What comes to mind when one thinks about building? One may envision constructions with blocks or engineering activities. Yet, constructing and building a number system requires the same sort of imagination, creativity, and perseverance as building a block city or engaging in engineering design. We know that children invent their own notation for…

This article describes students learning to build their own numbering system by recognizing and identifying patterns with interlocking cubes in different place values. The students used the Egyptian hieroglyphic numeral system in conjunction with this activity to connect learning in other subjects. Students used prior knowledge of place value to…

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…

This paper reports on an experiment involving 108 sixth-grade French students in which the history of Chinese numeration was used in mathematics education. The aim of this study is to evaluate the relevance of using ancient number systems in mathematics education, which is common in French textbooks. In the experiment a pre-test with usual…

In this paper, we explore student solutions to a free response mathematical assessment task which had opportunities for students to notice structural properties in the context of number systems. In total, 308 students aged between 10 years to 13 years participated in the study. Their responses were analysed to determine whether they noticed…

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…

We performed a meta-analysis of approximate number system (ANS) training studies to investigate the strength of the causal effects of practicing ANS related tasks on symbolic math performance. Across 33 effect sizes from 11 studies involving 754 participants, for which neither the treatment nor control group received symbolic training, we found a…

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 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…

The "Reasoning Algebraically about Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The thirty-four cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation into the…

Thomas Kuhn (1962/2012) introduced the term "paradigm shift" to the scientific literature to describe how knowledge in science develops. The aims of this article are to identify paradigm shifts, or revolutions, that have occurred in mathematics, and to discuss their relevance to teaching mathematics in schools. The authors argue that…

While conceptual understanding of properties, operations, and the base-ten number system is certainly associated with the ability to access math facts fluently, the role of math fact memorization to promote conceptual understanding remains contested. In order to gain insight into this question, this study looks at the results when one of three…

Recent research has supported and extended earlier research on how and for how long Indigenous people of Australasia have been counting. This history values the long history of Indigenous knowledge and re-writes the limited and sometimes false history that many Australian teachers accept and teach about number systems. The current views on the…

To increase opportunities for students to take more advanced math courses in high school, many school districts enroll grade 8 students in Algebra I, a gateway course for advanced math. But students who take Algebra I in grade 8 and skip other math courses, such as grade 8 general math, might miss opportunities to develop the foundational…

To increase opportunities for students to take more advanced math courses in high school, many school districts enroll grade 8 students in Algebra I, a gateway course for advanced math. But students who take Algebra I in grade 8 and skip other math courses, such as grade 8 general math, might miss opportunities to develop the foundational…