Teacher and Student Choices of Generalising Strategies: A Tale of Two Views? (original) (raw)

Types of Generalization Made by Pupils Aged 12–13 and by Their Future Mathematics Teachers

Scientia in educatione, 2021

This paper seeks to establish what kind of arguments pupils (aged 12–13) use and how they make their assumptions and generalizations. Our research also explored the same phenomenon in the case of graduate mathematics teachers studying for their masters’ degrees in our faculty at that time. The main focus was on algebraic reasoning, in particular pattern exploring and expressing regularities in numbers. In this paper, we introduce the necessary concepts and notations used in the study, briefly characterize the theoretical levels of cognitive development and terms from the Theory of Didactical Situations. We set out to answer three research questions. To collect the research data, we worked with a group of 32 pupils aged 12–13 and 19 university students (all prospective mathematics teachers in the first year of their master’s). We assigned them two flexible tasks to and asked them to explain their findings/formulas. Besides that, we collected additional (supportive) data using a short...

Promoting and Assessing Mathematical Generalising

Australian primary mathematics classroom, 2015

The difficulties faced by young children when generalising are highlighted and ideas about how teachers can select and design tasks that promote generalisation are provided.

Pre-service primary teachers' approaches to mathematical generalisation

2017

In our teaching with primary pre-service teachers (PSTs), each of us includes generalising tasks in the context of mathematical reasoning. We set out to explore the value of such activity from the perspective of PSTs and their approaches to generalisation. In this paper, we focus on one PST’s mathematical reasoning when working on the ‘flower beds’ problem. We analyse the ways in which this PST attends to: looking for a relationship; seeing structure within a single figure in a sequence; and seeing sameness and difference between figures in a sequence. We consider what motivates shifts in attention, we reflect on the significance of students’ prior experience, and of student collaboration in our teaching sessions.

Generalization in the Learning of Mathematics

Generalization is one of the fundamental activities in the learning of mathematics. The growing and improving mathematics is depended to applications of generalization from beginning until now. It seems that generalization needs to be introduced more among people who deal with mathematics. The main aim of this conceptual study is to explore the meanings of generalization from psychological and mathematical perspectives. In addition, the importance of generalization to support students in the learning of mathematical concepts is put forward.

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.

Enhancing students’ generalizations: a case of abductive reasoning

The aim of this paper is to understand how a path of teacher's actions leads to students' generalization. Generalization, as a main process of mathematical reasoning, may be inductive, abductive, or deductive. In this paper, we focus on an abductive generalization made by a student. The study is carried out in the third cycle of design of a design-based research involving lessons about linear equations in a grade 7 class. Data is gathered by classroom observations, video and audio recorded, and by notes made in a researcher's logbook. Data analysis focus on students' generalizations and on teacher's actions during whole-class mathematical discussions. The results show a path of teacher's actions, with a central challenging action, that allowed an extending abductive generalization, and also a subsequent deductive generalization.

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.

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.

Th e Strategies of Using the Generalizing Patterns of the Primary School 5th Grade Students

Th e main purpose of this study is to determine the strategies of using the generalizing patterns of the primary fifth grade students. Th e practice of this research is conducted on twelve students, which have high, middle and low success levels. Task-based interviews and students journals are used as the tools for data collection. For the analysis of the data, a classification method including “data reduction”, “data display” and “drawing conclusion and verification” are used. At the end of the research, it is seen that the visual and numerical approaches are adopted in the generalization of patterns and the visual approach is made easy for generalization, as well. In generally, the present strategies in the generalizing of patterns are also taken into account of near or far generalizing. Th e recursive strategies are used in the near generalizing. However, the explicit strategies are determined in using far generalizing.