it should be performed to determine the displacement in a straight-line velocitytime graph over an interval starting at the origin. Only 51.7% had P\_A6 correct, and a large number of students (30%) indicated that they would calculate the slope making the same slope-area mistake as in P\_A5. Both these additional questions indicate that a lack of physics conceptual knowledge contributed to participants' failure in the kinematics questions on area.

#### 6.2.4 Slope qualitative and quantitative (tasks 4a and 4b)

Mathematics item M\_S1 and physics item P\_S1 required students' judgement of intervals where the slope and the instantaneous velocity (on a position-time graph), respectively, are the highest. In both questions, <50% of the students chose the correct answer. According to the small w-value (0.18), the majority of students were unsure in both the mathematics and physics questions. It seems as if a lack of mathematics knowledge and understanding of the concept of slope is transferred from mathematics to physics. This deduction was confirmed in the additional physics questions P\_S4 and P\_S5, in which the participants had to identify the intervals on a velocity-time graph, where the gradient and acceleration, respectively, are negative. The w-value for these two questions is 0.17, indicating that the participants who did not know where the slope is negative, did not also know where the acceleration of the v-t graph is negative. In both questions, the option chosen by the second-most participants was DE, the interval with both negative function values and negative slope. This shows that students struggle to discriminate

For all four tasks, the results (Table 5) show that participants performed better in the mathematics questions than the corresponding physics questions. Comparison of the average percentages and w-values between the tasks shows differences in how participants performed. Their responses thus seem to depend on the charac-

In task 1, the reading of coordinate values from the given graphs in the mathematics and physics contexts was assessed (questions M\_C1, P\_C1 and P\_C2). The participants performed well in this task (>80% correct), and the low w-values (0.04 and 0.22) indicate consistency in responses, that is, the majority of participants answered correctly in both pairs of questions. It therefore, seems that the participants effectively transferred their mathematics knowledge about coordinates in a Cartesian plane to the kinematics domain. The lowest average performance (84.2%) obtained in the second kinematics question (P-C2) is probably due to the need to estimate the position (y) value by using the scale, which seems to be more difficult than reading values from intersections of grid lines as is the case in the

In both sets of mathematics and physics questions on the representation task, five graphs of different forms were given (see Appendix). In the three pairs of questions, the participants had to match a straight-line, hyperbolic and quadratic function to one of the given mathematics graphs and linear motion equations to

The vast majority of participants knew that the mathematics function in item M\_R1 is a straight-line graph and chose either the correct one, option 1 (67.6%), or the additional straight-line, option 2 (21.2%). With regard to the hyperbolic and parabolic functions g(x) in item M\_R2 and h(x) in item M\_R3, respectively, more than 70% of participants related each to the correct graphs. In both latter cases, the second largest contingent of participants (about 20%) connected the hyperbolic function to the parabolic graph or vice versa. These participants seem to confuse the

With regard to the physics items on this task, the largest correct percentage (65.1%) was also obtained for the straight-line representation (P\_R1). The small wvalue of 0.03 indicates transfer of these participants' mathematics knowledge to kinematics. For the hyperbolic and parabolic equations only, small percentages of participants succeeded (about 38 and 29%, respectively). The large w-values (0.58 and 0.44) imply medium to practically significant differences in responses to the

representations of hyperbola and parabola in the mathematics contexts.

teristics of the tasks, as discussed below:

Theorizing STEM Education in the 21st Century

6.2.1 Coordinates (task 1)

Additional physics questions.

Table 6.

other questions.

kinematic graphs.

174

6.2.2 Representations (task 2)

between function values (velocity) and slope (acceleration), which corresponds to the height-slope confusion reported by McDermott et al. [13] and Beichner [1].

the participants could not perform the tasks successfully. It is therefore deduced that characteristics of tasks had an influence on the students' graph comprehension. With regard to the task on reading coordinates, the participants successfully performed the required visual decoding skill in both contexts. In the physics context, they attached conceptual meanings (position and time) to the x and y coordinates on the Cartesian plane. This elementary task underlies all other kinematics graph tasks. A problem that a minority of participants experienced was to estimate a

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

In the mathematics questions on representation tasks, most students successfully performed the visual decoding task of identifying and connecting the form and the equation of the three types of graphs. However, some experienced problems to correctly judge which one of the two given straight-line graphs resembles the hyperbolic function f(x) and which of the two parabolas are represented by the quadratic equation h(x). Hyperbolic-parabolic confusion that occurred amongst a

