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Advanced Mathematical Thinking and the Learning of Numerical

Analysis in a Context of Investigation Activities

Ana Henriques

Naval Academy, Portuguese Navy

Doctoral student at the University of Lisbon

Introduction

The use of investigation activities, at different school levels, influences the evolution of

students conceptions about mathematics and may provide an alternative route for stu-

dents mathematical learning. Besides, a continuous experience in carrying out mathe-

matical investigations may support the development of students mathematical reason-

ing and may become an important element of the teaching process (Ponte, 2003, 2007).

Nevertheless, research results are still insufficient to guide an efficient implementation

of this kind of activities, mainly at university level (Henriques, 2006).

This research concerns the implementation of an educational innovation process sup-

ported by investigation activities, in a university level Numerical Analysis course. The

main aim of the study is to understand the mathematical thinking processes used by

university students when exploring this kind of activity and to know how it may influ-

ence the learning of concepts and procedures and the development of problem solving

skills.

Such problems lead to formulate the initial research questions:

1) What kind of representations do students construct when carrying out

investigation activities in Numerical Analysis?

2) What kind of reasoning strategies do students use when they carry out

investigation activities in Numerical Analysis?

3) What are the differences between formal definitions that students

know and the criteria they use to verify the proprieties?

4) How does the realization of investigation activities influence the

undestanding of concepts and procedures in Numerical Analysis?

5) How does the realization of investigation activities influence students

problem solving skills?

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Theoretical background

The theoretical framework includes general theories regarding advanced mathematical

thinking, problem solving and investigation activities.

The teaching and learning of mathematics at university level is a matter of concern for

educational research. The educational community has been carrying out studies using

different methodologies and focusing on different aspects in order to understand the

learning processes and the development of knowledge at tertiary level. Artigue (1998)

and Cornu (1991) refer some results of research work performed at this level. There is

substantial evidence that university students often have negative attitudes towards

mathematics courses and considerable difficulties with some mathematical processes

such as reasoning, non-routine problem solving and proving (Artigue, 2007; Lithner,

2008; Yusof & Tall, 1994).

Research also focuses the development of students mathematical thinking from

elementary to terciary level (Dreyfus, 1991; Gray; Pinto, Pitta & Tall, 1999; Tall, 1991,

1992, 1997). The recognised mathematical difficulties faced by students in their transi-

tion to university could be explained by a lack of understanding of many mathematical

concepts taught in previous school years as they have a greater level of complexity and

require advanced mathematical thinking.

Plya (1945, 1981) highlights the importance of solving problems in mathematics

teaching. In his view, the student ought to learn mathematics as he/she is invited to de-

velop creative investigation work, through appropriate problems that the author desig-

nates as investigation problems. Abrantes (1988, 1994) refers to open problems as

tasks with a greater potential in teaching and learning, bringing to discussion the actual

interest in investigation activities. In his view, both problem solving and investigation

activities appeal to imagination and creativity, and require several skills besides calcula-

tion and memorization of definitions and procedures. These higher order thinking

skills are related with the capacity to communicate, critical thinking, modelling, prov-

ing, and other metacognitive processes. Ponte and Matos (1996) share the same idea and

state that mathematical investigations, as problem solving, require complex thinking

processes and students creativity.

In addition, Henriques (2006) and Ponte (2007) show that the use of investigation ac-

tivities at different school levels influences the evolution of students conceptions in

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relation to mathematics and provides an alternative route for their mathematical learn-

ing. However, much has to be learned about the conditions of using such an approach,

mainly at university level.

Methodology

This study is an investigation on students thinking processes and mathematical con-

structions as they undertake investigation activities in their natural environment (the

classroom). Thus, the focus is on qualitative aspects of the students construction of

mathematical knowledge, which characterises this study as a qualitative and interpreta-

tive research based on case studies of groups of students (Yin, 2003).

The study also aims to connect the research with teaching practice. To do so, a teaching

experiment using investigation activities in an Numerical Analysis course will be car-

ried out. The whole study was designed to be divided into two experiments an ex-

ploratory study and a main study involving students from the Naval Academy.

Exploratory study

The exploratory study is designed to gain information on several aspects which will be

important later in the main study. The first aim of the exploratory study is to evaluate

the initial research questions. Preliminary research questions were chosen suggestedfrom a review of the relevant literature, from my previous work (Henriques, 2006;

Henriques & Ponte, in press) and from my own professional teaching experience of

Numerical Analysis at university level. These questions address a large spectrum of

issues which must be delimitated to enable a deeper analysis.

Previous research (e. g. , Nardi, 1996; Pinto, 1998) refer to grounded theory as research

strategy to identify trends related to students thinking processes. The exploratory study

may provide a conection between the researchs theoretical framework and the main

study. As a result of such an attempt at progressive analysis of data, based on the

literature, a refinement of the questions is expected to be followed in the main study.

At the same time, the exploratory study aims to evaluate the investigation activities

which will be used during the teaching experiment of the main study. The investigation

activities are the way I choose to gain information on students performance on thinking

strategies and problem solving and on some aspects which represent a shift from ele-

mentary to advanced mathematics which could be followed later in the main study.These aspects are: awareness and use of formal definitions of the concepts, use of for-

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mal theory, logical reasoning and formal language. Particular attention is paid to the

students concept images. The exploratory study provides an opportunity to confirm if

the investigation activities are designed in a suitable way to gain access to the students

mathematical cognition and to work out the technical literature in an empirical context,

which may lead to a new focus of interest to be followed in the main study.

Hence, three investigation activities were designed and proposed to the students as a

means to activate their thinking processes, to promote their contact with mathematical

processes typical of investigation activities and to provide a basis for learning the con-

cepts, processes and strategies of Numerical Analysis:

1. Working with real number intervals. This activity tackle both concepts and

rules of interval arithmetic;

2. Solving equations. This activity is a way of approaching the non-linear

equations methods;

3. Deep Waters. This activity is about numerical integration and curve fitting.

The exploratory study was conducted in April and May of the 2007/08 academic year as

an analysis of the Naval Academy students performance, at the end of a traditional

course on this subject. The participants are four groups of 3 elements of 2nd

year stu-

dents at the Naval Academy, formed spontaneously and voluntarily organised.

As a researcher, I mostly observed the students of the selected groups during their on-

going work on the exploration of the investigation activities. Occasionally, I commented

upon their activity and asked questions to encourage them to reflect upon the tasks and

to explain, both in written and orally, the ideas and strategies that they followed. The

aim is to identify the students understanding of the investigation process. These meet-

ings were audio recorded and transcribed. Some unexpected issues arose from the stu-

dents work on these activities. Some conjectures on observed facts and episodes which

were significant to the research questions, were also recorded in a descriptive form.

The on-going analysis seeks instances of relevance to research questions in terms of

patterns of responses from the students.

Main study

The exploratory study was restricted to an analysis of students performance at the end

of a Numerical Analysis course. In contrast, the main study will be designed to be a

long-term investigation based on a teaching experiment supported by investigation ac-

tivities in a Numerical Analysis course. This study will be conducted during the 1stse-

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mester of the academic year of 2008/09 and will take into account the experience of the

exploratory study.

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