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.
Sunday 29 March 2015
Sunday 22 March 2015
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.
Sunday 1 March 2015
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
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.
Saturday 21 February 2015
Is There a Geometric Imperative? by Dick Tahta
This weeks reading, "Is There a Geometric Imperative?" by Dick Tahta I found fascinating and challenging. It stretched my thinking past what I have experienced and thought of as Geometry and for that I am grateful. Tahta begins by talking about "time and space as basic areas of mathematical experience," and that "perceptions of the external world are internalised as mental images and these are the 'stuff' of geometry." I had never thought of mental images being considered as geometry, which lead me to believe there was a lot I needed to learn.
The main points made throughout the article are that geometry contains three powers: imagining, construing and figuring. I will endeavour to give you a glimpse into what each is.
The main points made throughout the article are that geometry contains three powers: imagining, construing and figuring. I will endeavour to give you a glimpse into what each is.
- Imagining - The power of taking something that is described by someone and seeing it in your mind. Each individual is unique and therefore will have a unique form of imagining.
- Construing - The power of seeing something that is drawn and then describing what you saw. "All seeing involves a 'seeing', an interpreting of what our eyes are conveying to the brain." Having the context of a image/scene allows for a more accurate construing of it. Tahta acknowledges that it is a skill to take a two-dimensional picture and see it with three-dimensional depth, is something that should be practiced to experience improvement. Once again, individuals are unique and therefore construe images in different ways, or through a different "point of view."
- Figuring - The power of "drawing what is seen." This is obviously something that takes practice and should not be underestimated as a difficult skill. As with the other two powers, figuring can be viewed differently by different people given where they are when observing the object or situation.
Friday 13 February 2015
Learning from Learners: Robust Counterarguments in Fifth Graders' Talk about Reasoning and Proving by Vicki Zack
This week I read "Learning from Learners: Robust Counterarguments in Fifth Graders' Talk about Reasoning and Proving" by Vicki Zack. I was initially drawn to reading this article, given it's approximate grade level to the class I teach, hoping to make some connections or gain some insights. In this article, Zack uses her fifth grade class as a subject group for her research surrounding the "Chessboard Problem." Her students are well versed in problem solving, as it is at the "core of their mathematics curriculum." Zack has her students' follow a distinct process when discussing their strategies for problem solving; firstly, they keep track of the thinking in Math Logs, they then share their solutions to a partner, a group of 4 or 5 and finally, a group of half the class (thirteen students.)
Zack then moves on to speaking directly about the "Chessboard problem." The students go through three tasks:
Task 1: Find all the squares (in a four by four grid given as a figure). Can you prove you have found them all?
Task 2: What if ... this was a 5 by 5 square? How many squares would you have? Extensions were subsequently posed.
Task 3: What if this was a 10 by 10 square? What if this was a 60 by 60 square? How many square would there be?
Zack provided the class with a Ted-Ross strategy that came up with the following answers:
" The answer for the 10 by 10 grid is 385 squares. So take the answer for the 10 by 10 square (385) and 10 x 6 = 60 so multiply 385 x 6 = 2310 and you have the answer for the 60 by 60. What would you say?"
There are five counterarguments that are found as trends when students are coming up with reasons why the Ted-Ross strategy does not work. I was quite confused in the article whether the counterarguments were presented for the students to use or whether their counterarguments were merely looked at to see if they fit the previous counterarguments found.
I like the idea of the students' looking at a possible solution and trying to diagnose why it may be wrong. I think this shows true understanding of the task and may lead to possible correct solutions. A quote that resonated with me was "The students who have generalised a rule after testing it, use what they know and trust it." I believe this is so powerful; the feeling of self-belief! I do not think that if the counterarguments were provided that this would be as beneficial or meaningful.
I am also left wondering if the final task of finding the 60 by 60 squares is one that is too lofty. I would love to have a better understanding of time frame surrounding this task and if there were any students who were stuck on the 10 by 10 square for extended periods of time. My last desire is that there were work samples shown, I think it would lead to a deeper understanding of the students' thinking and process.
Zack then moves on to speaking directly about the "Chessboard problem." The students go through three tasks:
Task 1: Find all the squares (in a four by four grid given as a figure). Can you prove you have found them all?
Task 2: What if ... this was a 5 by 5 square? How many squares would you have? Extensions were subsequently posed.
Task 3: What if this was a 10 by 10 square? What if this was a 60 by 60 square? How many square would there be?
