For the next three Thursdays (March 16, 23 & 30), we will not be meeting in person (sob!)
March 16 counts as our Reading Break week, and March 23 as our virtual class.
On March 30, I will be away at a conference in Berlin, so we will be adding an extra class at the end of our course (on April 13) to make up for missing this class. Your long papers and presentations will be due on the April 13 class -- more on the details in another blog post!
MORE ON THE VIRTUAL CLASS:
Here is a link to the article we will all read for our virtual class:
Oi-Lam Ng & Nathalie Sinclair on "Young children reasoning about symmetry in a dynamic
geometry environment"
Here's what to do to participate:
• First, read the article in depth by March 23 at the latest, and make your own notes on it.
• Then go to the "comments" section of this blog post, and join the discussion. Read what others have posted, and then add a comment with your thoughts, ideas, quotes, stops, arguments, agreements, speculations, alternate versions, questions, etc. around the article and the ongoing discussion of it.
• Dates: The discussion of the article can, in practice, stretch from tomorrow (March 10) to our next face-to-face class (April 6). I expect most of the discussion to take place between March 16 and 30.
• How much is enough? I will be monitoring the discussion throughout the period from March 10 - April 6. I expect that each person in the class will post at least 4 substantive postings, spread out over at least 7 days.
• A substantive posting should not be super-long (keep your postings to a maximum of two paragraphs each). However, it should be responsive to the article (and should refer back to the article frequently), and it should be responsive to the ongoing class discussion. It is not enough to say, "Great!", or "Yes, I agree". (You can certainly add those kinds of comments, but they will not count as substantive postings.) A substantive posting should be an interesting, engaging, in-depth comment, like the best of your regular blog posts.
• You cannot do all your postings in a clump. This is meant to be a class discussion or conversation on-line -- so you can't do four postings in a row, or do them all on the first or the last day. You should plan to space your postings over the course of a week or more, and take time to read and consider what others have written before posting.
In my experience, our virtual class is often one of the best discussions of the course! With this great group, I have no doubt that this tradition can be continued...
Enjoy!
March 16 counts as our Reading Break week, and March 23 as our virtual class.
On March 30, I will be away at a conference in Berlin, so we will be adding an extra class at the end of our course (on April 13) to make up for missing this class. Your long papers and presentations will be due on the April 13 class -- more on the details in another blog post!
MORE ON THE VIRTUAL CLASS:
Here is a link to the article we will all read for our virtual class:
Oi-Lam Ng & Nathalie Sinclair on "Young children reasoning about symmetry in a dynamic
geometry environment"
Here's what to do to participate:
• First, read the article in depth by March 23 at the latest, and make your own notes on it.
• Then go to the "comments" section of this blog post, and join the discussion. Read what others have posted, and then add a comment with your thoughts, ideas, quotes, stops, arguments, agreements, speculations, alternate versions, questions, etc. around the article and the ongoing discussion of it.
• Dates: The discussion of the article can, in practice, stretch from tomorrow (March 10) to our next face-to-face class (April 6). I expect most of the discussion to take place between March 16 and 30.
• How much is enough? I will be monitoring the discussion throughout the period from March 10 - April 6. I expect that each person in the class will post at least 4 substantive postings, spread out over at least 7 days.
• A substantive posting should not be super-long (keep your postings to a maximum of two paragraphs each). However, it should be responsive to the article (and should refer back to the article frequently), and it should be responsive to the ongoing class discussion. It is not enough to say, "Great!", or "Yes, I agree". (You can certainly add those kinds of comments, but they will not count as substantive postings.) A substantive posting should be an interesting, engaging, in-depth comment, like the best of your regular blog posts.
• You cannot do all your postings in a clump. This is meant to be a class discussion or conversation on-line -- so you can't do four postings in a row, or do them all on the first or the last day. You should plan to space your postings over the course of a week or more, and take time to read and consider what others have written before posting.
In my experience, our virtual class is often one of the best discussions of the course! With this great group, I have no doubt that this tradition can be continued...
Enjoy!
Here is a demo of how to write a comment.
