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Keith
Mar 6, 2008, 2:40:33 PM3/6/08
to MthEd608Winter2008
"Geometry Turned On" discusses 7 different answers to the question,
"What is dynamic geometry software good for?" Compare and contrast
these 7 different answers. Here are a few questions to help get the
juices flowing:
Are the answers all equally good?
If you were to place these answers in two different piles, what would
go where and why?
Can you think of anything that might be missing from these answers?
"What is dynamic geometry software good for?" Compare and contrast
these 7 different answers. Here are a few questions to help get the
juices flowing:
Are the answers all equally good?
If you were to place these answers in two different piles, what would
go where and why?
Can you think of anything that might be missing from these answers?
Janellie
Mar 11, 2008, 12:01:07 AM3/11/08
to MthEd608Winter2008
When I thought about all the reasons dynamic software was good, I
looked back with fond memories of all the times I was able to use
Geometer's sketchpad to help me explore a problem that would have been
essentially impossible without it. Thus, it was hard for me to say
that any of these 7 answers was not as good as another, but I do want
to say that I think "Exploration and Discovery" is the best answer
(from my experience). As far as proof goes, I think that, unless a
teacher helps a student understand proof is necessary, many students
may not have the desire to do a formal proof after they have convinced
themselves using the dynamic geometry software. I do believe that it
can be a basis of a good proof and good tool to use when you are doing
geometry proofs, but I'm not sure that is transparent to a student. I
was not as impressed with the "Loci" and "Simulation" answers, but
that is probably because I am not as familiar with those uses as I am
with the others.
As far as categories missing, I think that under proof I would
emphasize a power of dynamic geometry software is students' increased
capabilities to come up with counter-examples to prove something is
false.
I originally thought to put the answers into the following categories:
1. Answers that allow students to take mathematics they know and
develop representations and reasoning with the software...and...2.
Answers that allow students to take the software and develop
mathematical ideas they have never seen before. As I was doing this,
I realized there was a lot of overlap. I'm anxious to see what others
are going to do in answer to this question.
looked back with fond memories of all the times I was able to use
Geometer's sketchpad to help me explore a problem that would have been
essentially impossible without it. Thus, it was hard for me to say
that any of these 7 answers was not as good as another, but I do want
to say that I think "Exploration and Discovery" is the best answer
(from my experience). As far as proof goes, I think that, unless a
teacher helps a student understand proof is necessary, many students
may not have the desire to do a formal proof after they have convinced
themselves using the dynamic geometry software. I do believe that it
can be a basis of a good proof and good tool to use when you are doing
geometry proofs, but I'm not sure that is transparent to a student. I
was not as impressed with the "Loci" and "Simulation" answers, but
that is probably because I am not as familiar with those uses as I am
with the others.
As far as categories missing, I think that under proof I would
emphasize a power of dynamic geometry software is students' increased
capabilities to come up with counter-examples to prove something is
false.
I originally thought to put the answers into the following categories:
1. Answers that allow students to take mathematics they know and
develop representations and reasoning with the software...and...2.
Answers that allow students to take the software and develop
mathematical ideas they have never seen before. As I was doing this,
I realized there was a lot of overlap. I'm anxious to see what others
are going to do in answer to this question.
Shawn
Mar 11, 2008, 1:24:37 PM3/11/08
to MthEd608Winter2008
Geometry is something that I'm not very strong in, I believe, compared
to the rest of the class, so what I say comes from the perspective of
a novice. When I read the article I was excited at all the things
that a dynamic geometry program can do.
With regard to the question about if they're all equally good, I think
that they all looked great and I'm excited to learn more. If I had to
critique any answer, I'm not sure which one it would be. One could
say that the first answer about the Accuracy of Construction mentioned
some limitations to these types of programs. Before I read this
article I thought that Geometer's Sketchpad could do anything. But
when these limitations were mentioned, I realized that there were
some. However, I can't think of any situation where the accuracy of a
program like this will not be enough for the audience who uses it.