The participants' mathematics knowledge and understanding of matching expressions to types of graphs were only transferred to the physics domain in the case of straight-line graphs. With regard to the physics questions on parabolic and hyperbolic graphs, the majority of participants probably did not recognise correspondences in the kinematics expressions or graphs with the standard mathematical formats. This visual decoding problem may be based on the contextual task error, namely lack of understanding that the given kinematic equations are indeed functions, that is, s(t) and v(t). Consequently, their responses in the physics questions differed with medium to practical significance from those in the mathematics

The results on the area tasks indicate that the majority of participants have mathematical contextual knowledge related to areas of geometric forms and can execute the tasks of visual decoding (know what part on the graph is the area under the graph) and judgement (comparing the areas). In the corresponding physics questions, the participants firstly had to take the kinematics context of the questions into account before deciding what visual decoding and judgement tasks had to be done. The poor performance of the participants in the physics tasks indicated that they encountered problems in accomplishment of the contextual tasks. They seemed to lack knowledge and conceptual understanding of kinematics quantities and graphs, namely how to obtain the change in velocity from an acceleration-time and the change in position from a velocity-time graph. This knowledge deficiency was confirmed in the additional items on the area. Contextual difficulties in interpretation of the area under kinematics graphs were also found by Beichner [1],

Although participants' responses to questions on calculations of the slope of a straight line starting at the origin were correct, the other questions revealed deficiencies in the basic conceptual understanding of slopes in mathematics, namely that slope is the ratio of the change in y-values to the change in x-values. This hindered success in both contexts (with practical significance) in the tasks on the qualitative comparison of magnitudes of gradients as well as the understanding and application of negative and zero gradients. In these tasks, function value/slope confusion occurred, which was also reported by Beichner [1] and McDermott et al. [13]. This can be a contextual task error, but since the same confusion was encountered in the corresponding mathematics and physics questions, it is here also con-

Comparison of the performances in the corresponding mathematics and physics

tasks shows the following main trends causing success or failure in the physics

value using a scale.

domain.

minority of students also reveals judgement errors.

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

McDermott et al. [13] and Palmquist [16].

sidered as a judgement error.

questions:

177

In both the mathematics and physics quantitative contexts, the students performed much better in calculating the positive slopes starting at the origin (M\_S2 and P\_S2) than the zero slopes in later intervals (M\_S3 and P\_S3). According to the w-values, these pairs of questions were answered differently with small to medium effect, that is, similar mistakes were made. A reason for the very weak performances (16.5 and 7.9% correct) in items with zero slopes may be that the students do not understand that slope is the ratio of the change in y-values to the change in x-values. This is evident from the result that the majority of students (66.4% in M\_S3 and 57.7% in P\_S3) chose option 3 in these items where y/x instead of Δy/Δx is used for the slope. In the first pair of quantitative items (i.e., M\_S2 and P\_S2), y/x = Δy/Δx is valid, and the majority of participants (82.9 and 64.5%) consequently chose the correct option, even though they might have made the same error. Furthermore, area-slope confusion and slope/height confusion occurred amongst some of the participants. It thus seems that deficiencies in understanding the concept of gradient in mathematics has been transferred to the physics graphs.

## 7. Discussion of results

#### 7.1 Reader characteristics

Of the three reader characteristics evaluated (gender, last school year and faculty), none showed a practically significant difference in how the groups of students performed in the mathematics or the physics sections of the questionnaire. With regard to gender, male students outperformed female students in both the mathematics and physics sections with medium effect. Although the effects of the last school year were smaller, a larger effect was obtained for mathematics than physics. This result implies that students who had a gap of one or more year since their previous studies of mathematics performed observably weaker than those who did mathematics at school the previous year, although both groups performed badly in physics. An interesting result is the indifference of the faculty the students were enrolled in; engineering students performed similar to students from the humanities as well as from health and environmental sciences faculty.

#### 7.2 Task characteristics

The characteristics, namely context, visual decoding and judgement, of the tasks in the questionnaire are analysed in Table 1, and the results of the empirical investigation thereof are given in Tables 2 and 5. The main trends that were revealed are now discussed.

In the mathematics questions, the majority of participants were successful on reading coordinates (>90% correct), connecting representations (70% correct) and on qualitative and quantitative area tasks (65% correct). These participants showed conceptual understanding and effectively performed visual decoding and judgement tasks in the mathematics contexts. However, the majority of participants struggled with the tasks on slope, seemingly due to lack of conceptual understanding of the mathematical concept and calculation of slope.