ReplyDeleteOne of my first stops in this article was because of the mention of how relatively “late” symmetry is first explored in elementary. I was astonished to find that the “new” BC curriculum only mentions line symmetry is Grade 4: “using concrete materials such as pattern blocks to create designs that have a mirror image within them.” As this article discusses, students in Grades 1 and 2 were aware of the symmetry in the world around them and could give examples, some of horizontal lines of symmetry, followed by vertical lines of symmetry. Considering the success and evidence of young students being able to discuss symmetry and learn some of the vocabulary after only 1 lesson, why do you think that the curriculum waits to introduce it? There are patterns and relationships explored in earlier grades, which would seem to be a natural place to introduce symmetry.
ReplyDeleteI also noticed and agreed with “our findings confirm with previous studies that the teacher’s use of language plays a powerful role in shaping how students think about mathematics at hand.” (p. 432) I have always seen this strong connection in my classroom. Although I do not insist upon students always using the correct mathematical terms, particularly with ELL students, I am conscious of modelling the correct terminology. I find with modelling that students subconsciously notice the vocabulary and bring it into their discussions. Have you noticed this connection? Do you insist upon correct mathematical language or allow it to develop organically?
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ReplyDeleteThis study investigates the influence of using “dynamic visual mediators” in the form of DGEs in student learning process. It mainly aims to understand to what extent student’s thinking process might change when they use the DGE as a semiotic mediator in their learning. The results show that in an interactive environment student’s way of thinking has been changed and they have developed a new discourse that led them to generalize about properties of symmetry. The interesting point of this study is the role of teacher alongside the role of using DGE in the classroom. Teacher, as a mediator, leads students by using proper words and gestures. At first, students did not know which vocabulary must be used to describe the symmetry and squares’ move, so, they have used simple words, however, after a while teacher’s guides help them to improve their understanding. Besides using DGE makes it possible for students to observe the effect of moving squares or symmetrical line on its symmetrical square. Since in DGE changing colors, distance and number of squares were easy and fast, students had this opportunity to observe the several changes at the same time. This DGE’s ability gives students a holistic view of symmetry that pen and paper cannot provide easily.
ReplyDeleteI believe, nowadays, with using tablets, smartphone, and computers, children’s expectation of learning tools is changed. They would like to use dynamic tools to get a better understanding of a topic. I do not have many experiences using dynamic tools in my classroom; however, I have this experience asking my students to use IXL.CA website to solve mathematical questions. As you might know this website, it provides a dynamic environment to solve the questions. For instance, students can draw shapes or graph to find the answers. All of my students find this website interesting and intend to do their homework via this site. It was interesting to me that although many of mathematical operations such as addition, subtraction or finding area were the same as using the pen and paper, students prefer to use the website. They explained that it shows their scores, time and number of right and wrong answers. As the result of using this site, most of my students have done more homework and tried to get more correct answers.
Both of you (Nancy and Malihe) have commented on the teacher’s use of language. I also found it interesting that the teacher (who was also an author) carefully chose language and gestures “that tune with those of her students, while also moving towards mathematical meanings” (p. 431). This attention to language seems so important when considering the role that is plays in supporting the students’ reasoning and as a tool for their internalizing process. I agree that modelling language can be an effective way to direct student mathematical language and thinking. It requires a consciousness on the teachers’ part that would be engaging for the students. As well, the view of language as thinking and the changing in student discourse and use of words as a reflection on learning mentioned in the article is significant for me. Taking the time or making the space to listen to students work in groups can give insight into their process and understanding.
ReplyDeleteThis article made me appreciate the role of discussion about the shared visual process. At first I was skeptical about the necessity of the digital mediator, but I can’t think of any other ways to show dynamic symmetry. I have often used dynamic or interactive computer mathematics with students who need a lot of support when I don’t have enough time to give individual attention. I can see it’s benefit for homework as well, especially if it is then discussed. This work with students’ symmetry abilities at a primary grade level and the potential increase “spatial flexibility” made me think of how it could help my grade 7s. Graphing coordinates and performing transformations can be extremely difficult for some students. I would also love to find an interactive program for transforming graphing coordinates. Do you think that increased “spatial flexibility” is an important emphasis for early mathematics education?
When I was a pupil, I had little occasion to experience dynamic contents in the mathematics classrooms with cold textbooks and drills. I am curious about how current students will develop their mathematics knowledge and skill through the dynamic activities with technologies in the Digital Age. Because conventional and static textbooks limit students’ activities, the contents cannot be easily applied to cover students’ dynamic interests. However, the current IT devices can expand the mathematical contents to match students’ curiosity (but not always). It may cause more processes of both internal and external activities of students, so the dynamic tools help to build their creativity, flexibility, and diversity in addition to mathematical reasoning and meaning.
