History Lessons

[Keith]

"We learn from history that we do not learn from history." --Georg
Wilhelm Friedrich Hegel

As mathematics teachers, mathematics teacher educators, or mathematics
education researchers, what should we learn from this history of the
emergence of technology in mathematics education?

[Tenille]

For me, this article prompted one major lesson: technology has and
will continue to change mathematics, and as members of the mathematics
education community we need to be willing to consider how those
changes will impact what and how mathematics is taught. First of all,
the emergence of technology has changed the discipline of mathematics
itself, exemplified in the concept of proof and the four color
theorem. It seems only natural that if we are to let the disciple of
mathematics guide the instruction, that (through the use of the
transitive property) the emergence of technology should change
mathematics instruction. Technology has removed the need more
individuals to perform complex arithmetic problems; it seems to follow
that instruction should focus more on problem solving, I think that
these changes to the discipline of mathematics resulting in part from
technology have also prompted changes in research in mathematics
education. Not only has technology changed how mathematics should be
taught, but it has also changed what mathematics should be taught in
the school curriculum. In my lifetime, I have already seen changes in
the recommended school curriculum (although I am not sure that I have
seen many actual changes). I wonder what other changes will be
recommended and whether I will be willing to accept those changes.

I really enjoyed this article and didn't take issue with it. Am I
alone in this? Perhaps I didn't read it critically enough. I would
love to hear from others. It seems that teachers who are not
comfortable with technology and learned mathematics without it seem
very unwilling to use technology in their classroom (as more than just
a tool for checking answers). Do you think that professional
development is enough to get teachers to change? Or do we have to
wait for all of these teachers to retire?

[Rachelle]

I think the major lesson we should learn from this article is that
technology will continue to evolve and so the mathematics needed in
society will continue to evolve. I find it interesting that not only
has technology changed the field of mathematics (with computers
becoming used in proofs!) but technology has also changed the skills
needed in everyday life. For example, in the past, sales clerks needed
to be able to add a lot of numbers quickly and acurately. Now with
calculators and computers doing the adding, it is much more important
to know when to use addition rather than some other operation and to
be capable to quickly check the reasonableness of the answer. The
essential lesson I took from the article is that mathematics teachers
and mathematics teacher educators need to be willing to change their
beliefs about what skills should be learned in the mathematics
classroom, based on the technology available to the society.
Additionally, mathematics education researchers could make it their
task to continually research what skills are necessary with the
availability of new technology.

As a sidenote, something else that kept jumping out at me is that we
need to learn patience when trying to include technology in the
classroom and change curriculum because of it. At every technological
jump, the public opinion was slow to accept new ideas, schools were
slow to afford the hardware, and teachers were slow to learn how to
incorporate technology in their practice. It seems to me that major
changes in mathematics education will always take some time. Of
course, if research can support the inclusion of new technology more
rapidly, then maybe teacher educators can provide more opportunities
for teachers to learn how to use technology, and eventually the whole
process could be quicker. Who knows? I'm wondering if anyone else
feels like the process will always be slower than we'd like, or if
that's a thing of the past and new technology will be included in the
mathematics classroom more easily in the future. Let me know what you
think!

Keith, I was wondering if you picked this article because it had so
many references to Dewey? I kept distracting myself by trying to
analyze if the author was representing Dewey's ideas accurately. . .
Did anyone else notice themselves reverting back to seminar? :)

On Feb 14, 12:27 pm, Keith <LethaLeat...@gmail.com> wrote:

[erin...@byu.edu]

I really enjoyed this article or chapter and felt like the author did
a very thorough job preparing for it and compliling data. I walked
away with six major thoughts:

1- It is vitally important that we as educators be open-minded. From
the time in the early 1950's when a tiny sliver of silicon replaced a
vacuum tube to the present day, innovative thinkers have propelled the
development of technology. At any point in time I would bet that
there were those who scoffed at these dreamers, but with persistence
(and funding) they were able to turn their ideas into realities and
ultimately alter the direction of technology development.