In the physics domain, the majority of participants transferred and integrated their correct mathematics knowledge and skills on the reading coordinate task (>80% correct) as well as the representation of straight-line graphs (65% correct). In all other tasks, the average percentage was 50% or below, that is, the majority of

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

the participants could not perform the tasks successfully. It is therefore deduced that characteristics of tasks had an influence on the students' graph comprehension.

With regard to the task on reading coordinates, the participants successfully performed the required visual decoding skill in both contexts. In the physics context, they attached conceptual meanings (position and time) to the x and y coordinates on the Cartesian plane. This elementary task underlies all other kinematics graph tasks. A problem that a minority of participants experienced was to estimate a value using a scale.

In the mathematics questions on representation tasks, most students successfully performed the visual decoding task of identifying and connecting the form and the equation of the three types of graphs. However, some experienced problems to correctly judge which one of the two given straight-line graphs resembles the hyperbolic function f(x) and which of the two parabolas are represented by the quadratic equation h(x). Hyperbolic-parabolic confusion that occurred amongst a minority of students also reveals judgement errors.

The participants' mathematics knowledge and understanding of matching expressions to types of graphs were only transferred to the physics domain in the case of straight-line graphs. With regard to the physics questions on parabolic and hyperbolic graphs, the majority of participants probably did not recognise correspondences in the kinematics expressions or graphs with the standard mathematical formats. This visual decoding problem may be based on the contextual task error, namely lack of understanding that the given kinematic equations are indeed functions, that is, s(t) and v(t). Consequently, their responses in the physics questions differed with medium to practical significance from those in the mathematics domain.

The results on the area tasks indicate that the majority of participants have mathematical contextual knowledge related to areas of geometric forms and can execute the tasks of visual decoding (know what part on the graph is the area under the graph) and judgement (comparing the areas). In the corresponding physics questions, the participants firstly had to take the kinematics context of the questions into account before deciding what visual decoding and judgement tasks had to be done. The poor performance of the participants in the physics tasks indicated that they encountered problems in accomplishment of the contextual tasks. They seemed to lack knowledge and conceptual understanding of kinematics quantities and graphs, namely how to obtain the change in velocity from an acceleration-time and the change in position from a velocity-time graph. This knowledge deficiency was confirmed in the additional items on the area. Contextual difficulties in interpretation of the area under kinematics graphs were also found by Beichner [1], McDermott et al. [13] and Palmquist [16].

Although participants' responses to questions on calculations of the slope of a straight line starting at the origin were correct, the other questions revealed deficiencies in the basic conceptual understanding of slopes in mathematics, namely that slope is the ratio of the change in y-values to the change in x-values. This hindered success in both contexts (with practical significance) in the tasks on the qualitative comparison of magnitudes of gradients as well as the understanding and application of negative and zero gradients. In these tasks, function value/slope confusion occurred, which was also reported by Beichner [1] and McDermott et al. [13]. This can be a contextual task error, but since the same confusion was encountered in the corresponding mathematics and physics questions, it is here also considered as a judgement error.

Comparison of the performances in the corresponding mathematics and physics tasks shows the following main trends causing success or failure in the physics questions:

between function values (velocity) and slope (acceleration), which corresponds to the height-slope confusion reported by McDermott et al. [13] and Beichner [1]. In both the mathematics and physics quantitative contexts, the students performed much better in calculating the positive slopes starting at the origin (M\_S2 and P\_S2) than the zero slopes in later intervals (M\_S3 and P\_S3). According to the w-values, these pairs of questions were answered differently with small to medium effect, that is, similar mistakes were made. A reason for the very weak performances (16.5 and 7.9% correct) in items with zero slopes may be that the students do not understand that slope is the ratio of the change in y-values to the change in x-values. This is evident from the result that the majority of students (66.4% in M\_S3 and 57.7% in P\_S3) chose option 3 in these items where y/x instead of Δy/Δx is used for the slope. In the first pair of quantitative items (i.e., M\_S2 and P\_S2), y/x = Δy/Δx is valid, and the majority of participants (82.9 and 64.5%) consequently chose the correct option, even though they might have made the same error. Furthermore, area-slope confusion and slope/height confusion occurred amongst some of the participants. It thus seems that deficiencies in understanding the concept of gradient in mathematics has been transferred to the physics graphs.