ReplyDeleteI agree with the stop point Nancy mentions above. It invites me to check the Japanese national curriculum, and it says line symmetry is Grade 6 in Japan! I suppose the curriculum was built to follow system of mathematics (how we can teach mathematics easier) rather than development of children (what student can understand). Additionally, in the article, the students believe the word “symmetry” can be particular objects and associating with “folding” before their introduction of the symmetry machine. As far as I think about my teaching approach of symmetry, I might use only particular objects and a “folding” expression to teach my students what symmetry is. However, the results of activities the authors did in this research show me a variety of expressions from the students, so these creativity of the students could be observed because of these activities. It implies that teachers sometime might restrict students’ potentials without consciousness. Do you agree with it? Have you ever heard any cases that teachers misestimate and limit students’ abilities?
Tsubasa: I do think that, unfortunately, teachers do misestimate and limit their students' abilities. It is usually not a purposeful decision, however. Often I have seen it occur with teachers who have been teaching for over 5 years and feel comfortable teaching the age level. They feel confident with what students "should" know and how far to push them. These same teachers often feel that it is important to accelerate any gifted students, which is not the same as enrichment. Many elementary teachers, being "generalists", are often not expert at teaching mathematics and are not well prepared to enrich students, nor understand developmentally what mathematical level students will be progressing to. I think it still comes down to elementary teachers needing more training in mathematics.
ReplyDeleteTsubasa’s reflect makes me think of my experience as a student in the school and realize I had a little dynamic mathematic lesson too. However, my experience is more complicated because when I was in elementary school my country was in a war, so, we were studying in fear of bombing and using the cold textbook. As the student, we just were thankful for having math class and never expected to have dynamic class or lesson on that situations!!! However, several years later when I was in university studying industrial designing, I had to use pen and pencil for designing new products which were so difficult because the design must be accurate, but pen could not draw accurate line and curve. Besides changing a small part of the design needed to change the whole plan. But after a while when we started using computer everything was changed. There were several dynamic software that makes this opportunity for us to create 3d-model, to modify products and to get a new 2d- drawing very fast. Using dynamic space of computer was a fascinating experience for me to quickly draw my new ideas and modify my designs. So, now, I can understand to what extent using dynamic equipment in teaching a lesson can be helpful for students to (As Tsubasa mentioned) build and improve their creativity and imagination.
ReplyDeleteThe author argues that the children could tell visually that there was something wrong if the line of symmetry does not bisect the distance between the squares but not able to explain the the findings due to their lack of knowledge of perpendicular lines (p. 433). It links to my class experience. There are many times that students "sense" something in robotics activities(such as the correlation between the radius and perimeter) before they actually learn the concepts.However, one of the major obstacles of integrating robotics in classroom is the cost of the technology. In contrast, gesturing is totally free and requires barely nothing to employ in classrooms.
ReplyDeleteAdmittedly, geometry can be challenging for many elementary students. I have some experience in LEGO activities with students who cannot tell left from right, gears from circles. This study is interesting as it not only indicates that gesturing can be used to change other peoples' thoughts ( teacher's usage of gesture), but also shows that gesturing is part of cognitive process itself ( students' usage of gestures in discourse). It is quite encouraging to see the positive outcome of using gestures and dynamic environment to engage students in the learning.
Ting, I agree with you that cost of the technology is much more than gesturing (as it is free), however, I believe that gesture is limited, but technology gives us a variety of options and possibilities to help students. The virtual environment that technology can provide for students is new and exciting for them. They can do and sense many things that usually they could not do or show by the gesture. I think for elementary school students using 3d- software to show and work with shapes and dimensions is somehow magical. Moreover, nowadays, most of the classrooms are equipped with computer systems, so the possibility of using technology is higher and easier now.
DeleteMy experiences show that in learning geometry using computer system is very helpful. Usually, it is difficult for students to imagine a 3d- object. This problem is more severe in calculating the surface area of 3d objects such as a prism or pyramid. Also, it is difficult to build all these 3d objects with paper or clays, it is time-consuming and expensive, so using computer software is very helpful and applicable. The computer makes it possible to create all 3d- objects and show them to students quickly. I know all teachers do not have good computer skills to draw 3d objects, but it is possible to use the pre-design shapes and make a little change on them.
