What was your greatest 'learning' this semester with regard to teaching children mathematics? How has your thinking shifted?
Through this semester, with my mathematics course, I have really grasped the concept that children need to learn for themselves, how to work things out, rather than simply being told what to do and reciting rules to solve a problem.
There are various ways to approach a problem. Neither of which are better than the other, yet one of which may be better for an individual than it is for another individual. Supplying various manipulatives and guidance towards various methods to approach a subject is the best way to teach mathematics, rather than telling and showing students how to 'do' mathematics.
They will after all, be faced with situations in which they can apply what they have learned to various other problems and situations. Even in real life. Being able to approach such real life situations with an idea of how to solve it rather than a learned mathematical formula, will be much more useful and applicable to many problems that they can face in their lives.
This idea has helped me to strengthen my belief that mathematics should be taught from a realistic standpoint, in which problems are offered based on real life situations, and that the application of these problems should be as interactive and hands-on as possible. Use the world around your students to apply mathematics, allowing them to see for themselves, that math exists around them at all times. This course has opened up my eyes to the possibilities and variety of ways in which to do these things.
Saturday, April 5, 2014
Wednesday, February 26, 2014
K-6 Teacher's Resources
Today, I had the chance to view some of the teachers resources for K-6 Mathematics in Newfoundland. Being a very visual person and personally enjoying working in the primary grades, I was pleased to see the variety of books offered through large lap books and small reading books used in most of the primary grades. I think it was grade 3 however where there was an absence of these books. I feel that some sort of visual, helping to guide children would be a great resource. Even at my age now, I would still benefit from visuals, especially when introduced to something new! Ever try to put together some piece of furniture without the instructions? Now imagine that was a product from Ikea! Not happening... not for me anyways.
There will be kids that won't need or want visuals, but there most certainly will be the visual learners. We can as teachers, incorporate visuals and use manipulatives in the classroom, but I really like the idea of incorporating reading into mathematics. The simpleness of the Kindergarten book that depicted 10's by pictures of bears in train carts can be recreated and then manipulated by the students, sort of like a set of instructions for that piece of furniture. Then there was a pattern book showing pictures of those flat, colored, wooden shapes put into patterns in which the child has to guess the next shape needed to continue the pattern. Given the manipulatives of these wooden shapes to follow along with the book, children can be hands-on and do it themselves to show you what they know. They can then even extend the pattern to show any knowledge they have that extends beyond the question in the book.
Incorporating this idea into older classrooms, making things simple so that students can extend on visual problems with the use of manipulatives and/or working out problems on paper from visuals in a book could really help with their mathematics skills and make learning less cumbersome and boring. For instance, in a higher grade, the resources had a mass amount of writing, but the things that would catch my eye were the problems that were presented in pictures, such as a triangular set of frames that the child had to figure out the amount of ways that this set up could be rearranged. This visual I think, was still in the primary grades, in grade probably.
I understand that problems get more abstract as the grades progress but some sort of visual representation as a springboard into the deeper level of a concept, I feel, would be super helpful in many ways!
There will be kids that won't need or want visuals, but there most certainly will be the visual learners. We can as teachers, incorporate visuals and use manipulatives in the classroom, but I really like the idea of incorporating reading into mathematics. The simpleness of the Kindergarten book that depicted 10's by pictures of bears in train carts can be recreated and then manipulated by the students, sort of like a set of instructions for that piece of furniture. Then there was a pattern book showing pictures of those flat, colored, wooden shapes put into patterns in which the child has to guess the next shape needed to continue the pattern. Given the manipulatives of these wooden shapes to follow along with the book, children can be hands-on and do it themselves to show you what they know. They can then even extend the pattern to show any knowledge they have that extends beyond the question in the book.
Incorporating this idea into older classrooms, making things simple so that students can extend on visual problems with the use of manipulatives and/or working out problems on paper from visuals in a book could really help with their mathematics skills and make learning less cumbersome and boring. For instance, in a higher grade, the resources had a mass amount of writing, but the things that would catch my eye were the problems that were presented in pictures, such as a triangular set of frames that the child had to figure out the amount of ways that this set up could be rearranged. This visual I think, was still in the primary grades, in grade probably.