Of the three reader characteristics evaluated (gender, last school year and faculty), none showed a practically significant difference in how the groups of students performed in the mathematics or the physics sections of the questionnaire. With regard to gender, male students outperformed female students in both the mathematics and physics sections with medium effect. Although the effects of the last school year were smaller, a larger effect was obtained for mathematics than physics. This result implies that students who had a gap of one or more year since their previous studies of mathematics performed observably weaker than those who did mathematics at school the previous year, although both groups performed badly in physics. An interesting result is the indifference of the faculty the students were enrolled in; engineering students performed similar to students from the humani-

The characteristics, namely context, visual decoding and judgement, of the tasks

In the mathematics questions, the majority of participants were successful on reading coordinates (>90% correct), connecting representations (70% correct) and on qualitative and quantitative area tasks (65% correct). These participants showed conceptual understanding and effectively performed visual decoding and judgement tasks in the mathematics contexts. However, the majority of participants struggled with the tasks on slope, seemingly due to lack of conceptual understand-

In the physics domain, the majority of participants transferred and integrated their correct mathematics knowledge and skills on the reading coordinate task (>80% correct) as well as the representation of straight-line graphs (65% correct). In all other tasks, the average percentage was 50% or below, that is, the majority of

in the questionnaire are analysed in Table 1, and the results of the empirical investigation thereof are given in Tables 2 and 5. The main trends that were

ties as well as from health and environmental sciences faculty.

ing of the mathematical concept and calculation of slope.

7. Discussion of results

Theorizing STEM Education in the 21st Century

7.1 Reader characteristics

7.2 Task characteristics

revealed are now discussed.

176


as well as the physics concepts and principles. Only then they may be able to blend

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

In the physics classroom, the students have to be taught how to use their existing mathematics to solve the problems at hand. The instructor has to revise the relevant existing mathematics as well as physics knowledge and draw analogies between aspects such as geometric figures, expressions and graphic representations of functions, etc. Differences in discipline characteristics need to be discussed with the students so that they understand the purpose and applications of graphs in the two

Further research can be conducted for follow-up years after specific interventions have been done to specifically address the problems identified. This questionnaire can also be used by other lecturers for research purposes or to test their students' abilities and identify areas of concern and come up with intervention

It is thus recommended that lecturers of undergraduate introductory physics should emphasise the knowledge and skills of algebraic graphs in teaching and learning of kinematics, especially kinematics graphs. This will enable these students to collect data, analyse it, plot graphs and interpret graphs based on this knowledge

The researcher would like to acknowledge that this study is a result of Masters Research Study conducted and completed at the North West University, Potchefstroom campus. The researcher further acknowledges the influence of Dr Miriam Lemmer and Dr Mariette Hitge in their supervision, guidance, and leadership towards successful completion of Masters and compilation of this

and relate it to and show physics understanding and knowledge.

the knowledge effectively.

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

9. Recommendation

contexts.

strategies thereof.

Acknowledgements

chapter.

179


#### 7.3 Discipline characteristics

The results indicated that the majority of participants have an understanding of the physics discipline characteristics with regard to the use of kinematics concepts as variables that can be presented as coordinates on Cartesian planes. In the physics tasks on reading coordinates, they attached symbolic meanings (position and time) to the x and y coordinates. They also recognised correspondences between a linear motion equation and the standard mathematical format for straight lines in a representation task. However, they seem not to have the insight that kinematics relationships can be represented as functions, especially with regard to quadratic (parabolic) and hyperbolic functions. In addition, students failed to attach physical meaning to the area under graphs and slopes of graphs in the kinematics contexts.

In order for the participants to solve the physics questions correctly, they did not only have to know the discipline characteristics concerning kinematics graphs but also the discipline characteristics of graphs in mathematics. There are practices that are similar for mathematics and physics, for example, using the Cartesian coordinate system and placing the dependent variable on the vertical axis. Also, concepts such as slope and area are calculated the same in both contexts. Discipline characteristics that differ are, for example, that in mathematics, variables are abstract and have no units, whilst in physics variables, area under graphs and gradients all have physical meanings and units. Another difference is that in mathematics, the horizontal axis has a positive and negative side, whereas in kinematics, the concept time as the independent variable is on the horizontal axis and starts from zero only. The latter difference probably contributed to the significant differences in students' responses on the hyperbolic and parabolic representations. The kinematics graphs only showed the parts of the hyperbola or parabola for which the x-coordinate (time) is positive, which might have prevented students from recognising the graph form.

From the results of this study, it is clear that if students know the underlying mathematics, it does not imply that they can use it in another context. There is no automatic transfer from the mathematics domain to the physics domain when using mathematics to solve a physics problem. For a student to be able to solve a certain physics problem, he/she has to know and understand the underlying mathematics

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

as well as the physics concepts and principles. Only then they may be able to blend the knowledge effectively.