Ting, you have brought up two ideas having to do with cognition that are interesting for me. One is the “feeling” or “sensing” and the other is diverse abilities, including dyslexia and dysgraphia. I wonder if early work with dynamic geometry can help to develop embodied sense of not only symmetry, but left and right.
ReplyDeleteThe gestures and word use associated with a dynamic geographic environment takes students out of static symmetry ideas and requires students to be communicative. As a teacher and as a student, I know that I have sometimes been frustrated with the time consuming and verbally oriented processes of group learning. I sympathize with the introverts and the visual spatial learners. I am interested in experiences of students who are learning the language (verbal, cultural norms) of instruction and how they relate to social learning situations. I often wonder how much I should scaffold language connections to mathematics and promote social interactions, and how much I should provide for alternate routes of mathematics expression (creative or textbook) to build confidence and tap into previously learned skills. It seems like a balancing act.
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ReplyDeleteWhen I read this article, I wondered the significance of dynamic spatial reasoning at a young age. There is a problem solving ability test called the Cognitive Abilities Test (CogAT) that is commonly used in gifted education. One section is dedicated to paper folding-students predict what a figure would look like if it were folded. Essentially, students use visual-spatial reasoning to identify figures with various lines of symmetry. Interestingly, while in other sections scratch paper is used etc., in this section there are no instructions about providing paper to fold. This reminds me of a previous class discussion that we had about how binaries in math education may value the 'thinking or abstract' over the use of hand-held manipulatives or gestures. Also, this part of the CogAT test is for grade 2 students. Like you, Nancy and Tsubasa, I was also surprised when reminded that symmetry is only introduced in grade 4 and so this intuitive understanding of symmetry is assumed by the CogAT.
ReplyDeleteOne of the roles of the teacher mentioned in the article is that teachers 'exploit opportunities with corresponding tools' to provide meaningful social interactions (discussion) about this (p.425). This article made me think about how we can use embodiment to teach symmetry with 3 -D figures-it reminded me of a time where I failed to 'exploit' an opportunity for learning. I teach a SOMA Cube lesson where students combine blocks (1,2,3, or 4) to make all the possible 3-D figures (there are twelve). I usually focus on regular/irregular polygons, isometric drawings and puzzles as using the seven 'irregular' figures can be connected to create a cube in hundreds of ways. We talk about how if a 3-D figure is rotated, it is still the same. I gloss over symmetry because students often get stuck with two figures that are symmetrical but are distinct. I realize now though, that students have very little experience with talking about symmetry in grade 4. It makes me see this lesson with a whole new lens. It challenges me to think of ways to use embodiment and gestures to teach concepts-with regards to symmetry I was reminded of a mirror game where students face each other and mirror each other's actions.
Hi, Sharon, I am glad that you brought up CogAT assessment in relation to the article. I have the CogAt Grade 7/8 at hand and your comments made me read the practice questions again. At this level, students are asked to determine what a piece of paper will look like once folded. Since there are no physical manipulatives available to students, the general strategy to solve the problem is to visualize the folding. The ability to mirror objects in 2D and 3D is essential to the challenge. And I agree with your idea that the assessment seems to prefer "abstract thinking" over "concrete thinking". Such preference can easily be found in many math competitions and other advanced programs. I was wondering what makes gesturing less valued in these programs.
DeleteSharon and Ting: I am not familiar with the CogAT test at any age level, but I am very familiar with math contests, as I administer many from Grades 3 through to Grade 12. This is similar to what we discussed in our previous class with gesturing: the "accepted" progression for mathematics is concrete, pictorial, abstract. In all our readings and discussions, I have really started to wonder what it is that makes abstraction the goal in mathematics. Why is it considered beneficial to do all your work in your head or on paper? Some topics, such as the aforementioned geometry, are considered an exception to this "rule", and manipulatives are considered "normal" in older grades when learning geometry. We encourage children not to use fingers to count. Why? To me, it seems to further prove Sir Ken Robinson's point that we are preparing students for a world that no longer exists. The education system needs an overhaul. That being said, I'm not sure how that would look....