I understand that problems get more abstract as the grades progress but some sort of visual representation as a springboard into the deeper level of a concept, I feel, would be super helpful in many ways!
Thursday, January 30, 2014
On YouCubed, a site offerring free information on research and resources in the area of k-12 mathematics
I was intrigued by the site YouCubed, a non-profit organization, set up by Jo Boaler, that offers teachers and parents access to up-to-date research and resources in the area of k-12 mathematics, via the internet.
It discusses a concept that I have believed in for a long time: the idea that learning truly only
happens when a person is truly invested or enveloped in something. As with any subject or concept, in mathematics, this comes when there is a keen interest involved. Therefore, connections should be made between the mathematical content or concept being taught, and the children it is being taught to, via what motivates or interests them.
Just as the children in this video, found under the window into the classroom link on YouCubed indicated, math:
“doesn’t always have to be numbers, it can be letters
sometimes”
“can be like a story”
“is like a game”
“could be creative”
is used “if you’re making a house”.
Or, “you can use anything for math”
This short video shows what happens when children are opened
up to what math really is; when it is taken out of the bounds of a textbook,
and brought out into the real world where it actually exists and thrives. In this video, there was a point to capture the smiles, interactions, discussions, interconnectedness, enthusiasm, teamwork, thinking, exploration, etc. that were occurring among this group of children. This all took place after they had been asked (and I would assume, motivated) to solve a math problem that they had been presented with. Also, they seemed to be given the reins where they were allowed to make their own decisions and mistakes so that they could solve the problems on their own through deep thought, and discussion.
With the idea of children making mistakes, this is another concept that is brought up in various places throughout this site: the idea that children should even be encouraged to make mistakes. When a mistake is made, if one analyzes exactly what it was that made it a mistake, that one can truly learn the mechanics of the problem. For instance, as a new car owner, my car runs perfectly right now. I accept that and expect that it will run again tomorrow. But what if, when I place the key in in the morning, it does not start? It is only then that I begin to wonder why. If I leave a door open a crack and that runs down the battery then I will investigate and learn that this is what has happened and why it has happened. I will learn to make sure not to do this again, but also, will have learned a little bit more about the mechanics of how a car works.
This concept we have of not wanting children to make mistakes is therefore the opposite of the mindset that we want to set for ourselves and for our students then. Similarly, as is discussed in the various links, there is this idea that only certain people are `good`at math. This is a myth that has been around for so long that we are led to believe it to be true. The first link I opened on YouCubed brought me to a page that used the analogy of
mathematics as being an elephant in the classroom. This ‘elephant’ is this very myth. Much research has been
done to prove that this is not the case, and in fact, everyone has the
potential to be high achievers in math. One study even documented that the
hippocampus grew substantially when required to encompass a large amount of
spatial knowledge, shrinking only when this information was no longer needed,
such as when taxi drivers are required to learn large mapped areas, complete
with main roads and side roads. It is when these drivers retired years later that their
hippocampus’ apparently had shrunk back to normal size. I found this research to be quite interesting. It is worth pondering over, for what harm would there be in believing in each and every child having the potential to achieve highly in mathematics? Encouragement in itself can go a long way.
Giving them such encouragement and motivation to learn offers them the chance to excel. Without it, they may not reach the potential that they have. As teachers, this is something that we should be motivated to do for every child anyways. If someone is told that they will not do well, or isn’t given
the opportunity to excel, then they wont. Its as simple as that. As teachers we
should be providing the scaffolding required to help students reach their
potential, to be constantly challenging them and gently pushing them forward
and upwards through their learning. This should be the same with mathematics. It
is this widespread myth that only a select few are innately good at mathematics, that is holding people back. Myths, stereotypes, etc.
have too long made us think a certain way. We really need to think for
ourselves and to break away from such widespread notions. We should try to teach children to be critical thinkers. Why then should we remain uncritical ourselves?
