A post-WWII assignment from the U.S. Navy analyzing data from a test of a harbor mine-detecting algorithm leads to a new construction of the symmedian point.

The well known property of quadratic functions, that the tangents at either end of a chord of a parabola meet in a point aligned vertically with the midpoint of the chord is extended to polynomials of degree d. Given two distinct points on a polynomial of degree d, the Taylor polynomials of degree d - 1 at those points meet in d - 1 points whose…

Analysis of the patterns of signs of infinitely differentiable real functions shows that only four patterns are possible if the function is required to exhibit the pattern at all points in its domain and that domain is the set of all real numbers. On the other hand all patterns are possible if the domain is a bounded open interval.

Historical studies on the development of mathematical concepts will help mathematics teachers to relate their students' difficulties in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers, the number line, may not be fit for early teaching of operations involving negative…

This study investigates how teacher attention to student thinking informs adaptations of challenging tasks. Five teachers who had implemented challenging mathematics curriculum materials for three or more years were videotaped enacting instructional sequences and were subsequently interviewed about those enactments. The results indicate that the…

The evolution of indexing sequences and series has been little studied so far, although this topic presents difficulties for students. This symbolisation signified, however, an important means for operating more generally with finite and infinite series. This paper investigates the first steps during the eighteenth century and then the different…

This paper shows how the variance and standard deviation can be represented graphically by looking at each squared deviation as a graphical object--in particular, as a square. A series of displays show how the standard deviation is the size of the average square.

Formulae for row-sum of M-bonomial coefficients [image omitted] where G is an mth g-gonal number is developed from a study of the ratio between consecutive terms of the sequence of row-sum. The result generalizes the formula for row-sum of binomial coefficients: [image omitted].

Analysis of integer structure and right-end-digits can illustrate why 3 and 5 are features of primitive Pythagorean triples. The results also utilize the triangular and pentagonal numbers. (Contains 5 tables.)

This paper describes a pedagogical framework that teachers can use to support students who are engaged in solving open-ended problems, by explaining how two Japanese expert teachers successfully apply open-ended problems in their mathematics class. The Open-Ended Approach (OPA) framework consists of two main sections: Understanding Mathematical…

The introduction of linear functions is the turning point where many students decide if mathematics is useful or not. This means the role of parameters and variables in linear functions could be considered to be "threshold concepts". There is recognition that linear functions can be taught in context through the exploration of linear…

In this article, the authors begin by considering symbolic literacy in mathematics. Next, they examine the origins of misuse of the equals sign by primary and junior secondary students, where "=" has taken on an operational meaning. They explain that in algebra, students need both the operational and relational meanings of the equals…

Although fraction operations are procedurally straightforward, they are complex, because they require learners to conceptualize different units and view quantities in multiple ways. Prospective secondary school teachers sometimes provide an algebraic explanation for inverting and multiplying when dividing fractions. That authors of this article…

Linear functions are an important part of the middle school mathematics curriculum. Students in the middle grades gain fluency by working with linear functions in a variety of representations (NCTM 2001). Presented in this article is an activity that was used with five eighth-grade classes at three different schools. The activity contains 15 cards…

The purpose of this study was to replicate similar instructional techniques of Jimenez, Browder, and Courtade (2008) using a single-subject multiple-probe across participants design to investigate the effects of task analytic instruction coupled with semi-concrete representations to teach linear algebraic equations to middle school students with…