Just when you think you know everything about everything, it comes a new discovery about something as mundane as the minus sign. In Praise of the Minus Sign (FLM, 40, 2, July 2020) shows why the minus sign is the right sign for denoting negative numbers! The version posted here is a post-print version: PDF-file of the improved article after peer-review. the copyright is held by FLM :https://flm-journal.org/

It took me ages to find a framework for describing the historical uses and misuses of signed numbers. If you have ever wondered it is better to say “minus five” or “negative five”, Signed numbers and signed letters in Algebra (2019, For the Learning of Mathematics 39 :13-16) is a paper for you.

Big Blocks of Proof is based on several observations I’ve been having over the years. It is easier for students to prove that the sum of three odd numbers is odd, rather than to prove that the sum of two odd numbers is even. The former can be proved using big “factual” blocks, one of them the sum of two odd numbers is even!

Equivalence: an attempt at a history of the idea is the result of 13 years on and off research.

Making Associativity Operational is the result of a long long journey. It received many praises and at the same time, too many rejections! I believe such a mixed reaction is a sign of a good paper.

Using Untouchables to Make Arithmetic Structures Touchable has been written for teachers. But, if for any reason you are interested in possible connections between arithmetic and algebra, this is a good paper to read. There, you can find some theoretically attractive and practically novel ideas.

Symbols in early algebra: To be or not to be? is the first paper of a very productive journey with my PhD student, Leyla Khosroshahi. We explore the possibilities of doing algebra in preschools and in the absence of alpha-numeric symbols.

I have written several mathematical -historical comic strips. Our irrational Brother is the only one translated into English. I translated it for a presentation that I gave as an invited speaker at Irvine (University of California). The tittle of the presentation was “The use of historical comic strips to engage students in mathematics and to help them to appreciate mathematics as a human endeavour”.

Specularity in Algebra was written and then published in a situation when I hadn’t got access to For the Learning of Mathematics and many other journals. Thus, though I had the main idea of the paper for a long time, it took me another long time to write the initial draft. Moreover, any simple citing suggestion from the reviewers could cost me a few extra months to just find what I had been advised to cite! Fortunately, for this particular paper I received a manageable number of such suggestions.

Moore and Less! proposes a phenomenographic foundation for the so-called problem method in teaching mathematics. It also breaks away one of the main myths about Inquiry Based Teaching: “class size should be small.” The article tells the story of a very non-standard, absolutely student-centered multivariable calculus course with 136 students. The version of the paper shared here, though very similar to the final version is not the final version. If you cannot access the published version from the publisher’s site, please do not hesitate to e-mail me.

Experiencing Equivalence, but Organizing Order investigates the history of the idea of equivalence relation by using a variational approach; that is an investigation of the variation in the ways that some prominent mathematicians of the past have tackled certain situations that from the vantage point of today’s mathematics embody the idea of equivalence relation. The main result of the paper is given in the title! I called the approach used for the study “*historical variations*“, having in mind that one day I am going to use the same method to study the history of other mathematical concepts.

Examples, A Missing Link is a paper worth reading. In particular, the conclusion sets a framework for further research on example-generating and example-checking and how they might be related. However, apart from a small (and unpublished) research that I did with one of my master students years after this PME paper, I never seriously came back to the ideas discussed in the paper.

Examples: Generating versus Checking is part of a series of papers I wrote about examples. The series ended with a PME paper titled “Examples, a Missing Link”.

A Mad Dictator Partitions His Country is my first journal paper (in English). I wrote it when I was a Ph.D. student. It describes how interviews brought a change in my perception of the role of definitions. It, based on one of the interviews, also suggests an alternative (non-standard and new) definition of the notion of the equivalence relation. It tells how I learned to see a concept through learners’ eyes; that was the start of my turn to phenomenography. If the paper has any importance now, it is what was nicely summarised by one of the reviewers:

Recognizing that definitions are chosen not pre-existing and that a researcher who is awake and not simply looking for what they expect are important lessons for other researchers, particularly graduate students, to learn.

Students’ Experience of Equivalence Relation: A Phenomenographic Approach was written when I was a Ph.D. student. I believe it was one of the first papers that used phenomenography for studying people’s understanding of an advanced mathematics concept. Perhaps that is why it’s been cited more than my other papers; okay, I try to ignore this fact that the paper was co-written by David Tall 🙂

ORGANIZING WITH A FOCUS ON DEFINING, A PHENOMENOGRAPHIC APPROACH written when I didn’t know I am more interested in investigating the processes of defining, or investigating the result of defining.

STUDENTS’ EXPERIENCES OF ‘EQUIVALENCE RELATIONS’ is the very first of a number of papers written based on my doctoral study.

This one is only one page and the start of a journey that didn’t come to an end.

Based on the dissertation of my master student, Hosein Ghaffari, this one page reports the gem of an idea.

#### Papers in Farsi

The papers below are written in Farsi. I have translated the titles in English to give you a sense of my interests and directions. I also try to provide a brief description for each paper.

#### مقاله های فارسی

##### عنوان مقاله ها و خلاصه ی داده شده به انگلیسی است ولی لینک داده شده به اصل فارسی مقاله است

Written with my PhD student, Leyla Khosroshahi, the paper suggests a framework for connecting arithmetic to algebra, or better to say, doing algebra within arithmetic, in primary schools.