Another myth is that children who work faster at mathematics
are the more capable at mathematics. This was a belief of my own, and as I’ve
witnessed, is a belief of many, young and old alike. It is another notion that
has been put in our minds through the years in our western society. It makes
sense to me though, that, as this site addresses, slow learners are just as capable.
They are probably even more successful at math because they are taking their time. What is truly
important, over speed, is the learning that is taking place, or when, as Boaler states, students “deeply
understand things and their relations to each other. This is where intelligence
lies.”
Just like the child earlier who stated that “math is like a
game” when referring to the many ways in which math is all around us. With this
notion, we can see the math that exists in such things as a game of chess. Now
think about this game. It is one of high concentration. Would you expect the
players to speed through the game to get to the end? No. Rash decisions are
made in such a rush. It is a game that takes time and patience. All moves are
analyzed before each player makes a move. Like chess, mathematics has many
different ‘moves’ that can be made and we too, should be analyzing each move
and to try and gain a better understanding of the ‘game’ before we rush towards
the end. This is the way we should be thinking about mathematics and how we should be presenting it to students.
To make math fun, and to provide teachers and parents with
ideas, there are various links to activities and even a type of lesson plan
with step by step instructions to involve students in mathematics. The link
provided under content and tasks brought me to a lesson plan in geometry for grades 3-5. It is provided by the Crazy 8`s club. This
activity involves an item that many kids love: glow sticks! The fact that students would be working with manipulatives in the dark that glow, is a starting motivator right there! It is this type of creative thinking that engages
students to think creatively themselves
The amount of links provided
through various games and information on this site is invaluable. I can see the
potential of this site, especially if it is updated often. As of right now, it has a vast
amount of information even though it is still in its early stages.
Wednesday, January 22, 2014
"You can think of almost everything as a math problem" -from Math Curse, by Jon Scieszka + Lane Smith.
What is math then?
To take a definition from the internet, it is the abstract science of number, quantity, and space. But math is much broader than this.
It is an abstract way of thinking that allows us to make sense of and to understand various life experiences. Such a definition though, can be applied to other areas. For instance, the same could be said about science, and just as Scieszka's book states: "You can think of almost everything as a science problem" too. Therefore, it is the mind frame with which we apply our thoughts to a problem we are presented with, that determines the 'type' of problem we are focused on solving, and therefore, the outcome or answer that we are striving towards.
So Math, to me, comes down to thinking mathematically.
So, what does it mean to DO mathematics? To me, 'doing' mathematics is typically thought of as being presented with a problem and working our way through it using a set of rules around numbers taught to us from within a book.
Coming back to THINKING mathematically,
Math can be something we think about, without even thinking that we are thinking about it!
If we are to concentrate on thinking about something presented to us, in a mathematical way, then we can take our knowledge of math that we learned in a book, outside of this book, and apply this knowledge, expanding on it, to solve various problems.
Scieszka presents this way of thinking in Math Curse, as the main character finds problems in every aspect of her everyday life. Problems that can be thought of mathematically and can therefore be solved by thinking mathematically.
What is math then?
To take a definition from the internet, it is the abstract science of number, quantity, and space. But math is much broader than this.
It is an abstract way of thinking that allows us to make sense of and to understand various life experiences. Such a definition though, can be applied to other areas. For instance, the same could be said about science, and just as Scieszka's book states: "You can think of almost everything as a science problem" too. Therefore, it is the mind frame with which we apply our thoughts to a problem we are presented with, that determines the 'type' of problem we are focused on solving, and therefore, the outcome or answer that we are striving towards.
So Math, to me, comes down to thinking mathematically.
So, what does it mean to DO mathematics? To me, 'doing' mathematics is typically thought of as being presented with a problem and working our way through it using a set of rules around numbers taught to us from within a book.
Coming back to THINKING mathematically,
Math can be something we think about, without even thinking that we are thinking about it!
