Ali Khan Nazem al-Oloom, the top graduate of the engineering class in the second cohort of Dar al-Fonun and later a teacher at the same institution, was among the companions of Naser al-Din Shah during his first trip to Europe. He remained in France, where he studied mathematics under prominent figures such as Joseph Bertrand at the École Polytechnique. After roughly two years, he returned to Iran, where he authored Hekmat-e Tabee’i, the first physics textbook written in Persian, and later Hekmat-e Riyaziyat: Osoul-e Elm-e Hesab, known as Hesab-e Ali Khan. This latter work became the most widely used mathematics textbook in Iran’s modern educational history. Despite the originality of his pedagogical approach — which emphasized mathematical meaning over rote procedure — he did not live to witness the success of his work. Shaped by the socio-political constraints of his time, his life ended in self-inflicted death. This article offers, for the first time, a detailed portrait of his life, writings, and the historical trajectory of both.

The paper uses Mohandes al-Mulk as a vantage point from which to examine one of the most important transitional moments in the history of mathematics in Iran: the passage from older traditions of rhetorical mathematics to the newer world of symbolic mathematics. Mohandes al-Mulk is important precisely because his life connects several layers of that transition. He studied at Dar al-Fonun, later taught there, wrote influential textbooks, and stood at a point where earlier Iranian mathematical traditions, Dar al-Fonun mathematics, and more modern currents could still be seen together

Mirza Nezam, a mathematician of the Qajar era educated in France, endeavored to Persianize the mathematical knowledge he acquired by devising a Farsiinspired notation system. This task was daunting, given Iran’s significant mathematical lag compared to Europe. Did he succeed? The answer may be known, but not the extent of his attempt. Now, recent discoveries of his extensive mathematical writings provide, for the first time, an opportunity to assess his efforts comprehensively. These findings enable a closer examination of his attempts and their implications within Iran’s mathematical progress. This paper not only recounts Mirza Nezam’s life but also delves into his mathematical contributions, arguing that his work represents a unique anomaly that defies comparison with the prevailing mathematical practices of his era in Iran.

The aim of this short article is to compare the methods of Khayyam and Cardano in studying and solving cubic equations, as well as to compare the paths that their methods opened up for later mathematicians

A short article based on a teaching episode and historical evidence to show why the minus sign is the right sign for denoting negative numbers.

.If you have ever wondered whether it is better to say “minus five” or “negative five,” then “Signed Numbers and Signed Letters in Algebra” is well worth reading

Big Blocks of Proof” is based on several observations I have made over the years. It is easier for students to prove that the sum of three odd numbers is odd than to prove that the sum of two odd numbers is even. The former can be proved using large “factual” blocks, one of which is that the sum of two odd numbers is even.

This paper proposes a reading of the history of equivalence in mathematics. The paper has two main parts. The first part focuses on a relatively short historical period when the notion of equivalence is about to be decontextualized, but yet, has no commonly agreed-upon name. The method for this part is rather straightforward: following the clues left by the others for the ‘first’ modern use of equivalence. The second part focuses on a relatively long historical period when equivalence is experienced in context. The method for this part is to strip the ideas from their set-theoretic formulations and methodically examine the variations in the ways equivalence appears in some prominent historical texts. The paper reveals several critical differences in the conceptions of equivalence at different points in history that are at variance with the standard account of the mathematical notion of equivalence encompassing the concepts of equivalence relation and equivalence class.

The purpose of this paper is to propose an operational idea for developing algebraic thinking in the absence of alphanumeric symbols. The paper reports on a design experiment encouraging preschool children to use the associative property algebraically. We describe the theoretical basis of the design, the tasks used, and examples of algebraic thinking in 5–6-year-old children. Theoretically, the paper makes a critical distinction between operational and structural meanings of the notion of equality. We argue that mathematical thinking involving equality among young learners can comprise both an operational and a structural conception and that the operational conception has a side that is productively linked to the structural conception. Using carefully designed hands-on tasks, the crux of the paper is the realization of algebraic thinking (in verbal mathematics) as operationally experienced in the ability to transform one number structure, with a quantity that is subject to change, into another through equality-preserving transformations.

The story of the discovery of magnitudes that could not be expressed as fractions, and the tragic story of their discoverer

The notion of equivalence relation is arguably one of the most fundamental ideas of mathematics. Accordingly, it plays an important role in teaching mathematics at all levels, whether explicitly or implicitly. Our success in introducing this notion for its own sake or as a means to teach other mathematical concepts, however, depends largely on our own conceptions of it. This paper considers various conceptions of equivalence, in history, in mathematics today, and in mathematics education. It reveals critical differences in the notion of equivalence at different points in history and a meaning for equivalence proposed by mathematicians and mathematics educators that is at variance with the ways that learners may think. These differences call into question the most popular view of the subject: that the mathematical notion of equivalence relation is the result of spelling out our experience of equivalence. Moreover, the findings of this study suggest that the standard definition of an equivalence relation is ill-chosen from a pedagogical point of view but well-crafted from a mathematical point of view.

Written when I was a PhD student. This paper fcouses on ‘lay’ students understanding of equivalence relations through individual task-based interviews. We report a conceptual gap between “the everyday functioning of intelligence and mathematics” as to equivalence relations.