Theodore Eisenberg

Intuitions on Functions

A theoretical model is presented for commencing a systematic assessment of students’ intuitions on the mathematical notion of functions. This model considers functions in terms of a three-dimensional block, with specific function settings running along its x-axis, function topics running along its y-axis, and levels of abstraction running along its z-axis. Questionnaire booklets assessing specific notions of a function were written and administered to 127 junior high school students. It was found that high ability students tended toward a graphical approach to the notions, while low ability students were attracted to pictorial and tabular presentations of the notions. The model and its implications are discussed.

The Effect of Two Years of Tutoring on Mathematics and Reading Achievement

AbstractThis study investigated the cognitive impact of two years of tutoring on disadvantaged children in a “big-brother” type program. Although tutors were not specifically required to upgrade achievement, almost all assumed this to be a major thrust of their activities. Mathematics and reading skills were measured over a two-year period for children tutored two years, one year, and not at all. Findings indicate that one year of tutoring yielded some cognitive gains; however, a second year of tutoring did not increment them. Less emphasis on achievement goals during the second year of tutoring appeared to account for these results.

On an unknown algorithm for computing square roots

A relatively unknown algorithm for computing square roots is presented. In a survey of more than 50 teachers of mathematics, not one of them recalled having ever encountered it before. The algorithm is first illustrated and then proved.

On divisibility by 7 and other low valued primes

A group of preservice teachers could not recall or devise criteria for determining when 7 (or any higher prime) divides N. Tests for divisibility, a topic once studied by students, seems to have disappeared from the curriculum, with teachers themselves having only a pedestrian knowledge of this topic. This paper presents several different ways to construct tests of divisibility for low valued divisors.

Some of My Pet-Peeves with Mathematics Education

For nearly half-a-century I have been a mathematics-educator, and recently retired because of a mandatory retirement age for state workers in my country. As I think back over the years as to how the profession has changed, I am simultaneously proud and disillusioned. I am proud that there are so many different facets to our discipline, but at the same time I am disillusioned that there are so many different facets to our discipline, because we have seemingly lost sight of what our profession should be all about. Whereas many of us used to have appointments in departments of mathematics, the majority of us are now in departments of education, science teaching, cognitive science, and educational technology, where the teaching and learning of mathematics per se are attended to peripherally, if at all. Some colleagues claim we are discipline that has matured from it roots in mathematics; others however say we are a discipline that has lost its way. I am very much a member of this latter camp, a group that is shrinking in size daily. In an effort to inform the larger mathematical community of this state of affairs, I would like to put forth some of my pet-peeves on mathematics-education today.

Polynomials: a theme which should not be forgotten

There is a definite movement to remove many of the topics concerning polynomials from the school curriculum. The purpose of this paper is to point out why this movement is misguided and should be reversed.

Prime knowledge about primes

Several proofs demonstrating that there are infinitely many primes, different types of primes, tests of primality, pseudo primes, prime number generators and open questions about primes are discussed in Section 1. Some of these notions are elaborated upon in Section 2, with discussions of the Riemann zeta function and how algorithmic complexity enters into tests for primes. Readers may know segments of what follows, but hopefully this work will help them place their knowledge into richer landscapes.

On the solvability of x2= b(mod m)

Eulers criterion is the usual procedure used for assessing the solvability of the equation x2 = b(mod m). This criterion is applicable only when b and m are relatively prime and must be applied to each b (0≤ b < m). Described in this paper is an alternative criterion which characterizes and counts, for a general modulus m, all numbers b for which x2 = b(mod m) is solvable.