If we are to concentrate on thinking about something presented to us, in a mathematical way, then we can take our knowledge of math that we learned in a book, outside of this book, and apply this knowledge, expanding on it, to solve various problems.
Scieszka presents this way of thinking in Math Curse, as the main character finds problems in every aspect of her everyday life. Problems that can be thought of mathematically and can therefore be solved by thinking mathematically.
Monday, January 20, 2014
A response to Ken Robinson's 2006 TED Talk, with regards to ADHD
We were asked to respond to Ken Robinson's 2006 TED Talk. While viewing the video, what caught my attention most, was Robinson's example of the child, Jillian, whose teachers had wrote home to inform her parents that they thought she had a learning disorder.
This brings to mind, my own thoughts on this subject of children as being diagnosed with ADHD. We are taught in our education program, that all children learn differently, and that our task is to find a way to teach each and every child the various curriculum components in various ways, in an attempt to reach each child. Some children are visual learners, others kinesthetic, and others yet, may learn by reading or seeing. Yet, when faced with a child who can't sit still or concentrate for long periods of time, instead of finding out how they would learn best, and employing such techniques with which to teach them, we are forced to label them with a learning disorder and, as Robinson says, to "put {them] on medication and [telling them] ... to calm down". Maybe something like Robinson's statement that they "have to move to think," is true. Why then, can we incorporate visual aides to accommodate visual learners, and not do what we can to accommodate learners who maybe need to move to learn? What would be the harm in attempting such methods of teaching? I would very much like to know how Thom's Hartmann's Hunter school, which was for children diagnosed as having ADHD and Aspergers, went about in their daily lessons. For Thom looked at ADHD, not as a mental disease, or difference, but as a context disorder. He says that:
"Left-handedness is another, for example. If a left-handed person were put in a room with nothing but right-handed-required tools, she would have a problem 'succeeding'."
He then drives his point home as he continues, saying that:
"In the years when my parents were in school, it was common to tie the left arm of left-handed children to their bodies, so they'd be "forced to learn to be normal" and use their right hands. Enormous psychological wounding was done in the name of enforcing 'normalcy'."
This is eye-opening. Would you say today that someone who is left-handed is not normal? That there is something wrong with them? Do you think that there should be a drug created to correct this 'problem'? I sure don't!
Jillian, the dancer, went on to succeed immensely in life. If we constrain all of these children with 'learning problems' to sit still in class and to listen to what is being taught, not to experience the world as they should, we are, as Ken stresses, killing their creativity, and these children deserve the right to harness their overzealous energy to become productive members of society in their own areas of expertise and interest, just as any child should be given the opportunity. They may even succeed immensely, just as Jillian was able to do, when she was given the opportunity to explore her interests and to express herself by channelling her energy in a positive way.
Ken Robinson's 2006 TED Talks video:
http://www.ted.com/talks/ken_robinson_says_schools_kill_creativity.html
Thom Hartmann's 'Reinventing Our Schools'
http://www.wrightslaw.com/advoc/guest/hartmann_reinventing.htm
This brings to mind, my own thoughts on this subject of children as being diagnosed with ADHD. We are taught in our education program, that all children learn differently, and that our task is to find a way to teach each and every child the various curriculum components in various ways, in an attempt to reach each child. Some children are visual learners, others kinesthetic, and others yet, may learn by reading or seeing. Yet, when faced with a child who can't sit still or concentrate for long periods of time, instead of finding out how they would learn best, and employing such techniques with which to teach them, we are forced to label them with a learning disorder and, as Robinson says, to "put {them] on medication and [telling them] ... to calm down". Maybe something like Robinson's statement that they "have to move to think," is true. Why then, can we incorporate visual aides to accommodate visual learners, and not do what we can to accommodate learners who maybe need to move to learn? What would be the harm in attempting such methods of teaching? I would very much like to know how Thom's Hartmann's Hunter school, which was for children diagnosed as having ADHD and Aspergers, went about in their daily lessons. For Thom looked at ADHD, not as a mental disease, or difference, but as a context disorder. He says that:
"Left-handedness is another, for example. If a left-handed person were put in a room with nothing but right-handed-required tools, she would have a problem 'succeeding'."