For example, I don't think that it's used to make plans for a building
by architects. In all, I think that it is a good thing that they
mentioned the limitations. The reason being along the lines of the
activity with the graphing calculators and the limitations we saw with
them. If you don't know the limitations then you will assume that all
answers produced by the technology are correct.
Another quality that I liked from these descriptions was that fact
that they briefly described what they meant by the title and cited
sources of people who put these topics into action and got the desired
result of one thing that the program what created for. I'm sure this
book goes into more detail about these activities, but it's great to
see that teachers have made it successful. After having just said
this, I thought of a possible limitation. In our degree program, we
have often critiqued lesson plans or activities teachers do with their
students in math classes. I think that before I praise the program or
these qualities described in this article, we need to see how
effective those lessons were that were mentioned by the descriptions.
Perhaps we can improve upon them, but for now that's the best we have.
to the rest of the class, so what I say comes from the perspective of
a novice. When I read the article I was excited at all the things
that a dynamic geometry program can do.
With regard to the question about if they're all equally good, I think
that they all looked great and I'm excited to learn more. If I had to
critique any answer, I'm not sure which one it would be. One could
say that the first answer about the Accuracy of Construction mentioned
some limitations to these types of programs. Before I read this
article I thought that Geometer's Sketchpad could do anything. But
when these limitations were mentioned, I realized that there were
some. However, I can't think of any situation where the accuracy of a
program like this will not be enough for the audience who uses it.
For example, I don't think that it's used to make plans for a building
by architects. In all, I think that it is a good thing that they
mentioned the limitations. The reason being along the lines of the
activity with the graphing calculators and the limitations we saw with
them. If you don't know the limitations then you will assume that all
answers produced by the technology are correct.
Another quality that I liked from these descriptions was that fact
that they briefly described what they meant by the title and cited
sources of people who put these topics into action and got the desired
result of one thing that the program what created for. I'm sure this
book goes into more detail about these activities, but it's great to
see that teachers have made it successful. After having just said
this, I thought of a possible limitation. In our degree program, we
have often critiqued lesson plans or activities teachers do with their
students in math classes. I think that before I praise the program or
these qualities described in this article, we need to see how
effective those lessons were that were mentioned by the descriptions.
Perhaps we can improve upon them, but for now that's the best we have.
CJ
Mar 12, 2008, 10:17:51 PM3/12/08
to MthEd608Winter2008
I would have to say . . . are there eight? Is "Microworlds" an answer?
If so, then I see (1) Accuracy of construction, (2) Visualization, (3)
Exploration and Discovery, (4) Proof, (5) Transformations, (6) Loci,
(7) Simulation, and (8) Microworlds.These eight answers I don't think
are all on the same "level" but can be organized hierarchally, as
follows:
Accuracy of construction, Transformations, Loci, Simulation, and
Microworlds ALL contribute to Visualization, which can contribute to
Exploration and discovery AND Proof
Exploration and discovery can also contribute to proof, but proof can
also contribute to exploration and discovery. These two are top tier
mathematical processes, that can encompass many different aspects of
mathematics. For some people, mathematics IS Exploration and
discovery. For others, mathematics IS Proof. However, few people would
say that mathematics IS Visualization. Even fewer would say that
mathematics IS Accuracy of construction (at least I hope not). So
Exploration and discovery and Proof are top processes that can each
encompass all the other answers given. They are the most general
categories.
Visualization is almost as general as Exploration or Proof, and yet we
wouldn't say that Visualization is a goal in itself. It also isn't
essential for Exploration or Proof, but it sure can be a powerful
component of both.
I see the other five answers as characteristics of dynamic geometry
that carry visualization to a new level that allows us to spend less
time creating representations and more time interacting with those
representations. (I think we might call dynamic geometries
transformational tools because they do change the nature of
representations in this way.) Although we could probably create all of
these things pretty well on paper or in a thought experiment, having
accurate and dynamic constructions, transformations, loci,
simulations, and even microworlds at the push of a button likely
heightens the level of mathematical activity. It allows us to
visualize things quicker and better, which can then propel our
explorations and proof creation activities.