DeleteTing and Sharon, I am wondering to what extent your students correctly answered these questions?
DeleteMy experience with normal (not gifted) high school students shows me that they usually have problem imagining the shapes especially 3d shapes. As I mentioned in my other comments this difficulty gets worse when they tried to find surface area of combined 3d objects. In fact in many cases I had to make the shapes with paper or clay to help them imagine the shapes.
Ting and Sharon, I am wondering to what extent your students correctly answered these questions?
DeleteMy experience with normal (not gifted) high school students shows me that they usually have problem imagining the shapes especially 3d shapes. As I mentioned in my other comments this difficulty gets worse when they tried to find surface area of combined 3d objects. In fact in many cases I had to make the shapes with paper or clay to help them imagine the shapes.
From the perspective of constructionism, physical manipulatives (including fingers) are mainly beneficial for younger kids when they are initially exposed to newer concepts; but less valued when they are involved in building relatively advanced skills upon previous experiences. My understanding is that since geometry comes later after algebra, it means students will need some exposure to manipulatives even in highschool ages.
ReplyDeleteTing: With regards to the CogAT assessment, in the instructions there is nothing to prompt actual folding or gesturing or anything other than looking at a problem. I suppose it is a logical fallacy to assume that because physical manipulatives are not encouraged in this section, that gesturing is not valued. What I found interesting about symmetry on the CogAT is how it is considered an major indicator of visual-spatial problem solving ability for grade 2 students. The article mentioned that young students are capable of recognizing symmetry at young ages and so I wonder on a cognitive level what it is about recognizing symmetry that would indicate to test makers that this is a major predictor of mathematical ability? Also I wonder about young students experiences with symmetry as well; as they look at the world (perhaps with curiosity or wonder) do they notice symmetry in informal environments on their own? How can we 'exploit' this as educators?
ReplyDeleteThose are great questions, especially the one of "do they notice symmetry on their own?". I read an article years ago about the development of symbols and paintings. I vaguely remember that children as young as 4 can build symmetrical structures from sand and mud. It is not a surprise for me when I have pre-schoolers in lego class build all kinds of castles and plants with symmetry.
Delete:)
DeleteThe first thing that I think of about symmetry for young students is their bodies, the bodies of animals and some plants. There must be a biological feeling of comfort that comes from seeing or sensing symmetry and that starts from an early age. I read recently about symmetry that the fact that we place things directly in front of us is always aligning ourselves symmetrically. It makes sense then that preschool students experiment with building blocks and legos symmetrically. Minecraft would be a great way to work digitally with symmetrical 3D building. I am also thinking of clapping and dance moves with students, and of balancing mass, like on a teeter totter in terms of other or informal learning environments.
ReplyDeleteSo, if the students are noticing symmetry on their own, and we want to exploit these opportunities, it seems the article is suggesting that we make the system dynamic through some sort of play, and that we facilitate “meaningful social exchange”. “The teacher tunes with the students’ semiotic resources and uses them to guide the evolution of mathematical meanings”. For me this means playing with the students, rearranging blocks and asking them to spot a difference or make something symmetrical again. It also means introducing new words or systems of communicating. How important is spontaneity and reading the moment in this exchange? Will it be different every time?
I think you are right, Amanda, that as educators we need to "exploit" this curiosity with symmetry. I have also read that symmetry seems to come somewhat naturally to young children. I wonder if they connect it to reflections, as in mirrors, or if they start noticing symmetry in nature, such as in sunflowers, animals, spider webs, snowflakes, etc. There are many crafts that young students do, such as making snowflakes with paper, positive and negative drawings, doing a reflection of your name in cursive writing. I wonder what it is about us as humans that draws us naturally to symmetry? Is it our bodies? Or the world around us?
ReplyDeleteI do think and have seen that with spontaneity you will get different results every time, as some groups might be more risk takers and some more cautious. Of course as the teacher, there are ways to lead them down the path you may want if there are certain ideas you are hoping they learn. It is hard to be spontaneous when you have so many IRPs to cover!!
Three are many objects that involve symmetry around us. For example, I think origami can be a good tool for pupils to manipulate and experience symmetry with less cost. In fact, I can see a lot of origami textbooks which intend to 2-year-old children in Japan, so this supports the idea that people can recognize symmetry in very early stage. In addition to many examples Nancy mentions above, there are many artificial and symmetry objects such as playing cards. Additionally, it reminds me many traditional clothes in several countries are made with symmetry design when we visited the Museum of Anthropology in this class. Symmetry has history and is art. Thus, it might be able to connect some cultures that students belong to.