2- As mathematics educators it is vitally important that we be aware
of what is going on in the field of mathematics. On pg 1044 the
author discusses how the impact of technology on math ed is
"contingent on an awareness of the developments in mathematics that
have changed the very fabric of the discipline itself." Later, on pg
1076 the author declares that we must be aware of/ anticipate
"important mathematical concepts that will be needed [for the
mathematical skills that are in turn needed for technology]" and that
to determine what those concepts are, one only needs to look as far as
"mathematics itself." We NEED to be involved in and aware of the
mathematics driving today.

3- Kind of along the lines of #1, it is valuable to follow new leads
or be flexible. By this I mean that through exploration of "tangent"
ideas, graph theory and other such areas of study were given a name
and explored further. I'm sure we've all had the experience where we
plan a lesson and during class it becomes evident that our students
misunderstand such-and-such a topic. It would be stupid not to follow
up on that and help students come to a better understanding of that
difficult subject. Such "pioneering ventures" were evident throughout
this entire paper and are valuable models of what we as educators can
do in our classrooms.

4- Did anybody else think that the PLATO program was kind of a front-
runner or different version of task-based learning? To me at least it
seemed familiar and like such a great idea.

5- on pg 1076 the author mentioned that "there were indications that
the graphing calculator was also exerting an influence on the tests
and examinations that drove the curriculum." The part I want to bring
to your attention is the phrase "...tests and examinations...drove the
curriculum." Gross! It seems like once we started inventing these
tests to assess and compare student learning the curriculum began to
depend/ be driven by said tests. Just in case you missed my
sentiments on the matter: disgusting.

6- The biggest theme I saw repeated over and over again throughout the
summary of history is that when great ideas are implemented poorly,
you get problems. (pg 1051, 1055, 1068). I'm sure we've all had
experience with this in our own classrooms as we get some amazing idea
of a lesson to do or a topic to cover, but then we get stuck. Or, for
example, the whole investigations curriculum has proved a controversy
primarily (in my opinion) because of poor implementation. Either
teachers were not trained well enough or did not believe in the
program or something along those lines. My big big big question is
what one could do to implement ideas in a better way so that they
actually take shape and go somewhere? I think if anybody can answer
this, a LOT of people will gladly give you a million bucks. Good
luck. The race starts... NOW

On Feb 14, 12:27 pm, Keith <LethaLeat...@gmail.com> wrote:

[CJ]

What can we learn from this history? Well, as I was reading I had a
thought and formed a conjecture about mathematics and I invite you to
discuss the quality of my logic. It seems that new technology is
rapidly accepted in the "real world" but we have to fight to get it
into schools. For example, the mainframe computer was integrated into
graph theory, probability, decoding, chaos, and geometry branches of
mathematics quite quickly. Nobody hesitates to buy a calculator to
help them with their taxes in the home or business, but getting
teachers to use them in their classroom is quite a struggle (example:
yours truly). So based on this evidence, my conjecture is that
"mathematics" as defined in the "real world" is very different from
"school mathematics." In the real world, we use what we can to solve
our problems. In schools, we think that there is just one way, or a
preferred way to solve problems, and then we practice and become
fluent in those ways. These seem like two different things, and it
partly makes sense because schools should teach you good ways to do
things, but can we really call that kind of knowledge mathematics?
I . . . don't know. However, I think that a strong message of this
history is that technology is very useful in the field of mathematics,
but we are having a hard time agreeing on how technology should be
used in instruction. Perhaps it's due to a fundamental difference
between learning and doing. Or perhaps it's because we don't know what
mathematics is anymore.

Another thought that I had while reading this chapter is maybe
technology already has changed the face of arithmetic in schools, but
it hasn't done much for algebra yet. I say this because when I looked
at the sample problems for high school admission in 1885, the
arithmetic problems made me cringe. The algebra problems, on the other
hand seemed quite natural. So although I claim that I am good at PPA,
maybe I don't really understand how intense PPA has been in the past.
Maybe it is already dropping off the map without our knowing it. Of
course, there are many possible reasons for my feeling that the
arithmetic problems are way harder than the algebra problems. However,
for the sake of discussion, I suggest that calculators have had a much
larger impact on my learning than I realize. Did you guys feel this
way? Or am I just bad at PPA?