If so, then I see (1) Accuracy of construction, (2) Visualization, (3)
Exploration and Discovery, (4) Proof, (5) Transformations, (6) Loci,
(7) Simulation, and (8) Microworlds.These eight answers I don't think
are all on the same "level" but can be organized hierarchally, as
follows:
Accuracy of construction, Transformations, Loci, Simulation, and
Microworlds ALL contribute to Visualization, which can contribute to
Exploration and discovery AND Proof
Exploration and discovery can also contribute to proof, but proof can
also contribute to exploration and discovery. These two are top tier
mathematical processes, that can encompass many different aspects of
mathematics. For some people, mathematics IS Exploration and
discovery. For others, mathematics IS Proof. However, few people would
say that mathematics IS Visualization. Even fewer would say that
mathematics IS Accuracy of construction (at least I hope not). So
Exploration and discovery and Proof are top processes that can each
encompass all the other answers given. They are the most general
categories.
Visualization is almost as general as Exploration or Proof, and yet we
wouldn't say that Visualization is a goal in itself. It also isn't
essential for Exploration or Proof, but it sure can be a powerful
component of both.
I see the other five answers as characteristics of dynamic geometry
that carry visualization to a new level that allows us to spend less
time creating representations and more time interacting with those
representations. (I think we might call dynamic geometries
transformational tools because they do change the nature of
representations in this way.) Although we could probably create all of
these things pretty well on paper or in a thought experiment, having
accurate and dynamic constructions, transformations, loci,
simulations, and even microworlds at the push of a button likely
heightens the level of mathematical activity. It allows us to
visualize things quicker and better, which can then propel our
explorations and proof creation activities.
erin...@byu.edu
Mar 13, 2008, 1:07:03 AM3/13/08
to MthEd608Winter2008
FROM RACHELLE (her internet is down and she is dictating to me over
the phone):
I just wanted to point out that there's few connections to Doerr and
Zanger's tools for the calculator. It seemed like the accuracy of
construction section really related to the transformational tool
because it made the constructions quick, easy, and accurate instead of
focusing on the technicality of the construction - students could
focus more on the interpretation. Also, the visualization section had
obvious connections to the calculator as a visualization tool. Did
anybody notice any other connections? I'd love to hear!
In comparing/ contrasting the seven answers I decided to group them
into two categories. The first is "facilitating justification or
problem-solving skills." I felt that accuracy of construction and
proof were two sections that fit into this category. The authors
argued that these capabilities of dynamic geometry software encouraged
students to make their own conjectures, find evidence, and extend into
detail/ justification. I felt that the other sections fell into a
category I call 'developing geometric intuition." Most of these
sections help students see geometry in a dynamic way so they can
acquire a better intuition about the subject. The only section I had
difficulty categorizing into one of these two groups was exploration &
discovery. I would love to know what y'all think about where it
should fit?!?
"Thanks to Erin for typing this up and being so helpful" - really, she
said that!
the phone):
I just wanted to point out that there's few connections to Doerr and
Zanger's tools for the calculator. It seemed like the accuracy of
construction section really related to the transformational tool
because it made the constructions quick, easy, and accurate instead of
focusing on the technicality of the construction - students could
focus more on the interpretation. Also, the visualization section had
obvious connections to the calculator as a visualization tool. Did
anybody notice any other connections? I'd love to hear!
In comparing/ contrasting the seven answers I decided to group them
into two categories. The first is "facilitating justification or
problem-solving skills." I felt that accuracy of construction and
proof were two sections that fit into this category. The authors
argued that these capabilities of dynamic geometry software encouraged
students to make their own conjectures, find evidence, and extend into
detail/ justification. I felt that the other sections fell into a
category I call 'developing geometric intuition." Most of these
sections help students see geometry in a dynamic way so they can
acquire a better intuition about the subject. The only section I had
difficulty categorizing into one of these two groups was exploration &
discovery. I would love to know what y'all think about where it
should fit?!?
