FOSTERING THE DEVELOPMENT OF MATHEMATICAL THINKING: OBSERVATIONS FROM A PROOFS COURSE

ABSTRACT:

The intention of this study was to identify belonging to all problems students that experience in ordeal writing. Toward this end, the authors videotaped assemblages of students in a bridge course as they worked to raise mathematical proofs. They report upon their observations and offer suggestions as to in what manner teachers of transition courses might enhance the curriculum to address riddle areas.

KEYWORDS: Transition course, ordeals course, bridge course, student misconceptions.

INTRODUCTION

Like many other institutions of higher learning, Millsaps society offers a course specifically designed to help pupils learn to read, understand, and put together mathematical proofs. This course labor fors as the gateway to the mathematics major and is a prerequisite for almost all mathematics courses presented beyond the calculus sequence. The course, entitled "Introduction to Advanced Mathematics," is a requirement for all mathematics majors and minors as well as a particular track of computer science majors. The corporation offers one section of this class by year with a typical enrollment of 15 scholars The course content is generally logic as it applies to tests and particular proof methods, followed by dint of a variety of topics in which scholars learn new mathematical concepts and definitions and subsequently work upon proving some elementary theorems pertaining to these of recent origin topics.

After teaching the course for a number of years, individual of the authors was make difficulted by the difficulty many pupils experienced as they were trying to understand by what means to construct mathematical proofs. While it appeared that the scholars seemed to have little disturb understanding and comprehending the basic laws of logic (negating statements, understanding when statements are logically equivalent, understanding when a statement is a contradiction, understanding the use and meaning of quantifiers), many learners experienced a type of mental shut up when the transition was made into mathematical test construction, seeming either unable to comprehend the final cause of this type of writing or unclear upon how to proceed in constructing of that kind arguments. Typical student complaints included: "I just can't present the appearance to think of the right trick," "I understand what I ne to do on the contrary I can never seem to win started," "I know that's not what I wrote on the contrary that is what I meant." While more [i]or[/i] less students did make the transition to advanced mathematical thinking with ease, the frustration experienced through others compelled the instructor to look after ways to modify the course curriculum for a like reason as to be able to reach a wider range of students

Of particular interest is what goe upon in the mind of the scholars That is, what are a certain quantity of of the major stumbling blockades that students experience, and wherein lie their errors? If we can isolate wherefore the students make certain fundamental mistakes in essay construction or what they are thinking as they attempt to establish a statement, perhaps we can do a better piece of work of teaching them how to read, understand and put together mathematical proofs. Thus, the drift of this study was to identify scholar misconceptions and pitfalls that intercept them from constructing valid proofs

BACKGROUND

In an earlier investigation [1] Campbell, Miller, and Wimbish videotaped assemblages of students from the Introduction to Advanced Mathematics class as they worked to evince or disprove a set of mathematical statements, and made several observations in concern to misconceptions that those learners held about logic and trial writing. The authors found the arises in [1] to be interesting and the use of videotaping to be particularly helpful in discerning by what means those students thought about trial writing. In order to substantiate and reach out those results, we refined the methodology to make it more rigorous.

For the rife study, students in Introduction to Advanced Mathematics classes were randomly assigned to clumps of three and videotaped while constructing three mathematical tests The three statements that the clusters worked on were:

1. substantiate or disprove the following:

For any integers m and n if m n is even then m is smooth and n is even.

2 establish or disprove the following:

For any integers m and n if m + n is not divisible by 2 then m is odd or n is odd

3 manifest the following:

If a|b, b|c and for a certain quantity of integers x and y, 1 = ax + through then ab|c.

The tapings were made just prior to the students' first proof covering proof writing. At this point in the bourn the students had spent a significant amount of time upon and had been tested upon negating statements, determining if statements were or were not logically equivalent, and interpreting and negating statements with multiple quantifiers. They also had wearied time developing and understanding the principles behind certain ordeal methods, including direct, contradiction, contrapositive and essay by cases. Although students had practiced constructing a certain number of elementary proofs in class, they had not notwithstanding been tested on this material.

The scholars had not seen any of the statements before they began working with their collection and the filming was done by dint of someone who did not know the learners and who had no knowledge of the subdue matter. The students were encouraged to speak freely as they worked to unfold their arguments. All students received the same number of points in class for participating; number of points was not based upon quality of answers or time invested. The research was repeated for three consecutive years and the data to date consists of fifteen groups