Zack provided the class with a Ted-Ross strategy that came up with the following answers:
" The answer for the 10 by 10 grid is 385 squares. So take the answer for the 10 by 10 square (385) and 10 x 6 = 60 so multiply 385 x 6 = 2310 and you have the answer for the 60 by 60. What would you say?"
There are five counterarguments that are found as trends when students are coming up with reasons why the Ted-Ross strategy does not work. I was quite confused in the article whether the counterarguments were presented for the students to use or whether their counterarguments were merely looked at to see if they fit the previous counterarguments found.
I like the idea of the students' looking at a possible solution and trying to diagnose why it may be wrong. I think this shows true understanding of the task and may lead to possible correct solutions. A quote that resonated with me was "The students who have generalised a rule after testing it, use what they know and trust it." I believe this is so powerful; the feeling of self-belief! I do not think that if the counterarguments were provided that this would be as beneficial or meaningful.
I am also left wondering if the final task of finding the 60 by 60 squares is one that is too lofty. I would love to have a better understanding of time frame surrounding this task and if there were any students who were stuck on the 10 by 10 square for extended periods of time. My last desire is that there were work samples shown, I think it would lead to a deeper understanding of the students' thinking and process.
Saturday 7 February 2015
Our Culture, Geometrical Thinking and Mathematics Education by Paulus Gerdes
This week I read "Our Culture, Geometrical Thinking and Mathematics Education" by Paulus Gerdes. Gerdes identifies the need of engaging Colonized countries in mathematics through culture. He makes particular note of Africa and the fact that although there has been more attention to education in the country, there are still real concerns with regards to class sizes, qualified teachers and accessible information. A hasty curriculum has been implemented to the developing countries, leading to mathematics being seen as useless unless you intend on pursuing education past high school. He quotes the president of the Interamerican Committee on Mathematics Education saying "... mathematics has been used as a barrier to social access, reinforcing the power structure which prevails in the societies (of the Third World)." Gerdes mentions results of studies that have proved that although children learn mathematics in real world scenarios in their every day lives, when they reach school, no matter how similar the mathematics skills are that they are learning, they are confused. "D'Ambrosio concludes that 'learned matheracy eliminates the so-called spontaneous matheracy," and that "the early stages of mathematics education(offer) a very efficient way of instilling the sense of failure, of dependency in the children." D'Ambrosio furthermore was able to acknowledge "the need for incorporation of ethnomathematics into the curriculum in order to avoid a psychological blockade."
Now for the part of the article I found most fascinating and wonderful! Gerdes identifies areas of the African culture that uncover "hidden or frozen mathemathics." He identifies ways in which to see growth in mathematical confidence through cultural-mathematics. He links peasants' houses to rectangular axioms, artisans' funnels to polygon constructions, woven button patterns to pythagoras theorems, and finally, the woven fishing traps and woven baskets to the soccer ball patterns, both using "regular polyhedra." Gerdes realizes that pupils can reinvent artisan techniques, consequentially, doing and learning mathematics, "only if teachers themselves are conscious of hidden mathematics, are convinced of the cultural, educational, and scientific value of rediscovering and exploring hidden mathematics, are aware of the potential of 'unfreezing' this 'frozen mathematics.'" He acknowledges the need to educate those who are educating others.
When reading, I wondered if this has been looked at as a way to engage the First Nations children in mathematics education. In grade 4, we teach a lot about the First Nations' culture and I would love to incorporate some of their artisan work into math.
Lastly, my biggest hope is that since 1988, when this article was written, changes have been made! These significant findings I feel could widely influence the populations of many countries and have a huge impact on societies and mathematical engagement.
Now for the part of the article I found most fascinating and wonderful! Gerdes identifies areas of the African culture that uncover "hidden or frozen mathemathics." He identifies ways in which to see growth in mathematical confidence through cultural-mathematics. He links peasants' houses to rectangular axioms, artisans' funnels to polygon constructions, woven button patterns to pythagoras theorems, and finally, the woven fishing traps and woven baskets to the soccer ball patterns, both using "regular polyhedra." Gerdes realizes that pupils can reinvent artisan techniques, consequentially, doing and learning mathematics, "only if teachers themselves are conscious of hidden mathematics, are convinced of the cultural, educational, and scientific value of rediscovering and exploring hidden mathematics, are aware of the potential of 'unfreezing' this 'frozen mathematics.'" He acknowledges the need to educate those who are educating others.