"Thanks to Erin for typing this up and being so helpful" - really, she
said that!
erin...@byu.edu
Mar 13, 2008, 1:47:39 AM3/13/08
to MthEd608Winter2008
This paper flowed really well and was easy on my fried brain, which I
definitely appreciated. As far as discussing what dynamic geometry
software is good for, I felt like the authors did a very fair job of
answering this question. In response, I want to know where they came
up with their seven answers: did they develop these categories by
looking at data from a study? What study? Were some of these given in
literature; what literature? Do the authors see some of these
categories as being more important than the others?
Once we were told that we might want to place these categories into
different piles, I started to look for commonalities/ differences
among the seven categories. It seemed to me that they easily fell
into two categories: visualization and exploration.
Visualization: Obviously the biggest section of this category is the
visualization section - it describes the overarching commonality of
this category: "dynamic geometry software can help students SEE what
is meant by a general fact" (no idea what page number). In this
category are the visualization section, the transformation section
("in witnessing the action of these transformations... students see
that functions are not synonymous with symbolic formulas" - self
explanatory), loci section ("show how a locus is generated and to
reveal the shape of its traced path" - help students understand and
visualize what loci are), simulation ("provaide many oportunities to
simulate a surprising variety of situations" - through simulation
students can visualize situations/ problems).
Exploration: Again I love to be obvious: the overheading section in
this category is the exploration & discovery sections - "allows
students to 'test their own mathematical ideas... in a visual,
efficient and dynamic manner and be more fully engaged'. " Under this
category fall the proof section ("experimental evidence [dynamic
geometry software] provides produces strong conviction which can
motivate the desire for proof" - this implies that through exploration
students are motivated to do proofs), and the microworld section
("produces an environment in which Euclidean geometry can be explored"
- it's pretty self-explanatory there that a particular environment is
needed in order to foster exploration). I had a difficult time
forcing the accuracy section to fit into one of these two categories,
but I think that it fits better under exploration than visualization.
Why? It would do no good to explore mathematical concepts, etc... if
the constructions/ images/ directions students use to explore are
inaccurate. Therefore, exploration using dynamic geometry software
depends on accuracy of construction.
For the life of me I could NOT figure out anything that was missing,
although I felt like there was an idea on the tip of my tongue...
Maybe the ability to share student work with other students/ classes/
places in the world? Technology is becoming universally accessible
across the world, and it makes classroom presentations easy to
follow. But that's the only thing I could really think of...
I'm not sure if I have questions... I'm really excited to read
everybody else's categories and I'm sure I'll be able to come up with
more stuff then!
Love you all,
Erin
PS - I just wanted to sign it like a letter :)
definitely appreciated. As far as discussing what dynamic geometry
software is good for, I felt like the authors did a very fair job of
answering this question. In response, I want to know where they came
up with their seven answers: did they develop these categories by
looking at data from a study? What study? Were some of these given in
literature; what literature? Do the authors see some of these
categories as being more important than the others?
Once we were told that we might want to place these categories into
different piles, I started to look for commonalities/ differences
among the seven categories. It seemed to me that they easily fell
into two categories: visualization and exploration.
Visualization: Obviously the biggest section of this category is the
visualization section - it describes the overarching commonality of
this category: "dynamic geometry software can help students SEE what
is meant by a general fact" (no idea what page number). In this
category are the visualization section, the transformation section
("in witnessing the action of these transformations... students see
that functions are not synonymous with symbolic formulas" - self
explanatory), loci section ("show how a locus is generated and to
reveal the shape of its traced path" - help students understand and
visualize what loci are), simulation ("provaide many oportunities to
simulate a surprising variety of situations" - through simulation
students can visualize situations/ problems).