On Feb 14, 12:27 pm, Keith <LethaLeat...@gmail.com> wrote:

[CJ]

About professional development: I think that there is professional
development, and then there is professional development. Now that
that's cleared up, let me see if I can explain why I think that. I
think that there is professional development in which teachers are
told how technology can help them to teach and how technology can help
students learn. On the other hand, I think that there is professional
development where teachers actually increase their own understanding
of the mathematics via technology. I think that the second type is
much more effective. Although I love reading research and observing
videos and stuff, some of the times when I am most thoughtful about my
teaching is when I am in a mathematics class playing the role of the
student. If technology is used in effective ways and it actually
changes how I understand math in those classes, then I am motivated to
make the same opportunities available for my students. I admit that
this is just my opinion, and some teachers probably won't change how
they teach no matter what. However, I think that many teachers base
their teaching very strongly in their learning, and when I learn
something new I can't wait to share it with my students. So to answer
your question, I think I want to say that professional development
could probably do a lot to get technology into the mathematics
classroom, but if you look at a lot of the professional development
going on these days then it might be quicker to just wait for a whole
new crop of teachers. We didn't get to talk about it a lot, but the
pendulum vignette standard from last week seemed to be a step in the
right direction toward more effective professional development.

[Janellie]

"The tool defines the skill." Brought up by the article numerous
times, this phrase describes what might be one of the most important
lessons we can learn from the emergence of technology in mathematics
education. Like Keith is always saying, why go to Salt Lake City in a
horse and buggy if you can drive there? In the future we might ask,
why go to Salt Lake City in a car if you can ___________ (insert
future technology...fly in your personal aircraft?...teleport?...).
As teachers, teacher educators, and researchers, we need to not
consider how we can use new technology to aid us in the old
objectives, but rather consider what new objectives are important in
the teaching of mathematics. Therefore, technology will not just
change how students learn the material, but what material is
important. It was interesting that when computers and calculators
first came out, many mathematicians, teachers, and community members
were reluctant and even morally outraged that such technology would be
used in the math classroom. However, these attitudes change as the
power is recognized in these tools.

This leads to another major point of the paper. Things are not going
to change unless teachers are committed to the change. Therefore, in
order for technology to be more beneficial in the math classsroom,
teachers need to have a belief that technology is useful and also a
knowledge (mathematical and pedagogical) of how to make best use of
the technology. This carries implications for teachers, researchers,
and teacher educators. Teachers should be ever expanding their
knowledge of technology and be willing to try new things. Researchers
should show how and why technology should be used. Finally teacher
educators have the responsibility to help teachers have experiences
that will help them use technology and shape their beliefs about its
use.

In light of this last responsibility of mathematics teacher educators,
what are some ways that such educators could best help prospective
(and current) math teachers effectively use technology in their own
classrooms?

On Feb 14, 12:27 pm, Keith <LethaLeat...@gmail.com> wrote:

[Tenille]

[Tenille]

So I pressed the discard button on my last post, but it still managed
to post an empty message. Why do we even bother with technology? :)
Here is what I wanted to post.

I don't think that I would have been accepted to Jersey City High
School if I had to pass that arithmetic test. Not only is the
arithmetic difficult (I don't even want to think about attempting II
without a calculator), but I think that problems VI and VIII are very
difficult for reasons that have nothing to do with the PPA I learned
in elementary school. Am I the only one who doesn't know an algorithm
for solving work problems? I guess if I had to solve a lot of work
problems, I would develop an algorithm. But I don't. So every time I
have to solve a work problem (I think maybe I did this on the GRE
test), it becomes a problem solving activity. When I first looked at
this test, I thought that the arithmetic I learned in elementary
school had been "dumbed-down" (and that is probably true). But on the
other hand, I was doing algebra in 6th grade, long before high
school. I think that not having to learn how to take the square root
of a number opened up some time. I think this is why people are so
resistent to technology use in schools . . . they perceive the change
in the mathematics as negative. Their kids or students are learning
mathematics that the parent or teacher feels is easier than what they
had to do.