When reading, I wondered if this has been looked at as a way to engage the First Nations children in mathematics education. In grade 4, we teach a lot about the First Nations' culture and I would love to incorporate some of their artisan work into math.
Lastly, my biggest hope is that since 1988, when this article was written, changes have been made! These significant findings I feel could widely influence the populations of many countries and have a huge impact on societies and mathematical engagement.
Sunday 1 February 2015
In Fostering Communities of Inquiry, Must it Matter That the Teacher Knows "The Answer"? by Alan H. Schoenfeld
In this article Schoenfeld compares and contrasts two different post-secondary classes that he conducts: an undergraduate mathematical problem solving course, and a graduate level mathematics education research group. Schoenfeld sets the stage by acknowledging how differently his two courses are structured. In the problem solving course, he has control over what is taught, learned, and ultimately the steps the students take to get the answers; whereas, in his research group it is more student directed, he and the students engage in joint work together to find answers to both their questions and his.
Ultimately his main point is that he builds community in both situations which allows for engaged inquiry. This could be easily seen for me in the graduate level mathematics course, where Schoenfeld often finds himself in an area of unfamiliar research and is working with his graduate students to discover new territories. I was more intrigued with how he did it in his problem-solving course. The three main things that Schoenfeld acknowledge as crucial are:
"a. There is a common understanding that we are all seeking a particular kind of knowledge, and that while some of us know more than others, 'answers' are not generally known in advance.
b. The real 'authority' is not the Professor - it's a communally accepted standard for the quality of explanations, and our sense of what's right ...
c. There is a feeling of trust, in that we must feel free to have our ideas (and not ourselves) compete ..."
I have always realized the need to build a problem solving environment in the classroom where the students feel it is safe to explore, experiment, and ultimately work together to find the solution to problems, but I had never thought really thought of my role as the teacher in the end when we had developed that community. I love the idea that the students ultimately don't need the teachers' authority to tell them if their answers are right or wrong, that they will be able to "internalize the standards of mathematical judgment for themselves." To do so, Schoenfeld makes reference to John Mason's framework: "First convince yourself. Then convince a friend. Finally, convince an enemy." Schoenfeld goes into great details of his ability to teach and release the students safely into this community atmosphere. The problems he chooses are clearly purposeful and allow for further questions when the "initial answer" is found. I would love to ask the author how he would approach the student who is reluctant to find multiple solutions or defend their answer. Ultimately, I guess you are hoping that the community atmosphere will prevail, however, I have found that it can often take only one to ruin the atmosphere in the room.
Lastly, I am left excited by the fact that lots of schools are currently making a conscious switch to trying to do more mathematical problem based learning and I feel that if this inquiry starts at a young age, the sky is the limits with where their thinking may go!
Ultimately his main point is that he builds community in both situations which allows for engaged inquiry. This could be easily seen for me in the graduate level mathematics course, where Schoenfeld often finds himself in an area of unfamiliar research and is working with his graduate students to discover new territories. I was more intrigued with how he did it in his problem-solving course. The three main things that Schoenfeld acknowledge as crucial are:
"a. There is a common understanding that we are all seeking a particular kind of knowledge, and that while some of us know more than others, 'answers' are not generally known in advance.
b. The real 'authority' is not the Professor - it's a communally accepted standard for the quality of explanations, and our sense of what's right ...
c. There is a feeling of trust, in that we must feel free to have our ideas (and not ourselves) compete ..."
I have always realized the need to build a problem solving environment in the classroom where the students feel it is safe to explore, experiment, and ultimately work together to find the solution to problems, but I had never thought really thought of my role as the teacher in the end when we had developed that community. I love the idea that the students ultimately don't need the teachers' authority to tell them if their answers are right or wrong, that they will be able to "internalize the standards of mathematical judgment for themselves." To do so, Schoenfeld makes reference to John Mason's framework: "First convince yourself. Then convince a friend. Finally, convince an enemy." Schoenfeld goes into great details of his ability to teach and release the students safely into this community atmosphere. The problems he chooses are clearly purposeful and allow for further questions when the "initial answer" is found. I would love to ask the author how he would approach the student who is reluctant to find multiple solutions or defend their answer. Ultimately, I guess you are hoping that the community atmosphere will prevail, however, I have found that it can often take only one to ruin the atmosphere in the room.
Lastly, I am left excited by the fact that lots of schools are currently making a conscious switch to trying to do more mathematical problem based learning and I feel that if this inquiry starts at a young age, the sky is the limits with where their thinking may go!
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