Generalisation of Linear Figural Patterns in Secondary School Mathematics (original) (raw)
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The Seventh Grade Students’ Generalization Strategies of Patterns
Journal of Education and Learning (EduLearn)
This article describes a generalization strategy on pictorial visual patterns. This explorative descriptive study involves 60 students of 7 Grade Student of private junior high school in Tuban East Java Indonesia. Data obtained through the pattern generalization task. The type of pattern used in this research is pictorial sequences with two non-consecutive terms. Selection of a pictorial sequences with two non-consecutive pattern to focus students' attention on visual stimuli. Based on the students answers of pattern generalization task, there are 33 students who answered correctly and 27 students answered wrong. From the correct answer, there are six different general formula representations. The visualization strategy used by the students begins by splitting the image into smaller elements. The way students break down into smaller elements is also diverse. Students divide the image in the form of V (2 matchsticks), U shape (3 matchsticks), square shape (4 matchsticks) and lastly divide in a unit additive consisting of 7 matchsticks.
Repeating patterns: Strategies to assist young students to generalise the mathematical structure
Australasian Journal of Early Childhood
THIS PAPER FOCUSES ON VERY young students' ability to engage in repeating pattern tasks and identifying strategies that assist them to ascertain the structure of the pattern. It describes results of a study which is part of the Early Years Generalising Project (EYGP) and involves Australian students in Years 1 to 4 (ages 5 – 10). This paper reports on the results from the early years' cohort (Year 1 and 2 students). Clinical interviews were used to collect data concerning students' ability to determine elements in different positions when two units of a repeating pattern were shown. This meant that students were required to identify the multiplicative structure of the pattern. Results indicate there are particular strategies that assist students to predict these elements, and there appears to be a hierarchy of pattern activities that help students to understand the structure of repeating patterns.
The study investigates students' strategies involved in the generalization of "linear patterns". The study followed the qualitative research approach by conducting task-based interviews with twenty-nine primary second grade students from different high, intermediate and low ability levels. Results of the study presented several strategies involved in the generalization of the patterns including visual, auditory, mental, finger counting, verbal counting, and traditional (paper and pencil) strategies. The findings revealed that the type of the assigned pattern (simple or complex) and the type of the structure of the pattern itself (increasing or decreasing) play a big role for students' strategies involved to either discover the rule of the pattern or to extend it. However, students in early ages could master several skills and choose appropriate procedures to deal with patterns, which indicate that they could develop their algebraic thinking from early stages. Findings of the study also revealed that using different senses, using the idea of coins, using the numbers line, recognizing musical sounds, using concrete materials like fingers, applying different visual and mental strategies, and even applying traditional calculations could help students to work with " linear patterns". It is recommended that teachers introduce different strategies and procedures in teaching patterns to meet the needs of students as different learners, give them the opportunities to develop their thinking strategies and explore their thoughts. More research is recommended to explore students' strategies involved in the generalization of different kinds of patters at different stages.
Using Pattern Tasks to Develop Mathematical Understandings and Set Classroom Norms
Mathematics Teaching in the Middle School
The capacity to reason algebraically is critical in shaping students' future opportunities and, as such, is a central theme of K–12 education (NCTM 2000). One component of algebraic reasoning is “the capacity to recognize patterns and organize data to represent situations in which input is related to output by well-defined functional rules” (Driscoll 1999, p. 2). Geometric pattern tasks can be a useful tool for helping students develop algebraic reasoning, because the tasks provide students with opportunities to build patterns with materials such as toothpicks or pattern blocks. These materials help students “focus on the physical changes and how the pattern is being developed” (Friel, Rachlin, and Doyle 2001, p. 10). Such work might help bridge students' earlier mathematical experiences and lay the foundation for more formal work in algebra (English and Warren 1998; Ferrini-Mundy, Lappan, and Phillips 1997; NCTM 2000). Finally, the relationships between the quantities in pa...