I appreciated your insights that you shared when comparing the
arithmetic and algebra tests to current standards. I wonder how
technology will impact the algebra test in the future.

On Feb 20, 5:56 pm, CJ <christine.johnson....@gmail.com> wrote:

[Tenille]

Five words: model effective use of technology (emphasis on the word
model).

[Rachelle]

I want to comment on Janelle's question and Tenille's response. I
agree with Tenille, but I think that some teacher educators (maybe
those with different perspectives on learning) might interpret her
response in the wrong way. I will describe a possible interpretation
and then give my own interpretation.

So here's my thinking. In seminar we are reading a paper called
"Learning to teach hard mathematics: Do novice teachers and their
instructors give up too easily?" (Borko et. al, 1992). In this paper,
the authors discuss a student teacher who is unable to give her
students a conceptual explanation or appropriate representation for
the "invert and multiply" algorithm for division of fractions. The
authors discuss this student teacher's mathematics methods course in
which the instructor presented problems, demonstrated the use of a
manipulative for the problem, and then allowed the student teachers to
work with the manipulatives and discuss with groups. This methods
course did not help the student teacher to use any manipulatives or
representations with her students in the division of fractions
context, but some might say that this methods course instructor was
modelling the use of effective manipulatives. Is this how we "model
effective use of technology"? By giving a problem, showing techers how
to do it on a calculator, and then giving them the chance to practice?
I think that this interpretation of the phrase "model effective use of
technology" is not what Tenille intended (but I'd love to hear what
you think, Tenille), and would not be an effective way to help
teachers shape their beliefs about technology and its use in their
classrooms. Instead, I would interpret that phrase to mean a teacher
educator, in a course with prospective or practicing teachers, should
run the class as he/she would run a class with students. The
instructor would give "good calculator problems" and then allow the
students (teachers) to use technology as they opt to, giving direct
instruction on calculator functions when the need for that function
arises naturally, etc. In this situation, the teacher educator is
modelling how a teacher actually uses technology effectively in the
classroom, rather than just modelling how to use a calculator
effectively. In a course like this, the teachers are really
participating as students and are able to learn what it's like to
experience a technology-rich classroom. Such experience will help them
shape their beliefs about technology in authentic ways, and will give
them realistic ideas and examples of how technology can or should be
implemented in their own classrooms.

On Feb 20, 11:16 pm, Janellie <janellepeter...@gmail.com> wrote:

> > emergence of technology in mathematics education?- Hide quoted text -
>
> - Show quoted text -

[Janellie]

Rachelle, I really liked your thoughts and question about the slow
process of adopting technology in the classroom. I wish I could say
that technology would be readily accepted in the future and people
would learn to appreciate how the technology can help them. Like you
mentioned, the article pointed out that it is usually expensive to get
new technology and professional development (the good kind) is often
hard to come by. I do, however, feel that the use of technology in
the classroom will not be stagnant. In other words, more and more
technology will be implemented in the classroom, but perhaps it will
always be a step behind what is being used in society. Suppose all
teachers eventually decide to make use of the TI-89 graphing
calculator and programs like Geometer's Sketchpad in their classroom.
By that time, new technology will have come about and a group of
graduate students will be sitting around wondering if teachers would
ever pull away from their archaic ways and utilize new technology or
whether we will have to wait for a new generation of teachers until
this new technology will be used.

And yes, I was thinking about Dewey and seminar as I read this
article.

[Shawn]

I believe that history can teach us many things. I think that I look
to history, personal or otherwise, for any precedents that I can use
as a base for any current decision I have to make. This perspective
on the history of technology in math education can serve as a guide
for what we can do as a field today. We can use it and make decisions
like they did in the past because it was fruitful or we can
deliberately go in another direction because what they did in the past
didn't work out so well.