Exploration: Again I love to be obvious: the overheading section in
this category is the exploration & discovery sections - "allows
students to 'test their own mathematical ideas... in a visual,
efficient and dynamic manner and be more fully engaged'. " Under this
category fall the proof section ("experimental evidence [dynamic
geometry software] provides produces strong conviction which can
motivate the desire for proof" - this implies that through exploration
students are motivated to do proofs), and the microworld section
("produces an environment in which Euclidean geometry can be explored"
- it's pretty self-explanatory there that a particular environment is
needed in order to foster exploration). I had a difficult time
forcing the accuracy section to fit into one of these two categories,
but I think that it fits better under exploration than visualization.
Why? It would do no good to explore mathematical concepts, etc... if
the constructions/ images/ directions students use to explore are
inaccurate. Therefore, exploration using dynamic geometry software
depends on accuracy of construction.
For the life of me I could NOT figure out anything that was missing,
although I felt like there was an idea on the tip of my tongue...
Maybe the ability to share student work with other students/ classes/
places in the world? Technology is becoming universally accessible
across the world, and it makes classroom presentations easy to
follow. But that's the only thing I could really think of...
I'm not sure if I have questions... I'm really excited to read
everybody else's categories and I'm sure I'll be able to come up with
more stuff then!
Love you all,
Erin
PS - I just wanted to sign it like a letter :)
Tenille
Mar 13, 2008, 1:53:26 AM3/13/08
to MthEd608Winter2008
Since we are only required to compare and contrast 7 different
answers, does that mean that we get to leave out one of the answers?
If that is the case, I choose to leave out proof. I think that Chris
described my initial organization of the answers and articulated it
much better than I ever could. So I tried to think of a different way
to organize the 7 answers. I decided to organize them similar to how
the NCTM standards are organized: process and content. I found the
making higher-level mathematics accessible theme running through many
of the answers. I think that a few of the answers cited specific
mathematics that is now more accessible. I categorized these as
content answers, and they include: transformations, loci, and
microworlds. I thought that they remaining answers (accuracy,
visualization, exploration and discovery, and simulation) either
enabled or described mathematical processes.
You may be wondering why I chose to omit proof from my categorization
when it is obviously one of the process standards. If I was asked the
question "What is dynamic geometry software good for?", my answer (as
opposed to answers) would have been proof. However, I felt that the
authors treatment of proof in the article did not warrant its
inclusion in the categorization (the proof paragraph was, for lack of
a better word, lame). Dynamic geometry illuminates different roles of
proof in the mathematics classroom from the traditional verification
role to explanation to communication to systematization to discovery
to intellectual challenge (de Villiers, 2004). Don't you think that
something as vital to mathematics as proof deserves more than a lame
paragraph? By the way, I did see the proof thread running through
ALL the other 7 answers.
Also, I can't believe that I am on campus at 11:45pm . . . I guess
that's what happens when you wait until the last minute to turn in
your assignment and find out your internet is down.
de Villiers, M. D. (2004). Rethinking proof with the Geometer's
Sketchpad Version 4. Emeryville, CA: Key Curriculum Press.
answers, does that mean that we get to leave out one of the answers?
If that is the case, I choose to leave out proof. I think that Chris
described my initial organization of the answers and articulated it
much better than I ever could. So I tried to think of a different way
to organize the 7 answers. I decided to organize them similar to how
the NCTM standards are organized: process and content. I found the
making higher-level mathematics accessible theme running through many
of the answers. I think that a few of the answers cited specific
mathematics that is now more accessible. I categorized these as
content answers, and they include: transformations, loci, and
microworlds. I thought that they remaining answers (accuracy,
visualization, exploration and discovery, and simulation) either
enabled or described mathematical processes.