He then drives his point home as he continues, saying that:
"In the years when my parents were in school, it was common to tie the left arm of left-handed children to their bodies, so they'd be "forced to learn to be normal" and use their right hands. Enormous psychological wounding was done in the name of enforcing 'normalcy'."
This is eye-opening. Would you say today that someone who is left-handed is not normal? That there is something wrong with them? Do you think that there should be a drug created to correct this 'problem'? I sure don't!
Jillian, the dancer, went on to succeed immensely in life. If we constrain all of these children with 'learning problems' to sit still in class and to listen to what is being taught, not to experience the world as they should, we are, as Ken stresses, killing their creativity, and these children deserve the right to harness their overzealous energy to become productive members of society in their own areas of expertise and interest, just as any child should be given the opportunity. They may even succeed immensely, just as Jillian was able to do, when she was given the opportunity to explore her interests and to express herself by channelling her energy in a positive way.
Ken Robinson's 2006 TED Talks video:
http://www.ted.com/talks/ken_robinson_says_schools_kill_creativity.html
Thom Hartmann's 'Reinventing Our Schools'
http://www.wrightslaw.com/advoc/guest/hartmann_reinventing.htm
Wednesday, January 15, 2014
My Math Autobiography
From the very little that I actually CAN recall of my own experiences with math in primary and elementary, I cannot recall having any issues with it. I think of myself as having been a very obedient student back then, being very meek and mild. I probably got frustrated but would never dare get upset with anyone. I always wanted to please everyone, which to some extent, I still do today.
What do I remember about my actual learning experiences with math at this time? Well I mostly only recall the use of manipulative's. I guess that's the kinesthetic learner in me. I recall counting apples. I also recall using those rod like creations that represent the tens and the little squares that represent the ones. I recall being able to piece these together, or at least I -think- I do. I can't really remember much of what we did with them. Used them for counting I suppose.
The thing I remember most vividly though, were these 3-D wooden shapes that fit neatly in the palm of your hand. Cylinders, cones, spheres... I recall building things with them, or rolling them around. That wasn't the mathematics objective I'm sure, but maybe these are some of the things we did as we waited for our turn during Christmas concerts (we all waited in our classrooms on such special nights and brought board games from home to play with our classmates. But I'm sure we could have gotten quite tired of them and came up with our own imaginary play with anything and everything we could get our hands on, just as any kid would). Yet, I do also recall learning from them. The idea of learning what a cone was, by relating it to what a drawn triangle was, or a sphere from a circle. Such learning brought to life with the tangible form of these 3-D shapes that i could hold in my hand and personally examine from all angles, was wonderful. This would be much easier than seeing these shapes and their angles represented by dashed lines on a paper, or at least I would assume. This makes sense to me anyways.
My son recently came home with homework involving such 3-dimensional shapes on paper. He had to cut them out and glue them on another paper, and then proceed to cut our their corresponding names spelled out on another sheet of paper, and then glue these names underneath the shapes. Thinking back now, I think I would much prefer the manipulatives. No. I know I would. A 3-D shape isn't the same when it is on paper. It's supposed to be 3-Dimensional after all.
I recall when, in Grade 3 or so, my teacher scolded me for not doing homework or something else, and then proceeded to embarrass me by digging through my book-bag (although he had no reason to do so, as I was not playing with anything that was in there during class. I was actually going to my cousins house after school to play barbies). I watched in horror as he began to drag out every barbie doll and pony that I had stuffed in there, counting them out loud as he went. When he got to ...say 23, I automatically knew that he had missed my favorite purple 'my little pony', so, I reached into the bag and pulled out one more, meekly announcing a "24". Hey, there's some math! I knew how to count! Anyways ... these are the things I recall the most, not how the teacher assessed our learning, or the role of the teacher. Maybe it isn't simply that I can't remember but that I really didn't like mathematics back then, because I love language arts. I always did, and I recall writing stores in class most of all, and I would be able to tell you that one form of assessment there was the taking in of our work and handing it back with a mark. As I ponder what I just wrote, I wonder if I'm even being truthful. I could be recounting my high-school years, which started in a school that held grades 7 through 12. Well, I am doing my best here...