Looking at what has happened in the past with respect to the emergence
of technology, society itself changes. The skills required to work in
society change as well. As a result, the mathematics that are used in
the top jobs change, too. For example, new fields of engineering and
computer science do not require only a working knowledge of algebra
any more. They use calculus, linear algebra, etc. They also use
computers and technology to do the "grunt work." An engineer or
computer scientist needs to also have a idea about whether their
results make sense. I believe that these are the types of
mathematical skills that students need to have when they graduate
today. So, what do we do as mathematics teachers, educators, and
researchers with the time we dedicate to our students for the learning
of mathematics? I don't think that having them do calculator tasks is
a bad thing, because working with technology is something that many
students will do in their professions, and a great number of
professions use math with technology. I don't think that doing PPA is
a bad thing either, unless there are better ways of giving students a
sense of whether their program in technology is doing things correctly
or an appreciation for it. (Additionally, a lot of times when I get
an answer that doesn't make sense was because I didn't input the
numbers correctly. Also, we like technology because we've done by
hand many of the things that technology does for us automatically. If
students learn with technology those things they won't have the same
sense or appreciation.) I believe that there is a balance that we
need to strike in our teaching between PPA and new technology. (I
should use new here because that's what society goes for and
incorporates for themselves and what we need to incorporate in our
classrooms.) These are the lessons we should learn because of how
history plays out and repeats itself.

On Feb 14, 12:27 pm, Keith <LethaLeat...@gmail.com> wrote:

[Shawn]

I'm not sure that many of us would have been accepted to Jersey City
High School if we were to use our math education from today and take
the test. I think that if we received the math education of 1885 we
would do just fine. (You might have said that, too.) I find it very
interesting that we don't do the same arithmetic that we did back in
1885. But I believe that this is true because our society is
different. On the arithmetic test they did several problems about the
amount of work one person can do alone and together with another,
merchant discounts, and sugar in bulk to name a few. These are the
kinds of things that their society focused on. They didn't have to
worry about guiding missiles or calculating the trajectory of a
satellite. It will be interesting as you say to think about what the
future will do to our algebra education. Will it be things akin to
making trips to the moon or Mars, knowing everything a shuttle pilot
does? Only time will tell...

[erin...@byu.edu]

Tenille - I enjoyed your questions the best because they parallel some
of the same thoughts I was having. I think that professional
development will only go so far in terms of getting teachers to change
- what really must happen is a "change of heart" as Alma would put it,
or some sort of experience where they see the benefit and value of
using technology or teaching for understanding. If this never
happens, when teachers try to implement technology their dislike for
it will definitely come through and their bias could potentially rub
off on their students. With any teacher I'm afraid there is no
forcing of comfortability with technology (is that a word?), and so we
probably will have to wait for technology. I see a potential problem
if these experienced teachers who are the mentors of the rising
generation of teachers display their prejudice against technology; it
is vital then that the rising generation of teachers have personal
experiences where they recognize the value of technology in the
classroom and are excited to implement it.

On Feb 19, 12:23 pm, Tenille <tenille.can...@gmail.com> wrote:

> I really enjoyed this article and didn't take issue with it.  Am I
> alone in this?  Perhaps I didn't read it critically enough.  I would
> love to hear from others.  It seems that teachers who are not
> comfortable with technology and learned mathematics without it seem
> very unwilling to use technology in their classroom (as more than just
> a tool for checking answers).  Do you think that professional
> development is enough to get teachers to change?  Or do we have to
> wait for all of these teachers to retire?
>
> On Feb 14, 12:27 pm, Keith <LethaLeat...@gmail.com> wrote:
>
>
>
> > "We learn from history that we do not learn from history." --Georg
> > Wilhelm Friedrich Hegel
>
> > As mathematics teachers, mathematics teacher educators, or mathematics
> > education researchers, what should we learn from this history of the