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Problem-solving can be understood as a cognitive process in which students know facts, processes, concepts, and procedures and then apply the knowledge to solve problems in real situations. Indonesia’s national average achievement of numeracy skills in 2021, the cognitive process of competence reasoning is higher than the competencies of knowing and applying. This study aims to analyze students' cognitive processes in solving numeration problems related to the algebraic domain. The algebraic domain in this study is limited to competencies in making generalizations from patterns in number sequences and object configuration sequences. This research was conducted qualitatively with a phenomenological design using three high-category and three low-category students to achieve data saturation. The supporting instruments are students' answers and interview results related to the algebraic domain. This study concluded that students' cognitive processes in solving numeracy problems associated with the algebraic domain in the high and low categories have different descriptions. This difference in intelligence has an impact when solving math problems. This research can help enrich the understanding of students' cognitive processes and contribute to the development of better mathematics learning strategies and curricula.

Giving a high level modification story problems with multistep routine and non-routine problems in cartesian coordinate learning is an effort to achieve optimal learning and also important because it has good effect for students. Less than optimal learning of cartesian coordinates for eight grade it was usually represented by the frequent of student errors occurrence on an ongoing basis in solving on mathematical problems in assignments and daily tests. This research purpose was to reveal the dominant factor were caused eighth grade student errors in solving on cartesian coordinate multistep routine and non-routine modification story problems. This research is a descriptive qualitative in type. Research data collection based on survey, interview, test and documentation. The research subjects in this research were eight grade students of SMPN 4 Wonosobo. In this research 18 students of eight grade students SMPN 4 Wonosobo as the research sample. The sampling technique it was used in this research is purposive sampling. The results of this research indicate the dominant factor were caused eighth grade student errors in solving on cartesian coordinate multistep routine and non-routine modification story problems is students difficulties. Students difficulties such as compiling steps to solve problems in the form of multistep story problems, understanding the position conception of starting point with a certain point, understanding the conception of a perpendicular lines and a parallel lines to the X and Y axis, determining the distance of a point to certain point. The implication of this research is can obtain information about the dominant factor were caused student errors in solving on cartesian coordinate multistep routine and non-routine modification story problems as the relation to the current idealization of mathematics learning optimization and can be able for planing another appropriate and solution steps for the implementation of mathematics learning in an effort to minimize student errors.

Let G(V,E) be a simple and connected graph which set of vertices is V and set of edges is E. Irregular reflexive k-labeling f on G(V,E) is assignment that carries the numbers of integer to elements of graph, such that the positive integer {1,2, 3,...,ke} assignment to edges of graph and the even positive integer {0,2,4,...,2kv} assignment to vertices of graph. Then, we called as edge irregular reflexive k-labelling if every edges has different weight with k = max{ke,2kv}. Besides that, there is definition of reflexive edge strength of G(V,E) denoted as res(G), that is a minimum k that using for labeling f on G(V,E). This paper will discuss about edge irregular reflexive k-labeling for sun graph and corona of cycle and null graph, denoted by Cn ⨀ N2 and make sure about their reflexive edge strengths.

Tujuan penelitian ini adalah untuk menganalisis kemampuan pemahaman konsep matematis siswa ditinjau dari kemampuan awal matematika.penelitian ini adalah penelitian deskriftif eksploratif dengan pendekatan kualitatif. Penelitian ini dilaksanakan di SMA Muhammadiyah 5 Yogyakarta Tahun Ajaran 2020/2021 pada kelas X.subjek penelitian ini terdiri dari dua siswa dengan kemampuan awal tinggi, dua siswa kemampuan awal sedang dan dua siswa kemampuan awal rendah. Sumber data penelitian ini adalah hasil tes dan hasil wawancara. Focus penelitian ini adalah untuk mendeskripsikan pemahaman siswa tentang logaritma berdasarkan kemampuan awal yang dimiliki. Hasil penelitian ini menunjukkan bahwa pemahaman konsep siswa dipengaruhi oleh kemampuan awal yang mereka miliki. Sehingga siswa dengan kemamapuan awal tinggi mendominasi lebih menguasai indikatoer pemahaman konsep dibandingkan dengan siswa berkemampuan awal sedang atau rendah.

Assume that G(V,E) is a graph with V and E as its vertex and edge sets, respectively. We have G is simple, connected, and undirected. Given a function λ from a union of V and E into a set of k-integers from 1 until k. We call the function λ as a totally irregular total k-labeling if the set of weights of vertices and edges consists of different numbers. For any u ∈ V, we have a weight wt(u)=λ(u)+ ∑{uy ∈ E} λ(uy). Also, it is defined a weight wt(e)= λ(u)+ λ(uv) + λ(v) for each e=uv ∈ E. A minimum k used in k-total labeling λ is named as a total irregularity strength of G, symbolized by ts(G). We discuss results on ts of some caterpillar graphs in this paper. The results are ts(S{p,2,2,q}) = ⌈ (p+q-1)/2 ⌉ for p, q greater than or equal to 3, while ts(S{p,2,2,2,p}) = ⌈(2p-1)/2 ⌉, p ≥ 4.

Given graph G(V,E). We use the notion of total k-labeling which is edge irregular. The notion of total edge irregularity strength (tes) of graph G means the minimum integer k used in the edge irregular total k-labeling of G. A cactus graph G is a connected graph where no edge lies in more than one cycle. A cactus graph consisting of some blocks where each block is cycle Cn with same size n is named an n-uniform cactus graph. If each cycle of the cactus graph has no more than two cut-vertices and each cut-vertex is shared by exactly two cycles, then G is called n-uniform cactus chain graph. In this paper, we determine tes of n-uniform cactus chain graphs C(Cnr) of length r for some n ≡ 0 mod 3. We also investigate tes of related chain graphs, i.e. tadpole chain graphs Tr(4,n) and Tr(5,n) of length r. Our results are as follows: tes(C(Cnr)) = ⌈(nr + 2)/3⌉ ; tes(Tr(4,n)) = ⌈((5+n)r+2)/3⌉ ; tes(Tr(5,n)) = ⌈((5+n)r+2)/3⌉.

Problem solving is significant skill in facing 21st century. Students have different levels of problem solving. The levels that explain individual abilities in solving mathematics problems are called as cognitive level. It consists of three levels; cognition, metacognition, and epistemic cognition. The cognitive level influences individual in understanding the problem and deciding the right strategies to solve it. The material used to obtain the data is three variables linear equation system, by involving two high ability students at the tenth grade from one of the state senior high schools in Pati. This research employs qualitative method, which the data are collected through task-based interview and time triangulation is applied to validate the data. The results reveal that at the cognition level, students with high ability have an understanding related to conceptual and procedural knowledge, but the ability of students in factual knowledge is low, especially in terminology skill; at the metacognition level, the students are able to write down each strategy in each method, but it is not detail; while at the epistemic cognition level, they are able to explain the weaknesses of a method, but they have not been able to provide a solution to overcome its weaknesses.