Integrating history of mathematics in the classroom: an analytic survey by Constantinos Tzanakis and Abraham Arcavi

When reading this article I was struck by a topic that, to be honest, I had never really put to much thought into. I had always known tidbits about mathematics history but had never really considered how I learnt it nor how I may incorporate it into my teaching practices. I found it interesting that Tzanakis and Arcavi bring up possible objections to the incorporation of the history of mathematics into curriculum so early on in their article. I am assuming this is to address any skeptics that may be engaging in the article. I found it powerful that they decided to state both "philosophical and practical" objections because by mentioning them, they are able to validate that they have been considered in the decision making process.
Tzanakis and Arcavi raise "five main areas in which mathematics teaching may be supported, enriched, and improved through the history of mathematics into the educational process:
a) the learning of mathematics;
b) the development of views on the nature of mathematics and mathematical activity;
c) the didactical background of teachers and their pedagogical repertoire;
d) the affective predisposition towards mathematics; and
e) the appreciation of mathematics as a cultural-human endeavour.
I found many relevant points that support the teaching of the history of mathematics. You are able to emphasize the imperfect nature of the discipline, and that many failures or attempts were made before formulas or proofs were achieved. This will hopefully lead your students to taking more risks and realizing that they do not always need to get the right answer right away. You are also able to make clear interdisciplinary links that have been around for centuries and are sometimes lost when getting stuck in formulas. You can include information about past notations and methods for doing calculations.
The article went on to give many examples of application; however, I was discouraged that many of the ideas were focused on high school and university level classes. I believe the teaching of the history of mathematics is valuable at all age levels. As mentioned in the article, it is all about making sure the teachers have the knowledge and resources in order to properly incorporate it into their practices. Currently the teacher's lack resources and it is not taught in the pre-service programs. In my opinion, I think we have a long way to go in this regard. I also think that the skeptics that worry about not having enough time to teach everything will definitely continue to worry about this and not see this inclusion as a feasible option.

Charting the microworld territory over time: design and construction in mathematics education by Lulu Healy and Chronis Kynigos

When choosing to read this article I must admit that I knew nothing about microworlds so I was quite excited to embark on a truly new thought provoking journey. I unfortunately feel left not completely satisfied. I took copious amounts of notes as I read (I am not sure if this is because I didn't have the novelty of highlighting it, I just had to read it on a small tablet) hoping to keep my attention focused and to find flow in the article.

Healy and Kynigos refer a lot to the work of Papert who started working with the idea of microworlds in 1972. This original idea has evolved into as the article states, "redescribed as computational environments embedding a coherent set of scientific concepts and relations designed so that with an appropriate set of tasks and pedagogy, students can engage in exploration and construction activity rich in the generation of meaning." My take away is that you create an environment that can change and grow as the student learns which I love as a concept. The authors place this belief in two main constructivist theories, body syntonic and ego syntonic. These two involve the student relating behaviours of the microworld object to their own and then in turn passing judgement with relation to their real world experiences. Healy and Kynigos link these base theories to Vygotsky's view of internalisation that later links the fact that individuals can adapt the environment to them personally to Vygotsky's zone of proximal development. I always wonder how you can optimise students' success if you are purely relying on their self-motivation? I feel it takes great practice. Healy and Kynigos refer to the blurring of socially constructed lines which makes the teacher the knower and the student the learner. This i feel helps in the answering of my above question. 

Even with the thorough theoretical grounding I was still unclear of what this looked like in a classroom. One example given which surprised me was the use of storytelling. "Current research is identifying regularities in the stories,

which emerge across learners working on the same activ-

ities, suggesting that storytelling for meaning making is not

random: it seems that we might be able to predict the

storylines associated with different microworld tasks and

hence highlight possible connections to the particular

mathematical relations, which these stories emphasise

(Sinclair et al., in press)." Then an example was presented with a more technological basing about a calculator that uses sounds and colours as well as just numbers. The authors pose this as an idea for adaptations for students who are blind or deaf. The proven could be individualised to fit each students' desires, creating their own microworld. 