The only assessment I can think of that I can positively say was in elementary, was in kindergarten. We had these books. I recall doing math in these books, coloring shapes and such. Coloring by numbers. coloring this, and coloring that. Drawing lines to matching pictures....and then coloring them. I'm assuming that the teacher looked at these to see where we were at with the various activities... or maybe she would simply check to see if we colored within the lines.
As for what mathematics looked like. Well in Kindergarten, the room had many different areas. Different centres I suppose. Maybe one of them was for math or manipulatives? Maybe not. I'm guessing there was a sandbox though! That's all I can dredge up from memory. I used to have a VHS of my kindergarten graduation. We were in the classroom before it began. Something tells me I recall these centres from watching that video, and not from my actual memory of that exact time and experience in my life.... and now I am left to wonder just where that VHS tape could be. That's how great my memory is. Not that the tape would do much good. I haven't anything to play such dated material on anyways.
I do recall enjoying those manipulatives and those kindergarten coloring books. These things brought a sort of 'fun' element to learning. One that I guess wasn't as evident in the other areas of learning about mathematics, such as the times tables and telling time.
Math in high-school. I can recall much more of those days. Most of it was negative. I had one teacher who was my Uncle and whom seemed to be trying to make the statement to the whole class. No, to the whole school, that just because he had children and nieces/nephews in his classrooms, that he wouldn't treat them any better. Well he sure didn't. To hit his point home, I swear he treated us worse. Well, me at least. I wasn't in the classrooms of any of his other family members, so I couldn't tell you about their experiences. Another homework not done incident (I swear these must have been the only two times in which I didn't have my homework done. I was so organized that many could have called me obsessive over it at one point in time, not to mention being a fairly goody two shoes when it came to homework and studying!). Anyways, he made me sit outside the classroom. Needless to say, no learning got done for me that day, only a resentment for my teacher, and maybe for math too. Then there was Mr....Brown? He was temporary. Thank God! I recall raising my hand on many different occasions even though I knew it was futile. He would never ask me my question or attempt at describing the problem to me in any other way than the way he wanted to explain it. Even when I would ask it outright. He would ignore me completely.
In my last year of high-school, I had the option of taking an advanced math course that was only offered through distance education. With the support from my classmates, I decided to give this a try and it was going well. But then, a few months into the year, I moved to a new community, and thus, a new school, and I lost this close-knit support and went back to academic mathematics. In this school, I gained astounding grades (seriously, I got 100% on EVERY exam. A HUGE change from my regular 70's and 80's). I wasn't naive and I didn't feel that I'd earned them - the teaching at this school was just so different. It was like they had given up on their students or their teaching, or both, and just started handing students answers and grades. They let students off with so much, and just brushed many problems under the carpet, neatly hidden out of sight.
Through these experiences, I managed to build an uneasiness with mathematics, and created a sort of love/hate relationship with it.
So when I came to University, 11 years after my graduation from high-school, I saw the option to either take ONE math course, or TWO math courses. Being out of practice so long, and with my background, I felt it only natural to choose to do the ONE math course! The notion of a math placement test was lost on me. I never even heard of it until after my calculus course was done and over with. So Math 1000, or calculus, is what I took one summer intersession, along with a psychology course. And I can tell you, it was hard work! My prof, on the first day, while writing out a calculus equation on the board, proceeded to work out within it, a problem using the foil method on the board. He announced that if we did not know how to do what he was doing, then we'd best go back to 1090. I knew WHAT he was doing, but had no recollection as to HOW he was doing it. Who would have thought that Math 1000 was more complicated than Math 1090? As a newcomer to MUN, not I.