درک دانش آموزان چهارم دبستان از علامت تساوی

Written with Sharareh Dastjerdi and Maryam Adeli, the paper is exactly what the title suggests. Not surprisingly, the Iranian fourth-graders had the same understanding as nearly all the other primary students mentioned in the relevant literature: they had an operational understanding of the equal sign.

گذر از تفکر حسابی به تفکر جبری

Written with Kaye Stacey, The paper was mainly based on Kays’s research. We wrote it for Iranian teachers, using examples from Australian and Iranian context.

Written as part of a series of papers about the different roles of letters in

algebra, “A is Free” is to show how letters in the absence of a relevant structure might loose their meaning, but at the same time, might find freedom. Consider n+n+1+n=3n+1, it might represents two different ways of counting the same structure or …

Written as part of a series of papers about the different roles of letters in

algebra, “A is Absent” warns teachers and textbook authors about such careless uses of the so-called blanks (in different forms and shapes) and how they might be a source of confusion in the already hard journey from arithmetic to algebra. Think of different possibilities that you might be able to interpret something like blank + blank =2 blank…

Written as part of a series of papers about the different roles of letters in

algebra, “A is Silent” discusses the unproductive misuses of the standard exercise of translating something from words to the letters, from English (Farsi) to the symbolic language of algebra.

Written as part of a series of papers about the different roles of letters in

algebra, “A is Delicious”, based on the dissertation of Maryam Abdolahpour, my master student, discusses the side effects of the so-called fruit-salad algebra.

تاثیر اریگامی بر توسعه ی تفکر جبری دانش آموزان

Written with Faezeh Falahat, one of my master students, the paper is exactly what the title suggests! In particular, our focus was on the hidden algebraic aspect of origami as the keeper of structures and relations between different elements of a figure while the size might change.

از یک تا ده، راهی برای درک واقعیت های عددی

Written with Somayeh Shabani, one of my master students, the paper becomes the backbone of my algebraic theory, *specularity.*

This not a paper per se, but I thought it is better to be seated alongside the algebraic papers.

داستان جبر: بخش اول؛ بخش دوم؛ بخش سوم

This is the translation of the paper written by Anna Sfard and Liora Linchevski*. It was translated to accompany a number of my algebraic writings for teachers.

*Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification—the case of algebra. In *Learning mathematics* (pp. 87-124). Springer, Dordrecht.

#### Recreational Mathematics

##### ریاضیات بانمک و ترویجی

A tribute to Martin Gardner, in fact, the whole journal was published in honour of Gardner. Martin and I, or I and Martin was is written in Gardner’s style, adopting one of his works.

It is intuitively obvious was written to show whether something is obvious or not is dependent on one’s own intuition that may differ from one person to the other.

This is the first comic strip of a series of historical comic strips that I created based on the work of mathematicians. I used to call them fake histories since I was only true the development of mathematical concepts rather than the true story of the mathematicians involved in the development of the concepts. So there are cases in which two mathematicians who lived hundreds years apart coexist in the same frame as part of the story of the development of the concept!

A mathematical-historical comic stripe about square root of two and the “person” who discovered it.

نقط ، خط، رنگ ؛ قسمت اول؛ قسمت دوم

A mathematical-historical comic stripe about a famous conjecture in graph theory (i.e. the total colouring) around the life of Mehdi Behzad, the person who conjectured it.

A mathematical-historical comic stripe around the work of Ebadollah Mahmoodian on mathematical aspects of *Sudoku* and Latin Squares.

#### Translations

##### ترجمه ها

از حساب به جبر، چگونه حساب را بهتر درس دهیم

Written by Hung-Hsi Wu, this is the tenth *WikiLetter. *

مسیر طولانی ملموس شدن حساب و جبر

Written by David Tall, this is the fifth *WikiLetter *translated by Yasaman Baghaei and I.

توجه معلم کجاست، توجه دانش آموز کجا

Written by John Mason, this is the second *WikiLetter ,*translated by Maryam Adeli and I.

Written by Carolyn Kieran, this was the start of *WikiLetters*, translated by Sharareh Dastjerdy and I.

تحقیق ریاضی چگونه انجام می شود

How research is carried out?, written by Michael Atiyah, that happens to be my academic grandfather 🙂 A very readable paper with a nice and unusual arguments in the favour of rigour in mathematics.

“A metaphor for mathematics education” is a very interesting paper written by Gerg McColm. In fact, the paper introduces more a metaphor for mathematics, rather than mathematics education. The paper is only four pages long, but it took a good time of Zahra Gooya and I to translate it.

داستان جبر: بخش اول؛ بخش دوم؛ بخش سوم

Written by Anna Sfard and Liora Linchevski, The gains and the pitfalls of reification, proved to be very hard for the translation. We needed to create Farsi terms, the first one, for “Reification”. It took a few months from Zahra Kamyab and I to translate it. I think, one day, I do translate it again!

When I translated this one I was a master student, and I was paid about 5 pounds for the translation! At the time I was perhaps proud, being a maths teacher with a translation was better than being a maths teacher with no translation!