I guess it is fair to say that I started my research as a phenomenographer. In my first papers which were mostly about the notion of equivalence, there were always a few lines about phenomenography. Gradually, the research areas that I worked on increased and my explicit reference to phenomenography decreased to zero. However, it seems that it was always with me in whatever I did. I’ve always believed in certain principles, that, any phenomenon can be understood (experienced) in different (though limited numbers of) ways, that, the way someone understands something is somehow related to the way (s)he uses it. And finally something of different nature but decisive in my research is the wish of helping learners to learn better and not just showing where they fail.
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.
APMC-2016 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.
ESM-2009 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.
PME31 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 However, apart from a small (and unpublished) research that I did with one of my master students years after this PME paper, I never seriously came back to the ideas discussed in the paper.
BSRLM-2005 is part of a series of papers I wrote about examples. The series ended with a PME paper titled “Examples, a Missing Link”.
PME29 was written when I was a PhD 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 🙂
PRIMUS-2012 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.
RME-2005 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 equivalence relation. It tells how I learnt 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:
Recognising 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.
FLM-2012 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.
PME-37-SO-2 is a short report of the master’s dissertation of one of my students, Hosein. He was one of the first students who helped me to direct my research towards empowering students to learn mathematics rather than just showing their difficulties.
PME-37-SO-1 is a very early report of a collaborative work with one of my Ph.D. students, Sepideh, on a design to cultivate learning of negative numbers in an algebraically useful way. The paper was accepted as Short Oral, not as Research Report since, at that time, we hadn’t got enough data to support our idea. It is here to show the origin of our extended experimental research done later on.
PME-37-RR is an early report of a collaborative work with one of my Ph.D. students, Leyla, in early algebra. It has the essence of what is about to be published as a journal paper.
چه خالی است a جای (“a” is greatly missed) is the third of a series of papers about the different roles of letters in algebra, the first one was What a delicious a! (a چه خوشمزه است) and the second one, What a silent a! (a چه ساکت است). This one-page paper gives some example of how different and mostly ambivalent uses of the so-called “blank square” or the like in arithmetic could be a source of confusion when children start to use letters in algebra.
a چه ساکت است (What a silent a!) is the second of a series of papers about the different roles of letters in algebra, the first one being “what a delicious a!”. The paper criticizes the use of algebra as a meaningless language. It starts with the following real dialogue between me and my friend:
Me (in a proud voice): I have a friend who speaks in six languages.
My friend: Wow! What does he say with those six languages?!!!
Roshd-2008 is the first of a series of papers about the different roles of letters in algebra. It is a jargon-free paper based on research done by my first master students (Maryam Abdollahpour). The article addresses the problem of wrongly crossing out (in Persian, eating!) letters from the top and/or bottom of an algebraic fraction. The article starts with , inviting the reader to guess what it is.