But, with the help of and many visits to the teachers aide, one of her very helpful textbooks (God love her!), and good ol' YouTube, I stayed up many nights until midnight and taught myself algebra, so that I could learn calculus. I came out with 80 in both my psychology and calculus courses, and was quite proud of this. Once I got the hang of how to do the math, and what was actually going on, with a teacher and a teachers aide who could explain it to me in a way that made sense, I grew a great fondness for math, one of which I'd never felt before.
I had debated a minor in math after this, but with more courses completed in psychology, I went with psychology and the calculus course remains the one and only University course that I have completed to date.
Today, I teach math to my children, mainly through the use of manipulatives. (I also bought them wooden blocks for building things, though I am yet to buy them wooden cones, cylinders and spheres). That's about as major a use of mathematics for me that I can think of at the moment.
What do I remember about my actual learning experiences with math at this time? Well I mostly only recall the use of manipulative's. I guess that's the kinesthetic learner in me. I recall counting apples. I also recall using those rod like creations that represent the tens and the little squares that represent the ones. I recall being able to piece these together, or at least I -think- I do. I can't really remember much of what we did with them. Used them for counting I suppose.
The thing I remember most vividly though, were these 3-D wooden shapes that fit neatly in the palm of your hand. Cylinders, cones, spheres... I recall building things with them, or rolling them around. That wasn't the mathematics objective I'm sure, but maybe these are some of the things we did as we waited for our turn during Christmas concerts (we all waited in our classrooms on such special nights and brought board games from home to play with our classmates. But I'm sure we could have gotten quite tired of them and came up with our own imaginary play with anything and everything we could get our hands on, just as any kid would). Yet, I do also recall learning from them. The idea of learning what a cone was, by relating it to what a drawn triangle was, or a sphere from a circle. Such learning brought to life with the tangible form of these 3-D shapes that i could hold in my hand and personally examine from all angles, was wonderful. This would be much easier than seeing these shapes and their angles represented by dashed lines on a paper, or at least I would assume. This makes sense to me anyways.
My son recently came home with homework involving such 3-dimensional shapes on paper. He had to cut them out and glue them on another paper, and then proceed to cut our their corresponding names spelled out on another sheet of paper, and then glue these names underneath the shapes. Thinking back now, I think I would much prefer the manipulatives. No. I know I would. A 3-D shape isn't the same when it is on paper. It's supposed to be 3-Dimensional after all.
I recall when, in Grade 3 or so, my teacher scolded me for not doing homework or something else, and then proceeded to embarrass me by digging through my book-bag (although he had no reason to do so, as I was not playing with anything that was in there during class. I was actually going to my cousins house after school to play barbies). I watched in horror as he began to drag out every barbie doll and pony that I had stuffed in there, counting them out loud as he went. When he got to ...say 23, I automatically knew that he had missed my favorite purple 'my little pony', so, I reached into the bag and pulled out one more, meekly announcing a "24". Hey, there's some math! I knew how to count! Anyways ... these are the things I recall the most, not how the teacher assessed our learning, or the role of the teacher. Maybe it isn't simply that I can't remember but that I really didn't like mathematics back then, because I love language arts. I always did, and I recall writing stores in class most of all, and I would be able to tell you that one form of assessment there was the taking in of our work and handing it back with a mark. As I ponder what I just wrote, I wonder if I'm even being truthful. I could be recounting my high-school years, which started in a school that held grades 7 through 12. Well, I am doing my best here...
The only assessment I can think of that I can positively say was in elementary, was in kindergarten. We had these books. I recall doing math in these books, coloring shapes and such. Coloring by numbers. coloring this, and coloring that. Drawing lines to matching pictures....and then coloring them. I'm assuming that the teacher looked at these to see where we were at with the various activities... or maybe she would simply check to see if we colored within the lines.
As for what mathematics looked like. Well in Kindergarten, the room had many different areas. Different centres I suppose. Maybe one of them was for math or manipulatives? Maybe not. I'm guessing there was a sandbox though! That's all I can dredge up from memory. I used to have a VHS of my kindergarten graduation. We were in the classroom before it began. Something tells me I recall these centres from watching that video, and not from my actual memory of that exact time and experience in my life.... and now I am left to wonder just where that VHS tape could be. That's how great my memory is. Not that the tape would do much good. I haven't anything to play such dated material on anyways.