ReplyDeleteOne thing I am curious in geometry relates to this article (but a little!) is when students can recognize figures of three dimensions on two dimensions. Unless a teacher brings physical objects, she or he has to explain solid figures by drawing them on the white board as two dimensions. I think this is very counterintuitive for the students who are not familiar with this approach. Although I acknowledge the power of computer, the dynamic objects on screen might have limitation compared to the real objects. Teachers might need to utilize both virtual dynamic figures and physical figures.
Tsubasa mentioned a very good example of symmetry in clothes and fabric that we have seen in our class in the Museum of Anthropology. In my country, we do have a lot of these symmetrical patterns in our carpets, fabrics, and clothes. As a mechanical designer when my colleagues and I were discussed a variety of possible design for automotive products we usually discussed symmetrical patterns. One day, one of my colleagues stated that why we always think of symmetry in our design? What would happen if we do not have any symmetry in our design? Would not it be beautiful? From that moment till now I am thinking of symmetrical patterns around us and the reasons that I like these patterns? I think one of the most important reasons behind this matter is those ancient symmetrical patterns that were around me in Iran. I think we (maybe Iranians only) used to these symmetrical patterns and now it 's hard for us to think in any other way. When I was a little girl, we had a carpet that was full of symmetrical patterns, and I used to play symmetrically around its patterns, on that time I thought every single carpet in the word must have symmetrical patterns!
ReplyDeleteI would like to know your point of view regarding symmetrical patterns around us, do you (like me) think every dual object must be symmetrical?
These are great suggestions! There are so many starting places. I had no idea that there were origami books for such young children. Really great connections with ethno-mathematics. Malihe, I have to agree that sometimes there is an uneasy feeling when we look at things that aren't symmetrical. It makes me wonder about design principles and how teaching aesthetics can be merged with math.
ReplyDeleteTsubasa-I think that your questions about 2-D and 3-D also quite fascinating. From this article, it looked like students were just looking at 2-D squares and so moving square tiles as a task could be a good complimentary activity. I can see how using a screen to show 3-D objects could be very different than actually tangibly holding 3-D objects. When I took chemistry as an undergraduate I had a hard time with dynamic virtual software used to rotate molecules-the hand-held molecules were so much easier to understand (for me). In my work with students, this is the link to the SOMA puzzle pieces I have students make: https://www.google.ca/search?q=soma+cubes&safe=strict&source=lnms&tbm=isch&sa=X&ved=0ahUKEwit7vjUhf3SAhVK6mMKHZC8AmkQ_AUICCgB#imgrc=BRr0-ZsUljv9zM
The two 3-D figures on the bottom left are symmetrical but they are not the same shape (they can't be rotated to make the same shape) which to many people seems counterintuitive. This article and our conversation makes me wonder about the thinking processes involved in this! I think I underestimated the task before.
Nancy, I agree that the pressure of learning outcomes or IRP or the curriculum sometimes can be directly against spontaneity. I think that the conversation often turns to if it is possible to teach with spontaneity and taking time to really delve deep. I think Ted Aoki might call this the tension between the curriculum as planned and curriculum as lived.
Sharon, you bring up another challenge in teaching geometry:the gap between sex when rotating objects is required in a task. It seems there are more females students have the trouble dealing with 3D rotations than male students. Some researchers pointed out that the gap can be attributed to certain biological differences. However,the research is widely misinterpreted as spatial intelligence is a fixed quality that cannot be improved with effort. This is especially harmful and it is not true. As demonstrated in the article, with a careful role shift from teaching to facilitating with spontaneity, the student can be cast as self-directed learner who will positively connect their experience with the problem solving process. There might be some difference at the beginning, but the right training can close the gap in the long run.
ReplyDeleteIn my teaching, I have not experienced a difference in spatial thinking/reasoning/manipulating skills when comparing males and females. I often find it very hard to predict, who has these abilities and interests. Perhaps there are more male gender stereotyped careers that have to do with large scale building and manipulating 3D objects so these skills are looked for in males more often than females.
