Michael J. Bossé

The NCTM standards in light of the new math movement: A warning!

Abstract Although many have assailed the failure of the New Math Movement of the 1950s and 60s and extolled the virtues of the National Council of Teachers of Mathematics (NCTM) Standards , little has been written from the perspectives of those most instrumental within each movement. It is the perspective of those who took part in each reform effort into which this paper investigated. Interviewees ably delineated the philosophical perspectives of their respective movements, reasons for its success or failure or both, and their prognosis for the future of mathematics education. The telephone interviews of some of the seminal people involved within the two movements were transcribed, coded, analyzed, and are here reported. Many of the results could not have been imagined by anyone outside of the inner circles within these reform efforts. This article offers many surprising opinions that have not been available anywhere else and which offer a stern warning to the NCTM.

Putting the Modern in Algebra.

Abstract Secondary school mathematics teachers often need to answer the “Why do we do that?” question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students’ conceptual knowledge of secondary mathematics. This paper investigates the number systems from K-12 mathematics, discusses the properties held in each, and diagrammatically establishes connections between the number systems and algebraic structures.

Real-world problem-solving, pedagogy, and efficient programming algorithms in computer education

Modern research and curricular reforms equate pedagogical soundness with the connection of instructional content with real-world problems. Software engineers facing real-world computer problems are continually concerned with the efficiency of the program that they write. Divorcing programming concerns from efficiency unsatisfactorily presents the responsibilities and full concerns of computer programmers. Therefore, when programming tasks are simplified to avoid concerns for efficiency, the assignments become antiseptic, lose the nature of real-world problems, and become inconsistent with the true nature of computer programming concerns. This brief investigation considers real-world problems, pedagogy within computer programming education, and the often-missed consideration of efficiency within instructional computer programming assignments. If-then-else algorithms are compared with algorithms using arrays in light of programming efficiency and pedagogy in computer education.

Reforming the NCTM standards in light of historical perspective: Premature changes

Abstract The mathematics education community finds itself in strange times; times of transition and continual change. The constant evolution of mathematics education reform is upon us. A full nine years into the publishing and partial implementation of the NCTM Curriculum and Evaluation Standards mathematics educators eagerly anticipate the evaluation, rewriting, and publication of reform Standards. However, a din of opposition has arisen: one, a call for reform from within, seeking to rewrite the current Standards, and call from without, calling for a return to “Back to Basics.” This parallels the reform antagonism and Back to Basics movement which followed the unsuccessful reform effort of the New Math Movement more than three decades ago. Therefore, since historical precedence exists, this present phenomenon must be thoroughly considered in light of past reform efforts. This paper defines some of the historically consistent events and situations surrounding both the New Math Movement and the NCTM Standards. These commonalities include: overestimating teachers; overly ambitious suggestions in curriculum modifications; delayed teacher training; philosophically inconsistent evaluatory methodologies; and insufficient programmatic assessment. Arguing from these commonalities, this paper reports that any immediate reform of the NCTM Standards is premature and not supported by empirical evidence.

Rounding the Regression

Abstract Rounding is a necessary step in many mathematical processes. We are taught early in our education about significant figures and how to properly round a number. So when we are given a data set and asked to find a regression line, we are inclined to offer the line with rounded coefficients to reflect our model. However, the effects are not as insignificant as they might seem at first. In this paper, we investigate some consequences of rounding the coefficients in a least squares linear regression with respect to the calculated value of R2, and consider ways to minimize the amount of error that can arise.

Finding Real Roots of Polynomials Using Sturm Sequences

Abstract Following a line of inquiry regarding the exact number of real roots of a real polynomial, this investigation considers: Descartes’ Rule of Signs, the Budan–Fourier Theorem, and versions of Sturm’s Method in contrast with the approximate root count gleaned from graphing utilities. Online applets are provided to allow the reader to freely experiment with different polynomial examples. Additionally, activities at the end of each section facilitate further investigations and deeper understanding of the topics.

Dynamic Boolean Mathematics

ABSTRACT Dynamic mathematical environments allow users to reify mathematical concepts through multiple representations, transform mathematical relations and organically explore mathematical properties, investigate integrated mathematics, and develop conceptual understanding. Herein, we integrate Boolean algebra, the functionalities of a dynamic mathematical environment, and different mathematical topics to demonstrate investigations associated with dynamic Boolean mathematics.

Searching for the Black Box: Misconceptions of Linearity

14 You have lost contact with an unmanned surveillance plane as it is flying over a large stretch of uninhabited desert. You send high-altitude reconnaissance aircraft to take pictures of a potential crash site in the middle of the desert. The pictures reveal larger pieces of what you believe are debris from the crashed aircraft. You hope to find the flight data recorder to gain some clues to what caused the crash. Searching for the Black Box: Misconceptions of Linearity

Does your graphing software real-ly work?

Abstract Many popular mathematical software products including Maple, Mathematica, Derive, Mathcad, Matlab, and some of the TI calculators produce incorrect graphs because they use complex arithmetic instead of real arithmetic. This article expounds on this issue, provides possible remedies for instructors to share with their students, and demonstrates that using complex arithmetic has some important advantages.