Understanding the concept of completeness for an ordered field is known to be difficult for many university mathematics students. We hypothesise that the variety of possible axioms of completeness for the set of real numbers is one of the sources of difficulties as is the lack of understanding of the "raison d'être" of these axioms. In…

When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially come across as more puzzling or even mysterious. In this article we explore three specific statements from abstract algebra that involve the number…

Generally, people respond faster to small numbers with left-sided responses and large numbers with right-sided responses, a pattern known as the SNARC (spatial numerical association of response codes) effect. The SNARC effect is interpreted as evidence for amodal automatic access of magnitude and its spatial associations, because it occurs in…

Most any students can explain the meaning of "a[superscript b]", for "a" [element-of] [set of real numbers] and for "b" [element-of] [set of integers]. And some students may be able to explain the meaning of "(a + bi)[superscript c]," for "a, b" [element-of] [set of real numbers] and for…

Prospective elementary school teachers (PTs) come to their mathematics courses fluent in using procedures for adding and subtracting multidigit whole numbers, but many are unaware of the essential features inherent in understanding the base-ten place-value system (i.e., grouping, place value, base). Understanding these features is crucial to…

This study investigated the impact of a chatbot-based micro-learning system on students' learning motivation and performance. A quasi-experiment was conducted with 99 first-year students taking part in a basic computer course on number system conversion. The students were assigned to a traditional learning group or a chatbot-based micro-learning…

This study seeks to analyse the awareness of the pre-service teachers on the counting methods, systems and tools used in the prehistoric method and the Ancient period and to examine the distribution of this awareness by gender. A total of 42 sophomore-level students studying at a university in the Western Black Sea region, Turkey, participated in…

Community college developmental math students (N = 657) from three math levels were asked to place five whole numbers on a line that had only endpoints 0 and 20 marked. How the students placed the numbers revealed the same three stages of behavior that Steffe and Cobb (1988) documented in determining young children's number sense. 23% of the…

Earlier institution-sponsored research revealed that about 20% of students in community college basic math and pre-algebra programs lacked a sense of part-whole relationships with whole numbers. Using the same tool with a group of 86 workforce students, about 75% placed five whole numbers on an empty number line in a way that indicated lack of…

In this paper, the use of guided hyperlearning, unguided hyperlearning, and conventional learning methods in mathematics are compared. The design of the research involved a quasi-experiment with a modified single-factor multiple treatment design comparing the three learning methods, guided hyperlearning, unguided hyperlearning, and conventional…

The representation of numbers is commonly viewed as an ordered continuum of magnitudes, referred to as the "mental number line." Previous work has repeatedly shown that number representations evoked by a given task can be easily altered, yielding an ongoing discussion about the basic properties of the mental number line and how malleable…

In two experiments, participants judged the average numerosity between two sequentially presented dot patterns to perform an approximate arithmetic task. In Experiment 1, the response was given on a 0-20 numerical scale (categorical scaling), and in Experiment 2, the response was given by the production of a dot pattern of the desired numerosity…

This paper presents guided classroom activities that showcase two classic problems in which a finite limit exists and where there is a certain charm to engage liberal arts majors. The two scenarios build solely on students' existing knowledge of number systems and harness potential misconceptions about limits and infinity to guide their thinking.…

A facility with signed numbers forms the basis for effective problem solving throughout developmental mathematics. Most developmental mathematics textbooks explain signed number operations using absolute value, a method that involves considering the problem in several cases (same sign, opposite sign), and in the case of subtraction, rewriting the…

This article confronts the issue of why secondary and post-secondary students resist accepting the equality of 0.999... and 1, even after they have seen and understood logical arguments for the equality. In some sense, we might say that the equality holds by definition of 0.999..., but this definition depends upon accepting properties of the real…