You may be wondering why I chose to omit proof from my categorization
when it is obviously one of the process standards. If I was asked the
question "What is dynamic geometry software good for?", my answer (as
opposed to answers) would have been proof. However, I felt that the
authors treatment of proof in the article did not warrant its
inclusion in the categorization (the proof paragraph was, for lack of
a better word, lame). Dynamic geometry illuminates different roles of
proof in the mathematics classroom from the traditional verification
role to explanation to communication to systematization to discovery
to intellectual challenge (de Villiers, 2004). Don't you think that
something as vital to mathematics as proof deserves more than a lame
paragraph? By the way, I did see the proof thread running through
ALL the other 7 answers.
Also, I can't believe that I am on campus at 11:45pm . . . I guess
that's what happens when you wait until the last minute to turn in
your assignment and find out your internet is down.
de Villiers, M. D. (2004). Rethinking proof with the Geometer's
Sketchpad Version 4. Emeryville, CA: Key Curriculum Press.
Shawn
Mar 13, 2008, 11:26:43 AM3/13/08
to MthEd608Winter2008
Chris, I liked your characterizations of mathematics in light of these
answers to what dynamic geometry is good for. I like to think of
mathematics as the combination of those answers. (I don't think it is
all of mathematics, but a good portion.) You say that to a few people
mathematics IS visualization. You, perhaps, aren't saying that it
isn't a part of math by this, but I want to emphasize that if students
can visualize what x squared plus y squared equals 1 is, then they've
done powerful mathematics at that level. Finally, I think that if
there is a program that claims to be able help with these areas, it
should be worth looking into. Good answer, Chris, anyway I thought
you said many good things.
> > Can you think of anything that might be missing from these answers?- Hide quoted text -
>
> - Show quoted text -
answers to what dynamic geometry is good for. I like to think of
mathematics as the combination of those answers. (I don't think it is
all of mathematics, but a good portion.) You say that to a few people
mathematics IS visualization. You, perhaps, aren't saying that it
isn't a part of math by this, but I want to emphasize that if students
can visualize what x squared plus y squared equals 1 is, then they've
done powerful mathematics at that level. Finally, I think that if
there is a program that claims to be able help with these areas, it
should be worth looking into. Good answer, Chris, anyway I thought
you said many good things.
>
> - Show quoted text -
Janellie
Mar 13, 2008, 2:24:16 PM3/13/08
to MthEd608Winter2008
I agree with Tenille that the proof section of the paper is "lame".
As I have listened to Valerie and Sharon (two part time students) talk
about their project this past semester in 591, I have gained a greater
belief about the important role proof and reasoning should play in the
geometry curriculum. Their project is to make a geometry curriculum
with proof not as a distinct unit, but to have proof and reasoning
being the overarching theme. Also, I feel that my geometry experience
would have been enhanced had reasoning played a bigger part. In my
geometry class here at BYU (362...) we used geometer's sketchpad in
some of our homework assignments that asked us to prove things and I
thought it was very beneficial and empowering.
As an end note, thanks to everyone for their categories, I was having
a hard time coming up with two good "piles" to put these answers in
and I enjoyed reading how everyone thought of it.
As I have listened to Valerie and Sharon (two part time students) talk
about their project this past semester in 591, I have gained a greater
belief about the important role proof and reasoning should play in the
geometry curriculum. Their project is to make a geometry curriculum
with proof not as a distinct unit, but to have proof and reasoning
being the overarching theme. Also, I feel that my geometry experience
would have been enhanced had reasoning played a bigger part. In my
geometry class here at BYU (362...) we used geometer's sketchpad in
some of our homework assignments that asked us to prove things and I
thought it was very beneficial and empowering.
As an end note, thanks to everyone for their categories, I was having
a hard time coming up with two good "piles" to put these answers in
and I enjoyed reading how everyone thought of it.
Tenille
Mar 13, 2008, 2:26:52 PM3/13/08
to MthEd608Winter2008
Shawn, I really appreciated your thoughts about the limitations of the
software and the connection to calculators. As I read the article, I
tried to compare dynamic software to calculators and spreadsheets.