I do recall enjoying those manipulatives and those kindergarten coloring books. These things brought a sort of 'fun' element to learning. One that I guess wasn't as evident in the other areas of learning about mathematics, such as the times tables and telling time.
Math in high-school. I can recall much more of those days. Most of it was negative. I had one teacher who was my Uncle and whom seemed to be trying to make the statement to the whole class. No, to the whole school, that just because he had children and nieces/nephews in his classrooms, that he wouldn't treat them any better. Well he sure didn't. To hit his point home, I swear he treated us worse. Well, me at least. I wasn't in the classrooms of any of his other family members, so I couldn't tell you about their experiences. Another homework not done incident (I swear these must have been the only two times in which I didn't have my homework done. I was so organized that many could have called me obsessive over it at one point in time, not to mention being a fairly goody two shoes when it came to homework and studying!). Anyways, he made me sit outside the classroom. Needless to say, no learning got done for me that day, only a resentment for my teacher, and maybe for math too. Then there was Mr....Brown? He was temporary. Thank God! I recall raising my hand on many different occasions even though I knew it was futile. He would never ask me my question or attempt at describing the problem to me in any other way than the way he wanted to explain it. Even when I would ask it outright. He would ignore me completely.
In my last year of high-school, I had the option of taking an advanced math course that was only offered through distance education. With the support from my classmates, I decided to give this a try and it was going well. But then, a few months into the year, I moved to a new community, and thus, a new school, and I lost this close-knit support and went back to academic mathematics. In this school, I gained astounding grades (seriously, I got 100% on EVERY exam. A HUGE change from my regular 70's and 80's). I wasn't naive and I didn't feel that I'd earned them - the teaching at this school was just so different. It was like they had given up on their students or their teaching, or both, and just started handing students answers and grades. They let students off with so much, and just brushed many problems under the carpet, neatly hidden out of sight.
Through these experiences, I managed to build an uneasiness with mathematics, and created a sort of love/hate relationship with it.
So when I came to University, 11 years after my graduation from high-school, I saw the option to either take ONE math course, or TWO math courses. Being out of practice so long, and with my background, I felt it only natural to choose to do the ONE math course! The notion of a math placement test was lost on me. I never even heard of it until after my calculus course was done and over with. So Math 1000, or calculus, is what I took one summer intersession, along with a psychology course. And I can tell you, it was hard work! My prof, on the first day, while writing out a calculus equation on the board, proceeded to work out within it, a problem using the foil method on the board. He announced that if we did not know how to do what he was doing, then we'd best go back to 1090. I knew WHAT he was doing, but had no recollection as to HOW he was doing it. Who would have thought that Math 1000 was more complicated than Math 1090? As a newcomer to MUN, not I.
But, with the help of and many visits to the teachers aide, one of her very helpful textbooks (God love her!), and good ol' YouTube, I stayed up many nights until midnight and taught myself algebra, so that I could learn calculus. I came out with 80 in both my psychology and calculus courses, and was quite proud of this. Once I got the hang of how to do the math, and what was actually going on, with a teacher and a teachers aide who could explain it to me in a way that made sense, I grew a great fondness for math, one of which I'd never felt before.
I had debated a minor in math after this, but with more courses completed in psychology, I went with psychology and the calculus course remains the one and only University course that I have completed to date.
Today, I teach math to my children, mainly through the use of manipulatives. (I also bought them wooden blocks for building things, though I am yet to buy them wooden cones, cylinders and spheres). That's about as major a use of mathematics for me that I can think of at the moment.
Welcome To My Blog!
I am beginning this blog for the purpose of communicating my experiences and ideas generated from my current MUN mathematics education course: ED 3940. Through this course and this blog, I hope to explore my love/hate relationship with math, and to shift more weight to the 'love' side! :)
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