Generalizing: The Core of Algebraic Thinking (original) (raw)
Related papers
Generalization strategies and representations used by final-year elementary school students
International Journal of Mathematical Education in Science and Technology
Recent research has highlighted the role of functional relationships in introducing elementary school students to algebraic thinking. This functional approach is here considered to study essential components of algebraic thinking such as generalization and its representation, and also the strategies used by students and their connection with generalization. This paper jointly describes the strategies and representations of generalisation used by a group of 33 sixth-year elementary school students, with no former algebraic training, in two generalisation tasks involving a functional relationship. The strategies applied by the students differed depending on whether they were working on specific or general cases. To answer questions on near specific cases they resorted to counting or additive operational strategies. As higher values or indeterminate quantities were considered, the strategies diversified. The correspondence strategy was the most used and the common approach when students generalised. Students were able to generalise verbally as well as symbolically and varied their strategies flexibly when changing from specific to general cases, showing a clear preference for a functional approach in the latter.
Continuity Between Students' Experiences Solving Problems in
We discuss opportunities to build better continuity between students' experiences in arithmetic and in algebra by examining ways that external representations can be used to solve problems. We use examples from our research on algebra learning to illustrate often overlooked complexities that arise when using a single representation to analyze relationships and patterns of change between two covarying quantities. We use the term adaptive interpretation to describe ways in which, in the course of solving problems about situations that contain covarying quantities, students must shift their perspective on a representation as they shift their thinking about the situation. One set of examples demonstrates students' difficulties shifting their perspective on equations when shifting their attention from a varying quantity in a situation to a specific unknown value of that quantity. A second demonstrates students' difficulties shifting their perspective on tables and graphs w...
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 strategies of beginning high school algebra students
2005
This is a qualitative study of 22 9 th graders performing generalizations on a task involving linear patterns. Our research questions were: What enables/hinders students' abilities to generalize a linear pattern? What strategies do successful students use to develop an explicit generalization? How do students make use of visual and numerical cues in developing a generalization? Do students use different representations equally? Can students connect different representations of a pattern with fluency? Twenty-three different strategies were identified falling into three types, numerical, figural, and pragmatic, based on students' exhibited strategies, understanding of variables, and representational fluency.
– Algebra is generally considered as manipulating symbols, while algebraic thin king is about generalization. Patterns can be used for generalizat ion to develop early graders' algebraic thinking. In the generalization of pattern context, the purpose of this study is to investigate middle school students' reasoning and strategies at different grades when their algebraic thin king begin s to develop. First, 6 open-ended linear growth pattern problems as numeric, pictorial, and tabular representations were asked to 154 middle g rade students. Next, two students from each grade (6 th , 7 th , and 8 th grade) were interviewed to investigate how they interpret the relationship in different represented patterns, and which strategies they use. The findings of this study showed that students tended to use algebraic symbolis m as their grade level was increased. However, the students' conceptions about 'variable' we re troublesome.
The role of representations in promoting the quantitative reasoning
2017
In this communication, we discuss the role of representations in the development of conceptual knowledge of 2nd grade students involved in additive quantitative reasoning through the analysis of the resolutions of two tasks that present transformation problems. Starting to discuss what is meant by additive quantitative reasoning and mathematical representation, we present after some empirical results in the context of a teaching experiment developed in a public school. The results show the difficulties with the inverse reasoning present in both situations proposed to students. Most students preferably use the symbolic representation, using also the written language as a way to express the meaning attributed to its resolution. The iconic representation was used only by a pair of students. Representations have assumed a dual role, that of being the means of understanding the students' thinking, and also supporting the development of their mathematical thinking.
Factors Influencing Students' Generalisation Thinking Processes
In this study we presented students with generalisation activities in which we varied the representation along several dimensions, namely the type of function, the nature of the numbers, the format of tables, and the structure of pictures. Our results show that varying these dimensions has little effect on children's thinking-as in our previous study, few children tried to find a functional relationship between the variables, but persisted with using the recursive relationship between function values, making many logical errors in the process.