Does the software address some of the limitations of calculators and
spreadsheets. I think that the software definitely addresses the
issue that calculators are often not dynamic. I also wondered how
many of the answers given in the article could also apply to
calculators and spreadsheets. I think that the answers of accuracy,
visualization, exploration and discovery, and simulation could be
applied to calculators. The answers of accuracy, visualization,
exploration and discovery, proof, simulation, and microworlds could be
applied to spreadsheets. Perhaps if I was better at using the tools I
would see more answers that applied to calculators and spreadsheets.
Anyway, I thought that you made a good point that we need to be aware
of the limitations of each tool we use so that was can choose the best
technology to solve a problem. Do you think that the strengths and
limitations of technological tools should be made explicit to our
students so that they can choose the best technology to solve a
problem?
software and the connection to calculators. As I read the article, I
tried to compare dynamic software to calculators and spreadsheets.
Does the software address some of the limitations of calculators and
spreadsheets. I think that the software definitely addresses the
issue that calculators are often not dynamic. I also wondered how
many of the answers given in the article could also apply to
calculators and spreadsheets. I think that the answers of accuracy,
visualization, exploration and discovery, and simulation could be
applied to calculators. The answers of accuracy, visualization,
exploration and discovery, proof, simulation, and microworlds could be
applied to spreadsheets. Perhaps if I was better at using the tools I
would see more answers that applied to calculators and spreadsheets.
Anyway, I thought that you made a good point that we need to be aware
of the limitations of each tool we use so that was can choose the best
technology to solve a problem. Do you think that the strengths and
limitations of technological tools should be made explicit to our
students so that they can choose the best technology to solve a
problem?
erin...@byu.edu
Mar 13, 2008, 2:44:39 PM3/13/08
to MthEd608Winter2008
As I was reading everybody's comments I realized I think I have
something to say to all of you!
Janelle - you mentioned the two categories that you would use to split
the 7 (or technically 8) sections, but you didn't describe which
sections would go into each or why. Maybe you'll do so in your
response or in class?
Shawn - you said "If you don't know the limitations then you will
said. I think it is vital for us to use our brains to educate
ourselves about the practical applications/ limitations, etc...
accompanying technology and then to proceed cautiously from that
point. I found myself trying to extend this premise to non-
technological issues but had a difficult time doing so. I feel like
there are great life lessons to be learned if one of you can do that.
maybe I'll even award another blue ribbon...?!?
Chris - I think your definition of visualization contrasts to my
definition. I think that the authors intended for visualization to be
more than literal seeing, but truly SEEING with understanding. Kind
of like how hearken means more than just listen or hear, but hear and
do. I agree with Shawn in that if a student can visualize x-squared
plus y-squared = 1, they have indeed done powerful mathematics.
Rachelle - 1- way to connect with Doerr and Zanger, 2- why did you
have a hard time placing exploration & discovery? What about this
section made it difficult to fit in one of the two great categories
you created?
Tenille - It seems to me like your idea of accessibility correlates
with my idea of visualization. What do you think? Also, you said
which sections would fall under your category "enable/ describe
processes," but I really wanted to know exactly HOW each section did
that, in your mind. Especially b/c we differed in which sections we
compared and which we contrasted... I just wanted more explanation
from you. Then again, since the internet was out and I'm sure you
were rushed, maybe you ran out of time to do this?
Keith - I didn't want you to feel left out so I'm writing to tell you
hello.
Lastly, does anybody know who to call if the air conditioner in our
office is broken? it's on +4 heat or turned off for the whole school
year and about a week ago started blasting cold air. I find it too
cold to work in there, which is very frustrating, and Tenille's told
the custodians three times and we've talked to maintenance once but
nothing's happened. Help!
something to say to all of you!
Janelle - you mentioned the two categories that you would use to split
the 7 (or technically 8) sections, but you didn't describe which
sections would go into each or why. Maybe you'll do so in your
response or in class?
Shawn - you said "If you don't know the limitations then you will
assume that all answers produced by the technology are correct."