Although I really loved what I learned in the article, I still feel like I am not fully clear on the framework of the microworld as an entity. A quote stated in the article i found very appropriate for my feelings was, "One of the challenges faced by the microworld community, then, is to find methods and avenues for

communicating these ideas to a wider audience in a form that makes sense to all." I believe there are some excellent applications and I look forward to further clarifying my understanding. 

for the learning of mathematics - Volume 17, number 2, The 50th Edition

First impressions:
From the very first time I came in contact with this journal, I have been struck by the fact that the title is written in all lower case letters. I tried to find out why, but I cannot find an answer to this pondering of mine; however, I wonder if it is to make the journal less intimidating and to set it apart from the rest of the mathematics journals around. The colours the founders chose for the covers would somewhat go along with this theme of accessibility and comfort, seeing as they are popular colours from the 1970s and the journal did not begin until 1980. This makes me think that the colours are meant to be inviting for people to come in for a discussion, much like a grandmother's living room. 

Journal specifics:
FLM began in the 1980s, it is a Canadian publication and is therefore found in both French and English, and it is published three times a year, March, July, and November. The front and back covers are very simple and clear. For the 50th edition, there is a clear 50 filled in with a pattern, which seen on other covers as well. On the back cover there is always the table of contents, making it easy for reader's to locate articles that interest them without even opening the journal itself. Most of the articles are quite brief, 2,500 - 5,000 words only, making them inviting to the reader. From my experience in reading articles from FLM, the language used tends to be very friendly, not riddled with the typical jargon found in other journal articles, that require a dictionary at hand in order to read and understand. Some articles have pictures and diagrams, some do not. Most articles tend to have sub-headings, which once again, makes the article easier to read, by breaking up the main ideas and clearly stating what they are. Many of the articles have references, but not a huge list; most of the ideas raised are personal and exploratory. The advisory board has members from all around the world; however, the board of directors remains Canadian. 

The aim: 
"The journal aims to stimulate reflection on mathematics education at all levels, and promote study of its practices and its theories: to generate productive discussion; to encourage enquiry and research; to promote criticism and evaluation of ideas and procedures current in the field. It is intended for the mathematics educator who is aware that the learning and teaching of mathematics are complex enterprises about which much remains to be revealed and understood."

When reading the articles, it becomes clear that the journal wants to encourage critical dialogue in which to stimulate discussion between the stakeholders involved in mathematics education. The journal is welcoming to everyone involved in the field, from teachers to researchers alike, and it allows for thoughtful reflection on past and present teaching practices and current issues affecting this educational practice. 

The 50th edition: 
This edition seems to have the usual current topics of interest within its covers, for example ethnomathematics, word problems etc.; however, in addition it has two special articles about FLM itself. I was especially drawn to the article by Lesley Lee, The "spirit" of FLM. She presents three of her favourite articles and then comments about when the original editor left it was imperative to keep the same "spirit in the journal." She reflects on the "spirit" of the journal by stating: "the three articles I have reviewed here are, I believe, exemplars of that 'spirit' - presenting voices from within, without, and on the fringes, questioning, surprising, shocking, asking us to 'think again,' allowing contributors to be provocative, to test new ideas, and readers around the

world to share in the excitement of mathematics education-and this reader to express hers 'enfin.'" I found this quote extremely powerful and ultimately a clear stating of the purpose of the journal, to invite critical criticism and reflection among those who's daily studies are affected by the issues in mathematics education. Between some of the articles I found quotes or interviews, serving as intermissions. These interludes had a general theme of being true to yourself, your thoughts, and your peers. They all brought up the fact that it is important to add value to your field of interest and that new ideas need to be shared. To me these quotes described the tone of the journal as a whole; that it welcomes personal, new, and innovative ideas, much like Lesley Lee says in her special article.  

I personally find this journal accessible and relevant to me as a teacher, student, and forever learner. It raises issues that are current, gives an open and safe forum for educated discussion and sometimes disagreement, and it invites people of varying experiences into a common assembly.