Problem - based scaffolding for prospective mathematics teachers in graph theory course

of scaffolding by teachers

incomplete comprehension; (4).Providing explain clarification for pupils who lack the understanding to accomplish the assigned assignment (Fisher and Frey, 2010;Qamar and Riyadi, 2016;Haryati et al., 2020;Tiaradipa et al., 2020).Problem-based learning (PBL) is an instructional approach that places the student at the center of the learning process to promote active engagement and enhance the meaningfulness of the learning experience (Sudarsana et al., 2019;Asrial et al., 2021;et al., 2022;Wulandari, Khasanah and Octavianti, 2022;Rafiq, Triyono and Djatmiko, 2023;Ssemugenyi, 2023).Problem-based learning has the potential to generate enthusiasm among students for the study of Graph Theory (Wulandari, Khasanah and Octavianti, 2022).
Based on the preliminary observation, it has been noted that certain pupils encounter challenges when attempting to ascertain the quantity of edges within a graph.The subsequent passage illustrates the discoveries made by scholars in the challenges encountered by students in comprehending the principles of graph theory.

Figure 1. Student's trouble
Thus, the researchers employed scaffolding techniques in this study to address the challenges faced by students, namely by introducing authentic graph theory issues.The aim was to facilitate the process of overcoming these difficulties.

RESEARCH METHOD
This type of study is Qualitative research.The study was conducted on third-year students at Wisnuwardhana University Malang.The Graph Theory class was held once every week.At the beginning of the Graph Theory course, a brief review of the set.From the learning process, students have difficulty with concepts in Graph representation, which will be given scaffolding.The participants in this study were lecturers teaching Graph Theory courses and students in one class/offering.One class consists of heterogeneous male and female students in terms of mathematical ability.Data were collected through practice problems consisting of student work, interviews with lecturers, and field notes (Bassey, 2000).In this study, the scaffolding given to students is based on contextual problems that have been designed in such a way as to assist students in solving a mathematical problem on Graph Theory material, especially the concept of Graph representation.This study discusses the scaffolding process of 4 different problems: the road system problem, food chain predator behavior, chemical isomers, and graph isomorphism.

Table 1. Forms of Questioning Scaffolding (Road System Problem)
According to the data presented in Table 1, the subject can provide answers.However, it is recommended to employ scaffolding techniques such as asking to ascertain the subject's comprehension of the answer.This will allow for a thorough assessment of the subject's understanding of the question and their response.According to the data presented in Table 2, it can be observed that the participant exhibited a delay in submitting his response, prompting the provision of questioning scaffolding.To provide scaffolding in the form of inquiry, it is seen that the subject's response still contains faults.Consequently, when scaffolding is provided through prompting, the subject becomes aware of the location of their mistake.Table 3. Form of Cueing Scaffolding (Chemical Bonding Problems) According to the data presented in Table 3, the participant has employed reasoning to address the problem.However, uncertainty is still expressed by the statement, " Still need to think of it, Mum" Therefore, it is necessary to provide scaffolding in the form of cueing, as illustrated in Table 3.

Table 4. Form of Explaining Scaffolding (Graph degree sequence problem)
According to the data presented in Table 4, the scaffolding approach that exhibits the highest level of support is the explained scaffolding form.This form is preceded by the questioning, prompting, and cueing scaffolding forms.This phenomenon may occur when the subject's comprehension remains uncertain, facilitating the acceptance of the process through explanatory scaffolding.
ResearcherDraw the graph representing the road system in the figure besides, and determine the number of vertices, the number of edges, and the degrees of each vertex.Can you draw it?(because the subject did not draw the graph, only mentioned the number of vertices, edges, and degree of each vertices).SubjectYes, ma'am.(subject starts to draw the graph)Researcher Try to label the graph according to the known image.SubjectHere you go, ma'am, labeling the graph.ResearcherCorrect.Now, explain how to determine the number of vertices, edges, and degree of each point.SubjectThere are 5 vertices, ma'am, 7 sides, and the degree of vertices P=2, Q=4, R=2, T=3, S=3.Researcher What are the vertices?What is the degree of the vertex?SubjectThe vertices are P, Q, R, S, T. If the vertex degree is obtained from the number of edges directly connected to the vertex.P=2 (Q,S); Q=4 (P,S,T,R); R=2 (Q,T); T=3 (R,Q,S); S=3 (P,Q,T)

Researcher
Snakes are known to consume frogs, whereas birds are known to prey on spiders.It is worth noting that both birds and spiders exhibit a dietary preference for insects.Additionally, frogs are observed to consume snails, spiders, and insects as part of their feeding behavior.Please create a digraph illustrating the predatory behavior described.Subject Bu Like this Mum Researcher Are you sure about the digraph?Subject Long silence.Sure Mum Researcher Explain the predatory behavior of the problem presented.Subject Snakes are known to consume frogs, while birds are known to prey on spiders.It is worth noting that birds and spiders have a dietary preference for insects.Additionally, frogs exhibit a feeding behavior that includes the consumption of snails, spiders, and insects.We have been tasked with creating a digraph depicting the predatory behavior.Researcher What do the vertices represent?And what do the directed edges represent?Subject The conclusion is that animals in problems and directed sides eat.Right ma'am?Researcher The vertices are the animals in the problem, and the directed side is eating.Right, mum? Subject (Observing) Snakes eat frogs, right Frogs eat spiders, insects, and snails.Correct Spiders eat insects, right Insect eats bird.Wrong The bird eats spider, right Something is wrong, Mum; it should be an insect, not an insect-eating bird.So, the direction of my directed side needs to be corrected.Researcher Right, now you understand.Then what's the keyword?Subject As you asked earlier, we must know what the vertex represents and the directed side.Researcher Correct.

Researcher
Figure beside depicts the chemical structures of methane (  ) and propane (    ).(i) When considering these diagrams as graphs, it is crucial to analyze the characteristics of the vertices that represent carbon atoms () and hydrogen atoms ().There exist two distinct chemical compounds denoted by the formula     .Please illustrate the molecular structures by plotting the related graphs.Subject Here's the thing, Mum Researcher Is there another form with the same molecular formula  4  10 ?Subject Long silence Researcher How is it?Are there any other forms?Subject Confused, Mum, that is it.Researcher Notice what the vertices and sides are?Subject The nodes are Hydrogen atoms and carbon atoms.Researcher Try to observe that carbon atoms have what degree?And hydrogen atoms have what degree?Subject (observe) Carbon atoms have degree 4 and hydrogen atoms have degree 1. Researcher Right now, try to make a graph of another molecular arrangement of  4  10 .Subject One second, mum (thinking) Researcher The key is that the Carbon atom has degree 4, and the hydrogen atom has degree 1. Subject Still need to think of it, Mum.Researcher Do Carbon atoms always have to be straight chains?Try to make the Carbon atoms branched.Can it be?Subject I'll try it first mum.Researcher ok, try to make it first.Subject Tell you what Mum.Researcher Yes, that's right.Do you understand?What is the keyword, then?Subject We refer to the degree of Carbon atoms and Hydrogen atoms.Mum Researcher Yes, that's right, so that we can create different graphs from the arrangement of molecules with the same molecular formula.These graphs are called isomers in chemistry.