Amen. That was so eloquently put and says exactly what needs to be
said. I think it is vital for us to use our brains to educate
ourselves about the practical applications/ limitations, etc...
accompanying technology and then to proceed cautiously from that
point. I found myself trying to extend this premise to non-
technological issues but had a difficult time doing so. I feel like
there are great life lessons to be learned if one of you can do that.
maybe I'll even award another blue ribbon...?!?
Chris - I think your definition of visualization contrasts to my
definition. I think that the authors intended for visualization to be
more than literal seeing, but truly SEEING with understanding. Kind
of like how hearken means more than just listen or hear, but hear and
do. I agree with Shawn in that if a student can visualize x-squared
plus y-squared = 1, they have indeed done powerful mathematics.
Rachelle - 1- way to connect with Doerr and Zanger, 2- why did you
have a hard time placing exploration & discovery? What about this
section made it difficult to fit in one of the two great categories
you created?
Tenille - It seems to me like your idea of accessibility correlates
with my idea of visualization. What do you think? Also, you said
which sections would fall under your category "enable/ describe
processes," but I really wanted to know exactly HOW each section did
that, in your mind. Especially b/c we differed in which sections we
compared and which we contrasted... I just wanted more explanation
from you. Then again, since the internet was out and I'm sure you
were rushed, maybe you ran out of time to do this?
Keith - I didn't want you to feel left out so I'm writing to tell you
hello.
Lastly, does anybody know who to call if the air conditioner in our
office is broken? it's on +4 heat or turned off for the whole school
year and about a week ago started blasting cold air. I find it too
cold to work in there, which is very frustrating, and Tenille's told
the custodians three times and we've talked to maintenance once but
nothing's happened. Help!
erin...@byu.edu
Mar 13, 2008, 2:46:42 PM3/13/08
to MthEd608Winter2008
And I started writing before Janelle/ Tenille posted their responses,
which makes me sad b/c they probably won't get on here again before
class to see what I wrote them :(
which makes me sad b/c they probably won't get on here again before
class to see what I wrote them :(
CJ
Mar 13, 2008, 3:25:59 PM3/13/08
to MthEd608Winter2008
Hi Rachelle,
Good job talking on the phone. I also saw connections between Doerr
and Zanger article and this article. The idea of transformational tool
fits quite well as you have described. Dynamic geometries can be
computational tool, at least in GSP you can have the computer measure
the length of segments and areas and perform computations with those
results. GSP can also collect data based on the computations that you
ask it to do, for example, as you vary the length of the side of
triangle, GSP will tabulate those lengths along with the area of the
triangle at regular intervals in a table. This can then become data
for an analysis or figuring out of a relationship between the side
length and the area. I think that checking can fit with
visualization . . . you can make a conjecture about something
geometrical, and then you can actually construct it and look at it to
check. This may not prove that your conjecture is correct, but it may
help to steer you in the right direction. I also like your two piles.
Judging by the titles of the piles, I feel like exploration and
discovery could reasonably fit into both of your piles because I
think that exploration has helped me to develop problem skills and
develop geometric intuition.
Good job talking on the phone. I also saw connections between Doerr
and Zanger article and this article. The idea of transformational tool
fits quite well as you have described. Dynamic geometries can be
computational tool, at least in GSP you can have the computer measure
the length of segments and areas and perform computations with those
results. GSP can also collect data based on the computations that you
ask it to do, for example, as you vary the length of the side of
triangle, GSP will tabulate those lengths along with the area of the
triangle at regular intervals in a table. This can then become data
for an analysis or figuring out of a relationship between the side
length and the area. I think that checking can fit with
visualization . . . you can make a conjecture about something
geometrical, and then you can actually construct it and look at it to
check. This may not prove that your conjecture is correct, but it may
help to steer you in the right direction. I also like your two piles.
Judging by the titles of the piles, I feel like exploration and
discovery could reasonably fit into both of your piles because I
think that exploration has helped me to develop problem skills and
develop geometric intuition.
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