9. Recommendation

1.Participants have the correct mathematics knowledge and conceptual understanding and transfer it to the physics domain, for example, when

2.Participants reveal the mathematics knowledge but lack the necessary physics knowledge and conceptual understanding, for example, in the kinematics tasks on area and slope, they seem not to know which kinematics relation to use and

3.Participants are unable to blend mathematics and physics knowledge, for example, they do not perceive the kinematic equations as quadratic and represented by a parabola or as a hyperbolic expression and graph.

4.Participants transferred their misconceptions or insufficient knowledge in mathematics to the physics domain. This is evident in the height-slope confusion, area-slope confusion and parabola-hyperbola confusions that occurred in both the mathematics and physics domains. Inaccurate knowledge of the slope as the ratio of change in variables was to a large extent transferred

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

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

reading coordinates.

Theorizing STEM Education in the 21st Century

what to calculate.

from mathematics to kinematics.

7.3 Discipline characteristics

graph form.

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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 contexts.

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 strategies thereof.

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 and relate it to and show physics understanding and knowledge.