ReplyDeleteMalihe, in response to your question about every dual object having symmetry, I have spent many years studying human anatomy and physiology and have often wondered why our organs are not arranged in symmetry. From the outside, humans and animals seem symmetrical, but on the inside we are not. Why did we evolve to by asymmetrical? Our muscles and bones and many parts of our nervous system are mainly symmetrical. And, some plants are symmetrical but others aren’t. In chemistry, there are symmetrical and asymmetrical arrangements of atoms in molecules. Are symmetrical objects most pleasing, maybe calming, to our visual processing and do we therefore associate symmetry with beauty? Sharon, I do think that design principals, aesthetics and mathematics can be linked.
Sharon and Tsubasa, I agree that having and touching 3d object is much better than using computer software, However, I believe that making or preparing all 3d objects that might be studied in the classroom is difficult or even impossible, so in these cases having 3d shapes in computer is better than having nothing. As a mechanical designer I worked with these software for more than 10 years so maybe my experiences of working with these software makes me feel comfortable to use them. I do remember that in the early months, working and using these software was difficult for me too. However, still, I believe it is beneficial to use these software to help students learn about 3d objects or making symmetric shapes .
ReplyDeleteAmanda, thank you so much for sharing your knowledge about symmetrical parts of human body. I have heard that outside parts of human body are not symmetrical. When I was younger I have tried to compare my left and right hands to understand are they really asymmetrical? and surprisingly I have found several differences in them at least in their shapes! In general, although, I have seen many asymmetrical shapes that were beautiful, I still like to find symmetrical line in any design.
I think that human bodies are not "really" symmetrical in any way, but that we like to see them that way. I practiced craniosacral therapy for a while and started to focus on all the asymmetries in peoples' faces and heads. I guess this brings up the question of whether anything can be truly symmetrical...at a cellular or particle level.
DeleteAnd, I also have memories of playing with symmetries in rug patterns as a child. I like the idea of mathematics education from art patterns in everyday objects.
The body symmetry topic is so interesting! I have never thought about our anatomy as asymmetrical. I have often just thought about modern conceptions of beauty as symmetrical.
DeleteSharon,
ReplyDeleteThank you for your comment and letting me know about SOMA puzzles, this is very interesting! And, regarding to your comment on Nancy’s post, my classmates and I opened the mini-symposium focusing on the curriculum-as-plan and curriculum-as-lived in another course last term. I introduced the open-end approach in mathematics education as curriculum improvisation there. What I strongly thought is students are much much much much more creative than our (teacher’s) expectations. In addition to their ability of math, it might be easy for math educators to underestimate students’ creativity.
Malihe,
I agree with “having 3d shapes in computer is better than having nothing” as well as that “it is beneficial to use these software to help students learn about 3d objects or making symmetric shapes.” Actually, I love to use computers and those IT devices because I was a systems engineer and often used software (to develop the new software!). Due to the rapid growth of technology, I guess, the system which smoothly creates dynamic and touchable 3-D objects by using the software with 3-D printer will be easily invented. Therefore, mathematics education might develop rapidly same as the technology in next decade. My other concern to using software is teacher’s lack of computer skills. In my country, many teachers cannot use computer well in their class, and they are afraid of the risks of using it. We have to consider about it.
I would love to hear more about this Aoki symposium next class. I also agree that students are more creative than we give them credit for.
ReplyDeleteIn terms of elementary education, there has been a move towards computing. We are currently using 3-D modelling software to create 3-D prints in my program for instance. SFU and Surrey school district are also collaborating to create a Maker certificate program-hopefully this can move forward (get approval so that teachers can get a pay raise through taking this program) and provide avenues for teachers to become more comfortable with computing.
In addition to Japanese mathematics teachers feel risky when using computers, there is also a kind of the atmosphere (I am not sure this is cultural thing or not) that Japanese mathematics classes do not want to use any calculation tools such as a calculator. People might believe using or relying on those technologies may lead students to be spoiled. The teachers might hesitate to use it too because other people might think the teachers these days are spoiled same as the students. Actually, although the Japanese government suggests using computers in any subjects in schools, teachers do not receive any related training programs by the government. The certificate program Sharon introduced here is quite interesting. While the society has been changing, teachers, schools, and mathematics classrooms also need to transform to fit into the current society and children.
ReplyDelete