The Thinking Process of Students Using Trial and Error Strategies in Generalizing Linear Patterns
Numerical: Jurnal Matematika dan Pendidikan Matematika, 2020
Patterns generalization learning at the junior high school is more emphasis on the generalization of linear patterns. One problem in generalizing linear patterns is that students do not know the process of using trial and error strategies to generalize linear patterns. For this reason, the purpose of this study was to analyze the thought processes of 2 junior high school students who succeeded in generalizing linear patterns using trial and error strategies. The results show that there are two trial and error strategies that can be used to generalize linear patterns, namely: (1) Trial and error strategy by looking at the relationship of quantity consists of three steps. The first step is called relating, namely, the subject connects between the first term, the term in question, and difference. The second step is called searching, where the subject finds similarities by using addition and subtraction operations to obtain the nth term formula. The third step is called extending; the s...
The examination of 7th grade students’ achievements in mathematical patterns
The Eurasia Proceedings of Educational & Social Sciences, 2016
The aim of this study is to determine the 7 th grade students' achievements in mathematical patterns presented by figures, tables, number sequences, and word problems. This research is a situation determination study where quantitative methodology is used. The sample of the study consisted of 47 female and 50 male students, totally 97 students from 7 th grades in Giresun city on 2015-2016 academic year. Pattern topic oriented 7 questions were used as the data gathering tool. The questions which focused on the attainments of pattern topic were prepared by the researcher. It was determined that students could perform specialization in figural patterns but they couldn't reach a generalization. In other words, it was observed that students could find the required steps according to a given rule and so they could easily reach the situation which involved operational knowledge. Also, it was seen that although students found the number of figures in the next step of the pattern, they couldn't find the general rule that represented the pattern. Another result of the study was that students could recognize patterns of number sequences but they couldn't find the general term of the pattern. Lastly, it was determined that students couldn't understand patterns which were represented as word problems and they failed at these kind of pattern questions but they had success in pattern questions represented by tables. In this context, it can be offered to give much place to representation forms of patterns by figures, tables, number sequences, and word problems while students are given experiences of patterns.
Factors Affecting Students' Performance in Generalizing Algebraic Patterns
Factors Affecting Students' Performance in Generalizing Algebraic Patterns, 2019
Pattern generalization is indispensable to the development of algebraic thinking; however, students in early and even in late middle school are struggling in generalizing patterns. This study identifies the students' perceived factors that affect their performance in generalizing algebraic patterns and describes how each factor affects their process of generalization. This study used the qualitative descriptive design. The students were given open-ended problems which require them to generalize patterns. Results show that the factor which greatly influences the students' performance was associated to the structure of the task such as the size of the values and problem presentation. The ability to derive algebraic symbol influences students' thinking of getting better result while lacking this ability despite their complete and accurate solution leaves them the feeling of arriving at incorrect answer. With these results, teachers should help students to structure and organize their informal methods rather than focus on how to use mathematical formulas. Teachers should expose them to pattern-based problem solving with considerations to the elements that influences their performance.
GENERALIZATION OF PATTERNS: THE TENSION BETWEEN ALGEBRAIC THINKING AND ALGEBRAIC NOTATION
This study explores the attempts of a group of preservice elementary school teachers to generalize a repeating visual number pattern. We discuss students' emergent algebraic thinking and the variety of ways in which they generalize and symbolize their generalizations. Our results indicate that students' ability to express generality verbally was not accompanied by, and did not depend on, algebraic notation. However, participants often perceived their complete and accurate solutions that did not involve algebraic symbolism as inadequate.
Differentiation of Students' Reasoning on Linear and Quadratic Geometric Number Patterns
There are two purposes in this study. One is to compare how 7th and 8th graders reason on linear and quadratic geometric number patterns when they have not learnt this kind of tasks in school. The other is to explore the hierarchical relations among the four components of reasoning on geometric number patterns: understanding, generalizing, symbolizing, and checking, and to differentiate them between linear and quadratic geometric number patterns. From the national survey results, we argue that reasoning on geometric number patterns is a proper initial activity for learning algebraic thinking in Grade 7, and the relations between the checking component and the other components appear to be different between linear and quadratic patterns. Therefore, we propose that checking can play two kinds of role in reasoning on geometric number patterns. One is to induce a strategy for generalizing, and the other is to initiate the development of symbolizing after it is integrated with generalizing.