Revisiting and refining relations between nonsymbolic ratio processing and symbolic math achievement

In their 2016 Psych Science article, Matthews, Lewis and Hubbard (2016, https://doi.org/10.1177/0956797615617799) leveled a challenge against the prevailing theory that fractions—as opposed to whole numbers—are incompatible with humans’ primitive nonsymbolic number sense. Their ratio processing system (RPS) account holds that humans possess a primitive system that confers the ability to process nonysmbolic ratio magnitudes. Perhaps the most striking finding from Matthews et al. was that ratio processing ability predicted symbolic fractions knowledge and algebraic competence. The purpose of the current study was to replicate Matthews et al.’s novel results and to extend the study by including a control measure of fluid intelligence and an additional nonsymbolic magnitude format as predictors of multiple symbolic math outcomes. Ninety-nine college students completed three comparison tasks deciding which of two nonsymbolic ratios was numerically larger along with three simple magnitude comparison tasks in corresponding formats that served as controls. The formats included were lines, circles, and dots. We found that RPS acuity predicted fractions knowledge for three university math placement exam subtests when controlling for simple magnitude acuities and inhibitory control. However, this predictive power of the RPS measure appeared to stem primarily from acuity of the line-ratio format, and that predictive power was attenuated with the inclusion of fluid intelligence. These findings may help refine theories positing the RPS as a domain-specific foundation for building fractional knowledge and related higher mathematics.

Grade 9 learners’ understanding of fraction concepts: Equality of fractions, numerator and denominator

Our research with Grade 9 learners at a school in Soweto was conducted to explore learners’ understanding of fundamental fraction concepts used in applications required at that level of schooling. The study was based on the theory of constructivism in a bid to understand whether learners’ transition from whole numbers to rational numbers enabled them to deal with the more complex concept of fractions. A qualitative case study approach was followed. A test was administered to 40 learners. Based on their written responses, eight learners were purposefully selected for an interview. The findings revealed that learners’ definitions of fraction were neither complete nor precise. Particularly pertinent were challenges related to the concept of equivalent fractions that include fraction elements, namely the numerator and denominator in the phase of rational number. These gaps in understanding may have originated in the early stages of schooling when learners first conceptualised fractions during the late concrete learning phase. For this reason, we suggest a developmental intervention using physical manipulatives to promote understanding of fractions before inductively guiding learners to construct algorithms and transition to the more abstract applications of fractions required in Grade 9.

Triangle of Triangular Numbers

Among several interesting number triangles that exist in mathematics, Pascal’s triangle is one of the best triangle possessing rich mathematical properties. In this paper, I will introduce a number triangle containing triangular numbers arranged in particular fashion. Using this number triangle, I had proved five interesting theorems which help us to generate Pythagorean triples as well as establish bijection between whole numbers and set of all integers.

A Field Guide to Whole Number Representations in Children’s Books

Trade books are a common resource used to teach children mathematical ideas. Yet, detailed analyses of the mathematics content of such books to determine potential impacts on learning are needed. This study investigated how trade books represent whole numbers. A two-pronged approach was used a) one team documented every way 197 books represented numerical ideas and b) another team used standards to identify ideal representations. A third team validated the traits on 67 books. Greater variation than expected was documented (103 traits identified) and organized into a field guide for researchers to consult to design studies about how particular traits influence number learning. Studies could investigate how a particular trait supports learning or experimentally compare a selected combination of the 45 pictorial, 45 written symbol, 10 tactile, 2 kinesthetic, and 1 auditory trait. Implications for practice include recognizing what representations are present or missing from books used in classrooms. The study also serves as an example of how the field of mathematics education would benefit from adopting structures from disciplinary science, such as field guides, to inform how we organize phenomena of mathematics learning.

On the Exponential Diophantine Equation (132m) + (6r + 1)n = z2

Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have discussed the existence of the solution of exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers. Results of the present paper show that the exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers, has no solution in the whole number.

The Properties of the Arithmetic Function as the Consecutive Sum of Digits in Natural Numbers

In this paper defines the consecutive sum of the digits of a natural number, so far as it becomes less than ten, as an arithmetic function called and then introduces some important properties of this function by proving a few theorems in a way that they can be used as a powerful tool in many cases. As an instance, by introducing a test called test, it has been shown that we are able to examine many algebraic equalities in the form of in which and are arithmetic functions and to easily study many of the algebraic and diophantine equations in the domain of whole numbers. The importance of test for algebraic equalities can be considered equivalent to dimensional equation in physics relations and formulas. Additionally, this arithmetic function can also be useful in factorizing the composite odd numbers.