Logarithms, pivotal in advanced mathematics and real-world applications, often present challenges to students who resort to rote memorization without grasping their essence. While the emotional dimensions of learning mathematics are increasingly recognized, research in this area remains limited. This dissertation addresses this gap by examining a…

The study identified the cognitive mechanisms that underly individual differences in mathematics and reading. There is little systematic evidence regarding the connections between cognitive capabilities, mathematics and reading among high-school students with typical development. Moreover, the study aimed to examine whether or not gender is…

The current paper aims to identify the mathematical connections activated by 10 Mexican high school students while solving mathematical tasks that involve the exponential and logarithmic function. We used the Expanded Mathematical Connections Model (EMCM) and the OntoSemiotic Approach of Cognition and Mathematical Instruction (OSA) as theoretical…

In a cross-sectional study, 160 students in Grades 2, 4, 7, and 11 were interviewed about their reasoning when solving integer addition and subtraction open-number-sentence problems. We applied our previously developed framework for 5 Ways of Reasoning (WoRs) to our data set to describe patterns within and across participant groups. Our analysis…

We interviewed 40 students each in grades 7 and 11 to investigate their integer-related reasoning. In one task, the students were asked to write and interpret equations related to a story problem about borrowing money from a friend. All the students solved the story problem correctly. However, they reasoned about the problem in different ways.…

There is a general perception that the South African curriculum statements for mathematics create polarity between the 'old' and the 'new', which does not benefit both the teachers and the learners. The new curricula demand a radical shift from the traditional teacher-led approaches that teachers are familiar with, yet does not provide a model of…

Composition of functions is one of the five big ideas identified in NCTM's "Developing Essential Understanding of Functions, Grades 9-12" (Cooney, Beckmann, and Lloyd 2010). Through multiple representations (another big idea) and the use of The Geometer's Sketchpad[R] (GSP), students can directly manipulate variables and thus see dynamic visual…

In two experiments we explored the instructional value of a cross-domain mapping between "number" and "line" in secondary school students' understanding of density. The first experiment investigated the hypothesis that density would be more accessible to students in a geometrical context (infinitely many points on a straight…

We present an empirical study that investigated seventh-, ninth-, and eleventh-grade students' understanding of the infinity of numbers in an interval. The participants (n = 549) were asked how many (i.e., a finite or infinite number of numbers) and what type of numbers (i.e., decimals, fractions, or any type) lie between two rational numbers. The…

Many authors have discussed the question "why" we should use the history of mathematics to mathematics education. For example, Fauvel ("For Learn Math," 11(2): 3-6, 1991) mentions at least fifteen arguments for applying the history of mathematics in teaching and learning mathematics. Knowing "how" to introduce history into mathematics lessons is a…

Understanding the meaning of rational numbers and how to perform mathematical operations with those numbers seems to be a perennial problem in the United States for both adults and children. Based on previous work, we hypothesized that giving students more time to practice using rational numbers in an environment that enticed them to apply their…

For over a decade, research studies of mathematics education in high-performing countries have pointed to the conclusion that the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on the promise of common standards, the standards…

To probe beyond declarative knowledge about real numbers in secondary school, the authors interviewed students in Grades 9, 10 and 12. The main question seems to be whether the length of a decimal expansion is "indefinite" or "infinite." It blurs the mental representation of rational numbers as well. (Contains 3 footnotes.) [For complete…